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Simplify: A Theorem Prover for Program Checking
DAVID DETLEFS, GREG NELSON, AND JAMES B. SAXE
Hewlett-Packard
Abstract. This article provides a detailed description of the automatic theorem prover Simplify, which
is the proof engine of the Extended Static Checkers ESC/Java and ESC/Modula-3. Simplify uses
the Nelson–Oppen method to combine decision procedures for several important theories, and also
employs a matcher to reason about quantifiers. Instead of conventional matching in a term DAG,
Simplify matches up to equivalence in an E-graph, which detects many relevant pattern instances that
would be missed by the conventional approach. The article describes two techniques, error context
reporting and error localization, for helping the user to determine the reason that a false conjecture is
false. The article includes detailed performance figures on conjectures derived from realistic program-
checking problems.
Categories and Subject Descriptors: D.2.4 [Software Engineering]: Software/Program Verification;
F.3.1 [Logics and Meanings of Programs]: Specifying and Verifying and Reasoning about Programs;
F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic
General Terms: Algorithms, Verification
Additional Key Words and Phrases: Theorem proving, decision procedures, program checking
1. Introduction
This is a description of Simplify, the theorem prover used in the Extended Static
Checking project (ESC) [Detlefs et al. 1998; Flanagan et al. 2002]. The goal of
ESC is to prove, at compile-time, the absence of certain run-time errors, such as
out-of-bounds array accesses, unhandled exceptions, and incorrect use of locks.
We and our colleagues have built two extended static checkers, ESC/Modula-3 and
ESC/Java, both of which rely on Simplify. Our ESC tools first process source code
with a verification condition generator, producing first-order formulas asserting
the absence of the targeted errors, and then submit those verification conditions
to Simplify. Although designed for ESC, Simplify is interesting in its own right
and has been used for purposes other than ESC. Several examples are listed in
the conclusions.
Authors’ addresses: D. Detlefs, Mailstop UBUR02-311, Sun Microsystems Laboratories, One Net-
work Drive, Burlington, MA 01803-0902, e-mail: david.detlefs@sun.com; G. Nelson (Mailstop 1203)
and J. B. Saxe (Mailstop 1250), Hewlett-Packard Labs, 1501 Page Mill Rd., Palo Alto, CA 94304,
e-mail: {gnelson,jim.saxe}@hp.com.
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C
2005 ACM 0004-5411/05/0500-0365 $5.00
Journal of the ACM, Vol. 52, No. 3, May 2005, pp. 365–473.
366 D. DETLEFS ET AL.
Simplify’s input is an arbitrary first-order formula, including quantifiers. Simplify
handles propositional connectives by backtracking search and includes complete
decision procedures for the theory of equality and for linear rational arithmetic,
together with some heuristics for linear integer arithmetic that are not complete
buthave been satisfactory in our application. Simplify’s handling of quantifiers
by pattern-driven instantiation is also incomplete but has also been satisfactory
in our application. The semantics of John McCarthy’s functions for updating and
accessing arrays [McCarthy 1963, Sec. 9] are also predefined. Failed proofs lead
to useful error messages, including counterexamples.
Our goal is to describe Simplify in sufficient detail so that a reader who reim-
plemented it from our description alone would produce a prover which, though
not equivalent to Simplify in every detail, would perform very much like Simplify
on program-checking tasks. We leave out a few of the things that “just grew”, but
include careful descriptions of all the essential algorithms and interfaces. Readers
who are interested in more detail than this article provides are free to consult the
source code on the Web [Detlefs et al. 2003b].
In the remainder of the introduction, we provide an overview of the Simplify
approach and an outline of the rest of the report.
When asked to check the validity of a conjecture G, Simplify, like many theorem
provers, proceeds by testing the satisfiability of the negated conjecture ¬ G.
To test whether a formula is satisfiable, Simplify performs a backtracking
search, guided by the propositional structure of the formula, attempting to find
a satisfying assignment—an assignment of truth values to atomic formulas that
makes the formula true and that is itself consistent with the semantics of the
underlying theories. Simplify relies on domain-specific algorithms for checking
the consistency of the satisfying assignment. These algorithms will be described
later; for now, we ask the reader to take for granted the ability to test the consistency
of a satisfying assignment.
Forexample, to prove the validity of the conjecture G:
x < y ⇒ (x − 1 < y ∧ x < y + 2),
we form its negation, which, for purposes of exposition, we will rewrite as
x < y ∧ (x − 1 ≥ y ∨ x ≥ y + 2).
The literals that appear in ¬ G are
x < y
x − 1 ≥ y
x ≥ y + 2.
Any assignment of truth values that satisfies ¬ G must have x < y true, so the
backtracking search begins by postulating x < y. Then, the search must explore
two possibilities, either x − 1 ≥ y or x ≥ y +2 must be true. So the search proceeds
as follows:
assume x < y.
case split on the clause x − 1 ≥ y ∨ x ≥ y + 2
first case, assume x − 1 ≥ y
discover the inconsistency of x < y ∧ x − 1 ≥ y
backtrack from the first case (discard the assumption x − 1 ≥ y)
Simplify: A Theorem Prover for Program Checking 367
second case, assume x ≥ y + 2
discover the inconsistency of x < y ∧ x ≥ y + 2
backtrack from the second case
(having exhausted all cases and finding no satisfying assignment, ...)
report that ¬ G is unsatisfiable, hence G is valid
In summary, the basic idea of the backtracking search is that the set of paths
to explore is guided by the propositional structure of the conjecture; the test
for consistency of each path is by domain-specific algorithms that reflect the
semantics of the operations and predicates of particular theories, such as arithmetic.
Simplify handles quantified formulas with a matcher that heuristically chooses
relevant instances.
Section 2 describes Simplify’s built-in theory and introduces notation and
terminology. Section 3 describes the backtracking search and the heuristics that
focus it. Section 4 gives a high-level description of the domain-specific decision
procedures. Section 5 describes the additional machinery for handling quanti-
fied formulas, including some modifications to the search heuristics described in
Section 3. Section 6 describes the methods used by Simplify to report the reasons
that a proof has failed, an important issue that is often neglected. Sections 7 and 8
give details of the two most important domain-specific decision procedures, the E-
graph and Simplex modules. Section 9 presents various measurements of Simplify’s
performance. Sections 10 through 12 discuss related and future work, summarize
our experiences, and offer conclusions.
To help the reader find definitions of technical terms, we annotate some (not all)
uses of technical terms with terse cross-references of the form (
§1) to the section or
subsection containing the term’s definition. There is also a short index of selected
identifiers in Appendix A.
2. Simplify’s Built-in Theory
This section has two purposes. The first is to define Simplify’s underlying theory
more precisely. The second is to introduce some terminology that will be useful in
the rest of the article.
The input to Simplify is a formula of untyped first-order logic with function
and relations, including equality. That is, the language includes the propositional
connectives ∧ , ∨ , ¬, ⇒, and ⇔; the universal quantifier ∀, and the exis-
tential quantifier ∃. Simplify requires that its input be presented as a symbolic
expression as in Lisp, but, in this article, we will usually use more conventional
mathematical notation.
Certain function and relation symbols have predefined semantics. It is convenient
to divide these function and relation symbols into several theories.
First is the theory of equality, which defines the semantics of the equality rela-
tion =. Equality is postulated to be a reflexive, transitive, and symmetric relation
that satisfies Leibniz’s rule: x = y ⇒ f (x) = f (y), for any function f .
Second is the theory of arithmetic, which defines the function symbols +, −, ×
and the relation symbols >, <, ≥, and ≤. These function symbols have the usual
meaning; we will not describe explicit axioms. Simplify makes the rule that any
terms that occur as arguments to the functions or relations of the arithmetic theory
are assumed to denote integers, so that, for example, the following formula is
368 D. DETLEFS ET AL.
considered valid:
(∀x : x < 6 ⇒ x ≤ 5).
Third is the theory of maps, which contains the two functions select and store
and the two axioms:
(∀ a, i, x : select(store(a, i, x), i ) = x)
(∀a, i, j, x : i = j ⇒ select(store(a, i, x), j) = select(a, j )).
These are called the unit and non-unit select-of-store axioms respectively. In our
applications to program checking, maps are used to represent arrays, sets, and object
fields, for example. We write f [x]asshorthand for select( f, x).
Fourth, because reasoning about partial orders is important in program-checking
applications, Simplify has a feature to support this reasoning. Because Simplify’s
theory of partial orders is somewhat different from its other theories, we postpone
its description to Section 4.7.
Simplify’s theory is untyped, so that expressions that are intuitively mistyped,
like select(6, 2) or store(a, i, x) + 3 are legal, but nothing nontrivial about such
terms is deducible in Simplify’s built-in theory.
While our theory is untyped, we do draw a distinction between propositional
values and individual values. The space of propositional values has two members,
denoted by the propositional literal constants true and false. The space of indi-
vidual values includes integers and maps. The individual literal constants of our
language are the integer literals (like 14 and −36) and a special constant @true
that we sometimes use to reflect the propositional value true into the space of
individual values.
Since there are two kinds of values, there are two kinds of variables, namely
propositional variables, which range over {true, false}, and individual variables,
which range over the space of individual values. All bound variables introduced by
quantifiers are individual variables.
A term is an individual variable, an individual literal constant, or an application
of a function symbol to a list of terms. A term is a ground term if it contains no
quantified variables.
An atomic formula is a propositional variable, a propositional literal constant, or
the application of a relation to a list of terms. In Sections 3.3 and 5.3.1, we will define
special propositional variables called proxies. Although a proxy is not represented
as an identifier, in the syntactic taxonomy that we are currently describing, a proxy
is an atomic formula, like any other propositional variable.
The strict distinction between terms and formulas (and thus also between func-
tions and relations and between individual and propositional variables) is a feature
of the classical treatment of first-order logic, so maintaining the distinction seemed
a safe design decision in the early stages of the Simplify project. In fact, the strict
distinction became inconvenient on more than one occasion, and we are not sure if
we would make the same decision if we could do it over again. But we aren’t sure
of the detailed semantics of any alternative, either.
When the strict distinction between functions and relations enforced by
Simplify is awkward, we circumvent the rules by modelling a relation as a
function whose result is equal to @true iff the relation holds of its arguments.
We call such a function a quasi-relation.For example, to say that the unary
quasi-relation f is the pointwise conjunction of the unary quasi-relations g and h,
Simplify: A Theorem Prover for Program Checking 369
we could write
(∀x : f (x) = @true ⇔ g(x) = @true ∧ h(x) = @true).
As a convenience, Simplify accepts the command (
DEFPRED ( fx)), after which
occurrences of f (t) are automatically converted into f (t) = @true if they occur
where a formula is expected.
The DEFPRED facility has another, more general, form:
(
DEFPRED (rargs) body), (1)
which, in addition to declaring r to be a quasi-relation, also declares the meaning
of that relation, the same meaning as would be declared by
(∀args : r (args) = @true ⇔ body). (2)
However, although (1) and (2) give the same meaning to r, there is a heuristic
difference in the way Simplify uses them. The quantified form 2 is subjected to
pattern-driven instantiation as described in Section 5.1. In contrast, the formula 1
is instantiated and used by Simplify only when an application of r is explicitly
equated to @true. This is explained further in one of the fine points in Section 7.
An equality is an atomic formula of the form t = u where t and u are terms. An
inequality is an atomic formula of one of the forms
t ≤ u, t < u, t ≥ u, t > u
where t and u are terms. A binary distinction is an atomic formula of the form
t = u, where t and u are terms. A general distinction is an atomic formula of the
form
DISTINCT(t
1
,...,t
n
) where the t’s are terms; it means that no two of the t’s
are equal.
The atomic formulas of Simplify’s theory of equality are the equalities and the
distinctions (binary and general). We allow general distinctions because (1) in our
applications they are common, (2) expressing a general distinction in terms of
binary distinctions would require a conjunction of length O(n
2
), and (3) we can
implement general distinctions more efficiently by providing them as a primitive.
The atomic formulas of Simplify’s theory of arithmetic are the inequalities
and equalities.
Simplify’s theory of maps is characterized by the postulated semantics of the
function symbols store and select.Ithas no relation symbols of its own, and its
atomic formulas are simply the atomic formulas of the theory of equality.
A formula is an expression built from atomic formulas, propositional connec-
tives, and quantifiers.
The formula presented to Simplify to be proved or refuted is called the conjec-
ture. The negation of the conjecture, which Simplify attempts to satisfy, is called
the query.
Some particular kinds of formulas are of special importance in our exposi-
tion. A literal is an atomic formula or the negation of an atomic formula. This
atomic formula is called the atom of the literal. A clause is a disjunction of literals,
and a monome is a conjunction of literals. A unit clause is a clause containing a
single literal.
A satisfying assignment for a formula is an assignment of values to its free
variables and of functions to its free function symbols, such that the formula is
370 D. DETLEFS ET AL.
true if its free variables and function symbols are interpreted according to the
assignment, and the built-in functions satisfy their built-in semantics.
A formula is satisfiable (or consistent)ifithas a satisfying assignment, and valid
if its negation is not satisfiable.
An important fact on which Simplify relies is that a formula such as
(∀x :(∃y : P(x, y))
is satisfiable if and only if the formula
(∀x : P(x, f (x))))
is satisfiable, where f is an otherwise unused function symbol. This fact allows
Simplify to remove quantifiers that are essentially existential—that is, existential
quantifiers in positive position and universal quantifiers in negative position—from
the query, replacing all occurrences of the quantified variables with terms like f (x)
above. The function f is called a Skolem function, and this process of eliminating
the existential quantifiers is called Skolemization. The arguments of the Skolem
function are the essentially universally quantified variables in scope at the point of
the quantifier being eliminated.
An important special case of Skolemization concerns free variables of the
conjecture. All free variables of the conjecture are implicitly universally quantified
at the outermost level, and thus are implicitly existentially quantified in the query.
Simplify therefore replaces these variables with applications of nullary Skolem
functions (also called Skolem constants).
In addition to providing decision procedures for a built-in theory used for all
conjectures, Simplify allows users to supply an arbitrary formula as a background
predicate that is given once and then used as an implicit antecedent for a number of
different conjectures. Users can supply a background predicate containing axioms
for a theory that is useful in their application. An example of the use of this facility is
provided by ESC: many facts relating to the verification of a procedure are common
to all the procedures in a given module; ESC assumes those facts in the background
predicate and then checks the verification conditions (
§1) of the procedures in the
module one by one.
3. The Search Strategy
In this section, we describe Simplify’s backtracking search strategy. Since the search
strategy is essentially concerned with the propositional structure of the conjecture,
we assume throughout this section that the conjecture is a propositional formula all
of whose atomic formulas are propositional variables. Compared to recent advances
in propositional SAT solving [Zhang 1997; Guerra e Silva et al. 1999; Silva and
Sakallah 1999; Moskewicz et al. 2001], the backtracking search described in this
section is simple and old-fashioned. We include this material not because it is a
contribution by itself, but because it is the foundation for the later material in which
domain-specific decision procedures and quantifiers are incorporated.
3.1. T
HE INTERFACE TO THE CONTEXT. Simplify uses a global resettable data
structure called the context which represents the conjunction of the query (
§2)
together with the assumptions defining the current case.
The context has several components, some of which will be described in later
sections of the report. To begin with, we mention three of its components: the public
Simplify: A Theorem Prover for Program Checking 371
boolean refuted, which can be set any time the context is detected to be inconsistent;
the literal set, lits, which is a set of literals; and the clause set, cls, which is a set
of clauses, each of which is a set of literals. A clause represents the disjunction
of its elements. The literal set, on the other hand, represents the conjunction of
its elements. The entire context represents the conjunction of all the clauses in
cls together with lits. When there is no danger of confusion, we shall feel free to
identify parts of the context with the formulas that they represent. For example,
when referring to the formula represented by the clause set, we may simply write
“cls” rather than
c∈cls
(
l∈c
l).
The algorithm for determining satisfiability operates on the context through the
following interface, called the satisfiability interface:
proc AssertLit(P : Literal) ≡ add the literal P to lits and possibly
set refuted if this makes lits inconsistent
proc Push() ≡ save the state of the context
proc Pop() ≡ restore the most recently saved, but
not-yet-restored, context
AssertLit(P) “possibly” sets refuted when lits becomes inconsistent, because
some of Simplify’s decisions procedures are incomplete; it is always desirable to
set refuted if it is sound to do so.
In addition, the context allows clauses to be deleted from the clause set and
literals to be deleted from clauses.
Viewed abstractly, Push copies the current context onto the top of a stack, from
which Pop can later restore it. As implemented, Simplify maintains an undo stack
recording changes to the context in such a way that Pop can simply undo the changes
made since the last unmatched call to Push.
At any point in Simplify’s execution, those changes to the context that have
been made since the beginning of execution but not between a call to Push and
the matching call to Pop are said to have occurred “on the current path,” and such
changes are said to be currently in effect. For example, if a call AssertLit(P) has
occurred on the current path, then the literal P is said to be currently asserted.
The data structures used to represent the context also store some heuristic
information whose creation and modification is not undone by Pop,sothat Simplify
can use information acquired on one path in the backtracking search to improve its
efficiency in exploring other paths. In the sections describing the relevant heuristics
(scoring in Section 3.6 and promotion in Section 5.2.1) we will specifically note
those situations in which changes are not undone by Pop.
Whenever the conjunction of currently asserted literals becomes inconsistent,
the boolean refuted may be set to true.Asweshall see in more detail later, this will
cause the search for a satisfying assignment to backtrack and consider a different
case. Were refuted to be erroneously set to true, Simplify would become unsound.
Were it to be left false unnecessarily, Simplify would be incomplete. For now, we
are assuming that all atomic formulas are propositional variables, so it is easy for
AssertLit to ensure that refuted is true iff lits is inconsistent: a monome (
§2) is
inconsistent if and only if its conjuncts include some variable v together with its
negation ¬ v. The problem of maintaining refuted will become more challenging
as we consider richer classes of literals.
372 D. DETLEFS ET AL.
3.2. T
HE
Sat PROCEDURE
.Totest the validity of a conjecture G, Simplify
initializes the context to represent ¬ G (as described in Sections 3.3 and 3.4),
and then uses a recursive procedure Sat (described in this section) to test whether
the context is satisfiable by searching exhaustively for an assignment of truth values
to propositional variables that implies the truth of the context. If Sat finds such a
satisfying assignment for the query ¬ G, then Simplify reports it to the user as a
counterexample for G. Conversely, if Sat completes its exhaustive search without
finding any satisfying assignment for ¬ G, then Simplify reports that it has proved
the conjecture G.
The satisfying assignments found by Sat need not be total. It is often the case
that an assignment of truth values to a proper subset of the propositional variables
of a formula suffices to imply the truth of the entire formula regardless of the truth
values of the remaining variables.
It will be convenient to present the pseudo-code for Sat so that it outputs a set of
satisfying assignments covering all possible ways of satisfying the context, where
each satisfying assignment is represented as a monome, namely the conjunction
of all variables made true by the assignment together with the negations of all
variables made false. Thus the specification of Sat is:
proc Sat()
/* Outputs zero or more monomes such that (1) each monome is consistent, (2) each
monome implies the context, and (3) the context implies the disjunction of all the monomes.
*/
Conditions (1) and (2) imply that each monome output by Sat is indeed a
satisfying assignment for the context. Conditions (2) and (3) imply that the context
is equivalent to the disjunction of the monomes, that is, Sat,asgiven here, com-
putes a disjunctive normal form for the context. If Simplify is being used only to
determine whether the conjecture is valid, then the search can be halted as soon
as a single counterexample context has been found. In an application like ESC, it
is usually better to find more than one counterexample if possible. Therefore, the
number of counterexamples Simplify will search for is configurable, as explained
in Section 6.3.
We implement Sat with a simple backtracking search that tries to form a consistent
extension to lits by including one literal from each clause in cls.Toreduce the
combinatorial explosion, the following procedure (Refine)iscalled before each
case split. The procedure relies on a global boolean that records whether refinement
is “enabled”(has some chance of discovering something new). Initially, refinement
is enabled.
proc Refine() ≡
while refinement enabled do
disable refinement;
for each clause C in cls do
RefineClause(C);
if refuted then
return
end
end
end
end
Simplify: A Theorem Prover for Program Checking 373
proc RefineClause(C : Clause) ≡
if C contains a literal l such that [lits ⇒ l] then
delete C from cls;
return
end;
while C contains a literal l such that [lits ⇒¬l] do
delete l from C
end;
if C is empty then
refuted := true
else if C is a unit clause {l} then
AssertLit(l);
enable refinement
end
end
We use the notation [P ⇒ Q]todenote that Q is a logical consequence of
P.With propositional atomic formulas, it is easy to test whether [lits ⇒ l]: this
condition is equivalent to l ∈ lits. Later in this article, when we deal with Simplify’s
full collection of literals, the test will not be so easy. At that point (Section 4.6),
it will be appropriate to consider imperfect tests. An imperfect test that yielded a
false positive would produce an unsound refinement, and this we will not allow.
The only adverse effect of a false negative, on the other hand, is to miss the heuristic
value of a sound refinement, and this may be a net gain if the imperfect test is much
more efficient than a perfect one.
Evidently Refine preserves the meaning of the context so that the sequence
Refine(); Sat() meets the specification for Sat. Moreover, Refine has the heuris-
tically desirable effects of
—removing clauses that are already satisfied by lits (called clause elimination),
—narrowing clauses by removing literals that are inconsistent with lits and thus
inconsistent with the context (called width reduction), and
—moving the semantic content of unit clauses from cls to lits, (called unit assertion).
The pseudo-code above shows Refine employing its heuristics in a definite order,
attempting first clause elimination, then width reduction, then unit propagation. In
fact, the heuristics may be applied in any order, and the order in which they are
actually applied by Simplify often differs from that given above.
Here is an implementation of Sat:
proc Sat() ≡
enable refinement;
Refine();
if refuted then return
else if cls is empty then
output the satisfying assignment lits;
return
else
let c be some clause in cls, and l be some literal of c;
Push();
AssertLit(l);
delete c from cls;
Sat()
374 D. DETLEFS ET AL.
Pop();
delete l from c;
Sat()
end
end
The proof of correctness of this procedure is straightforward. As noted above
calling Refine preserves the meaning of the context. If refuted is true or cls contains
an empty clause then the context is unsatisfiable and it is correct for Sat to return
without emitting any output. If cls is empty, then the context is equivalent to the
conjunction of the literals in lits,soitiscorrect to output this conjunction (which
must be consistent, else the context would have been refuted) and return. If cls is
not empty, then it is possible to choose a clause from cls, and if the context is not
already refuted, then the chosen clause c is nonempty, so it is possible to choose a
literal l of c. The two recursive calls to Sat are then made with contexts whose
disjunction is equivalent to the original context, so if the monomes output by the
recursive calls satisfy conditions (1)–(3) for those contexts, then the combined set
of monomes satisfies conditions (1)–(3) for the original context.
Here are some heuristic comments about the procedure.
(1) The choice of which clause to split on can have an enormous effect on perfor-
mance. The heuristics that govern this choice will be described in Sections 3.5,
3.6, and 5.2 below.
(2) The literal set is implemented (in part) by maintaining a status field for each
atomic formula indicating whether that atomic formula’s truth value is known
to be true, known to be false,orunknown. The call AssertLit(l) normally sets
the truth status of l’s atom (
§2) according to l’s sense, but first checks whether
it is already known with the opposite sense, in which case it records detection
of a contradiction by setting the refuted bit in the context. The refuted bit is, of
course, reset by Pop.
(3) A possible heuristic, which we refer to as the subsumption heuristic,isto
call Context.AssertLit(¬ l) before the second recursive call to Sat (since the
first recursive call to Sat has exhaustively considered cases in which l holds,
subsuming the need to consider any such cases in the second call).
The preceding pseudo-code is merely a first approximation to the actual algorithm
employed by Simplify. In the remainder of this report, we will describe a number
of modifications to Sat,tothe procedures it calls (e.g., Refine and AssertLit), to
the components of the context, and to the way the context is initialized before a
top-level call to Sat. Some of these changes will be strict refinements in the technical
sense—constraining choices that have so far been left nondeterministic; others will
be more radical (e.g., weakening the specification of Sat to allow incompleteness
when quantifiers are introduced).
In the remainder of this section, we describe the initialization of the context (at
least for the case where the conjecture is purely propositional) and some heuristics
for choosing case splits.
3.3. E
QUISATISFIABLE CNF. We now turn to the problem of initializing the
context to represent the query (
§2), a process we sometimes refer to as interning
the query.
Simplify: A Theorem Prover for Program Checking 375
The problem of initializing the context to be equivalent to some formula F is
equivalent to the problem of putting F into conjunctive normal form (CNF), since
the context is basically a conjunction of clauses.
It is well known that any propositional formula can be transformed into logically
equivalent CNF by applying distributivity and DeMorgan’s laws. Unfortunately, this
may cause an exponential blow-up in size. Therefore, we do something cheaper:
we transform the query Q into a formula that is in CNF, is linear in the size of Q
and is equisatisfiable with Q.Wesay that two formulas F and G are equisatisfiable
if “F is satisfiable” is equivalent to “G is satisfiable”. In summary, we avoid
the exponential blow-up by contenting ourselves with equisatisfiability instead of
logical equivalence.
To do this, we introduce propositional variables, called proxies, corresponding
to subformulas of the query, and write clauses that enforce the semantics of these
proxy variables. For example, we can introduce a proxy X for P ∧ R by introducing
the clauses
¬ X ∨ P
¬ X ∨ R
X ∨¬P ∨¬R
whose conjunction is equivalent to
X ⇔ (P ∧ R).
We refer to the set of clauses enforcing the semantics of a proxy as the definition
of the proxy. Note that these clauses uniquely determine the proxy in terms of the
other variables.
Given a query Q,ifweintroduce proxies for all nonliteral subformulas of Q
(including a proxy for Q itself) and initialize the context to contain the definitions
of all the proxies together with a unit clause whose literal is the proxy for Q, then the
resulting context will be satisfiable if and only if Q is. The history of this technique
is traced to Skolem in the 1920s in an article by Bibel and Eder [1993].
During the interning process, Simplify detects repeated subformulas and repre-
sents each occurrence of a repeated subformula by the same proxy, which need
be defined only once. Simplify makes a modest attempt at canonicalizing its input
so that it can sometimes recognize subformulas that are logically equivalent even
when they are textually different. For example, if the formulas
R ∧ P
P ∧ R
¬ P ∨¬R
all occurred as subformulas of the query, their corresponding proxy literals would
all refer to the same variable, with the third literal having the opposite sense from the
first two. The canonicalization works from the bottom up, so if P and P
are canon-
icalized identically and R and R
are canonicalized identically, then, for example,
P ∧ R will canonicalize identically with P
∧ R
.However, the canonicalization is
not sufficiently sophisticated to detect, for example, that (P ∨¬ R)∧ R is equivalent
to P ∧ R.
The Sat procedure requires exponential time in the worst case, even for purely
propositional formulas. When combined with matching to handle formulas with
quantification (as discussed in Section 5), it can fail to terminate. This is not
376 D. DETLEFS ET AL.
surprising, since the satisfiability problem is NP-complete even for formulas in
propositional calculus, and the validity of arbitrary formulas in first-order predicate
calculus is only semidecidable. These observations don’t discourage us, since in
the ESC application the inputs to Simplify are verification conditions (
§1), and if a
program is free of the kinds of errors targeted by ESC, there is almost always a short
proof of the fact. (It is unlikely that a real program’s lack of array bounds errors
would be dependent on the four-color theorem; and checking such a program would
be beyond our ambitions for ESC.) Typical ESC verification conditions are huge
but shallow. Ideally, the number of cases considered by Simplify would be similar
to the number of cases that would need to be considered to persuade a human critic
that the code is correct.
Unfortunately, experience with our first version of Simplify showed that if we
turned it loose on ESC verification conditions, it would find ways to waste inordinate
amounts of time doing case splits fruitlessly. In the remainder of Section 3, we will
describe techniques that we use to avoid the combinatorial explosion in practice.
3.4. A
VOIDING
EXPONENTIAL MIXING WITH LAZY CNF. We discovered early
in our project that Simplify would sometimes get swamped by a combinatorial
explosion of case splits when proving a conjecture of the form P ∧ Q,even though
it could quickly prove either conjunct individually. Investigation identified the prob-
lem that we call “exponential mixing”: Simplify was mixing up the case analysis
for P with the case analysis for Q,sothat the total number of cases grew multi-
plicatively rather than additively.
We call our solution to exponential mixing “lazy CNF”. The idea is that instead
of initializing the clause set cls to include the defining clauses for all the proxies of
the query, we add to cls the defining clauses for a proxy p only when p is asserted
or denied. Thus, these clauses will be available for case splitting only on branches
of the proof tree where they are relevant.
Lazy CNF provides a benefit that is similar to the benefit of the “justification
frontier” of standard combinatorial search algorithms [Guerra e Silva et al. 1999].
Lazy CNF is also similar to the “depth-first search” variable selection rule of
Barrett et al. [2002a].
Introducing lazy CNF into Simplify avoided such a host of performance problems
that the subjective experience was that it converted a prover that didn’t work into
one that did. We did not implement any way of turning it off, so the performance
section of this paper gives no measurements of its effect.
3.4.1. Details of Lazy CNF. In more detail, the lazy CNF approach differs from
the nonlazy approach in five ways.
First, we augment the context so that in addition to the clause set cls and the literal
set lits,itincludes a definition set, defs, containing the definitions for all the proxy
variables introduced by equisatisfiable CNF and representing the conjunction of
those definitions. Simplify maintains the invariant that the definition set uniquely
specifies all proxy variables in terms of the nonproxy variables. That is, if v
1
,...,v
n
are the nonproxy variables and p
1
,..., p
m
are the proxy variables, defs will be such
that the formula
∀v
1
,...,v
n
: ∃! p
1
,..., p
m
: defs (3)
is valid (where ∃! means “exists uniquely”). The context as a whole represents
Simplify: A Theorem Prover for Program Checking 377
the formula
∀ p
1
,..., p
m
:(defs ⇒ (cls ∧ lits))
or equivalently (because of (3))
∃ p
1
,..., p
m
:(defs ∧ cls ∧ lits).
Second, we change the way that Simplify initializes the context before invoking
Sat. Specifically, given a query Q, Simplify creates proxies for all non-literal sub-
formulas of Q (including Q itself) and initializes the context so lits is empty, cls
contains only a single unit clause whose literal is the proxy for Q, and defs contains
definitions making all the proxy variables equivalent to the terms for which they
are proxies (that is, defs is initialized so that
∀v
1
,...,v
n
, p
1
,..., p
m
: defs ⇒ ( p
i
⇔ T
i
)
is valid whenever p
i
is a proxy variable for term T
i
). It follows from the conditions
given in this and the preceding paragraph that (the formula represented by) this
initial context is logically equivalent to Q.Itiswith this context that Simplify
makes its top-level call to Sat.
Third, we slightly modify the specification of Sat,asindicated by italics:
proc Sat()
/* Requires that all proxy literals in lits be redundant. Outputs zero or more monomes such
that (1) each monome is a consistent conjunction of non-proxy literals, (2) each monome
implies the context, and (3) the context implies the disjunction of the monomes. */
When we say that the proxy literals in lits are redundant, we mean that the
meaning of the context would be unchanged if all proxy literals were removed
from lits. More formally, if plits is the conjunction of the proxy literals in lits and
nlits is the conjunction of the nonproxy literals in lits then
(∀ p
1
,..., p
m
:(defs ∧ cls ∧ nlits) ⇒ plits).
Fourth, we modify the implementation of AssertLit so that proxy literals are
treated specially. When l is a nonproxy literal, the action of AssertLit(l) remains
as described earlier: it simply adds l to lits, and possibly sets refuted. When l
is a proxy literal, however, AssertLit(l) not only adds l to lits,but also adds the
definition of l’s atom (which is a proxy variable) to cls.Asanoptimization, clauses
of the definition that contain l are not added to the clause set (since they would
immediately be subject to clause elimination (
§3.2)) and the remaining clauses are
width-reduced by deletion of ¬ l before being added to the clause set. For example,
suppose that s is a proxy for S, that t is a proxy for T , and p is a proxy for S ∧ T ,
so that the definition of p consists of the clauses:
¬ p ∨ s
¬ p ∨ t
p ∨¬s ∨¬t
Then, the call AssertLit(p)would add p to lits and add the unit clauses s and t to
cls, while the call AssertLit(¬ p)would add the literal ¬ p to lits and add the clause
¬ s ∨¬t to cls.Itmay seem redundant to add the proxy literal to lits in addition
to adding the relevant width-reduced clauses of its definition to cls, and in fact, it
would be unnecessary—if performance were not an issue. This policy of adding
378 D. DETLEFS ET AL.
the proxy literal to lits is called redundant proxy assertion. Its heuristic value will
be illustrated by one of the examples later in this section.
Finally, we change Sat so that when it finds a satisfying assignment (
§2) (i.e.,
when cls is empty and lits is consistent), the monome it outputs is the conjunction of
only the non-proxy literals in lits. (Actually this change is appropriate as soon as we
introduce proxy variables, regardless of whether or not we introduce their defining
clauses lazily. Deleting the proxy literals from the monomes doesn’t change the
meanings of the monomes because the proxies are uniquely defined in terms of the
nonproxy variables.)
To see how use of lazy CNF can prevent exponential mixing, consider proving
a conjecture of the form P
1
∧ P
2
, where P
1
and P
2
are complicated subformulas.
Let p
1
be a proxy for P
1
, p
2
be a proxy for P
2
, and p
3
be a proxy for the entire
formula P
1
∧ P
2
.
In the old, nonlazy approach, the initial clause set would contain the unit clause
¬ p
3
; defining clauses for p
3
, namely
¬ p
3
∨ p
1
, ¬ p
3
∨ p
2
, p
3
∨¬p
1
∨¬p
2
;
and the defining clauses for p
1
, p
2
, and all other proxies for subformulas of P
1
and
P
2
. The Refine procedure would apply unit assertion to assert the literal ¬ p
3
, and
then would apply clause elimination to remove the first two defining clauses for p
3
and width reduction (§3.2) to reduce the third clause to
¬ p
1
∨¬p
2
.
This clause and all the defining clauses for all the other proxies would then be can-
didates for case splitting, and it is plausible (and empirically likely) that exponential
mixing would ensue.
In the new, lazy approach, the initial clause set contains only the unit clause ¬ p
3
.
The Refine procedure performs unit assertion, removing this clause and calling
AssertLit(¬ p
3
), which adds the clause
¬ p
1
∨¬p
2
to cls. At this point, Refine can do no more, and a case split is necessary. Suppose
(without loss of generality) that the first case considered is ¬ p
1
. Then, Sat pushes
the context, removes the binary clause, and calls AssertLit(¬ p
1
), adding ¬ p
1
to
the literal set and the relevant part of p
1
’s definition to the clause set. The refutation
of ¬ P
1
(recall that p
1
is logically equivalent to P
1
,givendefs) continues, perhaps
requiring many case splits, but the whole search is carried out in a context in which
no clauses derived from P
2
are available for case-splitting. When the refutation
of ¬ P
1
is complete, the case ¬ p
2
(meaning ¬ P
2
)isconsidered, and this case is
uncontaminated by any clauses from P
1
. (Note that if Simplify used the subsumption
heuristic (
§3.2) to assert p
1
while analyzing the ¬ p
2
case, the benefits of lazy CNF
could be lost. We will say more about the interaction between lazy CNF and the
subsumption heuristic in Section 3.7.)
As another example, let us trace the computation of Simplify on the conjecture
(s ∧ (s ⇒ t)) ⇒ t.
Simplify introduces proxies to represent the subformulas of the conjecture. Let us
Simplify: A Theorem Prover for Program Checking 379
call these p
1
, p
2
, p
3
, p
4
, and p
5
, where
p
1
is a proxy for s ⇒ t,
p
2
is a proxy for s ∧ (s ⇒ t), that is, for s ∧ p
1
, and
p
3
is a proxy for the entire conjecture, that is, for p
2
⇒ t.
The context is initialized to the following state:
defs:
definition of p
1
:
p
1
∨ s
p
1
∨¬t
¬ p
1
∨¬s ∨ t
definition of p
2
:
¬ p
2
∨ s
¬ p
2
∨ p
1
p
2
∨¬s ∨¬p
1
definition of p
3
:
p
3
∨ p
2
p
3
∨¬t
¬ p
3
∨¬p
2
∨ t
lits:
cls:
¬ p
3
The Refine procedure first performs unit assertion on the clause ¬ p
3
, removing the
clause ¬ p
3
from the clause set, adding ¬ p
3
to the literal set, and adding the unit
clauses p
2
and ¬ t to the clause set. These clauses are subjected to unit assertion in
turn: they are removed from the clause set, their literals are added to the literal set,
and the unit clauses s and p
1
(from the definition of p
2
) are added to the clause set.
Applying unit assertion to these clauses leaves the context in the following state:
defs:
(same as above)
lits:
¬ p
3
p
2
¬ t
s
p
1
cls:
¬ s ∨ t
The only clause in the clause set is ¬ s ∨ t (from the definition of p
1
). Since both
literals of this clause are negations of literals in the literal set, width reduction
can be applied to reduce this clause to an empty clause, thus refuting the context.
(Alternatively, the clause could be reduced to a unit clause—either to ¬ s or to
t—after which unit assertion would make the literal set inconsistent, refuting the
context). So we see that Sat can prove the conjecture
(s ∧ (s ⇒ t)) ⇒ t
entirely through the action of Refine, without the need for case splits.
380 D. DETLEFS ET AL.
Note that the proof would proceed in essentially the same way, requiring no case
splits, for any conjecture of the form
(S
1
∧ (S
2
⇒ T
1
)) ⇒ T
2
where S
1
and S
2
are arbitrarily complicated formulas that canonicalize (
§3.3) to the
same proxy literal s and where T
1
and T
2
are arbitrarily complicated formulas that
canonicalize to the same proxy literal t. But this desirable fact depends on redundant
proxy assertion, since without this policy, we would be left with the clause ¬ s ∨ t
in the clause set, and the literal proxies s and ¬ t would not have been introduced
into the literal set. This illustrates the value of redundant proxy assertion.
3.5. G
OAL PROXIES AND
ASYMMETRICAL IMPLICATION. Semantically, P ⇒ Q
is identical with ¬ P ∨ Q and with Q∨¬ P and with ¬ Q ⇒¬P. But heuristically,
we have found it worthwhile to treat the two arms of an implication differently.
The reason for this is that the verification condition generator (
§1), which pre-
pares the input to Simplify, may sometimes have reason to expect that certain case
splits are heuristically more desirable than others. By treating the consequent of an
implication differently than the antecedent, we make it possible for the verification
condition generator to convey this hint to Simplify.
Forexample, consider verifying a procedure whose postcondition is a conjunction
and whose precondition is a disjunction:
proc A()
requires P
1
∨ P
2
∨ ··· ∨ P
m
ensures Q
1
∧ Q
2
∧ ··· ∧ Q
n
This will eventually lead to a conjecture of the form
P
1
∨ P
2
∨···∨P
m
⇒ (Q
1
∧ ··· ∧ Q
n
),
(where each Q
is the weakest precondition of the corresponding Q with respect to
the body of A). This, in turn, will require testing the consistency of the query
(P
1
∨ P
2
∨···∨P
m
) ∧ (¬ Q
1
∨···∨¬Q
n
).
In this situation, it is heuristically preferable to split on the clause containing the
Q
’s, rather than on the precondition. That is to say, if there are multiple postcon-
ditions to be proved, and multiple cases in the precondition, it is generally faster
to prove the postconditions one at a time than to explore the various cases of the
precondition one at a time. (More generally, the procedure precondition could con-
tain many disjunctions, and even it contains none, disjunctions will appear in the
antecedent of the verification condition if there are conditionals in the procedure
body.)
A similar situation arises in proving the precondition for an internally called
procedure, in case the precondition is a conjunction. It is heuristically preferable to
prove the conjuncts one at a time, rather than to perform any other case splits that
may occur in the verification condition.
In order to allow the verification condition generator to give Simplify hints about
which case splits to favor, we simply adopt the policy of favoring splits in Q rather
than in P when we are proving a formula of the form P ⇒ Q. This preference
is inherited when proving the various elements of a conjunction; for example, in
Simplify: A Theorem Prover for Program Checking 381
FIG.1.Inone simple case, the scoring heuristic produces the tree at the right instead of the tree at
the left.
P ⇒ ((Q
1
⇒ R
1
) ∧ (Q
2
⇒ R
2
)), case splits in R
i
will be favored over case splits
in Q
i
or in P.
We implement this idea by adding a Boolean goal property to literals and to
clauses. When a goal proxy for P ⇒ Q is denied, the proxy for P is asserted, the
proxy for Q is denied, and the proxy for Q (only) is made a goal. When a goal
proxy for a conjunction is denied, producing a clause of two denied proxy literals,
the clause and each of its literals become goals. The proxy for the initial query
is also given the goal property. When choosing a case split, Simplify favors goal
clauses.
Instead of treating implication asymmetrically, it would have been possible to
alter the input syntax to allow the user to indicate the goal attribute in a more flexible
way, butwehave not done so.
3.6. S
CORING CLAUSES.Inour applications, we find that an unsatisfiable clause
set frequently contains many irrelevant clauses: its unsatisfiability follows from a
small number of relevant clauses. In such a case, if Simplify is lucky enough to split
on the relevant clauses first, then the proof search will go quickly. But if Simplify
is unlucky enough to split on the irrelevant clauses before splitting on the relevant
ones, then the proof search will be very slow.
To deal with this problem, we associate a score with each clause. When choosing
a case split, we favor clauses with higher scores. (This preference for high-scoring
clauses is given less priority than the preference for goal clauses.) Each time a
contradiction leads Simplify to backtrack, Simplify increments the score of the last
clause split on.
Figure 1 shows how this heuristic works in a particularly simple case, where
there are n binary clauses, only one of which is relevant. Let the clauses of the
context be
(P
1
∨ Q
1
) ∧ (P
2
∨ Q
2
) ∧···∧(P
n
∨ Q
n
)
and suppose that only the last clause is relevant. That is, each of P
n
and Q
n
is
inconsistent with the context, and none of the other literals have any relevant effect
on the context at all. Without scoring (and temporarily ignoring width reduction
(
§3.2)), if Simplify considers the clauses in the unlucky order in which they are
listed, the search tree has 2
n
leaves, as illustrated in the left of the figure. With
scoring, the proof tree has only 2n leaves, as illustrated in the right of the figure.
Since asserting a literal from an irrelevant clauses never leads to a contradiction,
the scores of these clauses will never be incremented. When the relevant clause
P
n
∨ Q
n
is considered, its score will be incremented by 2. For the rest of the proof,
the relevant clause will be favored over all irrelevant clauses.
382 D. DETLEFS ET AL.
The scoring heuristic is also helpful if there is more than one relevant clause. The
reader may wish to work out the proof tree in the case that two binary clauses are
relevant (in the sense that the four possible ways of choosing one literal from each
of the relevant clauses are all inconsistent) and n − 2 are irrelevant. In general, if
there are k relevant binary clauses and n − k irrelevant binary clauses, the scoring
heuristic produces a search tree with at most n2
k
leaves.
So much for the basic idea of scoring. In the actual implementation, there are
many details that need to be addressed. We will spare the reader most of them, but
mention two that are of some importance.
First, in order to be of any use, incrementing the score of a clause must not be
undone by Pop.However, scores do need to be reset periodically, since different
clauses are often relevant in different parts of the proof. Simplify resets scores
whenever it backtracks from a case split on a nongoal clause and the previous
case split on the current path (
§3.1) was on a goal clause. When Simplify resets
scores, it renormalizes all scores to be in the range zero to one, causing high
scoring clauses to retain some advantage but giving low scoring clauses a chance to
catch up.
Second, there are interactions between scoring and lazy CNF. When a clause C
contains a proxy literal P whose assertion leads to the introduction of another clause
D, then C is referred to as the parent clause of D. Suppose the child clause D is
relevant and acquires a high score. When Simplify backtracks high up in the proof
tree, above the split on the parent clause C, the clause D will no longer be present.
The only way to reintroduce the useful clause D is to split again on C.Wetake
the view that D’s high score should to some extent lead us to favor splitting on
C, thus reintroducing D. Therefore, when Simplify increases the score of a clause,
it also increases (to a lesser extent) the scores of the parent and grandparent clauses.
Of course, the score for the clause D is not reset each time its proxy causes it to
be introduced.
3.7. U
SING THE SUBSUMPTION HEURISTIC FOR PROXY LITERALS.InSection
3.4, we noted that the subsumption heuristic (
§3.2) may interact poorly with
lazy CNF. Specifically, applying the subsumption heuristic to proxy literals
could reintroduce the exponential mixing that lazy CNF was designed to avoid.
In fact, Simplify uses a modified version of the subsumption heuristic that
regains some of the benefits without the risk of reintroducing exponential
mixing.
Suppose Simplify does a case split on a proxy literal l of a clause c. After
backtracking from the case where l holds and deleting l from the clause c,itadds
¬ l to the literal set, but does not add the expansion of ¬ l to the clause set. Since
the expansion of l is not added to the clause set, it cannot be a source of exponential
mixing. However, if l is a proxy for a repeated subformula, other clauses containing
l or ¬ l may occur in the clause set, and the presence of ¬ l in the literal set will
enable width reduction or clause elimination (
§3.2).
A subtlety of this scheme is that Simplify must keep track of whether each proxy
has had its expansion added to the clause set on the current path. If a “never-
expanded” proxy literal l in the literal set is used to eliminate a clause (··· ∨l ∨ ···)
from the clause set, the expansion of l must be added to the clause set at that point.
Otherwise, Simplify might find a “satisfying assignment” (
§2) that does not actually
satisfy the query.
Simplify: A Theorem Prover for Program Checking 383
With the modification described here, Sat no longer maintains the invariant that
all proxy literals in lits are redundant. We leave it as an exercise for the reader to
demonstrate the correctness of the modified algorithm.
4. Domain-Specific Decision Procedures
In this section, we show how to generalize the propositional theorem-proving
methods described in the previous section to handle the functions and relations
of Simplify’s built-in theory.
The Sat algorithm requires the capability to test the consistency of a set of
literals. Testing the consistency of a set of propositional literals is easy: the set is
consistent unless it contains a pair of complementary literals. Our strategy for han-
dling formulas involving arithmetic and equality is to retain the basic Sat algorithm
described above, but to generalize the consistency test to check the consistency
of sets of arbitrary literals. That is, we implement the satisfiability interface from
Section 3.1:
var refuted: boolean
proc AssertLit(L : Literal)
proc Push()
proc Pop()
but with L ranging over the literals of Simplify’s predefined theory. The implemen-
tation is sound but is incomplete for the linear theory of integers and for the theory
of nonlinear multiplication. As we shall see, the implementation is complete for a
theory which is, in a natural sense, the combination of the theory of equality (with
uninterpreted function symbols) and the theory of rational linear arithmetic.
Two important modules in Simplify are the E-graph module and the Simplex
module. Each module implements a version of AssertLit for literals of a particular
theory: the E-graph module asserts literals of the theory of equality with unin-
terpreted function symbols; the Simplex module asserts literals of rational linear
arithmetic. When the AssertLit method of either module detects a contradiction, it
sets the global refuted bit. Furthermore each AssertLit routine must push sufficient
information onto the undo stack (
§3.1) so that its effects can be undone by Pop.A
fine point: Instead of using the global undo stack, Simplify’s theory modules actu-
ally maintain their own private undo stacks and export Push and Pop procedures,
which are called (only) by the global Push and Pop.We’re not sure we’d do it this
wayifwehad it to do over. In any case, in this paper, Pop always means to pop the
state of the entire context.
Because the context may include literals from both theories, neither satisfiability
procedure by itself is sufficient. What we really want is a satisfiability procedure for
the combination of the theories. The first decision procedure for linear arithmetic
combined with function symbols was invented by Robert Shostak [Shostak 1979],
but this method was specific to these two theories. Simplify employs a general
method for combining decision procedures, known as equality sharing.
The equality sharing technique was introduced in Nelson’s Ph.D. thesis [Nelson
1979]. Two more modern expositions of the method, including proofs of correct-
ness, are by Tinelli and Harandi [1996] and by Nelson [1983]. In this technique, a
collection of decision procedures work together on a conjunction of literals; each
working on one logical theory. If each decision procedure is complete, and the
384 D. DETLEFS ET AL.
individual theories are “convex” (a notion defined in the papers just cited), and if
each decision procedure shares with the others any equality between variables that
is implied by its portion of the conjunction, then the collective effort will also be
complete. The theory of equality with uninterpreted function symbols is convex,
and so is the theory of linear rational inequalities, so the equality sharing technique
is appropriate to use with the E-graph and Simplex modules.
We describe equality sharing in Section 4.1 and give high level descriptions of
the E-graph and Simplex modules in Sections 4.2 and 4.3. Sections 7 and 8 provide
more detailed discussions of the implementations, including undoing among other
topics. Sections 4.4 and 4.5 give further practical details of the implementation
of equality sharing. Section 4.6 describes modifications to the Refine procedure
enabled by the non-propositional literals of Simplify’s built-in theory. Section 4.7
describes the built-in theory of partial orders.
4.1. E
QUALITY SHARING.For a logical theory T ,aT -literal is a literal whose
function and relation symbols are all from the language of T .
The satisfiability problem for a theory T is the problem of determining the
satisfiability of a conjunction of T -literals (also known as a T -monome).
The satisfiability problem for a theory is the essential computational problem of
implementing the satisfiability interface (
§3.1) for literals of that theory.
Example. Let R be the additive theory of the real numbers, with function symbols
+, −, 0, 1, 2, 3,...
and relation symbols
=, ≤, ≥
and the axioms of an ordered field. Then the satisfiability problem for R is
essentially the linear programming satisfiability problem, since each R-literal is
a linear equality or inequality (
§2).
If S and T are theories, we define S ∪ T as the theory whose relation symbols,
function symbols, and axioms are the unions of the corresponding sets for S and
for T .
Example. Let E be the theory of equality with the single relation symbol
=
and an adequate supply of “uninterpreted” function symbols
f, g, h,...
Then the satisfiability problem for R ∪ E includes, for example, the problem of
determining the satisfiability of
f ( f (x) − f (y)) = f (z)
∧ x ≤ y
∧ y + z ≤ x
∧ 0 ≤ z.
(4)
The separate satisfiability problems for R and E were solved long ago, by Fourier
and Ackerman, respectively. But the combined problem was not considered until it
became relevant to program verification.
Simplify: A Theorem Prover for Program Checking 385
Equality sharing is a general technique for solving the satisfiability problem for
the theory S ∪ T ,given solutions for S and for T , and assuming that (1) S and T
are both first order theories with equality and (2) S and T have no other common
function or relation symbol besides =.
The technique produces efficient results for cases of practical importance, in-
cluding R ∪ E.
By way of example, we now describe how the equality-sharing procedure shows
the inconsistency of the monome (4) above.
First, a definition: in a term or atomic formula of the form f (...,g(...),...),
the occurrence of the term g(...)iscalled alien if the function symbol g does not
belong to the same theory as the function or relation symbol f .For example, in (4),
f (x) occurs as an alien in f (x) − f (y), because − is an R function but f is not.
To use the postulated satisfiability procedures for R and E ,wemust extract an
R-monome and an E-monome from (4). To do this, we make each literal homo-
geneous (alien-free) by introducing names for alien subexpressions as necessary.
Every literal of (4) except the first is already homogeneous. To make the first
homogeneous, we introduce the name g
1
for the subexpression f (x) − f (y), g
2
for f (x), and g
3
for f (y). The result is that (4) is converted into the following two
monomes:
E-monome R-monome
f (g
1
) = f (z) g
1
= g
2
− g
3
f (x) = g
2
x ≤ y
f (y) = g
3
y + z ≤ x
0 ≤ z
This homogenization is always possible, because each theory includes an inex-
haustible supply of names and each theory includes equality.
In this example, each monome is satisfiable by itself, so the detection of the
inconsistency must involve interaction between the two satisfiability procedures.
The remarkable news is that a particular limited form of interaction suffices to
detect inconsistency: each satisfiability procedure must detect and propagate to the
other any equalities between variables that are implied by its monome.
In this example, the satisfiability procedure for R detects and propagates the
equality x = y. This allows the satisfiability procedure for E to detect and propagate
the equality g
2
= g
3
.Now the satisfiability procedure for R detects and propagates
the equality g
1
= z, from which the satisfiability procedure for E detects the
inconsistency.
If we had treated the free variables “x” and “y”inthe example above as Skolem
constants “x()” and “y()”, then the literal “x ≤ y”would become “x() ≤ y()”,
which would be homogenized to something like “g
4
≤ g
5
” where g
4
= x() and
g
5
= y() would be the defining literals for the new names. The rule that equalities
between variables must be propagated would now apply to the g’s even though x()
and y() would not be subject to the requirement. So the computation is essentially
the same regardless of whether x and y are viewed as variables or as Skolem
constants.
Implementing the equality-sharing procedure efficiently is surprisingly subtle.
It would be inefficient to create explicit symbolic names for alien terms and to
introduce the equalities defining these names as explicit formulas. Sections 4.4 and
4.5 explain the more efficient approach used by Simplify.
386 D. DETLEFS ET AL.
The equality-sharing method for combining decision procedures is often called
the “Nelson–Oppen method” after Greg Nelson and Derek Oppen, who first imple-
mented the method as part of the Stanford Pascal Verifier in 1976–79 in a MACLisp
program that was also called Simplify but is not to be confused with the Simplify
that is the subject of this article. The phrase “Nelson–Oppen method” is often used
in contrast to the “Shostak method” invented a few years later by Rob Shostak at
SRI [Shostak 1984]. Furthermore, it is often asserted that the Shostak method is
“ten times faster than the Nelson–Oppen method”.
The main reason that we didn’t use Shostak’s method is that we didn’t (and
don’t) understand it as well as we understand equality sharing. Shostak’s original
paper contained several errors and ambiguities. After Simplify’s design was settled,
several papers appeared correcting and clarifying Shostak’s method [Ruess and
Shankar 2001; Barrett et al. 2002b; Barrett 2002]. The consensus of these papers
seems to be that Shostak’s method is not so much an independent combining method
butarefinement of the Nelson–Oppen method for the case when the theory admits a
solver and a canonizer. A number of other recent papers refine and discuss Shostak’s
method [Shankar and Ruess 2002; Conchon and Krsti´c 2003; Krsti´c and Conchon
2003; Ganzinger 2002; Ganzinger et al. 2004].
We still lack the firm understanding we would want to have to build a tool based on
Shostak’s ideas, but we do believe these ideas could be used to improve performance.
We think it is an important open question how much improvement there would be.
It is still just possible to trace the history of the oft-repeated assertion that Shostak’s
method is ten times faster than the Nelson–Oppen method. The source seems to be
a comparison done in 1981 by Leo Marcus at SRI, as reported by Steve Crocker
[Marcus 1981; Crocker 1988]. But Marcus’s benchmarks were tiny theorems that
were not derived from actual program checking problems. In addition, it is unclear
whether the implementations being compared were of comparable quality. So we
do not believe there is adequate evidence for the claimed factor of ten difference.
One obstacle to settling this important open question is the difficulty of measuring
the cost of the combination method separately from the many other costs of an
automatic theorem prover.
4.2. T
HE E-GRAPH MODULE.Wenow describe Simplify’s implementation of
the satisfiability interface for the theory of equality, that is, for literals of the forms
X = Y and X = Y , where X and Y are terms built from variables and applications
of uninterpreted function symbols. In this section, we give a high-level description
of the satisfiability procedure; Section 7 contains a more detailed description.
We use the facts that equality is an equivalence relation—that is, that it is reflexive,
symmetric, and transitive—and that it is a congruence—that is, if x and y are equal,
then so are f (x) and f (y) for any function f .
Before presenting the decision procedure, we give a simple example illustrating
how the properties of equality can be used to test (and in this case, refute) the
satisfiability of a set of literals. Consider the set of literals
1. f (a, b) = a
2. f ( f (a, b), b) = c
3. g(a) = g(c)
(5)
Simplify: A Theorem Prover for Program Checking 387
It is easy to see that this set of literals is inconsistent:
4. f ( f (a, b), b) = f (a, b) (from 1, b = b, and congruence)
5. f (a, b) = c (from 4, symmetry on 4, 2, and transitivity)
6. a = c (from 5, 1, symmetry on 1, and transitivity)
7. g(a) = g(c) (from 6 and congruence), which contradicts 3.
We now briefly describe the data structures and the algorithms used by Simplify
to implement reasoning of the kind used in the example above. Section 7 provides
a more detailed description.
A term DAG is a vertex-labeled directed oriented acyclic multigraph, whose
nodes represent ground terms (
§2). By oriented we mean that the edges leaving any
node are ordered. If there is an edge from u to v,wecall u a parent of v and v a
child of u.Wewrite λ(u)todenote the label of u,wewrite degree(u)todenote the
number of edges from u, and we write u[i]todenote the i th child of u, where the
children are ordered according to the edge ordering out of u.Wewrite children[u]
to denote the sequence u[1],...,u[degree(u)]. By a multigraph, we mean a graph
possibly with multiple edges between the same pairs of nodes (so that possibly
u[i] = u[ j ] for i = j ). A term f (t
1
,...,t
n
)isrepresented by a node u if λ(u) = f
and children[u]isasequence v
1
,...,v
n
where each v
i
represents t
i
.
The term DAG used by the satisfiability procedure for E represents ground terms
only. We will consider explicit quantifiers in Section 5.
Given an equivalence relation R on the nodes of a term DAG, we say that two
nodes u and v are congruent under R if λ(u) = λ(v), degree(u) = degree(v),
and for each i in the range 1 ≤ i ≤ degree(u), R(u[i], v[i]). The set of nodes
congruent to a given node is a called a congruence class.Wesay that equiva-
lence relation R is congruence-closed if any two nodes that are congruent under
R are also equivalent under R. The congruence closure of a relation R on the
nodes of a term DAG is the smallest congruence-closed equivalence relation that
extends R.
An E-graph is a data structure that includes a term DAG and an equivalence
relation on the term DAG’s nodes (called E-nodes). The equivalence relation relates
a node u to a node v if and only if the terms represented by u and v are guaranteed
to be equal in the context represented by the E-graph.
From now on, when we say that an E-node urepresents a term f (t
1
,...,t
n
), we
mean that the label of u is f and that each child u[i]ofu is equivalent to some
E-node that represents t
i
. That is, represents means “represents up to congruence”.
We can now describe the basic satisfiability procedure for E .Totest the satis-
fiability of an arbitrary E -monome M,weproceed as follows: First, we construct
an E-graph whose term DAG represents each term in M and whose equivalence
relation relates node(T )tonode(U) whenever M includes the equality T = U.
Second, we close the equivalence relation under congruence by repeatedly merging
the equivalence classes of any nodes that are congruent but not equivalent. Finally,
we test whether any literal of M is a distinction T = U where node(T ) and node(U)
are equivalent. If so, we report that M is unsatisfiable; otherwise, we report that M
is satisfiable.
Figure 2 shows the operation of this algorithm on the example (5) above. Note
that the variables a, b, and c are represented by leaf E-nodes of the E-graph. As
explained near the end of Section 2, the equivalence underlying the Skolemization
technique implies that it doesn’t matter whether we treat a, b, and c as variables or
388 D. DETLEFS ET AL.
FIG.2. Application of the congruence closure to example (5). (a) A term DAG for the terms in (5).
(b) The E-graph whose equivalences (shown by dashed lines) correspond to the equalities in (5).
(c) node( f ( f (a, b), b)) and node( f (a, b)) are congruent in (b); make them equivalent. (d) node(g(a))
and node(g(c)) are congruent in (c); make them equivalent. Since g(a) and g(c) are distinguished in
(5) but equivalent in (d), (5) is unsatisfiable.
as symbolic literal constants. The E-graph pictured in Figure 2 makes them labeled
nodes with zero arguments, that is, symbolic literal constants.
This decision procedure is sound and complete. It is sound since, by the
construction of the equivalence relation of the E-graph, two nodes are equivalent
in the congruence closure only if they represent terms whose equality is implied
by the equalities in M together with the reflexive, symmetric, transitive, and con-
gruence properties of equality. Thus, the procedure reports M to be unsatisfiable
only if it actually finds two nodes node(T
1
) and node(T
2
) such that both T
1
= T
2
and T
1
= T
2
are consequences of M in the theory E .Itiscomplete since, if it
reports monome M to be satisfiable, the equivalence classes of the E-graph provide
a model that satisfies the literals of M as well as the axioms of equality. (The fact
that the relation is congruence-closed ensures that the interpretations of function
symbols in the model are well defined.)
The decision procedure for E is easily adapted to participate in the equality-
sharing protocol: it is naturally incremental with respect to asserted equalities (to
assert an equality T = U it suffices to merge the equivalence classes of node(T )
and node(U ) and close under congruence) and it is straightforward to detect and
propagate equalities when equivalence classes are merged.
To make the E-graph module incremental with respect to distinctions (
§2), we
also maintain a data structure representing a set of forbidden merges. To assert
x = y,weforbid the merge of x’s equivalence class with y’s equivalence class
by adding the pair (x, y)tothe set of forbidden merges. The set is checked before
performing any merge, and if the merge is forbidden, refuted is set.
In Section 7, we describe in detail an efficient implementation of the E-graph,
including non-binary distinctions and undoing. For now, we remark that, by using
Simplify: A Theorem Prover for Program Checking 389
methods described by Downey et al. [1980], our implementation guarantees that
incrementally asserting the literals of any E -monome (with no backtracking)
requires a worst-case total cost of O(n log n)expected time, where n is the print
size of the monome. We also introduce here the root field of an E-node: v.root is
the canonical representative of v’s equivalence class.
4.3. T
HE SIMPLEX MODULE. Simplify’s Simplex module implements the satis-
fiability interface for the theory R. The name of the module comes from the Simplex
algorithm, which is the central algorithm of its AssertLit method. The module is
described in some detail in Section 8. For now, we merely summarize its salient
properties.
The Simplex method is sound and complete for determining the satisfiability
over the rationals of a conjunction of linear inequalities. Simplify also employs
some heuristics that are sound but incomplete for determining satisfiability over
the integers.
The space required is that for a matrix—the Simplex tableau—with one row for
every inequality (
§2) and one column for every Simplex unknown. The entries in
the matrix are integer pairs representing rational numbers.
In the worst case, the algorithm requires exponential time, but this worst case is
very unlikely to arise. In practice, the per-assertion cost is a small number of pivots
of the tableau, where the cost of a pivot is proportional to the size of the tableau
(see Section 9.15).
In our applications the unknowns represent integers. We take advantage of
this to eliminate strict inequalities, replacing X < Y by X ≤ Y − 1. This par-
tially compensates for the fact that the Simplex algorithm detects unsatisfiability
over the rationals rather than over the integers. Two other heuristics, described in
Section 8, offer additional compensation, but Simplify is not complete for integer
linear arithmetic.
4.4. O
RDINARY THEORIES AND THE SPECIAL ROLE OF THE E-GRAPH.In
Section 4.1, we described the equality-sharing procedure as though the roles of
the participating theories were entirely symmetric. In fact, in the implementation
of Simplify, the theory of equality with uninterpreted function symbols plays a spe-
cial role. Its decision procedure, the E-graph module, serves as a central repository
representing all ground terms in the conjecture.
Each of the other built-in theories is called an ordinary theory.
For each ordinary theory T , Simplify includes an incremental, resettable decision
procedure for satisfiability of conjunctions of T -literals (
§4.1). We use the name T
both for the theory and for this module. In this section we describe the interface
between an ordinary theory and the rest of Simplify. This interface is very like the
satisfiability interface of Section 3.1, but with the following differences:
The first difference is that the module for an ordinary theory T declares a
type T.Unknown to represent unknowns of T .Inorder to maintain the association
between E-nodes and unknowns, for each ordinary theory T , each E-node has a
T
unknown field whose value is a T.Unknown (or nil). It may seem wasteful of
space to include a separate pointer field in each E-node for each ordinary theory,
but the number of ordinary theories is not large (in the case of Simplify, the number
is two), and there are straightforward techniques (implemented in Simplify but not
described in this paper) that reduce the space cost in practice.
390 D. DETLEFS ET AL.
Each ordinary theory T introduces into the class T.Unknown whatever fields
its satisfiability procedure needs, but there is one field common to all the vari-
ous T.Unknown classes: the enode field, which serves as a kind of inverse to the
T
unknown field. More precisely:
(1) for any E-node e,ife.T
unknown = nil, then e.T unknown.enode is equivalent
to e, and
(2) for any T.Unknown u,ifu.enode = nil, then u.enode.T
unknown = u.
An unknown u is connected if u.enode = nil.
Each ordinary theory T must implement the following method for generating an
unknown and connecting it to an E-node.
proc T.UnknownForEnode(e : E-node):Unknown;
/* Requires that e.root = e. Returns e.T
unknown if e.T unknown = nil. Otherwise sets
e.T
unknown to a newly-allocated unconstrained unknown (with enode field initialized to
e) and returns it. */
The second difference is that the literals passed to AssertLit are not proposi-
tional unknowns but literals of T .Inparticular, AssertLit must accept literals of the
following kinds:
u
1
= u
2
u
0
= F(u
1
,...,u
n
) for each n-ary function symbol F of T
R(u
1
,...,u
n
) for each n-ary relation symbol R of T
¬ R(u
1
,...,u
n
) for each n-ary relation symbol R of T
where the u’s are unknowns of T . There must be procedures for building these
literals from unknowns, but we will not describe those procedures further here.
We introduce the abstract variable T .Asserted to represent the conjunction of
currently asserted T literals.
The third difference is that AssertLit must propagate equalities as well as check
consistency. We introduce the abstract variable T.Propagated to represent the con-
junction of equalities currently propagated from T . The T module is responsible
for maintaining the invariant:
Invariant (PROPAGATION FROM T ). For any two connected unknowns u and
v, the equality u = v is implied by Propagated iff it is implied by Asserted.
In summary, the specification for T .AssertLit is:
proc T.AssertLit(L : T -Literal);
/* If L is consistent with T .Asserted, set T.Asserted to L ∧ T.Asserted, propagating equal-
ities as required to maintain PROPAGATION FROM T . Otherwise set refuted to true.
*/
Each ordinary theory T can assume that the E-graph module maintains the
following invariant:
Invariant (PROPAGATION TO T ). Two equivalent E-nodes have non-nil
T
unknown fields if and only if these two T.Unknown’s are equated by a chain
of currently propagated equalities from the E-graph to the T module.
Section 7 describes the E-graph code that maintains this invariant.
Note that, while T.Asserted may imply a quadratic number of equalities between
T -unknowns, at most n − 1ofthese (where n is the number of T -unknowns) need
to be propagated on any path in order to maintain PROPAGATION FROM T .
Simplify: A Theorem Prover for Program Checking 391
Similarly, at most n − 1 equalities need be propagated from the E-graph to T in
order to maintain PROPAGATION TO T .
A fine point: It may be awkward for a theory to deal with an incoming equality
assertion while it is in the midst of determining what equalities to propagate as a
result of some previous assertion. To avoid this awkwardness, propagated equalities
(both from and to T ) are not asserted immediately but instead are put onto a work
list. Simplify includes code that removes and asserts equalities from the work list
until it is empty or the current case is refuted.
4.5. C
ONNECTING THE
E-GRAPH WITH THE ORDINARY THEORIES.Aliteral of
the conjecture may be inhomogeneous—that is, it may contain occurrence of func-
tions and relations of more than one theory. In our initial description of equality
sharing in Section 4.1, we dealt with this problem by introducing new variables
as names for the alien (
§4.1) terms and defining the names with additional equali-
ties. This is a convenient way of dealing with the problem in the language of first
order logic, but in the actual implementation there is no need to introduce new
variables.
Instead, Simplify creates E-nodes not only for applications of uninterpreted func-
tion symbols, but for all ground terms in the conjecture. For each E-node that is
relevant to an ordinary theory T because it is an application of or an argument to
a function symbol of T , Simplify allocates a T.Unknown and links it to the term’s
E-node. More precisely, for a term f (t
1
,...,t
k
) where f is a function symbol
of T , the E-nodes representing the term and its arguments are associated with
T.Unknown’s and the relation between these k + 1 unknowns is represented by an
assertion of the appropriate T -literal.
Forexample, if p is the E-node for an application of the function symbol +, and
q and r are the two children of p, then the appropriate connections between the
E-graph and the Simplex module are made by the following fragment of code:
var u := Simplex.UnknownForEnode( p.root),
v := Simplex.UnknownForEnode(q.root),
w := Simplex.UnknownForEnode(r.root) in
Simplex.AssertLit(u = v + w)
There are a variety of possible answers to the question of exactly when these
connections are made. A simple answer would be to make the connections eagerly
whenever an E-node is created that represents an application of a function of an
ordinary theory. But we found it more efficient to use a lazier strategy in which the
connections are made at the first point on each path where the Sat algorithm asserts
a literal in which the function application occurs.
Not only the function symbols but also the relation symbols of an ordinary theory
give rise to connections between E-nodes and unknowns. For each expression of the
form R(t
1
,...,t
k
), where R is a relation of an ordinary theory T , Simplify creates
a data structure called an
AF (atomic formula (§2)). The AF a for R(t
1
,...,t
k
)
is such that AssertLit(a) asserts R(u
1
,...,u
n
)toT , and AssertLit(¬ a) asserts
¬ R(u
1
,...,u
n
)toT , where u
i
is the T -unknown connected to the E-node for t
i
The use of unknowns to connect the E-graph with the Simplex tableau was
described in Nelson’s thesis [Nelson 1981, Sec. 13].
Building the collection of E-nodes and unknowns that represents a term is called
interning the term.
392 D. DETLEFS ET AL.
An implementation note: The
AF for R(t
1
,...,t
k
) may contain pointers either
to the E-nodes for the t’s or to the corresponding unknowns (u’s), depending on
whether the connections to T are made lazily or eagerly. The representation of a
literal is an
AF paired with a sense Boolean, indicating whether the literal is positive
or negative. The generic
AF has an Assert method which is implemented differently
for different subtypes of
AF each of which corresponds to a different relation R.
To assert a literal, its
AF’s Assert method is called with the sense boolean as a
parameter.
It is heuristically desirable to canonicalize (
§3.3) AF’s as much as possible, so
that, for example, if the context contains occurrences of x < y, y > x, ¬ x ≥ y,
and ¬ y ≤ x, then all four formulas are canonicalized to the same
AF. The E-graph
module exploits symmetry and the current equivalence relation to canonicalize
equalities and binary distinctions (
§2), but Simplify leaves it to each ordinary theory
to expend an appropriate amount of effort in canonicalizing applications of its own
relation symbols.
In accordance with our plan to distinguish functions from relations, we did not
make
AF a subtype of E-node: the labels in the E-graph are always function sym-
bols, never relations. In retrospect, we suspect this was a mistake. For example,
because of this decision, the canonicalization code for
AF’s in an ordinary theory
must duplicate the functionality that the E-graph module uses to produce canonical
E-nodes for terms. An even more unpleasant consequence of this decision is that
matching triggers (see Section 5.1) cannot include relation symbols. At one point
in the ESC project, this consequence (in the particular case of binary distinctions)
become so debilitating that we programmed an explicit exception to work around
it: we reserved the quasi-relation symbol neq and modified the assert method for
equality
AF’s so that neq(t, u) = @true is asserted whenever t = u is denied.
4.6. W
IDTH REDUCTION WITH DOMAIN-SPECIFIC LITERALS. The domain-
specific decision procedures create some new opportunities for the Refine procedure
to do width reduction and clause elimination (
§3.2). Suppose l is a literal in some
clause c in the clause set cls. The version of Refine in Section 3.2 deletes the c
from cls (clause elimination) if lits—viewedaaset of literals—contains l, and it
deletes l from c (width reduction) if lits contains ¬ l.Infact deleting l from cls
will leave the meaning of the context unchanged if lits—viewed as a conjunction of
literals—implies l (equivalently, if ¬ l is inconsistent with lits). Similarly, deleting
l from c will leave the meaning of the context unchanged if l is inconsistent with
(the conjunction of) lits (equivalently, if lits implies ¬ l).
Foraconsistent literal sets containing only propositional variables and their
negations, as in Section 3, containment (l ∈ lits) and implication ([lits ⇒ l])
are equivalent. For the larger class of literals of Simplify’s built-in theory, this
equivalence no longer holds. For example, the conjunction x < y ∧ y < z implies
the literal x < z,even though the set {x < y, y < z} does not contain the literal
x < z.
Since we have a decision procedure for equalities, distinctions, and inequalities
that is complete, incremental, and resettable, it is easy to write a procedure to test
a literal for consistency with the current literal set. Here are the specification and
implementation of such a procedure:
proc Implied(L : Literal): boolean
/* Returns true if and only if [lits ⇒ L]. */
Simplify: A Theorem Prover for Program Checking 393
proc Implied(L : Literal) ≡
Push();
AssertLit(¬ L);
if refuted then
Pop(); return true
else
Pop(); return false
end
end
We refer to this method of testing implication by lits as plunging on the lit-
eral. Plunging is somewhat expensive, because of the overhead in Push, Pop, and
especially AssertLit.
On the other hand, we can test for membership of a literal (or its complement)
in lits very cheaply by simply examining the sense of the literal and the status
(
§3.2) field of the literal’s
AF,but this status test is less effective than plunging
at finding opportunities for width reduction and clause elimination. The effective-
ness of the status test is increased by careful canonicalization of atomic formulas
into
AF’s.
For literals representing equalities and distinctions, Simplify includes tests (the
E-graph tests) that are more complete than the status test but less expensive than
plunging: the E-graph implies the equality T = U if the E-nodes for T and U are
in the same equivalence class, and it implies the distinction T = U if (albeit not
only if) the equivalence classes of the E-nodes for T and U have been forbidden to
be merged.
To compare the three kinds of tests, consider a context in which the following
two literals have been asserted:
i = j, f ( j ) = f (k).
Then
— j = i would be inferred by the status test because i = j and j = i are
canonicalized identically,
— f (i) = f (k)would be inferred by the E-graph test since f (i) and f ( j) are
congruent (
§4.2), hence equivalent but not by the status test if f (i) = f (k)was
canonicalized before i = j was asserted, and
— j = k would be inferred by plunging (since a trial assertion would quickly refute
j = k)but would not by inferred either the status test or the E-graph test.
In an early version of Simplify, we never did a case split without first applying
the plunging version of Refine to every clause. We found this to be too slow. In
the current version of Simplify, we apply the plunging version of Refine to each
non-unit clause produced by matching (see Section 5.2), but we do this just once,
immediately after the match is found. On the other hand, we continue to use E-graph
tests aggressively: we never do a case split without first applying the E-graph test
version of Refine to every clause.
4.7. T
HE THEORY OF PARTIAL ORDERS.Iff and g are binary quasirelations,
the syntax
(
ORDER fg)
394 D. DETLEFS ET AL.
is somewhat like a higher order atomic formula that asserts that f and g are the
irreflexive and reflexive versions, respectively, of a partial order. (We write “some-
what like” because Simplify’s logic is first order and this facility’s implementation
is more like a macro than a true higher-order predicate.)
Each assertion of an application of
ORDER dynamically creates a new instance
of a prover module whose satisfiability procedure performs transitive closure to
reason about assertions involving the two quasi-relations.
The orders facility is somewhat ad hoc, and we will not describe all its details in
this article. The interface to the dynamically created satisfiability procedures is
mostly like the interface described in Sections 4.4 and 4.5, but the procedures
propagate not just equalities but also ordering relations back to the E-graph, where
they can be used by the matcher to instantiate universally quantified formulas as
described in the next section. For example, this allows Simplify to infer
a
LT b ⇒ R(a, b)
from
(
ORDER LT LE) ∧ (∀x, y : x LE y ⇒ R(x, y)).
5. Quantifiers
So far, we have ignored quantifiers. But a treatise on theorem-proving that ignores
quantifiers is like a treatise on arithmetic that ignores multiplication: quantifiers are
near the heart of all the essential difficulties.
With the inclusion of quantifiers, the theoretical difficulty of the theorem-proving
problem jumps from NP-complete to undecidable (actually, semidecidable: an
unbounded search of all proofs will eventually find a proof if one exists, but no
bounded search will do so). Much of the previous work in automatic theorem-
proving (as described for example in Donald Loveland’s book [Loveland 1978])
has concentrated on strategies for handling quantifiers that are complete, that is, that
are guaranteed in principle eventually to find a proof if a proof exists. But for our
goal, which is to find simple proofs rapidly when simple proofs exist, a complete
search strategy does not seem to be essential.
5.1. O
VERVIEW OF MATCHING AND TRIGGERS. Semantically, the formula
(∀x
1
,...,x
n
: P)isequivalent to the infinite conjunction
θ
θ(P) where θ ranges
over all substitutions over the x’s. Heuristically, Simplify selects from this infinite
conjunction those instances θ(P) that seem “relevant” to the conjecture (as de-
termined by heuristics described below), asserts the relevant instances, and treats
these assertions by the quantifier-free reasoning methods described previously. In
this context, the quantified variables x
1
,...x
n
are called pattern variables.
The basic idea of the relevance heuristics is to treat an instance θ(P)asrelevant
if it contains enough terms that are represented in the current E-graph. The simplest
embodiment of this basic idea is to select a particular term t from P as a trigger,
and to treat θ(P)asrelevant if θ(t )isrepresented in the E-graph. (Simplify also
allows a trigger to be a list of terms instead of a single term, as described later in
this subsection).
Simplify: A Theorem Prover for Program Checking 395
The part of Simplify that finds those substitutions θ such that θ(t)isrepresented
in the E-graph and asserts the corresponding instance θ(P)iscalled the matcher,
since the trigger plays the role of a pattern that must be matched by some E-node.
The choice of a trigger is heuristically crucial. If too liberal a trigger is chosen,
Simplify can be swamped with irrelevant instances; if too conservative a trigger
is chosen, an instance crucial to the proof might be excluded. At a minimum, it is
important that every one of the pattern variables occur in the trigger, since otherwise
there will be infinitely many instances that satisfy the relevance criterion.
As an example of the effect of trigger selection, consider the quantified formula
(∀x, y : car(cons(x, y)) = x).
If this is used with the trigger cons(x, y), then, for each term of the form cons(a, b)
represented in the E-graph, Simplify will assert a = car(cons(a, b)), creating a
new E-node labeled car if necessary. If instead the formula is used with the more
restrictive trigger car(cons(x, y)), then the equality will be asserted only when the
term car(cons(a, b)) is already represented in the E-graph. For the conjecture
cons(a, b) = cons(c, d) ⇒ a = c,
the liberal trigger would allow the proof to go through, while the more conservative
trigger would fail to produce the instances necessary to the proof.
One of the pitfalls threatening the user of Simplify is the matching loop.For
example, an instance of a quantified assertion A might trigger a new instance
of a quantified assertion B which in turn triggers a new instance of A, and so
on indefinitely.
Simplify has features that try to prevent matching loops from occurring, namely
the activation heuristic of Section 5.2, and the trigger selection “loop test” of
Section 5.3. However, they don’t eliminate matching loops entirely, and Simplify
has a feature that attempts to detect when one has occurred, namely the “consecutive
matching round limit” of Section 5.2.
Sometimes we must use a set of terms as a trigger instead of a single term. For
example, for a formula like
(∀s, t, x : member(x, s) ∧ subset(s, t) ⇒ member(x, t)),
no single term is an adequate trigger, since no single term contains all the pattern
variables. An appropriate trigger is the set of terms {member(x, s), subset(s, t)}.A
trigger that contains more than one term will be called a multitrigger, and the terms
will be called its constituents.Atrigger with a single constituent will be called a
unitrigger.
Recall from Section 4.2 that when we say that an instance θ (t)isrepresented in
an E-graph, we mean that it is represented up to congruence. For example, consider
the E-graph that represents the equality f (a) = a.Ithas only two E-nodes, but it
represents not just a and f (a)but also f ( f (a)) and indeed f
n
(a) for any n.
Matching in the E-graph is more powerful than simple conventional pattern-
matching, since the matcher is able to exploit the equality information in the
E-graph. For example, consider proving that
g( f (g(a))) = a
396 D. DETLEFS ET AL.
follows from
(∀x : f (x) = x) (6)
(for which we assume the trigger f (x)) and
(∀x : g(g(x)) = x) (7)
(for which we assume the trigger g(g(x))). The E-graph representing the
query (
§2) contains the term g( f (g(a))). A match of (6) with the substitution
x := g(a) introduces the equality f (g(a)) = g(a) into the graph. The
resulting E-graph is shown in the figure to the right. By virtue of the equality,
the resulting E-graph represents an instance of the trigger g(g(x)), and the
associated instance of (7) (via the substitution x := a) completes the proof.
The standard top-down pattern-matching algorithm can be modified
slightly to match in an E-graph; the resulting code is straightforward, but
because of the need to search each equivalence class, the matcher requires
exponential time in the worst case. Indeed, the problem of testing whether
an E-node of an E-graph is an instance of a trigger is NP-complete, as has
been proved by Dexter Kozen [Kozen 1977]. More details of the matching
algorithm are presented below.
In practice, although the cost of matching is significant, the extra power derived
by exploiting the equalities in the matcher is worth the cost. Also, in our experi-
ence, whenever Simplify was swamped by a combinatorial explosion, it was in the
backtracking search in Sat, not in the matcher.
Although Simplify’s exploits the equalities in the E-graph, it does not exploit the
laws of arithmetic. For example, Simplify fails to prove
(∀x : P(x + 1)) ⇒ P(1 + a)
since the trigger x + 1 doesn’t match the term 1 + a.
There seem to be two approaches that would fix this.
The first approach would be to write a new matcher that encodes the partial match
to be extended by the matching iterators not as a simple binding of pattern variables
to equivalence classes but as an affine space of such bindings to be refined by the
iterator.
The second approach would be to introduce axioms for the commutativity and
associativity of the arithmetic operators so that the E-graph would contain many
more ground terms (
§2) that could be matched. In the example above, the equiv-
alence class of P(1 + a)would include P(a + 1). This method was used in the
Denali superoptimizer [Joshi et al. 2002], which uses Simplify-like techniques to
generate provably optimal machine code.
The first approach seems more complete than the second. Presumably, it would
find that P(a)isaninstance of P(x + 1) by the substitution x := a − 1, while
the second method, at least as implemented in Denali, is not so aggressive as to
introduce the E-node (a − 1) + 1 and equate it with the node for a.
But in the course of the ESC project, we never found that Simplify’s limitations in
using arithmetic information in the matcher were fatal to the ESC application. Also,
each approach contains at least a threat of debilitating combinatorial explosion, and
neither approach seems guaranteed to find enough matches to substantially increase
Simplify’s power. So we never implemented either approach.
Simplify: A Theorem Prover for Program Checking 397
Simplify transforms quantified formulas into data structures called matching
rules. A matching rule mr is a triple consisting of a body, mr.body, which is a
formula; a list of variables mr.vars;alist of triggers mr.triggers, where each trigger
is a list of one or more terms. There may be more than one trigger, since it may
be heuristically desirable to trigger instances of the quantified formula for more
than one ground term; a trigger may have more than one term, since it may be a
multi-trigger instead of a uni-trigger.
Simplify maintains a set of “asserted matching rules” as part of its context.
Semantically, the assertion of a matching rule mr is equivalent to the assertion
of (∀ mr.vars : mr.body). Heuristically, Simplify will use only those instances
θ(mr.body)ofthe matching rule such that for some trigger tr in mr.triggers, for
each constituent t of tr, θ (t)isrepresented in the E-graph.
There are three more topics in the story of quantifiers and matching rules.
—Matching and backtracking search: how the backtracking search makes use of
the set of asserted matching rules,
—Quantifiers to matching rules: how and when quantified formulas are turned into
matching rules and matching rules are asserted, and
—How triggers are matched in the E-graph.
These three topics are discussed in the next three sections.
5.2. M
ATCHING AND BACKTRACKING SEARCH.Inthis section, we describe
how the presence of asserted matching rules interacts with the backtracking search
in Sat.
We will make several simplifying assumptions throughout this section:
First, we assume that there is a global set of matching rules fixed for the whole
proof. Later, we will explain that the set of asserted matching rules may grow and
shrink in the course of the proof, but this doesn’t affect the contents of this section
in any interesting way.
Second, we assume that the body of every asserted matching rule is a clause, that
is, a disjunction of literals. We will return to this assumption in Section 5.3.
Third, we assume that we have an algorithm for enumerating substitutions θ that
are relevant to a given trigger tr. Such an algorithm will be presented in Section 5.4.1.
The high-level description of the interaction of searching and matching is very
simple: periodically during the backtracking search, Simplify performs a “round of
matching”, in which all relevant instances of asserted matching rules are constructed
and added to the clause set, where they become available for the subsequent search.
Before we present the detailed description, we make a few high-level points.
First, when Simplify has a choice between matching and case splitting, it favors
matching.
Second, Simplify searches for matches of rules only in the portion of the
E-graph that represents the literals that have been assumed true or false on the cur-
rent path (
§3.1). This may be a small portion of the E-graph, since there may be many
E-nodes representing literals that have been created but not yet been selected for a
case split. This heuristic, called the activation heuristic, ensures that Simplify will
to some extent alternate between case-splitting and matching, and therefore avoids
matching loops. Disabling this heuristic has a disastrous effect on performance
(see Section 9.8). To implement the heuristic, we maintain an active bit in every
E-node. When a literal is asserted, the active bit is set in the E-nodes that represent
398 D. DETLEFS ET AL.
the literal. More precisely, the bit is set in each E-node equivalent to any subterm
of any term that occurs in the literal. All changes to active bits are undone by Pop.
The policy of matching only in the active portion of the E-graph has one exception,
the “select-of-store” tactic described in Section 5.2.1 below.
Third, the first time a clause is created as an instance of a matching rule, we find
it worthwhile to refine it aggressively, by plunging (
§4.6). If a literal is found to
be untenable by plunging, it is deleted from the clause and also explicitly denied
(a limited version of the subsumption heuristic (
§3.2)). We also use refinement on
every clause in the clause set before doing any case split, but in this case we use
less aggressive refinement, by status and E-graph tests only.
Fourth, Simplify maintains a set of fingerprints of matches that have been found
on the current path. To see why, consider the case in which the matcher produces a
clause G and then deeper in the proof, in a subsequent round of matching, rediscov-
ers G.Itwould be undesirable to have two copies of G in the clause set. Therefore,
whenever Simplify adds to the clause set an instance θ(R.body)ofamatching rule
R by a substitution θ,italso adds the fingerprint of the pair (R,θ)toaset matchfp.
To filter out redundant instances, this set is checked as each match is discovered.
Insertions to matchfp are undone by Pop.
In general, a fingerprint is like a hash function, but is computed by a CRC
algorithm that provably makes collisions extremely unlikely [Rabin 1981]. To fin-
gerprint an instance θ of a matching rule m,weuse the CRC of the integer sequence
(i,θ(m.vars[1]).root.id,...θ(m.vars[m.vars.length]).root.id)
where i is the index of the matching rule in the set of matching rules. (The id
field of an E-node is simply a unique numeric identifier; see Section 7.1.) This
approach has the limitation that the root of a relevant equivalence class may have
changed between the time a match is entered in the fingerprint table and the time
an equivalent match is looked up, leading to the instantiation of matches that are in
fact redundant. We don’t think such a false miss happens very often, but we don’t
know for sure. To the extent that it does happen, it reduces the effectiveness the
fingerprint test as a performance heuristic, but doesn’t lead to any unsoundness or
incompleteness.
If a fingerprint collision (false hit) did occur, it could lead to incompleteness, but
not unsoundness. We attempted once to switch from 64-bit fingerprints to 32-bit
fingerprints, and found that this change caused incompleteness on our test suite, so
we retracted the change. We have never observed anything to make us suspect that
any collisions have occurred with 64-bit fingerprints.
Here is the place for a few definitions that will be useful later.
First, a redefinition: we previously followed tradition in defining a substitution
as a map from variables to terms, but from now on we will treat a substitution as
a map from variables to equivalence classes in the E-graph. Thus, θ(v)isproperly
an equivalence class and not a term; in contexts where a term is required, we can
take any term represented by the equivalence class.
We say that a substitution θ matches a term t to an E-node v if (1) the equalities
in the E-graph imply that θ(t)isequal to v, and (2) the domain of θ contains only
variables free in t.
We say that a substitution θ matches a list t
1
,...,t
n
of terms to a list v
1
,...,v
n
of E-nodes if (1) the equalities in the E-graph imply that θ(t
i
)isequal to v
i
, for
each i, and (2) the domain of θ contains only variables free in at least one of the t’s.
Simplify: A Theorem Prover for Program Checking 399
We say that a substitution θ matches a term t to the E-graph if there exists some
active (
§5.2) E-node v such that θ matches t to v.
We say that a substitution θ matches a list t
1
,...,t
n
of terms to the E-graph if
there exists some list v
1
,...,v
n
of active E-nodes such that θ matches t
1
,...,t
n
to
v
1
,...,v
n
.
These definitions are crafted so that the set of substitutions that match a trigger
to the E-graph is (1) finite and (2) does not contain substitutions that are essentially
similar to one another. Limiting the domain of a substitution to variables that appear
in the term or term list is essential for (1), and treating substitutions as maps to
E-graph equivalence classes instead of terms is essential for both (1) and (2).
Matching is added to the backtracking search from within the procedure Refine
(
§3.2). Recall that the purpose of Refine is to perform tactics, such as width reduction
and unit assertion, that have higher priority than case splitting. In addition to using
a Boolean to keep track of whether refinement is enabled, the new version of this
procedure uses a Boolean to keep track of whether matching has any possibility of
discovering anything new. Here is the code:
proc Refine() ≡
loop
if refuted or cls contains an empty clause then exit end;
if cls contains any unit clauses then
assert all unit clauses in cls;
enable refinement;
enable matching;
else if refinement enabled then
for each clause C in cls do
refine C by cheap tests (possibly setting refuted)
end;
disable refinement
else if matching enabled then
for each asserted matching rule M do
for each substitution θ
that matches some trigger in M.triggers to the E-graph do
let fp = fingerprint((M,θ)) in
if not fp ∈ matchfp then
add fp to matchfp;
let C = θ (M.body) in
refine C by plunging (possibly setting refuted);
add C to cls
end
end
end
end
end;
disable matching
else
exit
end
end
end
When Refine returns, either the context has become unsatisfiable (in which case
Sat backtracks) or Simplify’s unconditional inference methods have been exhausted,
in which case Sat performs a case split, as described previously.
400 D. DETLEFS ET AL.
There are two additional fine points to mention that are not reflected in the code
above.
First, Simplify distinguishes unit matching rules, whose bodies are unit clauses,
from non-unit matching rules, and maintains separate enabling Booleans for the
two kinds of rules. The unit rules are matched to quiescence before the nonunit
rules are tried at all. In retrospect, we’re not sure whether this distinction was worth
the trouble.
The second fine point concerns the consecutive matching round limit, which is a
limit on the number of consecutive rounds of non-unit matching that Simplify will
perform without an intervening case split. If the limit is exceeded, Simplify reports
a “probable matching loop” and aborts the proof.
5.2.1. The Matching Depth Heuristic. Matching also affects the choice of
which case split to perform.
We have mentioned the goal property (
§
3.5) and the score (§3.6) as criteria for
choosing case splits; an even more important criterion is the matching depth of a
clause. We define by mutual recursion a depth for every clause and a current depth
at any point on any path of the backtracking search: The current depth is initially
zero and in general is the maximum depth of any clause that has been split on in
the current path. The depth of all clauses of the original query, including defining
clauses introduced by proxies (
§3.3), is zero. The depth of a clause produced by
matching is one greater than the current depth at the time it was introduced by the
matcher.
Now we can give the rule for choosing a split: favor low depths; break depth ties
by favoring goal clauses; break depth-and-goal ties by favoring high scores.
Implementation note: At any moment, Simplify is considering clauses only of the
current depth and the next higher depth. Therefore, Simplify does not store the depth
of a clause as part of the clause’s representation, but instead simply maintains two
sets of clauses, the current clause set containing clauses of the current depth, and the
pending clause set containing clauses of the next higher depth. Only clauses in the
current clause set are candidates for case splitting. Clauses produced by matching
are added to the pending clause set. When the current clause set becomes empty,
Simplify increases the matching depth: the current set gets the pending set, and the
pending set gets the empty set.
An additional complexity is that some clauses get promoted, which reduces their
effective depth by one. Given the two-set implementation described in the previous
note, a clause is promoted simply by putting it into the current clause set instead
of the pending clause set. Simplify performs promotion for two reasons: merit
promotion and immediate promotion.
Merit promotion promotes a limited number of high scoring clauses. A promote
set of fingerprints of high-scoring clauses is maintained and used as follows. When-
ever Pop reduces the matching depth, say from d + 1tod, the clauses of depth
d + 1 (which are about to be removed by Pop) are scanned, and the one with the
highest score whose fingerprint is not already in the promote set has its fingerprint
added to the promote set. When choosing a case split, Simplify effectively treats
all clauses whose fingerprints are in the promote set as if they were in the current
clause set, and also increases their effective scores. Insertions to the promote set
are not undone by Pop,but the promote set is cleared whenever scores are renor-
malized. Also, there is a bound on the size of the promote set (defaulting to 10 and
Simplify: A Theorem Prover for Program Checking 401
settable by an environment variable); when adding a fingerprint to the promote set,
Simplify will, if necessary, delete the oldest fingerprint in the set to keep the size
of the set within the bound.
Immediate promotion promotes all instances of certain rules that are deemed
a priori to be important. Simplify’s syntax for quantified formulas allows the user
to specify that instances are to be added directly to the current clause set rather than
to the pending clause set. In our ESC application, the only quantified formula for
which immediate promotion is used is the non-unit select-of-store axiom:
(∀a, i, x, j : i = j ∨ select(store(a, i, x), j) = select(a, j)).
There is a bound (defaulting to 10 and settable by an environment variable) limiting
the number of consecutive case splits that Simplify will perform on immediately
promoted clauses in preference to other clauses.
The nonunit select-of-store axiom is so important that it isn’t surprising that
it is appropriate to treat it to immediate promotion. In fact, when working on
a challenging problem with ESC/Modula-3, we encountered a proof obligation
on which Simplify spent an unacceptable amount of time without succeeding,
and analysis revealed that on that problem, even immediate promotion was an
insufficiently aggressive policy. The best strategy that we could find to correct the
behavior was to add the select-of-store tactic, which searches for instances of the
trigger select(store(a, i, x), j)) even in the inactive portion of the E-graph. For
each such instance, the tactic uses the E-graph tests (
§4.6) to check whether the
current context implies either i = j or i = j, and if so, the application of select is
merged either with x or select(a, j)asappropriate. Because of the memory of this
example, Simplify enables the select-of-store tactic by default, although on the test
suites described in Section 9 the tactic has negligible performance effects.
In programming Simplify, our policy was to do what was necessary to meet the
requirements of the ESC project. One unfortunate consequence of this policy is
that the clause promotion logic became overly complicated. Merit promotion and
immediate promotion work as described above, and they are effective (as shown by
the data in Sections 9.6 and 9.7). But we now report somewhat sheepishly that as we
write we cannot find any examples for which the bounds on promote set size and
on consecutive splits on immediately promoted clauses are important, although
our dim memory is that we originally added those bounds in response to such
examples.
In summary, we feel confident that ordering case splits by depth is generally a
good idea, but exceptions must sometimes be made. We have obtained satisfac-
tory results by promoting high-scoring clauses and instances of the select-of-store
axiom, but a clean, simple rule has eluded us.
5.3. Q
UANTIFIERS TO MATCHING RULES.Inthis section, we describe how and
when quantified formulas get turned into matching rules.
We begin with a simple story and then describe the gory details.
5.3.1. Simple Story. We define a basic literal to be a nonproxy literal.
The query (
§2) is rewritten as follows: ⇒ and ⇔ are eliminated by using the
following equations
P ⇔ Q = (P ⇒ Q) ∧ (Q ⇒ P)
P ⇒ Q = (¬ P ∨ Q).
402 D. DETLEFS ET AL.
Also, all occurrences of ¬ are driven down to the leaves (i.e., the basic literals),
by using the following equations:
¬ (P ∧ Q) = (¬ P) ∨ (¬ Q)
¬ (P ∨ Q) = (¬ P) ∧ (¬ Q)
¬ ((∀x : P)) = (∃x : ¬ P)
¬ ((∃x : P)) = (∀x : ¬ P)
¬¬P = P.
Then existential quantifiers are eliminated by Skolemizing. That is, we replace each
subformula of the form (∃y : Q) with Q(y := f (x
1
,...,x
n
)), where f is a uniquely
named Skolem function and the x’s are the universally quantified variables in scope
where the subformula appears.
Finally, adjacent universal quantifiers are collapsed, using the rule
(∀x :(∀y : P)) = (∀x, y : P).
Thus, we rewrite the query into a formula built from ∧, ∨, ∀, and basic literals.
We say that the formula has been put into positive form.
The elimination rule for ⇔ can potentially cause an exponential explosion, but
in our application we have not encountered deep nests of ⇔ , and the rule has not
been a problem. The other elimination rules do not increase the size of the formula.
Once the formula has been put into positive form, we apply Sat to the formula
as described previously. This leads to a backtracking search as before, in which
basic literals and proxy literals are asserted in an attempt to find a path of assertions
that satisfies the formula. What is new is that the search may assert a universally
quantified formula in addition to a basic literal or proxy literal. Technically, each
universally quantified formula is embedded in a quantifier proxy which is a new
type of literal that can occur in a clause. Asserting a quantifier proxy causes the
universally quantified formula embedded in it to be converted to one or more
matching rules, and causes these rules to be asserted.
Thus, it remains only to describe how universally quantified positive formulas
are turned into matching rules.
To turn (∀x
1
,...,x
n
: P) into matching rules, we first rewrite P into CNF (true
CNF, not the equisatisfiable CNF (
§3.3) used in Sat). The reason for this is that
clausal rules are desirable, since if the body of a rule is a clause, we can apply
width reduction and clause elimination (
§3.2) to instances of the rule. By rewriting
P into a conjunction of clauses, we can distribute the universal quantifier into the
conjunction and produce one clausal rule for each clause in the CNF for P.For
example, for the quantified formula
(∀x : P(x) ⇒ (Q(x) ∧ R(x))),
we rewrite the body as a conjunction of two clauses and distribute the quantifier
into the conjunction to produce two clausal rules, as though the input had been
(∀x : P(x) ⇒ Q(x)) ∧ (∀x : P(x) ⇒ R(x)).
Then, for each rule, we must choose one or more triggers.
Simplify’s syntax for a universal quantifier allows the user to supply an explicit
list of triggers. If the user does not supply an explicit list of triggers, then Simplify
makes two tries to select triggers automatically.
Simplify: A Theorem Prover for Program Checking 403
First try: Simplify makes a unitrigger (
§
5.1) out of any term that (1) occurs in the
body outside the scope of any nested quantifier, (2) contains all the quantified vari-
ables, (3) passes the “loop test”, (4) is not a single variable, (5) is not “proscribed”,
and (6) contains no proper subterm with properties (1)–(5).
The loop test is designed to avoid infinite loops in which a matching rule creates
larger and larger instances of itself—that is, matching loops involving a single rule.
A term fails the loop test if the body of the quantifier contains a larger instance of
the term. For example, in
(∀x : P( f (x), f (g(x)))),
the term f (x)fails the loop test, since the larger term f (g(x)) is an instance of f (x)
via x := g(x). Thus, in this case, f (x) will not be chosen as a trigger, and Simplify
will avoid looping forever substituting x := t, x := g(t), x := g(g(t)), ....
The proscription condition (4) is designed to allow the user to exclude certain
undesirable triggers that might otherwise be selected automatically. Simplify’s input
syntax for a universal quantifier also allows the user to supply an explicit list of
terms that are not to be used as triggers; these are the terms that are “proscribed”.
Second try: If the first try doesn’t produce any unitriggers, then Simplify tries
to create a multitrigger (
§5.1). It attempts to select as the constituents (§5.1) a
reasonably small set of nonproscribed terms that occur in the body outside the
scope of any nested quantifier and that collectively contain all the pattern variables
and (if possible) overlap somewhat in the pattern variables that they contain. If
the second try succeeds, it produces a single multitrigger. Because there could be
exponentially many plausible multitriggers, Simplify selects just one of them.
The ESC/Modula-3 and ESC/Java annotation languages allow users to in-
clude quantifiers in annotations since the expressiveness provided by quantifiers is
occasionally required. But these tools don’t allow users to supply triggers for the
quantifiers, since the ESC philosophy is to aim for highly automated checking.
However, the background predicates (
§2) introduced by the ESC tools include many
explicit triggers, and these triggers are essential. In summary, for simple quantified
assertions (like “all elements of array A are non-null” or “all allocated readers have
non-negative count fields”) automatic trigger selection seem to work adequately.
But when quantifiers are used in any but the simplest ways, Simplify’s explicit
trigger mechanism is required.
5.3.2. Gory Details. A disadvantage of the simple story is that the work of
converting a particular quantified formula into a matching rule will be repeated
many times if there are many paths in the search that assert that formula. To avoid
this disadvantage, we could be more eager about converting quantified formulas
into matching rules; for example, we could convert every quantified formula into a
matching rule as part of the original work of putting the query into positive form.
But this maximally eager approach has disadvantages as well; for example, it may
do the work of converting a quantified formula into a matching rule even if the
backtracking search finds a satisfying assignment (
§2) without ever asserting the
formula at all. Therefore, Simplify takes an intermediate approach, not totally eager
but not totally lazy, either.
When the query is put into positive form, each outermost quantifier is converted
into one or more matching rules. These matching rules are embedded in the quan-
tifier proxy, instead of the quantified formula itself. Thus, the work of building
404 D. DETLEFS ET AL.
matching rules for outermost quantifiers is performed eagerly, before the back-
tracking search begins. However, universal quantifiers that are not outermost, but
are nested within other universal quantifiers, are simply treated as uninterpreted
literals: their bodies are not put into positive form and no quantifier proxies are
created for them.
When a matching rule corresponding to an outer universal quantifier is instanti-
ated, its instantiated body is asserted, and at this point the universal quantifiers that
are outermost in the body are converted into quantifier proxies and matching rules.
Let us say that a formula is in positive seminormal form if the parts of it outside
universal quantifiers are in positive normal form. Then, in general, the bodies of all
matching rules are in positive seminormal form, and when such a body is instantiated
and asserted, the outermost quantifiers within it are turned into matching rules
(which work includes the work of putting their bodies into positive seminormal
form).
That was the first gory detail. We continue to believe that the best approach to
the work of converting quantifiers into matching rules is somewhere intermediate
between the maximally eager and maximally lazy approaches, but we don’t have
confidence that our particular compromise is best. We report this gory detail for
completeness rather than as a recommendation.
Next we must describe nonclausal rules, that is, matching rules whose bodies are
formulas other than clauses. In the simple story, the bodies of quantified formulas
were converted to true CNF, which ensured that all rule bodies would be clauses.
Clausal rules are desirable, but conversion to true CNF can cause an exponential
size explosion, so we need an escape hatch. Before constructing the true CNF for
the body of a quantified formula, Simplify estimates the size of the result. If the
estimate is large, it produces a matching rule whose body is the unnormalized
quantifier body. The trigger for such a rule is computed from the atomic formulas
(
§2) in the body just as if these atomic formulas were the elements of a clause.
When a nonclausal rule is instantiated, the instantiation of its body is asserted and
treated by the equisatisfiable CNF methods described previously.
Recall that Simplify favors case splits with low matching depth (
§5.2.1). This
heuristic prevents Simplify from fruitlessly searching all cases of the latest instance
of a rule before it has finished the cases of much older clauses. We have described
the heuristic for clausal rules. Nonclausal rules introduce some complexities; for
example, the equisatisfiable CNF for the instance of the rule body may formally be
a unit clause, consisting of a single proxy; Simplify must not let itself be tricked
into asserting this proxy prematurely.
Finally, Simplify’s policy of treating nested quantifiers as uninterpreted literals
can cause trigger selection to fail. For example, in the formula
(∀x, y : P(x) ⇒ (∀z : Q(x, y) ∧ Q(z, x)))
both tries to construct a trigger will fail. Instead of failing, it might be better to
in such a case to move ∀ z outwards or ∀ y inwards. But we haven’t found trigger
selection failure to be a problem in our applications. (Of course, if we didn’t collapse
adjacent universal quantifiers, this problem would arise all the time.)
In the case of nested quantifiers, the variables bound by the outer quantifier may
appear within the inner quantifier. In this case, when the matching rule for the
outer quantifier is instantiated, an appropriate substitution must be performed on
Simplify: A Theorem Prover for Program Checking 405
the matching rule corresponding to the inner quantifier. For example, consider
(∀x : P(x) ⇒ (∀y : Q(x, y))).
If the outer rule is instantiated with x := E, then the substitution x := E is
performed on the body of the outer rule, which includes the inner matching rule.
Thus, the body of the inner rule will be changed from Q(x, y)toQ(E, y).
Our view that nested quantifiers should produce nested matching rules should
be contrasted with the traditional approach of putting formulas into prenex form
by moving the quantifiers to the outermost level. Our approach is less pure, but
it allows for more heuristic control. For example, the matching rule produced by
asserting the proxy for
(∀x : ¬ P(x) ∨ (∀y : Q(x, y))) (8)
is rather different from the matching rule produced by asserting the proxy for the
semantically equivalent
(∀x, y : ¬ P(x) ∨ Q(x, y)). (9)
In the case of (8), the matching rule will have trigger P(x), and if it is instantiated
and asserted with the substitution x := E , the clause ¬ P(E) ∨ (∀y : Q(E , y)) will
be asserted. If the case analysis comes to assert the second disjunct, the effect will
be to create and assert a matching rule for the inner quantifier.
The semantically equivalent formula (9) affects the proof very differently: a single
matching rule with trigger Q(x, y)would be produced. It is a little disconcerting
when semantically equivalent formulas produce different behaviors of Simplify;
on the other hand, it is important that the input language be expressive enough to
direct Simplify towards heuristically desirable search strategies.
Two final details: First, some transformations are performed as the quanti-
fied symbolic expression is transformed into matching rules. One straightforward
transformation is the elimination of unused quantified variables. For example, if
x does not occur free in P, then (∀x, y : P)istransformed into (∀y : P). Less
straightforward are the one-point rules,ofwhich there are several. The simplest is
the replacement of (∀x : x = T ∨ P(x)) by P(x := T ). The one-point rules were
important to an early version of ESC/Modula-3, whose treatment of data abstrac-
tion produce fodder for the rules. They are also useful for proving trivialities like
(∃x : x = 3). But neither the later versions of ESC/Modula-3 nor ESC/Java seem
to exercise the one-point rules. Second, some of the transformations described in
this subsection may be reenabled by others. For example, distributing ∀ over ∧ may
create new opportunities for merging adjacent universal quantifiers or eliminating
unused quantified variables. Simplify continues performing these transformations
until none is applicable.
5.4. H
OW TRIGGERS ARE MATCHED. Recall that to use the asserted matching
rules, the backtracking search performs an enumeration with the following structure:
for each asserted matching rule M do
for each substitution θ
that matches (
§5.2) some trigger in M.triggers to the E-graph do
...
end
end
406 D. DETLEFS ET AL.
In this section, we will describe how the substitutions are enumerated.
5.4.1. Matching Iterators. We present the matching algorithms as mutually
recursive iterators in the style of CLU [Liskov et al. 1981]. Each of the iterators
takes as arguments one or more terms together with a substitution and yields all ways
of extending the substitution that match the term(s). They differ in whether a term
or term-list is to be matched, and in whether the match is to be to anywhere in the
E-graph or to a specific E-node or list of E-nodes. When the matches are to arbitrary
E-nodes, the terms are required to be proper (i.e., not to be single variables).
We say that two substitutions θ and φ conflict if they map some variable to
different equivalence classes.
Here are the specifications of the four iterators:
iterator MatchTrigger(t : list of proper terms,θ : substitution)
yields all extensions θ ∪ φ of θ such that
θ ∪ φ matches (
§5.2) the trigger t in the E-graph and
φ does not conflict with θ.
iterator MatchTerm(t : proper term,θ : substitution)
yields all extensions θ ∪ φ of θ such that
θ ∪ φ matches t to some active E-node in the E-graph and
φ does not conflict with θ.
iterator Match(t : term, v : E-node,θ : substitution)
yields all extensions θ ∪ φ of θ such that
θ ∪ φ matches t to v, and
φ does not conflict with θ.
iterator MatchList(t : list of terms, v : list of E-node’s,θ : substitution)
yields all extensions θ ∪ φ of θ such that
θ ∪ φ matches the term-list t to the E-node-list v, and
φ does not conflict with θ.
Given these iterators, a round of matching is implemented approximately as
follows:
for each asserted matching rule M do
for each trigger tr in M.triggers do
for each θ in MatchTrigger(tr, {}) do
...
end
end
end
where {}denotes the empty substitution.
We now describe the implementation of the iterators.
The implementation of MatchTrigger is a straightforward recursion on the list:
iterator MatchTrigger(t : list of proper terms,θ : substitution) ≡
if t is empty then
yield(θ)
else
for each φ in MatchTerm(hd(t),θ) do
for each ψ in MatchTrigger(tl(t),φ) do
Simplify: A Theorem Prover for Program Checking 407
yield(ψ)
end
end
end
end
(We use the operators “hd” and “tl” on lists: hd(l)isthe first element of the list
l, and tl(l)isthe list of remaining elements.)
The implementation of MatchTerm searches those E-nodes in the E-graph with
the right label, testing each by calling MatchList:
iterator MatchTerm(t : proper term,θ : substitution) ≡
let f, args be such that t = f (args) in
for each active E-node v labeled f do
for each φ in MatchList(args, children[v],θ) do
yield(φ)
end
end
end
end
The iterator MatchList matches a list of terms to a list of E-nodes by first
finding all substitutions that match the first term to the first E-node, and then
extending each such substitution in all possible ways that match the remaining
terms to the remaining E-nodes. The base case of this recursion is the empty list,
which requires no extension to the substitution; the other case relies on Match to
find the substitutions that match the first term to the first E-node:
iterator MatchList(t : list of terms, v : list of E-node’s,θ : substitution) ≡
if t is the empty list then
yield(θ)
else
for each φ in Match(hd(t), hd(v),θ) do
for each ψ in MatchList(tl(t), tl(v),φ) do
yield(ψ)
end
end
end
end
The last iterator to be implemented is Match, which finds all ways of matching
a single term to a single E-node. It uses recursion on the structure of the term.
The base case is that the term consists of a single pattern variable. In this case
there are three possibilities: either the substitution needs to be extended to bind the
pattern variable appropriately, or the substitution already binds the pattern variable
compatibly, or the substitution already contains a conflicting binding for the pattern
variable. If the base case does not apply, the term is an application of a function
symbol to a list of smaller terms. (We assume that constant symbols are represented
as applications of function symbols to empty argument lists, so constant symbols
don’t occur explicitly as a base case.) To match a proper term to an E-node, we
must enumerate the equivalence class of the E-node, finding all E-nodes that are in
the desired equivalence class and that have the desired label. For each such E-node,
we use MatchList to find all substitutions that match the argument terms to the list
408 D. DETLEFS ET AL.
of E-node arguments:
iterator Match(t : term, v : E-node,θ : substitution) ≡
if t is a pattern variable then
if t is not in the domain of θ then
yield(θ ∪{(t, v’s equivalence class)})
else if θ(t) contains v then
yield(θ)
else
skip
end
else
let f, args be such that t = f (args) in
for each E-node u such that u is equivalent to v and
f is the label of u do
for each φ in MatchList(args, children[u],θ) do yield(φ) end
end
end
end
end
We conclude this section with a few additional comments and details.
There are two places where E-nodes are enumerated: in Match, when enumerating
those E-nodes u that have the desired label and equivalence class, and in MatchTerm,
when enumerating candidate E-nodes that have the desired label. In both cases, it
is important to enumerate only one E-node from each congruence class (
§4.2),
since congruent E-nodes will produce the same matches. Section 7 shows how to
do this.
In Match, when enumerating those E-nodes u that have the desired label and
equivalence class, the E-graph data structure allows two possibilities: enumerating
the E-nodes with the desired label and testing each for membership in the desired
equivalence class, or vice-versa. A third possibility is to choose between these
two based on the whether the equivalence class is larger or smaller than the set
of E-nodes with the desired label. Our experiments have not found significant
performance differences between these three alternatives.
Simplify uses an optimization that is not reflected in the pseudo-code written
above: In MatchTerm(t,θ), it may be that θ binds all the pattern variables occurring
in t.Inthis case, MatchTerm simply checks whether an E-node exists for θ(t ),
yields θ if it does, and yields nothing if it doesn’t.
When Simplify constructs a trigger constituent from an S-expression, subterms
that contain no quantified variables (or whose quantified variables are all bound
by quantifiers at outer levels) are generally interned (
§4.5) into E-nodes at trigger
creation time. This produces an extra base case in the iterator Match:ift is an E-node
w, then Match yields θ if w is equivalent to v, and yields nothing otherwise.
5.4.2. The Mod-Time Matching Optimization. Simplify spends much of its time
matching triggers in the E-graph. The general nature of this matching process was
described above. In this section and the next, we describe two important optimiza-
tions that speed up matching: the mod-time optimization and the pattern-element
optimization.
To describe the mod-time optimization, we temporarily ignore multitriggers.
Simplify: A Theorem Prover for Program Checking 409
Roughly speaking, a round of matching performs the following computation:
for each matching rule with uni-trigger T and body B do
for each active E-node V do
for each substitution θ such that θ (T )isequivalent to V do
Assert(θ(B))
end
end
end
Consider two rounds of matching that happen on the same path in the search tree.
We find that in this case, it often happens that for many pairs (T, V ), no assertions
performed between the two rounds changed the E-graph in any way that affects
the set of instances of trigger T equivalent to E-node V , and consequently the set
of substitutions that are discovered on the first round of matching is identical to the
set discovered on the second round. In this case, the work performed in the second
round for (T, V )ispointless, since any instances that it could find and assert have
already been found and asserted in the earlier round.
The mod-time optimization is the way we avoid repeating the pointless work. The
basic idea is to introduce a global counter gmt that records the number of rounds of
matching that have occurred on the current path (
§3.1). It is incremented after each
round of matching, and saved and restored by Push and Pop.Wealso introduce a
field E.mt for each active E-node E, which records the value of gmt the last time
any proper descendant of E wasinvolved in a merge in the E-graph. The idea is to
maintain the “mod-time invariant”, which is
for all matching rules mr with trigger T and body B,
for all substitutions θ such that θ (T )isrepresented in the E-graph,
either θ(B) has already been asserted, or
the E-node V that represents θ(T ) satisfies V.mt = gmt
Given the mod-time invariant, the rough version of a matching round becomes:
for each matching rule with trigger T and body B do
for each active E-node V such that v.mt = gmt do
for each substitution θ such that θ (T )isequivalent to V do
Assert(θ(B))
end
end
end;
gmt := gmt + 1
The reason that the code can ignore those E-nodes V for which V.mt = gmt is
that the invariant implies that such an E-node matches a rule’s trigger only if the
corresponding instance of the rule’s body has already been asserted. The reason that
gmt := gmt + 1 maintains the invariant is that after a matching round, all instances
of rules that match have been asserted.
The other place where we have to take care to maintain the invariant is when
E-nodes are merged. When two E-nodes V and W are merged, we must enumerate
all E-nodes U such that the merge might change the set of terms congruent to U in
the graph. We do this by calling UpdateMT(V ) (or, equivalently, UpdateMT(W ))
410 D. DETLEFS ET AL.
immediately after the merge, where this procedure is defined as follows:
proc UpdateMT(V : E-node) ≡
for each P such that P is a parent (
§4.2) of some E-node equivalent to V do
if P.mt < gmt then
Push onto the undo stack the triple ("UndoUpdatemt", P, P.mt);
P.mt := gmt;
UpdateMT(P)
end
end
end
In the case of circular E-graphs, the recursion is terminated because the second
time an E-node is visited, its mod-time will already have been updated.
So much for the basic idea of the mod-time optimization. We have three details
to add to the description.
First, we need to handle the case of rules with multitriggers. This is an awkward
problem. Consider the case of a multitrigger p
1
,..., p
n
, and suppose that merges
change the E-graph so that there is a new instance θ of the multitrigger; that is,
so that each θ(p
i
)isrepresented in the new E-graph, but for at least one i, the
term θ ( p
i
)was not represented in the old E-graph. Then, for some i, the E-node
representing θ( p
i
) has its mod-time equal to gmt,but this need not hold for all i .
Hence, when the matcher searches for an instance of the multitrigger by extending
all matches of p
1
in ways that match the other p’s,itcannot confine its search to new
matches of p
1
. Therefore, the matcher considers each multi-trigger p
1
,..., p
n
n
times, using each constituent p
i
in turn as a gating term. The gating term is matched
first, against all E-nodes whose mod-times are equal to gmt, and any matches found
are extended to cover the other constituent in all possible ways. In searching for
extensions, the mod-times are ignored.
Second, it is possible for matching rules to be created dynamically during the
course of a proof. If we tried to maintain the mod-time invariant as stated above
when a new rule was created, we would have to match the new rule immediately
or reset the mt fields of all active E-nodes, neither of which is attractive. Therefore,
we give each matching rule a Boolean field new which is true for matching rules
created since the last round of matching on the current path, and we weaken the
mod-time invariant by limiting its outer quantification to apply only to matching
rules that are not new. During a round of matching, new matching rules are matched
against all active E-nodes regardless of their mod-times.
Third, as we mentioned earlier, the matcher restricts its search to the “active”
portion of the E-graph. We therefore must set V.mt := gmt whenever an E-node V
is activated, so that the heuristic doesn’t lead the matcher to miss newly activated
instances of the trigger.
Combining these three details, we find that the detailed version of the mod-time
optimization is as follows. The invariant is:
for all matching rules mr with trigger T
1
,...,T
n
and body B
either mr.new or
for all substitutions θ such that each θ (T
i
)isrepresented
by an active E-node in the E-graph,
either θ(B) has already been asserted, or
there exists an i such that the E-node V that represents
θ(T
i
) satisfies V .mt = gmt.
Simplify: A Theorem Prover for Program Checking 411
The algorithm for a round of matching that makes use of this invariant is:
for each matching rule mr do
if mr.new then
for each tr in mr.triggers do
for each θ such that (∀t ∈ tr : θ(t)exists and is active) do
Assert(θ(mr.body))
end
end;
mr.new := false
else
for each tr in mr.triggers do
for each t in tr do
for each active E-node V such that v.mt = gmt do
for each φ such that φ(t )isequivalent to V do
for each extension θ of φ such that
(∀q ∈ tr : θ (q)exists and is active) do
Assert(θ(mr.body))
end
end
end
end
end
end
end;
gmt := gmt + 1
To maintain the invariant, we call UpdateMT(V ) immediately after an equiva-
lence class V is enlarged by a merge, we set V .mt := gmt whenever an E-node V
is activated, and we set mr.new := true whenever a matching rule mr is created.
More details:
Although we have not shown this in our pseudo-code, the upward recursion in
UpdateMT can ignore inactive E-nodes.
Forany function symbol f , the E-graph data structure maintains a linked list of
all E-nodes that represent applications of f . The matcher traverses this list when
attempting to match a trigger term that is an application of f . Because of the mod-
time optimization, only those list elements with mod-times equal to gmt are of
interest. We found that it was a significant time improvement to keep this list sorted
in order of decreasing mod-times. This is easily done by moving an E-node to the
front of its list when its mod-time is updated. In this case, the record pushed onto the
undo stack must include the predecessor of the node whose mod-time is changing.
5.4.3. The Pattern-Element Matching Optimization. The mod-time optimiza-
tion speeds matching by reducing the number of E-nodes examined. The pattern-
elements optimization speeds matching by reducing the number of triggers
considered.
Consider again two rounds of matching that happen on the same path in the
search tree. We find that in this case, it often happens that for many triggers T ,
no assertions performed between the two rounds changed the E-graph in any way
that affects the set of instances of T present in the E-graph. In this case, the work
performed in the second round for the trigger T is pointless, since any instances
that it finds have already been found in the earlier round.
The pattern-element optimization is a way of avoiding the pointless work. The
basic idea is to detect the situation that a modification to the E-graph is not relevant
412 D. DETLEFS ET AL.
to a trigger, in the sense that the modification cannot possibly have caused there to be
any new instances of the trigger in the E-graph. A round of matching need consider
only those triggers that are relevant to at least one of the modifications to the E-graph
that have occurred since the previous round of matching on the current path.
There are two kinds of modifications to be considered: merges of equivalence
classes and activations of E-nodes. To begin with, we will consider merges only.
There are two ways that a merge can be relevant to a trigger. To explain these ways,
we begin with two definitions.
A pair of function symbols ( f, g)isaparent–child element of a trigger if the
trigger contains a term of the form
f (...,g(...),...),
that is, if somewhere in the trigger, an application of g occurs as an argument of f .
A pair of (not necessarily distinct) function symbols ( f, g)isaparent–parent
element of a trigger for the pattern variable x if the trigger contains two distinct
occurrences of the pattern variable x, one of which is in a term of the form
f (...,x,...),
and the other of which is in a term of the form
g(...,x,...),
that is, if somewhere in the trigger, f and g are applied to distinct occurrences of
the same pattern variable.
Forexample, ( f, f )isaparent–parent element for x of the trigger f (x, x), but
not of the trigger f (x).
The first case in which a merge is relevant to a trigger is “parent–child” relevance,
in which, for some parent–child element ( f, g)ofthe trigger, the merge makes some
active application of g equivalent to some active argument of f .
The second case is “parent–parent” relevance, in which for some parent–parent
element ( f, g)ofthe trigger, the merge makes some active argument of f equivalent
to some active argument of g.
We claim that a merge that is not relevant to a trigger in one of these two ways
cannot introduce into the E-graph any new instances of the trigger. We leave it to
the reader to persuade himself of this claim.
This claim justifies the pattern-element optimization. The basic idea is to maintain
two global variables, gpc and gpp, both of which contain sets of pairs of function
symbols. The invariant satisfied by these sets is:
for all matching rules mr with trigger P and body B
for all substitutions θ such that θ (P)isrepresented in the E-graph
θ(B) has already been asserted, or
gpc contains some parent-child element of P,or
gpp contains some parent-parent element of P.
To maintain this invariant, we add pairs to gpc and/or to gpp whenever a merge
is performed in the E-graph.
To take advantage of the invariant, a round of matching simply ignores those
matching rules whose triggers’ pattern elements have no overlap with gpc or gpp.
After a round of matching, gpc and gpp are emptied.
Simplify: A Theorem Prover for Program Checking 413
In addition to merges, the E-graph changes when E-nodes are activated. The
rules for maintaining gpc and gpp when E-nodes are activated are as follows: when
activating an E-node V labeled f , Simplify
(1) adds a pair ( f, g)togpc for each active argument of V that is an application of
g,
(2) adds a pair (h, f )togpc for each active E-node labeled h that has V as one of
its arguments, and
(3) adds a pair ( f, k)togpp for each k that labels an active E-node which has any
arguments in common with the arguments of V .
Activation introduces a new complication: activating an E-node can create a
new instance of a trigger, even though it adds none of the trigger’s parent–parent or
parent–child elements to gpp or gpc.Inparticular, this can happen in the rather trivial
case that the trigger is of the form f (x
1
,...,x
n
) where the x’s are distinct pattern
variables, and the E-node activated is an application of f . (In case of a multitrigger,
any constituent of this form can suffice, if the pattern variables that are arguments
to f don’t occur elsewhere in the trigger). To take care of this problem, we define
a third kind of pattern element and introduce a third global set.
A function symbol f is a trivial parent element of a trigger if the trigger consists
of an application of f to distinct pattern variables, or the trigger is a multitrigger
one of whose constituents is an application of f to distinct pattern variables that
do not occur elsewhere in the multitrigger.
We add another global set gp and add a fourth alternative to the disjunction in
our invariant:
either θ(B) has already been asserted, or
gpc contains some parent-child element of P,or
gpp contains some parent-parent element of P,or
gp contains some trivial parent element of P.
The related changes to the program are: we add f to gp when activating an
application of f ,weempty gp after every round of matching, and a round of
matching does not ignore a rule if the rule’s trigger has any trivial parent element
in common with gp.
5.4.4. Implementation of the Pattern-Element Optimization. We must imple-
ment the pattern-element optimization carefully if it is to repay more than it costs.
Since exact set operations are expensive, we use approximate sets, which are like
real sets except that membership and overlap tests can return false positives.
First, we consider sets of function symbols (like gp). We fix a hash function
hash whose domain is the set of function symbols and whose range is [0..63] (for
our sixty-four bit machines). The idea is that the true set S of function symbols is
represented approximately by a word that has bit hash(s) set for each s ∈ S, and
its other bits zero. To test if u is a member of the set, we check if bit hash(u)isset
in the word; to test if two sets overlap, we test if the bitwise AND of the bit vectors
is non-zero.
Next we consider sets of pairs of function symbols (like gpc and gpp). In this
case, we use an array of 64 words; for each ( f, g)inthe set, we set bit hash(g)of
word hash( f ). When adding an unordered pair ( f, g)togpp,weadd either ( f, g)
or (g, f )tothe array representation, but not necessarily both. At the beginning of
414 D. DETLEFS ET AL.
a round of matching, when gpp is about to be read instead of written, we replace it
with its symmetric closure (its union with its transpose).
For sparse approximate pair sets, like the set of parent–parent or parent–child
elements of a particular matching rule, we compress away the empty rows in the
array, but we don’t do this for gpp or gpc.
With each equivalence class root q in the E-graph, we associate two approximate
sets of function symbols: q.lbls and q.plbls, where q.lbls is the set of labels of active
E-nodes in q’s class and q.plbls is the set of labels of active parents of E-nodes in
Q’s class. The code for merging two equivalence classes, say the class with root r
into the class with root q,isextended with the following:
proc UpdatePatElems(q, r : E-node) ≡
gpc := gpc ∪ r.plbls × q.lbls ∪ r.lbls × q.plbls;
gpp := gpp ∪ r.plbls × q.plbls;
Push onto the undo stack the triple ("UndoLabels", q, q.lbls);
q.lbls := q.lbls ∪ r.lbls;
Push onto the undo stack the triple ("UndoPLabels", q, q.plbls);
q.plbls := q.plbls ∪ r.plbls
end
The operations on r.lbls and r.plbls must be undone by Pop. This requires saving
the old values of the approximate sets on the undo stack (
§3.1). Since Simplify does
a case split only when matching has reached quiescence, gpc and gpp are always
empty when Push is called. Thus, Pop can simply set them to be empty, and there
is no need to write an undo record when they are updated.
We also must maintain lbls, plbls, gpc, gpp, and gp when an E-node is activated,
butweomit the details.
In addition to function symbols and pattern variables, a trigger can contain
E-nodes as a representation for a specific ground term (
§2). Such E-nodes are
called trigger-relevant.For example, consider the following formula:
(∀x : P( f (x)) ⇒ (∀y : g(y, x) = 0)).
The outer quantification will by default have the trigger f (x). When it is matched,
let V be the E-node that matches x. Then a matching rule will be created and
asserted that represents the inner quantification. This rule will have the trigger
g(y, V ), which contains the pattern variable y, the function symbol g, and the
constant trigger-relevant E-node V .For the purpose of computing gpc and the
set of parent–child elements of a trigger, we treat each trigger-relevant E-node as
though it were a nullary function symbol. For example, after the matching rule
corresponding to the inner quantification is asserted, then a merge of an argument
of g with an equivalence class containing V would add the parent–child element
(g, V )togpc.Trigger-relevant E-nodes also come up without nested quantifiers, if
a trigger contains ground terms (terms with no occurrences of pattern variables),
such as
NIL or Succ(0).
5.4.5. Mod-Times and Pattern Elements and Multitriggers. There is one final
optimization to be described that uses pattern elements and mod-times. Recall from
Section 5.4.2 that, when matching a multitrigger, each constituent of the multitrigger
is treated as a gating term (to be matched only against E-nodes with the current mod-
time; each match of the gating term is extended to match the other constituents of the
multitrigger, without regard to mod-times). The need to consider each constituent
Simplify: A Theorem Prover for Program Checking 415
of the multitrigger as a gating term is expensive. As a further improvement, we will
show how to use the information in the global pattern-element sets to reduce the
number of gating terms that need to be considered when matching a multitrigger.
As a simple example, consider the multitrigger
f (g(x)), h(x).
An E-graph contains an instance of this multitrigger if and only if there exist E-nodes
U , V , and W (not necessarily distinct) satisfying the following conditions:
(1) f (U)isactive,
(2) g(V )isactive,
(3) h(W )isactive,
(4) U is equivalent to g(V ), and
(5) V is equivalent to W .
Now suppose that, when this multitrigger is considered for matching, gpc =
{( f, g)} and gpp is empty. For any new instance (U, V, W )ofthe multitrigger, one
of the five conditions must have become true last. It cannot have been condition
(2), (3), or (5) that became true last, since any modification to the E-graph that
makes (2), (3), or (5) become the last of the five to be true also introduces (g, h)
into gpp. Consequently, for each new instance of the multitrigger, the modification
to the E-graph that made it become an instance (the enabling modification) must
have made condition (1) or (4) become the last of the five to be true. However, any
modification to the E-graph that makes (1) or (4) become the last of the five to be
true also updates the mod-time of the E-node f (U). Consequently, in this situation,
it suffices to use only f (g(x)) and not h(x)asagating term.
As a second example, consider the same multitrigger, and suppose that when the
multitrigger is considered for matching, gpc and gp are empty and gpp ={(g, h)}.
In this case, we observe that the last of the five conditions to be satisfied for any
new instance (U, V, W )ofthe multitrigger cannot have been (1), (2), or (4), since
satisfying (1), (2), or (4) last would add ( f, g)togpc; nor can it have been (3) that
was satisfied last, since satisfying (3) last would add h to gp. Therefore, the enabling
modification must have satisfied condition (5), and must therefore have updated the
mod-time of both f (U) and g(V ). Consequently, in this situation, we can use either
f (g(x)) or h(x)asagating term; there is no reason to use both of them.
As a third example, consider the same multitrigger, and suppose that when the
multitrigger is considered for matching, gpc ={( f, g)}, gpp ={(g, h)}, and gp is
empty. In this case, the enabling modification must either have satisfied condition 4
last, updating the mod-time of f (U), or have satisfied condition (5) last, updating
the mod-times of both f (U ) and h(W ). Consequently, in this situation, it suffices
to use only f (g(x)) as the gating term; it would not suffice to use only h(x).
In general, we claim that when matching a multitrigger p
1
,..., p
n
,itsuffices to
choose a set of gating terms such that
(1) Each p
i
is included as a gating term if it contains any parent–child element in
gpc;
(2) For each x
i
, g, h such that (g, h)isingpp and the trigger has (g, h)asaparent–
parent element for x
i
, the set of gating terms includes some p
j
that contains
pattern variable x
i
; and
416 D. DETLEFS ET AL.
(3) Each p
i
is included as a gating term if it has the form f (v
1
,...,v
k
) where the
v’s are pattern variables, f is in gp, and for each v
j
, either the trigger has no
parent–parent elements for v
j
or some parent–parent element for v
j
is in gpp.
We now justify the sufficiency of the three conditions.
We begin with some definitions. We define a term to be of unit depth if it has no
parent–child elements. We define a multitrigger to be tree-like if none of its pattern
variables occurs more than once within it. We define the shape of a multitrigger
T as the tree-like multitrigger obtained from T be replacing the i th occurrence of
each pattern variable x with a uniquely inflected form x
i
.For example, the shape
of f (x, x)is f (x
1
, x
2
) Matching a multitrigger is equivalent to matching its shape,
subject to the side conditions that all inflections of each original pattern variable are
matched to the same equivalence class. The enabling modification to the E-graph
that creates a match θ to a multitrigger must either have created a match of its shape
or satisfied one of its side conditions. We consider these two cases in turn.
If the enabling modification created a match of the shape of the multitrigger, then
it must have created a match of some constituent (
§5.1) of that trigger. There are two
subcases. If the constituent is not of unit depth, then the enabling modification must
have added to gpc a parent–child element of some constituent of the multitrigger,
and updated the mod-time of the E-node matched by that constituent. By condition
(1), therefore, the instance will not be missed. If the constituent is of unit depth,
then the enabling modification must be the activation of the E-node matched by the
constituent. In this case, the enabling modification must have added the constituent’s
function symbol gp and, for each pattern variable x that occurs in the constituent
and that occurs more than once in the entire multitrigger, it must have added some
parent–parent element for x to gpp.Bycondition (3), therefore, the instance will
not be missed.
If the enabling modification created a match by satisfying one of the side con-
ditions, then it must have added to gpp a parent–parent element for some pattern
variable, and updated the mod-times of the all the E-nodes matched by constituents
containing that variable. By condition (2), therefore, the instance will not be missed.
To construct a set of gating terms, Simplify begins with the minimum set that
satisfies conditions (1) and (3), and then for each g, h, x
i
as required by condition
(2), the set of gating terms is expanded if necessary. The final result will depend on
the order in which the different instances of condition (2) are considered, and may
not be of minimum size, but in practice this approach reduces the number of gating
terms enough to more than pay for its cost on average.
Even with the mod-time and pattern-element optimizations, many of the matches
found by the matcher are redundant. Thus, we are still spending some time discov-
ering redundant matches. But in all or nearly all cases, the fingerprint (
§5.2) test
detects the redundancy, so the unnecessary clauses are not added to the clause set.
6. Reporting Errors
Many conjectures are not theorems. Simplify’s backtracking search very frequently
finds a context that falsifies the conjecture. But it is of little help to the user to know
only that some error exists somewhere, with no hint where it is. Thus, when a con-
jecture is invalid, it is critical to present the reason to the user in an intelligible way.
6.1. E
RROR CONTEXT REPORTING. Since Sat maintains a context representing
a conjunction of literals that characterizes the current case, one simple way of
Simplify: A Theorem Prover for Program Checking 417
reporting failed proofs is to print that conjunction of literals to the user. We call this
error context reporting.For example, the invalidity of the conjecture x ≥ 0 ⇒
x > 10, would be reported with the error context x ≥ 0 ∧ x ≤ 10. We also call this
error context a counterexample even though a specific value for x is not supplied.
We don’t include quantified formulas in the reported error context, but only basic
literals (
§5.3.1).
One difficulty with error context reporting is that of producing a readable textual
description of the error context, the context printing problem.Ifthe literals are
simply printed one by one, the list may become voluminous and include many
redundant literals. To avoid this, Greg Nelson suggested in his thesis [Nelson 1981]
that a succinct error context somehow be computed directly from the E-graph
and Simplex tableau. In the case of the E-graph, Nelson [1981, Sec. 11] gave an
attractive algorithm for finding a minimum-sized set of equalities equivalent to
agiven E-graph, and this algorithm is in fact implemented in Simplify. Simplify
also implements an algorithm for translating the Simplex tableau into a succinct
conjunction of literals, but it is not as elegant as the algorithm for the E-graph.
The difficulties of the context printing problem are exacerbated by the back-
ground predicate described in Section 2: the previously mentioned succinct repre-
sentations of the E-graph and the Simplex tableau may include many literals that
are redundant in the presence of the background predicate. Therefore, by default,
Simplify prunes the error context by testing each of its literals and deleting any that
are easily shown to be redundant in the presence of the background predicate and
the previously considered literals. This is expensive, and we often use the switch
that disables pruning. Indeed, now that we have implemented error localization, de-
scribed in the next section, we sometimes use the switch that disables error context
reporting altogether.
6.2. E
RROR
LOCALIZATION. The idea of error localization is to attribute the
invalidity of the conjecture to a specific portion of its text.
Foraconjecture like x ≥ 0 ⇒ x > 10, error context reporting works well. But
for the problems that occur in program checking, error contexts are often large even
after pruning. If a ten-page conjecture includes the unprovable conjunct i ≥ 0, a
ten-page error context that includes the formula i < 0isnot a very specific in-
dication of the problem. We would rather have an output like “Can’t prove the
postcondition i ≥ 0online 30 of page 10”. ESC relies on Simplify’s error local-
ization to report the source location and type of the program error that led to the
invalid verification condition.
At first, error localization may seem hopelessly underspecified: is the invalidity
of an implication to be blamed on the weakness of its antecedent or the strength
of its consequent? To make error localization work, we extended Simplify’s input
language with labels.Alabel is just a text string; if P is a first-order formula and
L is a label, then we introduce the notation L : P as a first-order formula whose
semantics are identical to P but that has the extra operational aspect that if Simplify
refutes a conjecture containing an occurrence of L : P, and P is true in the error
context, and P’s truth is “relevant” to the error context, then Simplify will include
the label L in its error report.
We also write L
∗
: P as short for ¬ (L : ¬ P), which is semantically equivalent
to P but causes L to be printed if P is false and its falsehood is relevant. (Simplify
doesn’t actually use this definition; internally, it treats L
∗
as a primitive, called a
negative label. But in principle, one primitive suffices.)
418 D. DETLEFS ET AL.
Forexample, suppose that a procedure dereferences the pointer p at line 10 and
accesses a[i]online 11. The obvious verification condition has the form
Precondition ⇒ p = null ∧ i ≥ 0 ∧··· .
Using labels (say,
|Null@10| and |IndexNegative@11|), the verification
condition instead has the form:
Precondition ⇒
|
Null@10
|
∗
: p = null ∧|
IndexNegative@11|
∗
: i ≥ 0 ∧ ··· .
Thus, if the proof fails, Simplify’s output will include a label whose name encodes
the source location and error type of a potential error in the source program. This
is the basic method ESC uses for error reporting. Todd Millstein has extended the
method by changing the ESC verification condition generator so that labels emitted
from failed proofs determine not only the source location of the error, but also the
dynamic path to it [Millstein 1999]. We have also found labels to be useful for
debugging failed proofs when using Simplify for other purposes than ESC.
Before we implemented labels in Simplify, we achieved error localization in ESC
by introducing extra propositional variables called error variables. Instead of the la-
bels
|Null@10| and |IndexNegative@11| in the example above, we would have
introduced error variables, say
|EV.Null@10| and |EV.IndexNegative@11| and
generated the verification condition
Precondition ⇒
(|
EV.Null@10|∨ p = null) ∧ (|EV.IndexNegative@11|∨i ≥ 0) ∧ ... .
If the proof failed, the error context would set at least one of the error variables
to false, and the name of that variable would encode the information needed to
localize the error. We found, however, that error variables interfered with the
efficacy of subsumption (
§3.2) by transforming atomic formulas (§2) into nonatomic
formulas, and interfered with the efficacy of the status test by transforming distinct
occurrences of identical formulas into distinct formulas.
We have said that a label will be printed only if the formula that it labels is
“relevant”. For example, suppose that the query is
(P ∧ (Q ∨ L
1
: R)) ∨ (U ∧ (V ∨ L
2
: R))
and that it turns out to be satisfiable by taking U and R to be true and P to be false.
We would like L
2
to be printed, but we consider that it would be misleading to
print L
1
.
Perhaps unfortunately, we do not have a precise definition of “relevant”.
Operationally, Simplify keeps labels attached to occurrences of literals (includ-
ing propositional proxy literals (
§3.3)) and quantifier proxies (§5.3.1)), and when
an error context is discovered, the positive labels of the asserted literals and the
negative labels of the denied literals are printed.
An ancillary benefit of the label mechanism that we have found useful is that
Simplify can be directed to print a log of each label as the search encounters it.
This produces a dynamic trace of which proof obligation Simplify is working on at
any moment. For example, if a proof takes a pathologically long time, the log will
reveal which proof obligation is causing the problem.
So much for the basic idea. We now mention some details.
Simplify: A Theorem Prover for Program Checking 419
First, labels interact with canonicalization (
§3.3). If two formulas are the same
except for labels, canonicalizing them to be the same will tend to improve efficiency,
since it will improve the efficacy of the status test.
Simplify achieves this in the case that the only difference is in the outermost
label. That is, L : P and M : P will be canonicalized identically even if L and
M are different labels. However, Simplify does not canonicalize two formulas that
are the same except for different labels that are nested within them. For example,
(L : P) ⇒ Q might be canonicalized differently than (M : P) ⇒ Q.Wedon’t
have any data to know whether this is hurting performance.
Second, labels must be considered when rewriting Boolean structure, for exam-
ple, when creating matching rules from quantified formulas. For example, we use
the rewriting rule
P ∧ L : true −→ L : P.
If a conjunct simplifies to true, the rewrite rules obviously delete the conjunct. If
the conjunct is labeled, we claim that it is appropriate to move the label to the other
conjunct. Lacking a precise definition of relevance, we can justify this claim only
informally: If, after the rewriting, P turns out to be true and relevant, then before
the rewriting, the occurrence of true was relevant; hence L should be printed. If,
after the rewriting, P turns out to be false or irrelevant, then before the rewriting,
the occurrence of true was irrelevant. Similarly, we use the rules
P ∨ L : true −→ L : true
P ∨ L
∗
: false −→ L
∗
: P
P ∧ L
∗
: false −→ L
∗
: false.
More difficult problems arise in connection with the treatment of labels when
Simplify puts bodies of quantified formulas into CNF and when it instantiates
matching rules. Consider what happens when Simplify asserts the formula
(∀x : P(x) ∨ L :((Q
1
(x) ∧ Q
2
(x)) ∨ (R
1
(x) ∧ R
2
(x)))), (10)
If formula (10) did not include the label L, Simplify would would produce four
matching rules, corresponding to the following four formulas:
(∀x : P(x) ∨ Q
1
(x) ∨ R
1
(x))
(∀x : P(x) ∨ Q
1
(x) ∨ R
2
(x))
(∀x : P(x) ∨ Q
2
(x) ∨ R
1
(x))
(∀x : P(x) ∨ Q
2
(x) ∨ R
2
(x)).
(11)
With the label L present, Simplify produces the same four matching rules, but with
portions of their bodies labeled in a manner that allows Simplify to output the label
L when appropriate. In particular, Simplify will output L when the (heuristically
relevant) formulas asserted in an error context include both Q
1
(T ) and Q
2
(T )
or both R
1
(T ) and R
2
(T ) for some term T .Onthe other hand, the inclusion of,
for example, Q
1
(T ) and R
2
(T )inanerror context will not be sufficient to cause
Simplify to output the label L, nor will the inclusion of Q
1
(T ) and Q
2
(U), where
U is a term not equivalent to T .
Simplify achieves this effect by synthesizing derived labels in addition to the
original labels present in the input conjecture. We will not describe derived labels
in full detail. As an example that gives the flavor of the approach, we mention the
420 D. DETLEFS ET AL.
rule for distributing ∨ into a labeled conjunction:
P ∨ L :(Q ∧ R) → (P ∨ Lα : Q) ∧ (P ∨ Lβ : R)
The label L is reportable if both of the conjunct derived labels Lα and Lβ are
reportable. (This is the clause of the recursive definition of “reportable” applicable
to conjunct derived labels.) The effect of the rewrite rule is thus that L is reported
if Q ∧ R is true and relevant, which is what is desired. Simplify also uses disjunct
derived labels and parametrized derived labels.
6.3. M
ULTIPLE
COUNTEREXAMPLES.Asremarked in Section 3.2, Simplify can
find multiple counterexamples to a conjecture, not just one. In practice, even a
modest-sized invalid conjecture may have a very large number of counterexamples,
many of them differing only in ways that a user would consider uninteresting.
Reporting them all would not only be annoying, but would hide any interesting
variability amidst an overwhelming amount of noise. Therefore, Simplify defines a
subset of the labels to be major labels and reports only counterexamples that differ
in all their major labels.
In more detail, Simplify keeps track of the set of all major labels already reported
with any counterexample for the current conjecture, and whenever a literal with
a label in this set is asserted (more precisely, asserted if the label is positive or
denied if the label is negative), Simplify backtracks, just as if the context had been
found to be inconsistent. In addition, an environment variable specifies a limit on
the number of error contexts the user is interested in, and Simplify halts its search
for more error contexts whenever the limit has been reached. Simplify also halts
after reporting any error context having no major labels, which has the effect of
limiting the number of error contexts to one in the case of unlabeled input.
When Simplify is run from the shell, the limit on the number of reported
counterexamples defaults to 1. ESC/Java changes the default to 10 and gener-
ates conjectures in such a way that each counterexample will have exactly one
major label, namely the one used to encode the type and location of a potential
program error, such as
|Null@10| or |IndexNegative@11| from the example
in Section 6.2. These labels all include the character “
@”, so, in fact, in general
Simplify defines a major label to be one whose name includes an “
@”.
7. The E-graph in Detail
In Section 4.2, we introduced the E-graph module with which Simplify reasons
about equality. In this section we describe the module in more detail. The key ideas
of the congruence closure algorithm were introduced by Downey et al. [1980], but
this section describes the module in more detail, including propagating equalities,
handling distinctions (
§2), and undoing merges. First, we introduce the features and
invariants of the basic E-graph data structure, and then we give pseudocode for the
main algorithms.
7.1. DATA STRUCTURES AND INVARIANTS. The E-graph represents a set of
terms and an equivalence relation on those terms. The terms in the E-graph include
all those that occur in the conjecture, and the equivalence relation of the E-graph
equates two nodes if the equality of the corresponding terms is a logical consequence
of the current equality assertions.
Simplify: A Theorem Prover for Program Checking 421
FIG.3. The term f (a, b)asrepresented in an abstract term DAG on the left and in a concrete E-graph
on the right.
We could use any of the standard methods for representing an equivalence
relation, which are generally known as union-find algorithms. From these meth-
ods we use the so-called “quick find” approach. That is, each E-node p contains a
field p.root that points directly at the canonical representative of p’s equivalence
class. Thus E-nodes p and q are equivalent exactly when p.root = q.root.Inad-
dition, each equivalence class is linked into a circular list by the next field. Thus,
to merge the equivalence classes of p and q,we(1) reroot one of the classes (say,
p’s)bytraversing the circular list p, p.next, p.next.next,...updating all root fields
to point to q.root and then (2) splice the two circular lists by exchanging p.next
with q.next.For efficiency, we keep track of the size of each equivalences class
(in the size field of the root) and reroot the smaller of the two equivalence classes.
Although the union-find algorithm analyzed by Tarjan [1975] is asymptotically
more efficient, the one we use is quite efficient both in theory and in practice
[Knuth and Sch¨onhage 1978], and merges are easy to undo, as we describe below.
Equality is not just any equivalence relation: it is a congruence. The algorithmic
consequence is that we must represent a congruence-closed equivalence relation.
In Section 4.2, we defined the congruence closure of a relation on the nodes of a
vertex-labeled oriented directed graph with nodes of arbitrary out-degree. In the
implementation, we use the standard reduction of a vertex-labeled oriented directed
graph of general degree to an oriented unlabeled directed graph where every node
has out-degree two or zero. That is, to represent the abstract term DAG (
§4.2), in
which nodes are labeled and can have any natural number of out-edges, we use a
data structure that we call the concrete E-graph,inwhich nodes (called E-nodes)
are unlabeled and either have out-degree two (binary E-nodes) or zero (atomic
E-nodes). We learned this kind of representation from the programming language
Lisp, so we will call the two edges from a binary E-node by their Lisp names, car
and cdr. The basic idea for limiting the outdegree to two is to represent the sequence
of children (
§4.2) of a DAG node by a list linked by the cdr fields, as illustrated
in Figure 3.
Here is a precise definition of the concrete E-graph representation:
Definition 1. An E-node can represent either a function symbol, a term, or a
list of terms. The rules are as follows: (1) Each function symbol f is represented
by a distinct atomic E-node, λ( f ). (2) Each term f (a
1
,...,a
n
)isrepresented by a
binary E-node e such that e.car = λ( f ) and e.cdr represents the list a
1
,...,a
n
. (3)
A nonempty term list (a
1
,...a
n
)isrepresented by a binary E-node e such that e.car
is equivalent to some node that represents the term a
1
and e.cdr represents the list
(a
2
,...,a
n
). The empty term list is represented by a special atomic E-node enil.
422 D. DETLEFS ET AL.
E-nodes of types (1), (2), and (3) are called symbol nodes, term nodes, and list
nodes, respectively. The words “is equivalent to” in (3) reflect our convention from
Section 4.2 that “represents” means “represents up to congruence”.
In Section 4.2, we defined the word “E-node” to refer to the nodes of an abstract
term DAG, which correspond to just the term nodes of the concrete E-graph. In this
section, we will use the word “E-node” to refer to all three types of concrete E-nodes.
In the concrete E-graph, we define a binary node p to be a parent of p.car and
of p.cdr. More precisely, it is a car-parent of the former and a cdr-parent of the
latter. A node p is a parent, car-parent,orcdr-parent of an equivalence class if it
is such a parent of some node in the class. Because every cdr field represents a list,
and every car field represents either a term or a function symbol, we also have the
following invariant
Invariant (CAR-CDR DICHOTOMY). A symbol node or term node may
have car-parents, but not cdr-parents; a list node may have cdr-parents, but not
car-parents.
In the concrete E-graph, we define two binary nodes to be congruent with respect
to a relation R if their car’s and cdr’s are both related by R.Wesay that equivalence
relation R is congruence-closed if any two nodes that are congruent under R are
also equivalent under R. The congruence closure of a relation R on the nodes of
the E-graph is the smallest congruence-closed equivalence relation that extends R.
We leave it to the reader to verify that if R is a relation on term nodes and Q the
congruence closure of R in the sense just defined, then Q restricted to term nodes
is the congruence closure of R in the sense of Section 4.2.
The matching iterators and optimizations of Section 5.4 were presented there in
terms of the abstract term DAG. Translating them to use the concrete E-graph is
for the most part straightforward and will be left to the reader. One point that we
do mention is that Simplify maintains mod-times (
§5.4.2) for list nodes as well as
for term nodes, and therefore UpdateMT iterates over concrete parents.
Because all asserted equalities are between term nodes, all congruences are
either between two term nodes or between two list nodes, and therefore we have
the following invariant:
Invariant (TERM-LIST-FSYM TRICHOTOMY). Every equivalence class in
the E-graph either (1) consists entirely of term nodes, (2) consists entirely of list
nodes, or (3) is a singleton of the form {λ( f )} for some function symbol f .
The singleton class {enil} is of the form (2).
Returning to the implementation of congruence closure, the reduction to the
concrete E-graph means that we need to worry only about congruences between
nodes of degree 2. We define the signature of a binary node p to be the pair
(p.car.root.id, p.cdr.root.id), where the id field is a numerical field such that
distinct nodes have distinct id’s. The point of this definition is that two binary
nodes are congruent if and only if they have the same signature.
Merging two equivalence classes P and Q may create congruences between
parents of nodes in P and parents of nodes in Q.Todetect such new congruences,
it suffices to examine all parents of nodes in one of the classes and test whether
they participate in any new congruences. We now discuss these two issues, the test
and the enumeration.
To support the test for new congruences, we maintain the signature table sigT ab,
a hash table with the property that, for any signature (i, j ), sigTab(i, j)isanE-node
Simplify: A Theorem Prover for Program Checking 423
with signature (i, j), or is nil if no such E-node exists. When two equivalence classes
merged, the only nodes whose signatures change are the parents of whichever class
does not give its id to the merged class. The key idea in the Downey–Sethi–Tarjan
algorithm is to use this fact to examine at most half of the parents of the merged
equivalence class. This can be done by keeping a list of parents of each equivalence
class along with the size of the list, and by ensuring that the id of the merged class
is the id of whichever class had more parents.
We represent the parent lists “endogenously”, that is, the links are threaded within
the nodes themselves, as follows: Each E-node e contains a parent field. For each
root node r , r.parent is some parent of r’s equivalence class, or is nil if the class has
no parent. Each binary E-node p contains two fields samecar and samecdr, such
that for each equivalence class Q, all the car-parents of Q are linked into a circular
list by the samecar field, and all the cdr-parents of Q are linked into a circular list
by the samecdr field.
With these data structures, we can implement the iteration
for each parent p of the equivalence class of the root r do
Visit p
end
as follows:
if r.parent = nil then
if r is a symbol node or term node then
for each node p in the circular list
r.parent, r.parent.samecar, r.parent.samecar.samecar,...do
Visit p
end
else
for each node p in the circular list
r.parent, r.parent.samecdr, r.parent.samecdr.samecdr,...do
Visit p
end
end
To support the key idea in the Downey–Sethi–Tarjan algorithm, we introduce the
parentSize field. The parentSize field of the root of an equivalence class contains
the number of parents of the class.
The facts represented in the E-graph include not only equalities but also
distinctions (both binary and general (
§2)). To represent distinctions in the E-graph,
we therefore add structure to the E-graph to represent that certain pairs of nodes are
unmergeable in the sense that if they are ever combined into the same equivalence
class, the context is unsatisfiable.
Simplify’s E-graph uses two techniques to represent unmergeability, one suited
to binary distinctions and the other to general distinctions.
To represent binary distinctions, we supply each E-node e with a list of E-nodes,
e.forbid, called a forbid list, and maintain the following invariant:
Invariant (FORBID LIST VALIDITY). If a binary distinction between two
E-nodes x and y has been asserted on the current path, then x.root.forbid contains
a node equivalent to y and y.root.forbid contains a node equivalent to x.
To assert a binary distinction X = Y ,wefind the E-nodes x and y that represent
X and Y .Ifthey have the same root, then the assertion produces an immediate
contradiction. Otherwise, we add x to y.root.forbid and also add y to x.root.forbid.
424 D. DETLEFS ET AL.
When asserting an equality X = Y ,wefind the E-nodes x and y that represent X
and Y and traverse the forbid list of either x.root or y.root (whichever list is shorter),
looking for nodes equivalent to the other. If one is found, then the assertion produces
an immediate contradiction. If no contradiction is found, we set the forbid list of
the root of the combined equivalence class to be the concatenation of the two lists.
Because an E-node may be on more than one forbid list, we represent forbid
lists “exogenously”: that is, the nodes of the list are distinct from the E-nodes
themselves. Specifically, forbid lists are represented as circular lists linked by a
link field and containing E-nodes referenced by an e field.
Since an n-way general distinction is equivalent to the conjunction of O(n
2
)
binary distinction, it seems unattractive to use forbid lists for representing gen-
eral distinctions, so for these we introduce a technique called distinction classes.
A distinction class is a set of E-nodes any two of which are unmergeable. A new
distinction class is allocated for each asserted general distinction. To represent
membership in distinction classes, we supply each E-node e with a bit vector
e.distClasses and maintain the invariant:
Invariant (DISTINCTION CLASS VALIDITY). For any root E-node r and
integer i , r.distClasses[i]istrue if some E-node in r’s equivalence class is a member
of the ith distinction class created on the current path.
To assert a general distinction
DISTINCT(T
1
,...,T
n
), we find an E-node t
i
representing T
i
.Ifany two of these E-nodes have the same root, then the asser-
tion produces an immediate contradiction. Otherwise, we allocate a new distinction
class number d and set bit d of each t
i
.root.distClasses to true.
When asserting an equality X = Y ,wefind the E-nodes x and y that represent X
and Y .Ifx.root.distClasses and y.root.distClasses contain any true bit in common,
then the assertion produces an immediate contradiction. If no contradiction is found,
we set the .distClasses field of the root of the combined equivalence class to be the
union (bitwise OR) of x.root.distClasses and y.root.distClasses
A fine point: Simplify allocates at most k distinction classes on any path in its
backtracking search, where k is the number of bits per word in the host computer.
Forany additional occurrences of general distinctions, it retreats to the expansion
into
n
2
binary distinctions. If this retreat had become a problem, we were prepared
to use multiword bit vectors for distClasses,but in our applications, running on our
64-bit Alpha processors, the retreat never became a problem.
In summary, here is the implementation of the test for unmergeability:
procUnmergeable(x, y : E-node):boolean ≡
var p, q, pstart, qstart, pptr, qptr in
p, q := x.root, y.root;
if p = q then return false end;
if p.distClasses ∩ q.distClasses ={}then
return true
end;
pstart, qstart := p.forbid, q.forbid;
if pstart = nil and qstart = nil then
pptr, qptr := pstart, qstart;
loop
if pptr.e.root = q or qptr.e.root = p then
return true
else
pptr, qptr := pptr.link, qptr.link
end
Simplify: A Theorem Prover for Program Checking 425
if pptr = pstart or qptr = qstart then exit loop end
end;
return false
end
end
Foravariety of reasons, we maintain in the E-graph an explicit representation
of the relation “is congruent to.” This is also an equivalence relation and again
we could use any standard representation for equivalence relations, but while the
main equivalence relation of the E-graph is represented in the root, next, and size
fields using the quick-find technique, the congruence relation is represented with
the single field cgPtr using simple, unweighted Fischer–Galler trees [Galler and
Fischer 1964; Knuth 1968, Sec. 2.3.3]. That is, the nodes are linked into a forest by
the cgPtr field. The root of each tree of the forest is the canonical representative of
the nodes in that tree, which form a congruence class. For such a congruence root
r, r.cgPtr = r.For a nonroot x, x.cgPtr is the tree parent of x.
In Section 5.4.1, which defined the iterators for matching triggers in the
E-graph, we mentioned that for efficiency it was essential to ignore nodes that
are not canonical representatives of their congruence classes, this is easily done by
testing whether x.cgPtr = x.Wewill also maintain the invariant that the congru-
ence root of a congruence class is the node that represents the class in the signature
table. That is,
Invariant (SIGNATURE TABLE CORRECTNESS). The elements of sigTab are
precisely the pairs ((x.car.root.id, x.cdr.root.id), x) such that x is a binary E-node
that is a congruence root.
The last subject we will touch on before presenting the implementations of the
main routines of the E-graph module is to recall from Section 4.4 the strategy for
propagating equalities from the E-graph to the ordinary theories. For each ordinary
theory T ,wemaintain the invariant:
Invariant (PROPAGATION TO T ). Two equivalent E-nodes have non-nil
T
unknown fields if and only if these two T.Unknown’s are equated by a chain
of currently propagated equalities from the E-graph to the T module.
When two equivalence classes are merged in the E-graph, maintaining this invari-
ant may require propagating an equality to T .Toavoid scanning each equivalence
class for non-nil T
unknown fields, the E-graph module also maintains the invariant:
Invariant (ROOT T
unknown). If any E-node in an equivalence class has a non-
nil T
unknown field, then the root of the equivalence class has a non-nil T unknown
field.
We now describe the implementations of some key algorithms for maintaining
the E-graph, namely those for asserting equalities, undoing assertions of equalities,
and creating E-nodes to represent terms.
7.2. A
SSERTEQ AND MERGE.Toassert the equality X = Y , Simplify calls
AssertEQ(x, y) where x and y are E-nodes representing the terms X and Y .
AssertEQ maintains a work list in the global variable pending. This is a list of
pairs of E-nodes whose equality is implicit in the input but not yet represented in
the E-graph. AssertEQ repeatedly merges pairs from the pending list, checking for
unmergeability before calling the procedure Merge.
426 D. DETLEFS ET AL.
proc AssertEQ(x, y : E-node) ≡
pending :={(x, y)};
while pending ={}do
remove a pair (p, q) from pending;
p, q := p.root, q.root;
if p = q then
// p and q are not equivalent.
if Unmergeable(p, q) then
refuted := true;
return
else
Merge(p, q)
end
end
end
// The E-graph is congruence-closed.
end
The call Merge(x, y) combines the equivalence classes of x and y, adding pairs
of nodes to pending as new congruences are detected.
Merge(x, y) requires that x and y are roots, that they are neither equivalent nor
unmergeable, and that either (1) they are both term nodes representing terms whose
equality is implicit in the current context, or (2) they are list nodes representing lists
of terms (X
1
,...,X
n
) and (Y
1
,...,Y
n
) such that each equality X
i
= Y
i
is implicit
in the current context.
proc Merge(x, y : E-node) ≡
// Step 1: Make x be the root of the larger equivalence class.
M1 if x.size < y.size then
M2
x, y := y, x
M3 end;
// x will become the root of the merged class.
// Step 2: Maintain invariants relating E-graph to ordinary theories.
M4 for each ordinary theory T do
// maintain PROPAGATION TO T
M5 if x .T unknown = nil and y.T unknown = nil then
M6 Propagate the equality of x .T unknown with y.T unknown to T
M7 end
// maintain ROOT T
unknown
M8 if x .T unknown = nil then x.T unknown := y.T unknown end;
M9 end;
// Step 3: Maintain sets for pattern-element optimizations (
§5.4.3).
M10 UpdatePatElems(x, y);
// Step 4: Make x unmergable with nodes now unmergable with y.
// Merge forbid lists.
M11 if y.forbid = nil then
M12 if x .forbid = nil then
M13 x .forbid := y.forbid
M14 else
M15 x .forbid.link, y.forbid.link := y.forbid.link, x.forbid.link
M16 end
M17 end;
// Merge distinction classes.
Simplify: A Theorem Prover for Program Checking 427
M18 x.distClasses := x.distClasses ∪ y.distClasses;
// Step 5: Update sigTab, adding pairs of newly congruent nodes to
// pending.
M19 var w in
// 5.1: Let w be the root of the class with fewer parents.
M20 if x .parentSize < y.parentSize then
M21 w := x
M22 else
M23 w := y
M24 end;
// (Only parents of w’s class will get new signatures.)
// 5.2: Remove old signatures of w’s class’s parents.
M25 for each parent p of w’s equivalence class do
M26 if p = p.cgPtr then // p is originally a congruence root.
M27 Remove ((p.car.root.id, p.cdr.root.id), p) from the signature table
M28 end;
// 5.3: Union the equivalence classes.
// 5.3.1: Make members of y’s class point to new root x.
M29
for each v in the circular list y, y.next, y.next.next,...do
M30 v.root := x
M31 end;
// 5.3.2: Splice the circular lists of equivalent nodes
M32 x .next, y.next := y.next, x.next;
M33 x .size := x.size + y.size;
// 5.4: Preserve signatures of the larger parent set by swapping
// id’s if necessary.
M34 if x .parentSize < y.parentSize then
M35 x .id, y.id := y.id, x.id
M36 end;
// 5.5: Put parents of w into sigTab with new signatures,
// and add pairs of newly congruent pairs to pending.
M37 for each parent p of w’s equivalence class do
M38 if p = p.cgPtr then // p is originally a congruence root.
M39 if the signature table contains an entry q under the key
M40 (p.car.root.id, p.cdr.root.id) then
// Case 1: p joins q’s congruence class.
M41 p.cgPtr := q;
M42 pending := pending ∪{( p, q)}
M43 else
// Case 2: p remains a congruence root.
M44 Insert ((p.car.root.id, p.cdr.root.id), p) into the signature table
M45 end
M46 end
M47 end
M48 end;
// Step 6: Merge parent lists.
M49 if y.parent = nil then
M50 if x .parent = nil then
M51 x .parent := y.parent
M52 else
// Splice the parent lists of x and y.
M53 if x and y are term nodes then
M54 x .parent.samecar, y.parent.samecar :=
M55 y.parent.samecar, x.parent.samecar
M56 else
M57 x .parent.samecdr, y.parent.samecdr :=
M58 y.parent.samecdr, x.parent.samecdr
428 D. DETLEFS ET AL.
M59 end
M60 end
M61 end;
M62 x.parentSize := x.parentSize + y.parentSize;
// Step 7: Update mod-times for mod-time matching optimization (
§5.4.2).
M63 UpdateMT(x)
// Step 8: Push undo record.
M64 Push onto the undo stack (§3.1) the pair ("UndoMerge", y)
M65 end
The union of distClasses fields on line M18 is implemented as a bitwise OR
operation on machine words. The for each loops starting on lines M25 and M37
are implemented as described in Section 7.1.
7.3. U
NDOM
ERGE. During backtracking, when we pop an entry of the form
(
"UndoMerge", y) from the undo stack, we must undo the effect of Merge by
calling UndoMerge(y).
Here is the pseudocode for UndoMerge, followed by some explanatory remarks.
In what follows, when we refer to “the merge” or to a step in the execution of
Merge,wemean the particular execution of Merge whose effect is being undone.
proc UndoMerge(y : E-node) ≡
// Compute the other argument of Merge.
U1 var x := y.root in
// Undo Step 6 of Merge.
U2 x .parentSize := x.parentSize − y.parentSize;
U3 if y.parent = nil then
U4 if x .parent = y.parent then
U5 x .parent := nil
U6 else
// Unsplice the parent lists of x and y.
U7 if x and y are term nodes then
U8 x .parent.samecar, y.parent.samecar :=
U9 y.parent.samecar, x.parent.samecar
U10 else
U11 x .parent.samecdr, y.parent.samecdr :=
U12 y.parent.samecdr, x.parent.samecdr
U13 end
U14
end
U15 end;
// Undo Step 5 of Merge.
U16 var w in
U17 if x .parentSize < y.parentSize then
U18 w := x
U19 else
U20 w := y
U21 end;
// w now has the value computed in lines M20–M24.
// Undo Case 2 iterations in Step 5.5 of Merge.
U22 for each parent p of w’s equivalence class do
U23 if p = p.cgPtr then
U24 Remove ((p.car.root.id, p.cdr.root.id), p) from the signature table
Simplify: A Theorem Prover for Program Checking 429
U25 end;
U26 end;
// Undo Step 5.4 of Merge
U27 if x .parentSize < y.parentSize then
U28 x .id, y.id := y.id, x.id
U29 end;
// Undo Step 5.3 of Merge
U30 x .size := x.size − y.size;
U31 x .next, y.next := y.next, x.next;
U32 for each v in the circular list y, y.next, y.next.next,...do
U33 v.root := y
U34 end;
// Undo Step 5.2 and Case 1 iterations of Step 5.5 of Merge.
U35 for each parent p of w’s equivalence class do
U36 var cg := p.cgPtr in
U37
if (p.car.root = cg.car.root or p.cdr.root = cg.cdr.root) then
// Undo a Case 1 iteration in Step 5.5 of Merge.
U38
p.cgPtr := p
U39 end;
U40 if p.cgPtr = p then
// Undo an iteration of Step 5.2 of Merge.
U41 Insert ((p.car.root.id, p.cdr.root.id), p) into the signature table
U42 end
U43 end
U44 end
U45 end;
// Undo Step 4 of Merge.
U46 x .distClasses := x.distClasses − y.distClasses;
U47 if x .forbid = y.forbid then
U48 x .forbid := nil
U49 else if y.forbid = nil then
U50 x .forbid.link, y.forbid.link := y.forbid.link, x.forbid.link
U51 end;
// Undo Step 2 of Merge.
U52 for each ordinary theory T do
U53 if x .T unknown = y.T unknown then x .T unknown := nil end
U54 end
U55 end
U56 end
Recall that the overall form of Merge is:
Step 1: Make x be the root of the larger equivalence class.
Step 2: Maintain invariants relating E-graph to ordinary theories.
Step 3: Maintain sets for pattern-element optimizations.
Step 4: Make x unmergable with nodes now unmergable with y.
Step 5: Update sigTab, adding pairs of newly congruent nodes to
pending.
Step 6: Merge parent lists.
Step 7: Update mod-times for mod-time matching optimization
Step 8: Push undo record.
Step 1 has no side effects; it merely sets x to the E-node that will be the root of the
merged equivalence class, and y to be the root of the other pre-merge equivalence
430 D. DETLEFS ET AL.
class. UndoMerge thus has no actual undoing code for Step 1. However, line U1
ensures that x and y have the values that they had after line M3.
UndoMerge contains no code to undo the calls to UpdatePatElems and UpdateMT
(Steps 3 and 7 of Merge). These routines create their own undo records, as described
in Sections 5.4.3 and 5.4.2. Since the undoing code for these records is independent
of the rest of UndoMerge (and vice-versa), it doesn’t matter that these undo records
are processed out of order with the rest of UndoMerge.
Finally, UndoMerge has no code for undoing Step 8, since Pop will have already
popped the undo record from the undo stack before calling UndoMerge.
The remaining side effects of Merge are undone mostly in the reverse order that
they were done, except that the substeps of Step 5 are undone slightly out of order,
as discussed below. The process is mostly straightforward, but we call attention to
afew points that might not be immediately obvious.
In the undoing code for Step 6, we rely on the fact that x.parent and y.parent
could not have been equal before the merge. The precondition for Merge(x, y)
requires that x and y must either both be term nodes or both be list nodes. If they
are term nodes, then x.parent.car is equivalent to x and y.parent.car is equivalent
to y. Since the precondition for Merge(x, y) requires that x and y not be equivalent,
it follows that x.parent = y.parent. The case where x and y are list nodes is similar.
In the undoing code for Step 4, we rely on the fact that x.distClasses and
y.distClasses must have been disjoint before the merge, which follows from the
specification that the arguments of Merge not be unmergeable. Thus, x.distClasses
can be restored (on line U46) by set subtraction implemented with bit vectors.
Similarly, in the undoing code for Step 2, we rely on the fact that x.T
unknown
and y.T
unknown could not have been equal before the merge.
The most subtle part of UndoMerge is the undoing of Step 5 of Merge. Recall
that the structure of Step 5 is:
5.1: Let w be the root of the class with fewer parents.
5.2: Remove old signatures of w’s class’s parents.
5.3: Union the equivalence classes.
5.4: Preserve signatures of the larger parent set by swapping
id’s if necessary.
5.5: Put parents of w into sigTab with new signatures,
and add pairs of newly congruent pairs to pending.
Step 5.1 has no nonlocal side effects, so needs no undoing code. However, lines
U17–U21 of UndoMerge ensure that w has the value that is had at the end of
Step 5.1 of Merge, since this is needed for the undoing of the rest of Step 5. The
remaining substeps of Step 5 are undone mostly in the reverse order that they were
done, the exception being that a part of Step 5.5—namely, the linking together of
congruence classes (line M41)—is undone somewhat later, for reasons that we now
explain.
Properly undoing the effects of Merge upon congruence pointers depends on
two observations. The only place that Merge updates a cgPtr field is at line M41,
p.cgPtr := q. When this is executed, (1) p is a congruence root before the merge,
and (2) p and q become congruent as a result of the merge. Because of (1), the
value that p.cgPtr should be restored to is simply p. Because of (2), UndoMerge
can locate those p’s whose cgPtr fields need to be restored by locating all parents
p of w’s equivalence class that are not congruent to p.cgPtr in the prestate of the
Simplify: A Theorem Prover for Program Checking 431
merge. (UndoMerge enumerates these p’s after restoring the equivalence relation
to its pre-merge state.)
We conclude our discussion of UndoMerge with an observation about the
importance of the Fischer–Galler trees in the representation of congruences
classes. In our original implementation of Simplify’s E-graph module, we did not
use Fischer–Galler trees. In place of the SIGNATURE TABLE CORRECTNESS
invariant given above, we maintained—or so we imagined—a weaker invariant
requiring only that the signature table contain one member of each congruence
class. The result was a subtle bug: After a Merge-UndoMerge sequence, all
congruence classes would be still be represented in the signature table, but not
necessarily with the same representatives they had before the Merge;asubsequent
UndoMerge could then leave a congruence class unrepresented.
Here is an example illustrating the bug. Swapping of id fields plays no role in
the example, so we will assume throughout that larger equivalence classes happen
to have larger parent sets, so that no swapping of id fields occurs.
(1) We start in a state where nodes w, x, y, and z are the respective roots of different
equivalence classes W , X , Y , and Z , and the binary nodes (p, w), (p, x), (p, y),
and (p, z) are present in the E-graph.
(2) W and X get merged. Node x becomes the root of the combined equivalence
class
WX , and node (p, w )isrehashed and found to be congruent to ( p, x), so
they are also merged.
(3)
WX and Y get merged. Node y becomes the root of the combined equivalence
class
WXY , and node ( p, x)isrehashed and found to be congruent to ( p, y),
so they are also merged.
(4) The merges of ( p, x) with (p, y) and of
WX with Y are undone. In the pro-
cess of undoing the merge of
WX with Y , the parents of WX are rehashed.
Nodes (p, w ) and (p, x) both have signature ( p.id, x.id). In the absence of the
Fischer–Galler tree, whichever one happens to be rehashed first gets entered in
the signature table. It happens to be (p, w).
(5) The merges of ( p, w) with (p, x) and of W with X are undone. In the process of
undoing the merge of W with X, the parents of W are rehashed. Node (p, w)is
removed from the signature table and re-entered under signature ( p.id, w.id).
At this point ( p, x) has signature ( p.id, x.id), but the signature table contains
no entry for that signature.
(6) X and Z get merged, x becomes the root of the combined equivalence class,
and the parents of Z are rehashed. The new signature of ( p, z)is(p.id, x.id).
Since the signature table lacks an entry for this signature, the congruence of
(p, z) and ( p, x) goes undetected.
7.4. C
ONS. Finally, we present pseudo-code for Cons(x, y), which finds or
constructs a binary E-node whose car and cdr are equivalent to x and y, respectively.
We leave the other operations to the reader.
proc Cons(x, y : E-node):E-node ≡
var res in
x := x.root;
y := y.root;
if sigTab(x.id, y.id) = nil then
res := sigTab(x .id, y.id)
432 D. DETLEFS ET AL.
else
res := new(E-node);
res.car := x;
res.cdr := y;
// res is in a singleton equivalence class.
res.root := res;
res.next := res;
res.size := 1;
// res has no parents.
res.parent := nil;
res.parentSize := 0;
res.plbls :={};
// Set res.id to the next available Id.
res.id := idCntr;
idCntr := idCntr + 1;
// res is associated with no unknown.
for each ordinary theory T do
res.T
unknown := nil
end;
// res is a car-parent of x .
if x .parent = nil then
x.parent := res;
res.samecar := res
else
// Link res into the parent list of x.
res.samecar := x.parent.samecar;
x.parent.samecar := res
end;
x.parentSize := x.parentSize + 1;
// res is a cdr-parent of y.
if y.parent = nil then
y.parent := res;
res.samecdr := res
else
// Link res into the parent list of y.
res.samecdr := y.parent.samecdr;
y.parent.samecdr := res
end;
y.parentSize := y.parentSize + 1;
// res is the root of a singleton congruence class.
res.cgPtr := res;
Insert ((x .id, y.id), res) into sigTab;
if x = λ( f ) for some function symbol f then
// res is an application of f .
res.lbls :={f }
else
res.lbls :={}
end;
// res is inactive.
res.active := false;
Push onto the undo stack (
§3.1) the pair ("UndoCons", res)
end;
Simplify: A Theorem Prover for Program Checking 433
return res;
end
end
In the case where x = λ( f ), the reader may expect f to be added to plbls of
each term E-node on the list y.Infact, Simplify performs this addition only when
the new E-node is activated (
§5.2), not when it is created.
7.5. T
HREE FINAL FINE POINTS.Wehave three fine points to record about the
E-graph module that are not reflected in the pseudo-code above.
First, the description above implies that any free variable in the conjecture will
become a constant in the query (
§2), which is represented in the E-graph by the
Cons of a symbol node (
§7.1) with enil.Infact, as an optimization to reduce the
number of E-nodes, Simplify represents such a free variable by a special kind of
term node whose representation is like that of a symbol node. Since the code above
never takes the car or cdr of any term nodes except those known to be parents, the
optimization has few effects on the code in the module, although it does affect the
interning (
§4.5) code and the code that constructs matching triggers (§5.1).
Second, because Simplify never performs matching during plunging (
§4.6), there
is no need to update mod-times (
§5.4.2) and pattern element (§5.4.3) sets for merges
that occur during plunging, and Simplify does not do so.
Third, recall from Section 2 that Simplify allows users to define quasi-relations
with its
DEFPRED facility. When an equivalence class containing @true is merged
with one containing an application of such a quasi-relation, Simplify instantiates
and asserts the body of the
DEFPRED command. Simplify also notices most (but not
all) cases in which an application of such a quasi-relation becomes unmergeable
(
§7.1) with @true, and in these cases it instantiates and asserts the negation of
the body. Simplify avoids asserting multiple redundant instances of quasi-relation
bodies by using a table of fingerprints (
§5.2) (in essentially the same way that it
avoids asserting redundant instances of matching rules).
8. The Simplex Module in Detail
In this section, we describe an incremental, resettable procedure for determining the
satisfiability over the rationals of a conjunction of linear equalities and inequalities
(
§2). The procedure also determines the equalities between unknowns implied by the
conjunction. This procedure is based on the Simplex algorithm. We also describe
afew heuristics for the case of integer unknowns. The procedure and heuristics
described in this section are the semantics of arithmetic built into Simplify.
The Simplex algorithm was invented by George Dantzig [Dantzig 1963] and is
widely used in Operations Research [Chvatal 1983]. Our procedure shares with the
Simplex algorithm a worst case behavior that would be unacceptable, but that does
not seem to arise in practice.
We will describe the Simplex algorithm from scratch, as it is used in Simplify,
rather than merely refer the reader to a standard operations research text. This
lengthens our paper, but the investment of words seems justified, since Simplify
requires an incremental, resettable, equality-propagating version of the algorithm
and also because Simplify unknowns are not restricted to be non-negative, as they
seem to be in most of the operations research literature on the algorithm. This section
434 D. DETLEFS ET AL.
is closely based on Section 12 of Nelson’s revised Ph.D. thesis [Nelson 1981], but
our presentation is smoother and corrects an important bug in Nelson’s account.
Our description is also somewhat discrepant from the actual code in Simplify, since
we think it more useful to the reader to present the code as it ought to be instead of
as it is.
Subsections 8.1, 8.2, 8.3, 8.4, 8.5, and 8.6 describe in order: the interface to the
Simplex module, the tableau data structure used in the implementation, the algo-
rithm for determining satisfiability, the modifications to it for propagating equalities,
undoing modifications to the tableau, and some heuristics for integer arithmetic.
Before we dive into all these details, we mention an awkwardness caused by the
limitation of the Simplex algorithm to linear equalities and inequalities. Because
of this limitation, Simplify treats some but not all occurrences of the multiplication
sign as belonging to the Simplex theory. For example, to construct the E-node for
2 × x we ensure that this E-node and x’s both have unknowns, and assert that
the first unknown is double the second, just as described in Section 4.5. The same
thing happens for c × x in a context where c is constrained to equal 2. However,
to construct the E-node for x × y where neither x nor y is already constrained to
equal a numerical constant, we give up, and treat × as an uninterpreted function
symbol. Even if x is later equated with a numerical constant, Simplify, as currently
programmed, continues to treat the occurrence of × as an uninterpreted function
symbol.
In constructing
AF’s (§4.5), the Simplex module performs the canonicalizations
hinted at in Section 4.5. For example, if the conjecture includes the subformulas
T
1
< T
2
, T
2
> T
1
, T
1
≥ T
2
, the literals produced from them will share the same
AF, with the literal for the last having opposite sense from the others. However,
2 × T
1
< 2 × T
2
will be canonicalized differently.
8.1. T
HE I
NTERFACE. Since Simplex is an ordinary theory, the interface it
implements is the one described in Section 4.4. In particular, it implements the
type Simplex.Unknown, representing an unknown rational number, and implements
the abstract variables Simplex.Asserted and Simplex.Propagated, which represent
respectively the conjunction of Simplex literals that have been asserted and the
conjunction of equalities that have been propagated from the Simplex module. (In
the remainder of Section 8, we will often omit the leading “Simplex.” when it is
clear from context.)
As mentioned in Section 4.4, the fundamental routine in the interface is AssertLit,
which tests whether a literal is consistent with Asserted and, if it is, conjoins the
literal to Asserted and propagates equalities as necessary to maintain
Invariant (PROPAGATION FROM Simplex). For any two connected (
§4.4)
unknowns u and v, the equality u = v is implied by Propagated iff it is implied by
Asserted.
Below, we describe two routines in terms of which AssertLit is easily imple-
mented. These two routines operate on formal affine sums of unknowns, that is
formal sums of terms, where each term is either a rational constant or the product
of a rational constant and an Unknown:
proc AssertGE(fas : formal affine sum of connected unknowns);
/* If fas ≥ 0isconsistent with Asserted, set Asserted to fas ≥ 0 ∧ Asserted and propagate
equalities as required to maintain PROPAGATION FROM Simplex. Otherwise, set refuted
to true. */
Simplify: A Theorem Prover for Program Checking 435
proc AssertZ(fas : formal affine sum of connected unknowns);
/* If fas = 0isconsistent with Asserted, set Asserted to fas = 0 ∧ Asserted and propagate
equalities as required to maintain PROPAGATION FROM Simplex. Otherwise, set refuted
to true. */
The Simplex module exports the following procedures for associating Simplex
unknowns (for the remainder of Section 8, we will simply say “unknowns”) with
E-nodes and for maintaining the abstract predicate Asserted over the unknowns:
proc UnknownForEnode(e : E-node):Unknown;
/* Requires that e.root = e. Returns e.Simplex
Unknown if it is not nil. Otherwise sets
e.Simplex
Unknown to a newly-allocated unknown and returns it. (The requirement that
e be a root causes the invariants PROPAGATION TO Simplex and ROOT SIMPLEX
UNKNOWN to be maintained automatically.) */
Of course, these routines must push undo records, and the module must provide
the undoing code to be called by Pop.
8.2. T
HE TABLEAU
. The Simplex tableau consists of
—three natural numbers m, n, and dcol, where 0 ≤ dcol ≤ m,
—two one-dimensional arrays of Simplex unknowns, x[1],...,x[m] and
y[0],...,y[n], and
—a two-dimensional array of rational numbers a[0, 0],...,a[n, m].
Each Simplex unknown u has a boolean field u.restricted.Ifu.restricted is true, the
unknown u is said to be restricted, which means that its value must be nonnegative.
We will present the tableau in the following form:
1 x[1] ··· x[m]
y[0] a[0, 0] a[0, 1] ··· a[0, m]
.
.
.
.
.
.
.
.
.
.
.
.
y[n]
a[n, 0] a[n, 1] ··· a[n, m]
In the displayed form of the tableau restricted unknowns will be superscripted with
the sign ≥, and a small ∇ will be placed to the right of column dcol.
The constraints represented by the tableau are the row constraints, dead column
constraints, and sign constraints. Each row i of the tableau represents the row
constraint
y[i] = a[i, 0] +
j:1≤ j ≤m
a[i, j] × x[ j].
For each j in the range 1,...,dcol, there is a dead column constraint, x[ j] = 0.
Columns with index 1,...,dcol are the dead columns; columns dcol + 1,...,m
are the live columns; and column 0 is the constant column. Finally, there is a
sign constraint u ≥ 0 for each restricted unknown. Throughout the remainder
of Section 8, RowSoln denotes the solution set of the row constraints, DColSoln
denotes the solution set of the dead column constraints, and SignSoln denotes the
solution set of the sign constraints. We refer to the row constraints and the dead
column constraints collectively as hyperplane constraints and denote their solution
set as HPlaneSoln.Werefer to the hyperplane constraints and the sign constraints
436 D. DETLEFS ET AL.
collectively as tableau constraints and denote their solution set as TableauSoln.We
will sometimes identify these solution sets with their characteristic predicates.
It will be convenient to have a special unknown constrained to equal 0. Therefore,
the first row of the tableau is owned by a special unknown that we call Zero, whose
row is identically 0.
y[0] = Zero
a[0, j] = 0, for j :0≤ j ≤ m
The unknown Zero is allocated at initialization time and connected to the E-node
for 0.
Here is a typical tableau in which dcol is 0.
1
∇
m
≥
n
≥
p
Zero 0000
w
≥
01−12
z
−1 −1 −30
(12)
The tableau (12) represents the tableau constraints:
m ≥ 0
n ≥ 0
w ≥ 0
Zero = 0
w = m − n + 2p
z =−m − 3n − 1.
The basic idea of the implementation is that the tableau constraints are the con-
crete representation of the abstract variable Asserted, which is the conjunction of
Simplex literals that have been asserted. There is one fine point: All unknowns that
occur in arguments to AssertLit are connected to E-nodes, but the implementation
allocates some unknowns that are not connected to E-nodes (these are the classical
“slack variables” of linear programming). These must be regarded as existentially
quantified in the representation:
Asserted = (∃unc : TableauSoln)
where unc is the list of unconnected unknowns.
The unknown x[ j]iscalled the owner of column j , and y[i]iscalled the owner
of row i . The representation of an unknown includes the Boolean field ownsRow
and the integer field index, which are maintained so that for any row owner
y[i], y[i].ownsRow is true and y[i].index = i, and for any column owner x[ j ],
x[ j ].ownsRow is false and x[ j].index = j.
Recall that we are obliged to implement the procedure UnknownForEnode, spec-
ified in Section 4.4:
proc UnknownForEnode(e : E-node):Unknown ≡
if e.Simplex
Unknown = nil then
return e.Simplex
Unknown
end;
m := m + 1;
x[m]:= new(Unknown);
Push ("Deallocate", x[m]) onto the undo stack (
§3.1);
Simplify: A Theorem Prover for Program Checking 437
x[m].enode := e
x[m].ownsRow := false;
x[m].index := m;
for i := 1 to n do
a[i, m]:= 0
end;
e.Simplex
Unknown := x[m];
return x[m]
end
The assert procedures take formal affine sums of unknowns as arguments, but
the hearts of their implementations operate on individual unknowns. We therefore
introduce a procedure that converts an affine sum into an equivalent unknown, more
precisely:
proc UnknownForFAS(fas : formal affine sum of unknowns) : Unknown;
/* Allocates and returns a new row owner constrained to be equal to fas. */
The implementation is a straightforward manipulation of the tableau data structure:
proc UnknownForFAS(fas : formal affine sum of unknowns) : Unknown ≡
n := n + 1;
y[n]:= new(Unknown);
Push ("Deallocate", y[n]) on the undo stack (
§3.1);
y[n].ownsRow := true;
y[n].index := n;
let fas be the formal affine sum k
0
+ k
1
× v
1
+ ···+k
p
× v
p
in
a[n, 0] := k
0
;
for j := 1 to m do
a[n, j]:= 0
end;
for i :=1 to p do
if v
i
.ownsRow then
for j := 0 to m do
a[n, j]:= a[n, j] + k
i
× a[v
i
.index, j]
end
else if v
i
.index > dcol then
a[n, v
i
.index]:= a[n, v
i
.index] + k
i
// else v
i
owns a dead column and is constrained to be 0
end
end
end;
return y[n]
end
8.3. TESTING CONSISTENCY. The sample point of the tableau is the point that
assigns 0 to each column owner and that assigns a[i, 0] to each row owner y[i].
Obviously, the sample point satisfies all the row constraints and dead column
constraints. If it also satisfies all the sign constraints, then the tableau is said to
be feasible. Our algorithm maintains feasibility of the tableau at all times.
Arow owner y[i]inafeasible tableau is said to be manifestly maximized if
every non-zero entry a[i, j], for j > dcol,isnegative and lies in a column whose
owner x[ j]isrestricted. It is easy to see that in this case y[i]’s maximum value
over TableauSoln is its sample value (i.e., its value at the sample point) a[i, 0].
438 D. DETLEFS ET AL.
Alive column owner x[ j ]inafeasible tableau is said to be manifestly unbounded
if every negative entry a[i, j ]inits column is in a row owned by an unrestricted
unknown. It is easy to see that in this case TableauSoln includes points that assign
arbitrarily large values to x[ j].
The pivot operation modifies the tableau by exchanging some row owner
y[ j ] with some column owner x[i] while (semantically) preserving the tableau
constraints. It is clear that, if a[i, j] = 0, this can be done by solving the y[ j ]
row constraint for x[i] and using the result to eliminate x[i] from the other row
constraints. We leave it to the reader to check that the net effect of pivoting is to
transform
Pivot
Column
Any
Other
Column
Pivot
Row
ab
Any
Other
Row
cd
into
Pivot
Column
Any
Other
Column
Pivot
Row
1/a −b/a
Any
Other
Row
c/ad− bc/a
We will go no further into the implementation of Pivot. The implementation in
Nelson’s thesis used the sparse matrix data structure described in Section 2.2.6 of
Knuth’s Fundamental Algorithms [Knuth 1968]. Using a sparse representation is
probably a good idea, but in fact Simplify doesn’t do so: it uses a dynamically
allocated two-dimensional sequential array of rational numbers.
We impose upon the caller of Pivot(i, j ) the requirement that j be the index of
alive column and that (i, j)issuch that feasibility is preserved.
The pivot operation preserves not only Asserted but also DColSoln, RowSoln,
and SignSoln individually.
An important observation concerns the effect of the pivot operation on the sample
point. Let u be the owner of the pivot column. Since all other column owners have
sample value 0 both before and after the pivot operation, while u is 0 before but not
necessarily after the pivot operation, it follows that the sample point moves along
the line determined by varying u while leaving all other column owners zero and
setting each row owner to the unique value that satisfies its row constraint. We call
this line the line of variation of u.
Here is the key fact upon which the Simplex algorithm is based:
If u is any unknown of a feasible tableau, then there exists a sequence
of pivots such that each pivot preserves feasibility, no pivot lowers the
sample value of u, and the net effect of the entire sequence is to pro-
duce a tableau in which u is either manifestly maximized or manifestly
unbounded. We call such a sequence of pivots a rising sequence for u.
Any of several simple rules will generate an appropriate sequence of pivots; no
backtracking is required. The simplest pivot selection rule, described by George
Dantzig in his original description of the Simplex Algorithm, is essentially to choose
any pivot that increases (or at least does not decrease) the sample value of u while
preserving feasibility. This is the rule that Simplify uses and we describe it in detail
below. The reason we use it is that it is simple. A disadvantage is that it is not
guaranteed to increase the sample value of u, and in fact it has been known for
Simplify: A Theorem Prover for Program Checking 439
many years that in the worst case this rule can lead to an infinite loop in which
the Simplex algorithm pivots indefinitely. Unfortunately, the “lexicographic pivot
selection rule”, which prevents infinite looping, does not prevent exponentially
many pivots in the worst case, which would be indistinguishable in practice from
infinite looping. We therefore choose the simple rule, and put our faith in the
overwhelming empirical evidence that neither infinite looping nor exponentially
long rising sequences have any appreciable chance of occurring in practice.
Using this fact, and temporarily ignoring the details of how the pivots are chosen
as well as the issue of propagating equalities, we can now describe an implemen-
tation of AssertGE(fas)inanutshell:
Create a row owner u whose row constraint is equivalent, given the
existing constraints, to u = fas, and then pivot the tableau until either
(1) u has a non-negative sample value or (2) u is manifestly maximized
with a negative sample value. In case (1), restricting u leaves the tableau
feasible and expresses the constraint fas ≥ 0, so restrict u.Incase (2),
the inequality fas ≥ 0isinconsistent with the existing constraints, so set
refuted.
Returning to the key fact, and the way in which the program exploits it, each
pivot of the rising sequence for an unknown u is located by the routine FindPivot.
The unknown u must be a row owner, and the argument to FindPivot is the index
of its row. More precisely, given a row index i, FindPivot(i) returns a pair (h, j)
for which one of the following three cases holds:
—i = h and pivoting on (h, j)isthe next pivot in a rising sequence for y[i],
—y[i]ismanifestly maximized and (h, j) = (−1, −1), or
—i = h and pivoting on (h, j) produces a tableau in which x[ j ] (formerly y[i]) is
manifestly unbounded.
Using FindPivot (and Pivot), it is straightforward to determine the sign of the
maximum of any given unknown. This task is performed by the routine SgnOfMax,
whose specification is
proc SgnOfMax(u : Unknown):integer ≡
/* Requires that u ownarow or a live column. Returns −1, 0, or +1 according as the
maximum value of u over the solution set of the tableau is negative, zero, or positive. (An
unbounded unknown is considered to have a positive maximum.) In case 0 is returned, the
tableau is left in a state where u is manifestly maximized at zero. In case 1 is returned, the
tableau is left in a state where u is a row owner with positive sample value or is a manifestly
unbounded column owner. */
and whose implementation is
proc SgnOfMax(u : Unknown):integer ≡
if u owns a manifestly unbounded column then
return 1
end;
ToRow(u); // pivots u to be a row owner, as described later
while a[u.index, 0] ≤ 0 do
( j, k):= FindPivot(u.index);
if j = u.index then
Pivot( j, k);
// u is manifestly unbounded.
return 1
else if ( j, k) = (−1, −1) then
440 D. DETLEFS ET AL.
// u is manifestly maximized.
return sign(a[u.index, 0])
else
Pivot( j, k)
end
end;
return 1
end
We now turn to the implementation of FindPivot. FindPivot(i) chooses the pivot
column first and then the pivot row.
Whatever column j is chosen, that column’s owner x[ j ] has current sample
value zero, and when pivoted into a row, it will in general then have non-zero
sample value. Since all other column owners will continue to be column owners
and have sample value zero, the row constraint for row i implies that the change
to the sample value of the row owner y[i] caused by the pivot will be a[i, j] times
the change in the sample value of the pivot column owner x[ j] (which is also
the post-sample-value of that unknown). In order for the pivot to have any chance
of increasing the sample value of y[i ], it is therefore necessary that a[i, j] = 0.
Furthermore, if x[ j]isrestricted, then we can only use pivots that increase its
sample value. In this case, the sample value of y[i] will increase only if a[i, j] > 0.
In summary, we choose any live pivot column with a positive entry in row i or with
anegative entry in row i and an unrestricted column owner. Notice that if there is
no such column, then row i’s owner is manifestly maximized, and FindPivot can
return (−1, −1).
Having chosen the pivot column j,wemust now choose the pivot row. To choose
the pivot row, we begin by recalling that each pivot in column j moves the sample
point along the line of variation of x[ j ]. Each restricted row owner y[h] whose entry
a[h, j]inthe pivot column has sign opposite to a[i, j] imposes a limit on how far
the sample point may be moved along the line of variation of x[ j]inthe direction
that increases y[i]. If there are no such restricted row owners, then the solution set
contains an infinite ray in which the pivot column owner takes on values arbitrarily
far from zero in the desired direction, and therefore y[i] takes on arbitrarily large
values. In fact, as the reader can easily show, pivoting on (i, j) will in this case
move y[i]tobeamanifestly unbounded column owner. On the other hand, if one
or more rows do impose restrictions on the distance by which the pivot column
owner can be moved, then we choose as the pivot row any row h that imposes the
strictest restriction. Pivoting on (h, j) makes y[h] into a column owner with sample
value 0 and therefore obeys the strictest restriction, and hence all the restrictions.
proc FindPivot(i : integer):(integer × integer) ≡
var j , sgn in
if there exists a k > dcol such that ¬ x[k].restricted and a[i, k] = 0 then
Choose such a k;
( j, sgn):= (k, a[i, k])
else if there exists a k > dcol such that
x[k].restricted and a[i, k] > 0 then
Choose such a k;
( j, sgn):= (k, +1)
else
return (−1, −1)
end;
// Column j is the pivot column, and the sign of sgn is
Simplify: A Theorem Prover for Program Checking 441
// the direction in which the pivot should move x[ j ].
var champ := i, score :=∞in
for each row h such that h = i and y[h].restricted do
if sgn × a[h, j ] < 0 ∧|a[h, 0]/a[h, j]| < score then
score :=|a[h, 0]/a[h, j]|;
champ := h
end
end;
return (champ, j)
end
end
end
We next give the specification and implementation of ToRow, which is similar
to the second half of FindPivot.
proc ToRow(u : Unknown) ≡
/* Requires that u not own a dead or manifestly unbounded column. A no-op if u already
ownsarow.Ifu owns a column, pivots u into a row. */
proc ToRow(u : Unknown) ≡
if u.ownsRow then
return
end;
var j := u.index, champ, score :=∞in
for each row h such that y[h].restricted ∧ a[h, j ] < 0 do
if −a[h, 0]/a[h, j] < score then
score :=−a[h, 0]/a[h, j];
champ := h
end
end;
Pivot(champ, j)
end
end
If u owns a column, then ToRow must pivot u into a row, while preserving
feasibility of the tableau. Recall that any pivot in u’s column leaves the sample
point somewhere on the line of variation of u.Ifu were manifestly unbounded,
then the entire portion of the line of variation of u for which u > 0would lie inside
the solution set of TableauSoln. But it is a precondition of ToRow that u not be
manifestly unbounded. That is, there is at least one restricted row owner whose row
has a negative entry in u’s column. The sign constraint of each such row owner
y[h] imposes a bound on how far the sample point can be moved along the line of
variation of u in the direction of increasing u while still remaining in the solution
set. Specifically, since a[h, u.index]isnegative, y[h] decreases as u increases on
the line of variation of u, and u therefore must not be increased beyond the point
at which y[h] = 0. ToRow chooses a restricted row owner whose sign constraint
imposes the strictest bound on the increasing motion of u along the line of variation
of u and moves the sample point to that bound by pivoting u’s column with that
row, thereby leaving the chosen row owner as a column owner with sample value 0.
We are now ready to give the implementation for AssertGE:
proc AssertGE(fas : formal affine sum of connected unknowns) ≡
var u := UnknownForFAS(fas) in
442 D. DETLEFS ET AL.
var sgmx := SgnOfMax(u) in
if sgmx < 0 then
refuted := true;
return
else
u.restricted := true;
Push ("Unrestrict", u) onto the undo stack (
§3.1);
if sgmx = 0 then
CloseRow(u.index)
end;
return
end
end
end
end
The fact that AssertGE propagates enough equalities in the case that SgnOfMax
returns 0 is a consequence of the specification for CloseRow:
proc CloseRow(i : integer);
/* Requires that y[i]bearestricted unknown u that is manifestly maximized at zero.
Establishes PROPAGATION CORRECTNESS. May modify dcol and thus modify the dead
column constraints, but preserves the solution set of the tableau. */
The fact that AssertGE is correct to propagate no equalities in the case that
SgnOfMax returns 1 is a consequence of the following lemma:
L
EMMA 1. Suppose that an unknown u has a positive value at some point in
TableauSoln. Then adding the sign constraint u ≥ 0 does not increase the set of
affine equalities that hold over the solution set.
P
ROOF. Let f be an affine function of the unknowns that is zero over the
intersection of TableauSoln with the half-space u ≥ 0. We will show that f (P) = 0
for all P ∈ TableauSoln.Bythe hypothesis, f (P) = 0ifu(P) ≥ 0. It remains to
consider the case that u(P) < 0.
By the assumption of the lemma, there is a point R ∈ TableauSoln such that
u(R) > 0. Since u is negative at P and positive at R, there is a point Q = R
on the line segment PR such that u(Q) = 0. Note that Q ∈ TableauSoln, since
TableauSoln is convex. Since u(R) > 0 and u(Q) = 0, both R and Q lie in the
half-space u ≥ 0. Since f = 0 holds over the intersection of TableauSoln with this
half-space, it follows that f (R) and f (Q) are both zero. Since the affine function
f is zero at two distinct points, Q and R,onthe line PR,itmust be zero on the
entire line, including at P.
It would be possible to implement AssertZ using two calls to AssertGE,butwe
will show a more efficient implementation below.
8.3.1. Simplex Redundancy Filtering. Simplify’s AssertGE incorporates an
optimization Simplex redundancy filtering not shown in the pseudo-code above.
It sometimes happens that an assertion is trivially redundant, like the second asser-
tion of AssertLit(n ≥ 0); AssertLit(n + 1 ≥ 0). AssertGE filters out such redundant
assertions. The technique is to examine the row owned by an unknown that is about
to be restricted. If every nonzero entry in the row is positive and lies either in the
constant column or in a column with a restricted owner, then the assertion for which
the row was created is redundant, so instead of restricting the row owner and testing
for consistency, AssertGE simply deletes the row.
Simplify: A Theorem Prover for Program Checking 443
8.4. P
ROPAGATING EQUALITIES.Itremains to implement CloseRow, whose job
is to handle the case where some unknown u has been constrained to be at least zero
in a tableau that already implies that u is at most zero. In this case, the dimensionality
of the solution set is potentially reduced by the assertion, and it is potentially
necessary to propagate equalities as required by the equality-sharing (
§4.1) protocol.
Two unknowns are said to be manifestly equal if either (1) both own rows, and
those rows are identical, excluding entries in dead columns; (2) one is a row owner
y[i], the other is a column owner x[ j], and row i,excluding entries in dead columns,
contains only a single nonzero element a[i, j] = 1; (3) both own dead columns;
or (4) one owns a dead column and the other owns a row whose entries, excluding
those in dead columns, are all zero. It is easy to see that if u and v are manifestly
equal, the equality u = v holds over TableauSoln (and, in fact, over HPlaneSoln).
Detecting equalities implied by the hyperplane constraints (
§8.2) alone is straight-
forward, as shown by the following lemma:
L
EMMA 2. An equality between two unknowns of the tableau is implied by the
hyperplane constraints iff the unknowns are manifestly equal.
P
ROOF. The hyperplane constraints imply that each unknown is equal to a
certain formal affine sum of the owners of the live columns, as follows:
For each row owner, y[i]: y[i] = a[i, 0] +
j:dcol< j ≤m
a[i, j] × x[ j].
For each live column owner, x[k]: x[k] = 0 +
j:dcol< j ≤m
δ
jk
× x[ j], where δ
ij
is1ifi = j and 0 otherwise.
For each dead column owner, x[k]: x[k] = 0 +
j:dcol< j ≤m
0 × x[ j ].
Since, in satisfying the hyperplane constraints, the values of the live column
owners can be assigned arbitrarily, two unknowns are equal over HPlaneSoln iff
the corresponding formal affine sums are identical in all coefficients. It is straight-
forward to check that coefficients are identical precisely when the unknowns are
manifestly equal.
The next lemma gives a simple condition under which the problem of finding
equalities that hold over TableauSoln reduces to that of finding equalities that hold
over HPlaneSoln:
L
EMMA 3. Suppose that for each restricted unknown u, either u is manifestly
equal to Zero or TableauSoln includes a point at which u is strictly positive. Then
for any affine function f of the unknowns,
f = 0 over TableauSoln ⇔ f = 0 over HPlaneSoln.
P
ROOF. The “⇐” part is a consequence of the fact that TableauSoln ⊆
HPlaneSoln.
The “⇒” part follows by induction on the number of restricted unknowns not
manifestly equal to Zero. The base case follows from the fact that if all restricted un-
knowns are manifestly equal to Zero, then HPlaneSoln coincides with TableauSoln.
The induction step is simply an instance of Lemma 1.
When each restricted unknown either is positive somewhere in TableauSoln or
is manifestly equal to Zero,wesay that the tableau is minimal. (This name comes
from the fact that a minimal tableau represents Asserted using the smallest possible
number of live columns, namely the dimension of the solution set.) An immediate
consequence of Lemmas 2 and 3 and the definition of Asserted is:
444 D. DETLEFS ET AL.
L
EMMA 4. Suppose that the tableau is consistent and minimal. Then an equality
between two connected unknowns is implied by Asserted iff it is manifest.
The procedure CloseRow uses SgnOfMax to find any restricted unknowns that
cannot assume positive values in TableauSoln, and which therefore must be equal
to zero over TableauSoln. When it finds such unknowns, it makes their equality to
Zero manifest, using the procedure KillCol, which is specified as follows:
proc KillCol( j : integer)
/* Requires that column j be owned by an unknown u = x[ j ]having value 0 over
TableauSoln, and that all manifest equalities of connected unknowns are implied by
Propagated. Adds the dead column constraint u = 0 (undoably), leaving Asserted un-
changed, and propagates equalities as necessary to ensure that all manifest equalities of
connected unknowns are still implied by Propagated. */
As CloseRow applies KillCol to more and more columns the tableau will eventually
become minimal. At this point, Lemma 4 and the postcondition of the last call to
KillCol will together imply PROPAGATION CORRECTNESS.
proc KillCol( j : integer) ≡
Push "ReviveCol" onto the undo stack (
§3.1);
swap column j with column dcol + 1;
dcol := dcol + 1;
propagate an equality between x[dcol] and Zero;
for each i such that y[i]isconnected and a[i, dcol] = 0 do
if y[i]ismanifestly equal to some connected unknown u then
propagate an equality between y[i] and some such u
end
end
end
proc CloseRow(i) ≡
for each j > dcol such that a[i, j] = 0 do
// a[i, j] < 0
KillCol( j)
end;
var R := the set of all restricted unknowns with
sample value 0 but not manifestly equal to Zero in
while R ={}do
var u := any element of R in
delete u from R;
if SgnOfMax(u) = 0 then
// u owns a row and every non-zero entry in that row is in a
// column whose owner has value 0 over TableauSoln.
for each j > dcol such that a[u.index, j] = 0 do
// a[u.index, j] < 0
KillCol( j)
end
// u is now manifestly equal to Zero
else
// SgnOfMax(u) =+1, so skip
end;
delete from R all elements that are manifestly equal to Zero or
have positive sample values
end
end
end
end
Simplify: A Theorem Prover for Program Checking 445
The reader might hope that the first three lines of this procedure would be enough
and that the rest of it is superfluous, but the following example shows otherwise:
1
∇
p
≥
qr
Zero 00 0 0
a
≥
0 −10 0
b
≥
01−11
c
≥
01 1−1.
In the tableau above, the unknown a is manifestly maximized at zero. When
CloseRow is applied to a’s row, the first three lines kill the column owned by p.
This leaves a nonminimal tableau, since the tableau then implies b = c = 0 and
that q =−r . The rest of CloseRow will restore minimality by pivoting the tableau
to make either b or c a column owner and killing the column.
As a historical note, the description in Nelson’s thesis [Nelson 1979] erroneously
implied that CloseRow could be implemented with the first three lines alone. This
bugwas also present in our first implementation of Simplify and went unnoticed
for a considerable time, but eventually an ESC example provoked the bug and we
diagnosed and fixed it. We note here that the fix we implemented years ago is dif-
ferent from the implementation above, so the discrepancy between paper and code
is greater in this section than in the rest of our description of the Simplex module.
As remarked above, AssertZ could be implemented with two calls to AssertGE.
But if it were, the second of the two calls would always lead to a call to CloseRow,
which would incur the cost of reminimizing the tableau by retesting the restricted
unknowns. This work is not always necessary, as shown by the following lemma:
L
EMMA 5. If an unknown u takes on both positive and negative values within
TableauSoln, and a restricted unknown v is positive somewhere in TableauSoln,
then (v > 0 ∧ u = 0) must hold somewhere in TableauSoln.
P
ROOF
. Let P be a point in TableauSoln such that v(P) > 0. If u(P) = 0,
we are done. Otherwise, let Q be a point in TableauSoln such that u(Q) has sign
opposite to u(P). Since TableauSoln is convex, line segment PQ lies entirely within
TableauSoln. Let R be the point on PQ such that u(R) = 0. Since v is restricted,
v(Q) ≥ 0. Since v(Q) ≥ 0, v(P) > 0, R ∈ PQ, and R = Q,itfollows that
v(R) > 0.
We will provide an improved implementation of AssertZ that takes advantage
of Lemma 5. The idea is to use two calls to SgnOfMax to determine the signs of
the maximum and minimum of the argument. If these are positive and negative
respectively, then by Lemma 5, we can perform the assertion by killing a single
column. We will need to modify SgnOfMax to record in the globals (iLast, jLast)
the row and column of the last pivot performed (if any). Since SgnOfMax(u)never
performs any pivots after u acquires a positive sample value, and since pivot-
ing is its own inverse, it follows that this modified SgnOfMax satisfies the addi-
tional postcondition:
If u’s initial sample value is at most zero, and u ends owning a row
and 1 is returned, then the tableau is left in a state such that pivoting at
(iLast, jLast)would preserve feasibility and make u’s sample value at
most zero.
446 D. DETLEFS ET AL.
Using the modified SgnOfMax,weimplement AssertZ as follows:
proc AssertZ(fas : formal affine sum of connected unknowns) ≡
var u := UnknownForFAS(fas) in
if u is manifestly equal to Zero then
return
end;
var sgmx := SgnOfMax(u) in
if sgmx < 0 then
refuted := true;
return
else if sgmx = 0 then
CloseRow(u.index);
return
else
// Sign of max of fas is positive.
if u.ownsRow then
for j := 0 to m do
a[u.index, j]:=−a[u.index, j]
end;
else
for i := 1 to n do
a[i, u.index]:=−a[i, u.index]
end
end;
// u now represents −fas.
sgmx := SgnOfMax(u) in
if sgmx < 0 then
refuted := true;
return
else if sgmx = 0 then
CloseRow(u.index);
return
else
// Sign of max of −fas is also positive, hence Lemma 5 applies.
if u.ownsRow then
Pivot(u.index, jLast);
end;
KillCol(u.index);
return
end
end
end
end
end
The main advantage of this code for AssertZ over the two calls to AssertGE
occurs in the case where both calls to SgnOfMax return 1. In this case, the unknown
u takes on both positive and negative values over the initial TableauSoln,sowe
know by Lemma 5 that if any restricted unknown v is positive for some point in the
initial TableauSoln, v is also positive at some point that remains in TableauSoln after
the call to KillCol adds the dead column constraint u = 0. Thus, the minimality
of the tableau is guaranteed to be preserved without any need for the loop over
the restricted unknowns that would be caused by the second of the two AssertGE
assertions.
We must also show that feasibility of the tableau is preserved in the case where
both calls to SgnOfMax return 1. In this case, the first call to SgnOfMax leaves u
Simplify: A Theorem Prover for Program Checking 447
with sample value at least 0. After the entries in u’s row or column are negated, u
then has sample value at most 0. If the second call to SgnOfMax then leaves u as
arow owner, the postconditions of SgnOfMax guarantee that u ≥ 0atthe sample
point, and that pivoting at (iLast, jLast)would move the sample point to a point
on the line of variation (
§8.3) of x[jLast] where u ≤ 0 while preserving feasibility.
Instead, we pivot at (u.index, jLast), which moves the sample point to the point
along the line of variation of x[jLast]tothe point where u = 0. By the convexity
of TableauSoln, this pivot also preserves feasibility.
A final fine point: The story we have told so far would require a tableau row for
each distinct numeric constant appearing in the conjecture. In fact, Simplify incor-
porates an optimization that often avoids creating unknowns for numeric constants.
A price of this optimization is that it becomes necessary to propagate equalities not
only between unknowns but also between an unknown and a numeric constant. For
example, when Simplify interns (
§4.5) and asserts a literal containing the term f (6),
it creates an E-node for the term 6 but does not connect it to a Simplex unknown.
On the other hand, if the Simplex tableau later were to imply u = 6, for some
connected unknown u, Simplify would then ensure that an equality is propagated
between u.enode and the E-node for 6. The changes to CloseRow required to detect
unknowns that have become “manifestly constant” are straightforward.
8.5. U
NDOING TABLEAU OPERATIONS. The algorithms we have presented push
undo records when they modify SignSoln, DColSoln, RowSoln,orthe set of con-
nected unknowns. In this section, we describe how these undo records are processed
by Pop. Since the representation of Asserted is given as a function of SignSoln,
DColSoln, RowSoln,itfollows from the correctness of these bits of undoing code
that Pop meets its specification of restoring Asserted.
Notice that we do not push an undo record for Pivot.Asaconsequence Pop
may not restore the initial permutation of the row and column owners. This
doesn’t matter as long as the semantics of SignSoln, DColSoln, RowSoln are
restored.
To process an undo record of the form (
"Unrestrict", u):
u.restricted := false.
To process an undo record of the form
"ReviveCol":
dcol := dcol − 1.
To process an undo record of the form (
"Deallocate", u):
if u.ownsRow then
copy row n to row u.index;
y[u.index]:= y[n];
n := n − 1
else if u’s column is identically 0 then
copy column m to column u.index;
x[u.index]:= x[m];
m := m − 1
else
perform some pivot in u’s column that preserves feasibility;
copy row n to row u.index;
y[u.index]:= y[n];
n := n − 1
end;
448 D. DETLEFS ET AL.
if u.enode = nil then
u.enode.Simplex
Unknown := nil;
u.enode := nil
end
In arguing the correctness of this undo action, we observe that there are two cases
in which an unknown is allocated: UnknownForEnode allocates a completely un-
constrained unknown and connects it to an E-node, and UnknownForFAS allocates a
constrained unknown and leaves it unconnected. If the unknown u to be deallocated
was allocated by UnknownForEnode, then at entry to the action above the tableau
will have been restored to a state where u is again completely unconstrained. In
this case, u must own an identically zero column. All that is required to undo the
forward action is to delete the vacuous column, which can be done by overwriting
it with column m and decrementing m.Ifu was allocated by UnknownForFAS, then
u will be constrained by RowSoln.Ifu happens to a row owner, we need only delete
its row. If u owns a column, we cannot simply delete the column (since that would
change the projection of RowSoln onto the remaining unknowns), so we must pivot
u to make it a row owner. The argument that it is always possible to do this while
preserving feasibility is similar to those for ToRow and for FindPivot, and we leave
the details to the reader.
8.6. I
NTEGER HEURISTICS.Asiswell known, the satisfiability problem for
linear inequalities over the rationals is solved in polynomial time by various
ellipsoid methods and solved rapidly in practice by the Simplex method, but the
same problem over the integers is NP-complete: Propositional unknowns can be
constructed from integer unknowns by adding constraints like 0 ≤ b ∧ b ≤ 1, after
which ¬ b is encoded 1 − b, and b ∨ c ∨ d is encoded b + c + d > 0. Thus, 3SAT
is encoded.
For integer linear inequalities that arise from such an encoding, it seems extremely
likely (as likely as P = NP) that the only way to solve the satisfiability problem will
be by some kind of backtracking search. A fundamental assumption of Simplify
is that most of the arithmetic satisfiability problems arising in program checking
don’t resemble such encoded SAT problems and don’t require such a backtracking
search.
In designing Simplify, we might have introduced a built-in monadic predicate
characterizing the integers. Had we done so, a conjecture could mention both
integer and rational unknowns. But rational unknowns did not seem useful in
our program-checking applications. (We did not aspire to formulate a theory of
floating-point numbers.) So Simplify treats all arithmetic arguments and results
as integers.
The Simplex module’s satisfiability procedure is incomplete for integer linear
arithmetic. However, by combining the complete decision procedure for rational
linear arithmetic described above with three heuristics for integer arithmetic, we
have obtained satisfactory results for our applications.
The first heuristic, the negated inequality heuristic,isthat to assert a literal of
the form ¬ a ≤ b, Simplify performs the call AssertGE(a − b − 1).
We found this heuristic alone to be satisfactory for a number of months. It is
surprisingly powerful. For example, it allows proving the following formula:
i ≤ n ∧ f (i) = f (n) ⇒ i ≤ n − 1.
Simplify: A Theorem Prover for Program Checking 449
The reason this formula is proved is that the only possible counterexample (
§6.1)
i ≤ n, f (i) = f (n), ¬ i ≤ n − 1,
and the heuristically transformed third literal i − (n − 1) − 1 ≥ 0 combines with
the first literal to propagate the equality i = n, contradicting the second literal.
Eventually, we encountered examples for which the negated inequality heuristic
alone was insufficient. For example:
2 ≤ i ∧ f (i) = f (2) ∧ f (i) = f (3) ⇒ 4 ≤ i.
We therefore added a second integer heuristic, the Tighten Bounds proof tactic.
If, for some term t that is not an application of +, −,or×, the Simplex tableau
contains upper and lower bounds on t: L ≤ t and t ≤ U , for integer constants L
and U , Simplify will try to refute the possibility t = L (not try so hard as to do a
case split, but Simplify will do matching on both unit and nonunit matching rules
(
§
5.2) in the attempted refutation), and if it is successful, it will then strengthen the
lower bound from L to L + 1 and continue. It would be perfectly sound to apply
the tactic to terms that were applications of +, −,or×,but for efficiency’s sake
we exclude them.
The Tighten Bounds tactic is given low priority. Initially, it had the lowest priority
of all proof tactics: it was only tried as a last resort before printing a counterexample.
However, we later encountered conjectures whose proof required the use of Tighten
Bounds many times. We therefore introduced a Boolean recording whether the tactic
had recently been useful. When this Boolean is true, the priority of the tactic is raised
to just higher than case splitting. The Boolean is set to true whenever the Tighten
Bounds tactic is successful, and it is reset to false when Simplify backtracks from a
case split on a non-goal clause and the previous case split on the current path (
§3.1)
wasonagoal (
§3.5) clause. That is, the boolean is reset whenever clause scores are
renormalized (as described in Section 3.6).
Simplify’s third integer arithmetic heuristic is the manifest constant heuristic. If
all live column entries in some row of the tableau are zero, then the tableau implies
that the row owner is equal to the entry in its constant column entry. If the constant
column entry in such a row is not an integer, then the Simplex module detects a
contradiction and sets refuted.
8.7. O
VERF LOW. Simplify implements the rational numbers in the simplex
tableau as ratios of 32-bit or 64-bit integers, depending on the platform. By
default, Simplify does no overflow checking. If an overflow does occur it can
produce a crash or an incorrect result. Several of the bug reports that we have re-
ceived have been traced to such overflows. Simplify accepts a switch that causes
it to check each tableau operation for overflow and halt with an error message if
an overflow occurs. When ESC/Java is applied to its own front end, none of the
proofs of the 2331 verification conditions overflows on a 64-bit machine; one of
them overflows on a 32-bit machine. Enabling overflow checking increases total
proof time by about four percent on the average.
9.Performance Measurements
In this section, we report on the performance of Simplify and on the performance
effects of heuristics described in the rest of the article.
450 D. DETLEFS ET AL.
All performance figures in this section are from runs on a 500-MHz Compaq
Alpha (EV6) with 5-GB main memory. The most memory used by Simplify on the
“front-end test suite”, defined below, seems to be 100 MB. The machine has three
processors, but Simplify is single-threaded. Simplify is coded in Modula-3 with
all bounds checking and null-dereference checks enabled. Simplify relies on the
Modula-3 garbage collector for storage management.
Our goal was to achieve performance sufficient to make the extended static
checker useful. With the released version of Simplify, ESC/Java is able to check
the 44794 thousand lines of Java source in its own front end (comprising 2331
routines and 29431 proof obligations) in 91 minutes. This is much faster than the
code could be checked by a human design review, so we feel we have succeeded.
We have no doubt that further performance improvements (even dramatic ones)
are possible, but we stopped working on performance when Simplify became fast
enough for ESC.
We used two test suites to produce the performance data in this section. The first
suite, which we call the “small test suite” consists of five verification conditions
(
§1) generated by ESC/Modula-3 (addhi, cat, fastclose, frd-seek, simplex),
eleven verification conditions generated by ESC-Java:
toString
isCharType
visitTryCatchStmt
binaryNumericPromotion
Parse
getRootInterface
checkTypeDeclOfSig
checkTypeDeclElem
main
getNextPragma
scanNumber
and two artificial tests (domino6x4x and domino6x6x) that reduce the well-known
problem of tiling a mutilated checkerboard with 1-by-2 dominos into test cases
that exercise case splitting and the E-graph. The second suite, which we call the
“front-end test suite” consists of the 2331 verification conditions for the routines
of ESC/Java’s front end. Both test suites contain valid conjectures only. These tests
are available on the web [Detlefs et al. 2003a].
Much of our performance work aimed not so much at improving the average case
as at steering clear of the worst case. Over the course of the ESC project, Simplify
would occasionally “go off the deep end” by diving into a case analysis whose
time to completion was wildly longer than any user would ever wait. When this
happened, we would analyze the problem and the computation into which Simplify
stumbled, and design a change that would prevent the problem from occurring. The
change might not improve average case behavior, but we took the view that going
off the deep end on extended static checking of realistic programs was worse than
simple slowness.
Of course, it could be that the next realistic program that ESC is applied to will
send Simplify off the deep end in a way that we never encountered and never took
precautions against. Therefore, when ESC uses Simplify to check a routine, it sets
a timeout. If Simplify exceeds the timeout, ESC notifies the user and goes on to
Simplify: A Theorem Prover for Program Checking 451
FIG.4. Baseline performance data for the small test suite.
check the next routine. The default time limit used by ESC/Java, and in the tests
reported in this section, is 300 seconds.
Many of the optimizations that Simplify employs can be disabled, and Sim-
plify has other options and configurable parameters. In Sections 9.2 through 9.13,
we evaluate the performance effects of the most important of these options. We
don’t, of course, test every combination of options, but we have tried to test each
option against the baseline behavior in which all options have their default values.
Sections 9.14 through 9.17 discuss more general performance issues.
One of the verifications condition in the front end test suite requires approxi-
mately 400 seconds with Simplify’s default options, and thus times out with our
300-second time limit. No other test in either suite times out.
Figure 4 gives the baseline performance data for the tests in the small suite. For
each benchmark, the figure presents the benchmark name, the time in seconds for
Simplify to prove the benchmark, the number of case splits performed, the number
of times during the proof that the matching depth (
§5.2.1) was incremented, and the
maximum matching depth that was reached.
9.1. S
IMPLIFY AND OTHER PROVERS.Aswegotopress, Simplify is a rather
old system. Several more recent systems that use decision procedures are substan-
tially faster than Simplify on unquantified formulas. This was shown by a recent
performance study performed by de Moura and Ruess [2004]. Even at its advanced
age, Simplify seems to be a good choice for quantified formulas that require both
matching and decision procedures.
9.2. P
LUNGING.Asdescribed in Section 4.6, Simplify uses the plunging heuris-
tic, which performs a Push-Assert-Pop sequence to test the consistency of a literal
with the current context, in an attempt to refine the clause containing the literal.
Plunging is more complete than the E-graph tests (
§4.6), but also more expensive.
By default Simplify plunges on each literal of each clause produced by matching a
non-unit matching rule (
§5.2): untenable literals are deleted from the clause, and if
any literal’s negation is found untenable, the entire clause is deleted.
452 D. DETLEFS ET AL.
FIG.5. Performance data for the small test suite with plunging disabled.
Early in Simplify’s history, we tried refining all clauses by plunging before doing
each case-split. The performance of this strategy was so bad that we no longer even
allow it to be enabled by a switch, and so cannot reproduce those results here.
We do have a switch that turns off plunging. Figure 5 compares, on the small
test suite, the performance of Simplify with plunging disabled to the baseline. The
figure has the same format as Figure 4 with an additional column that gives the
percentage change in time from the baseline. The eighteen test cases of the small
suite seem insufficient to support a firm conclusion. But, under the principle of
steering clear of the worst case, the line for
cat is a strong vote for plunging.
Figure 6 below illustrates the performance effect of plunging on each of the front
end test suite. The figure contains one dot for each routine in the front end. The
dot’s x-coordinate is the time required to prove that routine’s verification condition
by the baseline Simplify and its y-coordinate is the time required by Simplify with
plunging disabled. As the caption of the figure shows, plunging reduces the total
time by about five percent. Moreover, in the upper right of the figure, all dots that
are far from the diagonal are above the diagonal. Thus, plunging both improves
the average case behavior and is favored by the principle of steering clear of the
worst case.
The decision to do refinement by plunging still leaves open the question of how
much effort should be expended during plunging in looking for a contradiction.
If asserting the literal leads to a contradiction in one of Simplify’s built-in theory
modules, then the matter is settled. But if not, how many inference steps will
be attempted before admitting that the literal seems consistent with the context?
Simplify has several options, controlled by a switch. Each option calls AssertLit,
then performs some set of tactics to quiescence, or until a contradiction is detected.
Here’s the list of options (each option includes all the tactics the preceding option).
0 Perform domain-specific decision procedures and propagate equalities.
1 Call (a nonplunging version of) Refine.
2 Match on unit rules (
§5.2).
3 Match on non-unit rules (but do not plunge on their instances).
Simplify: A Theorem Prover for Program Checking 453
FIG.6.Effect of disabling plunging on the front-end test suite.
Simplify’s default is option 0, which is also the fastest of the options on the front-
end test suite. However, option 0, option 1, and no plunging at all are all fairly
close. Option 2 is about twenty percent worse, and option 3 is much worse, leading
to many timeouts on the front-end test suite.
9.3. T
HE MOD-TIME OPTIMIZATION.Asdescribed in Section 5.4.2, Simplify
use the mod-time optimization, which records modification times in E-nodes in
order to avoid fruitlessly searching for new matches in unchanged portions of
the E-graph. Figure 7 shows the elapsed times with and without the mod-time
optimization for Simplify on our small test suite.
For the two domino problems, there is no matching, and therefore the mod-time
feature is all cost and no benefit. It is therefore no surprise that mod-time updating
caused a slowdown on these examples. The magnitude of the slowdown is, at least
to us, a bit of a surprise, and suggests that the mod-time updating code is not as tight
as it could be. Despite this, on all the cases derived from program checking, the
mod-time optimization was either an improvement (sometimes quite significant)
or an insignificant slowdown.
Figure 8 shows the effect of disabling the mod-time optimization on the front-end
test suite. Grey triangles are used instead of dots if the outcome of the verification is
different in the two runs. In Figure 8, the two triangles in the upper right corner are
cases where disabling the mod-time optimization caused timeouts. The hard-to-see
454 D. DETLEFS ET AL.
FIG.7. Performance data for the small test suite with the mod-time optimization disabled.
FIG.8.Effect of disabling the mod-time optimization on the front-end test suite.
Simplify: A Theorem Prover for Program Checking 455
FIG.9. Performance data for the small test suite with the pattern-element optimization disabled.
triangle near the middle of the figure is an anomaly that we do not understand:
somehow turning off the mod-time optimization caused a proof to fail. In spite of
this embarrassing mystery, the evidence in favor of the mod-time optimization is
compelling, both for improving average performance and for steering clear of the
worst case.
9.4. T
HE PATTERN-ELEMENT OPTIMIZATION.Asdescribed in Section 5.4.3,
Simplify uses the pattern-element optimization to avoid fruitlessly trying to match
rules if no recent change to the E-graph can possibly have created new instances.
As shown in Figures 9 and 10, the results on the examples from program checking
are similar to those for the mod-time optimization—either improvements (some-
times quite significant) or insignificant slowdowns. The domino examples suggest
that the cost is significantly smaller than that for updating mod-times.
The mod-time and pattern-element optimizations have a common purpose—
saving matching effort that cannot possibly yield new matches. Figure 11 shows
what happens when both are disabled. About three quarters of the savings from
the two optimizations overlap, but the remaining savings is significant enough to
justify doing both.
9.5. S
UBSUMPTION.Asdescribed in Section 3.2, Simplify uses the subsumption
heuristic, which asserts the negation of a literal upon backtracking from the case in
which the literal was asserted. By default, Simplify uses subsumption by setting the
status (
§3.2) of the literal to false, and also, if it is a literal of the E-graph (except
for nonbinary distinctions (
§2)) or of an ordinary theory, denying it by a call to
the appropriate assert method. By setting an environment variable, we can cause
Simplify to set the status of the literal, but not to assert its negation. Figure 12 shows
that the performance data for this option are inconclusive. The data on the front end
test suite (not shown) are also inconclusive.
9.6. M
ERIT PROMOTION.Asdescribed in Section 5.2.1, Simplify uses merit
promotion: When a Pop reduces the matching depth, Simplify promotes the highest
scoring (
§3.6) clause from the deeper matching level to the shallower level.
456 D. DETLEFS ET AL.
FIG. 10. Effect of disabling the pattern-element optimization on the front-end test suite.
Figures 13 and 14 show the effect of turning off merit promotion. In our small
test suite, one method (
cat) times out, and two methods (getNextPragma and
scanNumber) show noticeable improvements. On the front-end suite, turning off
promotion causes two methods to time out and improves the performance in a way
that counts on only one (which in fact is
getNextPragma). The total time increases
by about ten percent.
We remark without presenting details that promote set size limits of 2, 10 (the
default), and a million (effectively infinite for our examples) are practically indis-
tinguishable on our test suites.
9.7. I
MMEDIATE PROMOTION.Asdescribed in Section 5.2.1, Simplify allows a
matching rule to be designated for immediate promotion, in which case any instance
of the rule is added to the current clause set instead of the pending clause set. In
our test suites, the only immediately promoted rule is the nonunit select-of-store
axiom (
§2).
Figure 15 shows the effect of turning off immediate promotion on the small
test suite. Immediate promotion provides a significant improvement on four test
cases (
cat, fastclose, simplex, and scanNumber), but a significant degradation
on another (
getNextPragma). The front-end test suite strengthens the case for
the heuristic: Looking at the upper right corner of Figure 16 which is where the
expensive verification conditions are, outliers above the diagonal clearly outnumber
Simplify: A Theorem Prover for Program Checking 457
FIG. 11. Effect of disabling both the mod-time and pattern-element optimizations on the front-end
test suite.
FIG. 12. Performance data for the small test suite with subsumption disabled.
458 D. DETLEFS ET AL.
FIG. 13. Performance data for the small test suite with merit promotion disabled.
FIG. 14. Effect of disabling merit promotion on the front-end test suite.
Simplify: A Theorem Prover for Program Checking 459
FIG. 15. Performance data for the small test suite with immediate promotion disabled.
FIG. 16. Effect of disabling immediate promotion on the front-end test suite.
460 D. DETLEFS ET AL.
FIG. 17. Performance data for the small test suite with the activation heuristic disabled (so that
triggers are matched against all E-nodes, rather than active E-nodes only).
outliers below the diagonal. Furthermore, the heuristic reduces the total time on the
front end test suit by about five percent.
We remark without presenting details that consecutive immediately promoted
split limits of 1, 2, 10 (the default), and a million (effectively infinite for our
examples) are practically indistinguishable on our test suites.
9.8. A
CTIVATION
.Asdescribed in Section 5.2, Simplify uses the activation
heuristic, which sets the active bit in those E-nodes that represent subterms of
atomic formulas (
§2) that are currently asserted or denied, and restricts the matcher
(
§5.1) to find only those instances of matching rule triggers that lie in the active
portion of the E-graph.
Figure 17 shows that the effect of turning off activation on the small test suite is
very bad. Five of the tests time out, two crash with probable matching loops (see
Section 5.1), several other tests slow down significantly, and only one (
checkType-
DeclOfSig
) shows a significant improvement. In the front-end test suite (not
shown) thirty-two methods time out, eighteen crash with probable matching loops,
and the cloud of dots vote decisively for the optimization.
9.9. S
CORING.Asdescribed in Section 3.6, Simplify uses a scoring heuristic,
which favors case splits that have recently produced contradictions. Figures 18 and
19 show that scoring is effective.
Of the problems in the small test suite derived from program checking, scoring
helps significantly on six and hurts on none. The results are mixed for the two
domino problems.
On the front-end test suite, scoring had mixed effects on cheap problems, but
wasvery helpful for expensive problems. With scoring off ten problems time out,
and the total time increases by sixty percent.
9.10. E-G
RAPH TESTS.Asdescribed in Section 4.6, Simplify uses E-graph tests,
which checks the E-graph data structure, instead of the literal status fields only, when
refining clauses. Figures 20 and 21 show that the E-graph test is effective, especially
on expensive problems.
Simplify: A Theorem Prover for Program Checking 461
FIG. 18. Performance data for the small test suite with scoring disabled.
FIG. 19. Effect of disabling scoring on the front-end test suite.
462 D. DETLEFS ET AL.
FIG. 20. Performance data for the small test suite with the E-graph status test disabled.
FIG. 21. Effect of disabling the E-graph status test on the front-end test suite.
Simplify: A Theorem Prover for Program Checking 463
F
IG. 22. Performance data for the small test suite with distinction classes disabled (so that an n-ary
distinction is translated into
n
2
binary distinctions).
9.11. DISTINCTION CLASSES.Asdescribed in Section 7.1, Simplify uses dis-
tinction classes, a space-efficient technique for asserting the pairwise distinctness
of a set of more than two items. Interestingly enough, Figures 22 and 23 show
the performance effects of distinction classes are negligible. We do not know what
causes the strange asymmetrical pattern of dots in the front end test suite.
9.12. L
ABELS
.Asdescribed in Section 6.2, Simplify uses labels to aid in error
localization. We have done two experiments to measure the cost of labels, one to
compare labels versus no error localization at all, and one to compare labels versus
error variables.
The first experiment demonstrates that Simplify proves valid labeled verification
conditions essentially as fast as it proves valid unlabeled verification conditions.
This experiment was performed on the small test suite by setting a switch that
causes Simplify to discard labels at an early phase. Figure 24 shows the results:
there were no significant time differences on any of the examples.
The second experiment compares labels with the alternative “error variable”
technique described in Section 6.2. It was performed on the small test suite by
manually editing the examples to replace labels with error variables. Figure 25
shows the results: in all but one of the examples where the difference was significant,
labels were faster than error variables.
9.13. S
IMPLEX REDUNDANCY FILTERING.Asdescribed in Section 8.3.1, Sim-
plify uses Simplex redundancy filtering to avoid processing certain trivially redun-
dant arithmetic assertions. Figure 26 shows the effect of disabling this optimization
on the front-end test suite. The aggregate effect of the optimization is an improve-
ment of about 10 percent.
The remaining subsections of this section do not evaluate particular options, but
present measurements that we have made that seem to be worth recording.
9.14. F
INGERPRINTING MATCHES.Asdescribed in Section 5.2, Simplify main-
tains a table of fingerprints of matches, which is used to filter out redundant matches,
that is, matches that have already been instantiated on the current path (
§3.1). The
464 D. DETLEFS ET AL.
FIG. 23. Effect of disabling distinction classes on the front-end test suite.
FIG. 24. Performance data for the small test suite with labels ignored.
Simplify: A Theorem Prover for Program Checking 465
FIG. 25. Performance data for the small test suite with labels replaced by error variables.
FIG. 26. Effect of disabling simplex redundancy filtering on the front-end test suite.
466 D. DETLEFS ET AL.
FIG. 27. Distribution of the 264630 calls to SgnOfMax in the baseline run of the front-end
test suite, according to tableau size (defined as max(m, n)) and number of pivots required.
mod-time (
§5.4.2) and pattern-element (§5.4.3) optimizations reduce the number of
redundant matches that are found, but even with these heuristics many redundant
matches remain. On the program-checking problems in the small test suite, the
fraction of matches filtered out as redundant by the fingerprint test ranges from
39% to 98% with an unweighted average of 71%. This suggests that there may be
an opportunity for further matching optimizations that would avoid finding these
redundant matches in the first place.
9.15. P
ERFORMANCE OF THE SIMPLEX MODULE.Torepresent the Simplex
tableau (
§8.2), Simplify uses an ordinary m by n sequentially allocated array of
integer pairs, even though in practice the tableau is rather sparse. It starts out sparse
because the inequalities that occur in program checking rarely involve more than
two or three terms. Although Simplify makes no attempt to select pivot elements
so as to preserve sparsity, our dynamic measurements indicate that in practice and
on the average, roughly 95% of the tableau entries are zero when Pivot is called.
So a change to a sparse data structure (e.g., the one described by Knuth [1968,
Sec. 2.2.6]) might improve performance, but more careful measurements would be
required to tell for sure.
One important aspect of the performance of the Simplex module is the question of
whether the number of pivots required to compute SgnOfMax in practice is as bad as
its exponential worst case. Folklore says that it is not. Figure 27 presents evidence
that the folklore is correct. We instrumented Simplify so that with each call to
SgnOfMax it would log the dimensions of the tableau and the number of pivots per-
formed. The figure summarizes this data. Ninety percent of the calls resulted in fewer
than two pivots. Out of more than a quarter of a million calls to SgnOfMax performed
on the front-end test suite, only one required more than 511 pivots (in fact 686).
9.16. P
ERFORMANCE OF THE E-GRAPH MODULE.Inthis section, we present a
few measurements we have made of the E-graph module. For each measurement, we
report its minimum, maximum, and unweighted average value over the benchmarks
of the small test suite. We included the domino benchmarks, although in some cases
they were outliers.
The crucial algorithm for merging two equivalence classes and detecting new
congruences is sketched in Section 4.2 and described in detail in Section 7. Sim-
plify’s Sat algorithm typically calls the Merge algorithm thousands of times per
second. Over the small test suite, the rate ranges from 70 to 46000 merges/second
and the unweighted average rate is 6100 merges/second.
Simplify: A Theorem Prover for Program Checking 467
Roughly speaking, the E-graph typically contains a few thousand active (
§5.2)
(concrete) (
§7.1)) E-nodes. Over the small test suite, the maximum number of active
E-nodes over the course of the proof ranged from 675 to 13000, and averaged 3400.
The key idea of the Downey–Sethi–Tarjan congruence-closure algorithm
[Downey et al. 1980] is to find new congruences by rehashing the shorter of two
parent lists, as described in Section 7.1. This achieves an asymptotic worst case
time of O(n log n). The idea is very effective in practice. Over the examples of the
small test suite, the number of E-nodes rehashed per Merge (ignoring rehashes in
UndoMerge) ranged from 0.006 to 1.4, and averaged 0.75.
We chose to use the “quick find” union-find algorithm [Yao 1976] rather than one
of the almost linear algorithms [Tarjan 1975] because the worst-case cost of this
algorithm (O(n log n)onany path) seemed acceptable and the algorithm is easier
to undo than union-find algorithms that use path compression. Over the examples
of the small test suite, the average number of E-nodes rerooted per merge (ignoring
rerooting during UndoMerge) ranged from 1.0 to 1.2, and averaged 1.05.
Over the examples of the small test suite, the fraction of calls to Cons that find
and return an existing node ranged from 0.4 to 0.9, and averaged 0.79.
9.17. E
QUALITY
PROPAGATIONS.For the benchmarks in the small test suite, we
counted the rate of propagated equalities per second between the E-graph and the
Simplex module, adding both directions. The rate varies greatly from benchmark
to benchmark, but is always far smaller than the rate of merges performed in the
E-graph altogether. For the six benchmarks that performed no arithmetic, the rate
is of course zero. For the others, the rate varies from a high of about a thousand
propagated equalities per second (for
cat, addhi, and frd-seek)toalowoftwo
(for
getRootInterface
).
10. Related and Future Work
Like Simplify, the constraint solver Cassowary [Badros et al. 2001] implements
aversion of the Simplex algorithm with unknowns that are not restricted to be
non-negative. The Cassowary technique, which the Cassowary authors attribute to
Marriott and Stuckey [1998], is to use two tableaus: one tableau containing only
the unknowns restricted to be non-negative, and a second tableau containing unre-
stricted unknowns. In their application (which doesn’t propagate equalities, but does
extract specific solutions), the use of two tableaus gives a performance advantage,
since they needn’t pivot the second tableau during satisfiability testing, but only
when it is time to compute actual solution values for the unrestricted unknowns.
Although Cassowary does not propagate equalities, Stuckey [1991] has described
methods for detecting implicit equalities in a set of linear inequalities. Even in
an application requiring equality propagation, the two-tableau approach might be
advantageous, since the second tableau could be pivoted only when the dimension
of the solution set diminished, making it necessary to search for new manifest
equalities. Cassowary also supports constraint hierarchies (some constraints
are preferences rather than requirements) and out-of-order (non-LIFO) removal
of constraints.
A recent paper by Gulwani and Necula [2003] describes a an efficient random-
ized technique for incrementally growing a set of linear rational equalities and
reporting any equalities of unknowns. Their technique, which reports two variables
as equivalent if they are equal at each of a small set of pseudo-random points in the
468 D. DETLEFS ET AL.
solution flat, has a small chance of being unsound. However, it could safely be used
as a filter tha might speed the search for new manifest equalities after a dimension
reduction in a Simplex tableau.
The idea of theorem proving with cooperating decision procedures was intro-
duced with the theorem prover of the Stanford Pascal Verifier. This prover, also
named Simplify, was implemented and described by Greg Nelson and Derek Op-
pen [Nelson and Oppen 1979, 1980].
In addition to Simplify and its ancestor of the same name, several other automatic
theorem provers based on cooperating decision procedures have been applied in
program checking and related applications. These include the PVS prover from SRI
[Owre et al. 1992], the SVC prover from David Dill’s group at Stanford [Barrett
et al. 1996], and the prover contained in George Necula’s Touchstone system for
proof-carrying code [Necula 1998; Necula and Lee 2000].
PVS includes cooperating decision procedures combined by Shostak’s method
[Shostak 1984]. It also includes a matcher to handle quantifiers, but the matcher
seems not to have been described in the literature, and is said not to match in the
E-graph [N. Shanker, 2003, Personal Communication].
SVC also uses a version of Shostak’s method. It does not support quantifiers.
It compensates for its lack of quantifiers by the inclusion of several new decision
procedures for useful theories, including an extensional theory of arrays [Stump
et al. 2001] and a theory of bit-vector arithmetic [Barrett et al. 1998].
Touchstone’s prover adds proof-generation capabilities to a Simplify-like core.
It doesn’t handle general quantifiers, but does handle first-order Hereditary Harrop
formulas.
During our use of Simplify, we observed two notable opportunities for perfor-
mance improvements. First, the backtracking search algorithm that Simplify uses
for propositional inference had been far surpassed by recently developed fast SAT
solvers [Silva and Sakallah 1999; Zhang 1997; Moskewicz et al. 2001]. Second,
when Simplify is in the midst of deeply nested case splits and detects an inconsis-
tency with an underlying theory, the scoring (
§3.6) heuristic exploits the fact that
the most recent case split must have contributed to the inconsistency, but Simplify’s
decision procedures supply no information about which of the other assertions in
the context contributed to the inconsistency.
In the past few years several systems hae been developed that exploit these
opportunities by coupling modern SAT solving algorithms with domain-specific
decision modules that are capable of reporting reasons for any inconsistencies that
they detect. When the domain-specific decision modules detect an inconsistency in a
candidate SAT solution, the reasons are recorded as new propositional clauses (var-
iously referred to as “explicated clauses” [Flanagan et al. 2003], “conflict clauses”
[Barrett et al. 2002a] and “lemmas on demand” [de Moura and Ruess 2002] that
the SAT solver then uses to prune its search for further candidate solutions. It is
interesting to note that a very similar idea occurs in a 1977 paper by Stallman and
Sussman [Stallman and Sussman 1977], in which newly discovered clauses are
called “NOGOOD assertions”.
11. Our Experience
We would like to include a couple of paragraphs describing candidly what we have
found it like to use Simplify.
Simplify: A Theorem Prover for Program Checking 469
First of all, compared to proof checkers that must be steered case by case through
a proof that the user must design herself, Simplify provides a welcome level of
automation.
Simplify is usually able to quickly dispatch valid conjectures that have simple
proofs. For invalid conjectures with simple counterexamples, Simplify is usually
able to detect the invalidity, and the label mechanism described in Section 6.2 is
usually sufficient to lead the user to the source of the problem. These two cases
cover the majority of conjectures encountered in our experience with ESC, and we
suspect that they also cover the bulk of the need for automatic theorem proving in
design-checking tools.
For more ambitious conjectures, Simplify may time out, and the diagnosis and
fix are likely to be more difficult. It might be that what is needed is a more ag-
gressive trigger (
§5.1) on some matching rule (to find some instance that is crucial
to the proof). Alternatively, the problem might be that the triggers are already too
aggressive, leading to useless instances or even to matching loops. Or it might be
that the triggers are causing difficulty only because of an error in the conjecture.
There is no easy way to tell which of these cases applies.
In the ESC application, we tuned the triggers for axioms in the background
predicate (
§2) so that most ESC users do not have to worry about triggers. But
ambitious users of ESC who include many quantifiers in their annotations may
well find that Simplify is not up to the task of choosing appropriate triggers.
12. Conclusions
Our main conclusion is that the Simplify approach is very promising (nearly prac-
tical) for many program-checking applications. Although Simplify is a research
prototype, it has been used in earnest by groups other than our own. In addition
to ESC, Simplify has been used in the Slam device driver checker developed at
Microsoft [Ball et al. 2001] and the KeY program checker developed at Karlsruhe
University [P. H. Schmitt, 2003, Personal Communication; Ahrendt et al. 2003].
One of us (Saxe) has used Simplify to formally verify that a certain feature of the
Alpha multiprocessor memory model is irrelevant in the case of programs all of
whose memory accesses are of uniform granularity. Another of us (Detlefs) has used
Simplify to verify the correctness of a DCAS-based lock-free concurrent double-
ended-queue. The data structure is described in the literature [Agesen et al. 2000]
and the Simplify proof scripts are available on the web [Detlefs and Moir 2000].
The Denali 1 research project [Joshi et al. 2002] (a superoptimizer for the Alpha
EV6 architecture) used Simplify in its initial feasibility experiments to verify that
optimal machine code could be generated by automatic theorem-proving methods.
The Denali 2 project (a superoptimizer for the McKinley implementation of IA-64)
has used Simplify to check that axioms for a little theory of prefixes and suffixes are
sufficient to prove important facts about the IA-64 extract and deposit instructions.
In particular, we conclude that the Nelson–Oppen combination method for rea-
soning about important convex theories works effectively in conjunction with the
use of matching to reason about quantifiers. We have discussed for the first time
some crucial details of the Simplify approach: undoing merge operations in the
E-graph, correctly propagating equalities from the Simplex algorithm, the iterators
for matching patterns in the E-graph, the optimizations that avoid fruitless matching
470 D. DETLEFS ET AL.
effort, the interactions between matching and the overall backtracking search, the
notion of an ordinary theory and the interface to the module that reasons about an
ordinary theory.
Appendix
A. Index of Selected Identifiers
We list here each identifier that is used in this paper outside the top-level section in
which it is defined, together with a brief reminder of its meaning and a reference
to its section of definition.
e.active:True if E-node e represents a term in a current assertion
(Sec. 5.2)
T.Asserted: Conjunction of the currently asserted T -literals (Sec. 4.4)
AssertEQ: Assert equality between terms represented as E-nodes
(Sec. 7.2)
AssertLit: Assert a literal (Sec. 3.1)
AssertZ: Assert that a formal affine sum of Simplex unknowns equals
0 (Sec. 8.4)
children[u]: Children of node u in the term DAG (Sec. 4.2)
cls: Clause set of the current context (Sec. 3.1)
Cons: Constructor for binary E-nodes (Sec. 7.4)
DEFPRED: Declare, and optionally define, a quasi-relation (Sec. 2)
DISTINCT: n-ary distinction operator (Sec. 2)
E-node: E-node data type (Sec. 4.2)
u.enode: E-node associated with unknown u of an ordinary theory
(Sec. 4.4)
e.id: Unique numerical identifier of E-node e (Sec. 7.1)
r.lbls: Set of labels (function symbols) of active E-nodes in the
class with root r (Sec. 5.4.4)
lits: Literal set of the current context (Sec. 3.1)
Merge: Merge equivalence classes in the E-graph (Sec. 7.2)
Pivot:Pivot the Simplex tableau (Sec. 8.3)
r.plbls: Set of labels of active (term DAG) parents of E-nodes in
the class with root r (Sec. 5.4.4)
Pop: Restore a previously saved context (Sec. 3.1)
T.Propagated: Conjunction of the equalities currently propagated from
theory T (Sec. 4.4)
Push:Save the current context (Sec. 3.1)
Refine: Perform width reduction, clause elimination, and unit as-
sertion (Sec. 3.2); extended to perform matching (Sec. 5.2)
refuted:Iftrue, the current context is inconsistent (Sec. 3.1)
e.root: Root of E-node e’s equivalence class (Sec. 4.2)
Sat: Determine whether the current context is satisfiable
(Sec. 3.2); respecified (Sec. 3.4.1)
select: Access operator for maps (arrays) (Sec. 2)
SgnOfMax: Determine sign of maximum of a Simplex unknown
(Sec. 8.3)
Simplify: A Theorem Prover for Program Checking 471
Simplex: Module for reasoning about arithmetic (Sec. 4.3)
store: Update operator for maps (arrays) (Sec. 2)
r.T
unknown: T.Unknown associated with root E-node r (Sec. 4.4)
@true: Constant used to reflect true as an individual value (Sec. 2)
UndoMerge: Undo effects of Merge (Sec. 7.3)
T.Unknown: Unknown of the theory T (Sec. 4.4)
T.UnknownForEnode: Find or create a T.Unknown associated with an E-node
(Sec. 4.4)
ACKNOWLEDGMENTS
. The work described in this article was performed at the Dig-
ital Equipment Corporation Systems Research Center, which became the Compaq
Systems Research Center, and then (briefly) the Hewlett-Packard Systems Research
Center.
We thank Wolfgang Bibel for information on the history of the proxy variable
technique, Alan Borning and Peter Stuckey for information about Cassowary, and
Rajeev Joshi, Xinming Ou, Lyle Ramshaw, and the referees for reading earlier
drafts and offering helpful comments. Also, we are thankful to many of you out
there for bug reports.
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RECEIVED AUGUST 2003; REVISED JULY 2004; ACCEPTED JANUARY 2005
Journal of the ACM, Vol. 52, No. 3, May 2005.
