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On the decision problem for the guarded fragment with transitivity
Abstract
The guarded fragment with transitive guards, [GF+TG], is an extension of GF in which certain relations are required to be transitive, transitive predicate letters appear only in guards of the quantifiers and the equality symbol may appear everywhere. We prove that the decision problem for [GF+TG] is decidable. This answers the question posed in (Ganzinger et al., 1999). Moreover, we show that the problem is 2EXPTIME-complete. This result is optimal since the satisfiability problem for GF is 2EXPTIME-complete (Gradel, 1999). We also show that the satisfiability problem for two-variable [GF+TG] is NEXPTIME-hard in contrast to GF with bounded number of variables for which the satisfiability problem is EXPTIME-complete
... (ii) What is the exact complexity of the monadic GF 2 ? The first question was answered positively in (Szwast & Tendera 2001) where using a heavy model-theoretic construction, it was shown that the guarded fragment with transitive guards GF[T G] is decidable in 2EXPTIME. Kiero´nskiKiero´nski (2003) has proved the matching 2EXPTIME lower bound for the monadic GF 2 with transitivity, answering hereby the second question. ...
... Kiero´nskiKiero´nski (2003) has proved the matching 2EXPTIME lower bound for the monadic GF 2 with transitivity, answering hereby the second question. A practical disadvantage of procedures based on enumeration of structures , like the one given for GF[T G] in (Szwast & Tendera 2001), is that without further optimizations, those methods exhibit the full worst-case complexity . Resolution-based approach, is a reasonable alternative to modeltheoretic procedures, as its goal-oriented nature and numerous refinements allow to scale well between " easy " and " hard " instances of problems. ...
... (i) N is closed under rules of OR Sel up to redundancy and (ii) N is a subset of (GT). Corollary 4.8 (Szwast & Tendera 2001) GF[T G] is decidable in double exponential time. Proof. ...
We show how well-known refinements of ordered resolution, in particular redundancy elimination and ordering constraints in combination with a selection function, can be used to obtain a decision procedure for the guarded fragment with transitive guards. Another contribution of the paper is a special scheme notation, that allows to describe saturation strategies and show their correctness in a concise form.
... The complexity of the epistemic logic SELAS is currently unknown to us though we conjecture it is decidable due to the very limited use of quantifiers. Under the translation in Section 2, the name-free fragment can be viewed as a guarded fragment of first-order logic with transtive guards [19], which implies decidability. However, the non-rigid names translate into function symbols in first-order language, which may cause troubles since the guarded fragment with function symbols in general yields an undecidable logic [9]. ...
In standard epistemic logic, agent names are usually assumed to be common knowledge implicitly. This is unreasonable for various applications. Inspired by term modal logic and assignment operators in dynamic logic, we introduce a lightweight modal predicate logic where names can be non-rigid. The language can handle various de dicto /de re distinctions in a natural way. The main technical result is a complete axiomatization of this logic over S5 models.
... There, GF 2 is the two-variable restriction of the guarded fragment GF [1], where all quantifiers are guarded by atoms, and GF+TG is the restriction of GF 2 with transitive relations, where the transitive relation symbols are allowed to appear only in guards. As shown [29,30] undecidability of FO 2 with transitivity transfers to GF 2 with transitivity; however, GF+TG is decidable not depending on the number of transitive symbols. Moreover, as noted in [11], the decision procedure developed for GF 2 +TG can be applied to GF 2 with one transitive relation that is allowed to appear also outside guards, giving 2-ExpTime-completeness of the latter fragment. ...
We study the satisfiability problem for the two-variable first-order logic over structures with one transitive relation. % We show that the problem is decidable in 2-NExpTime for the fragment consisting of formulas where existential quantifiers are guarded by transitive atoms. As this fragment enjoys neither the finite model nor the tree model property, to show decidability we introduce novel model construction technique based on the infinite Ramsey theorem. We also point out why the technique is not sufficient to obtain decidability for the complimentary fragment where existential quantifiers are 'guarded' by negated transitive atoms, i.e. the existential subformulas are of the form $\exists y(\neg Txy\wedge \neg Tyx \wedge \phi(x,y))$, where $T$ denotes the transitive relation. Hence, contrary to our previous claim, [31], the status of the satisfiability problem for the full two-variable logic with one transitive relation remains open.
... Number of special symbols in the signature 1 2 3 or more GF 2 Transitivity 2-EXPTIME undecidable undecidable Sat: [21] FinSat: [28,27] [ For GF 2 , it also makes sense to study variants in which the distinguished predicates may appear only in guards [11]. In this case, GF 2 with any number of equivalences appearing only as guards remains NEXPTIME-complete [21], while GF 2 with any number of transitive relations appearing only as guards is 2-EXPTIME-complete [51,20] (tight complexity bounds for the finite satisfiability problem are established in [27]). ...
Over the past two decades several fragments of first-order logic have been identified and shown to have good computational and algorithmic properties, to a great extent as a result of appropriately describing the image of the standard translation of modal logic to first-order logic. This applies most notably to the guarded fragment, where quantifiers are appropriately relativized by atoms, and the fragment defined by restricting the number of variables to two. The aim of this talk is to review recent work concerning these fragments and their popular extensions. When presenting the material special attention is given to decision procedures for the finite satisfiability problems, as many of the fragments discussed contain infinity axioms. We highlight most effective techniques used in this context, their advantages and limitations. We also mention a few open directions of study.
... Unfortunately, FO 2 has a drawback, which becomes significant when one thinks about practical applications: it cannot express transitivity of a binary relation. Moreover, in contrast to modal logic or to some variants of the guarded fragment [16,12], extending FO 2 by transitivity statements leads to undecidability [6,10]. ...
We prove that the satisfiability problem for the two-variable, universal fragment of first-order logic with constants (or, alternatively phrased, for the Bernays-Schönfinkel class with two universally quantified variables) remains decidable after augmenting the fragment by the transitive closure of a single binary relation. We give a 2-NExpTime-upper bound and a 2-ExpTime-lower bound for the complexity of the problem. We also study the cases in which the number of constants is restricted. It appears that with two constants the considered fragment has the finite model property and NExpTime-complete satisfiability problem. Adding a third constant does not change the complexity but allows to construct infinity axioms. A fourth constant lifts the lower complexity bound to 2-ExpTime. Finally, we observe that we are close to the border between decidability and undecidability: adding a third variable or the transitive closure of a second binary relation lead to undecidability.
... But transitivity axioms make the guarded fragment (GF) of first order logic undecidable [12]. On the other side, if transitive relations only occur in guards, GF is decidable [24], and consequently so is satisfiability in HL m (@, ↓, E, ✸ − , Trans) \ ✷↓✷. But the translation of relation inclusion axioms may have transitive relations out-side guards. ...
In previous works, a tableau calculus has been defined, which constitutes a
decision procedure for hybrid logic with the converse and global modalities and
a restricted use of the binder. This work shows how to extend such a calculus
to multi-modal logic enriched with features largely used in description logics:
transitivity and relation inclusion assertions.
The separate addition of either transitive relations or relation hierarchies
to the considered decidable fragment of multi-modal hybrid logic can easily be
shown to stay decidable, by resorting to results already proved in the
literature. However, such results do not directly allow for concluding whether
the logic including both features is still decidable. The existence of a
terminating, sound and complete calculus for the considered logic proves that
the addition of transitive relations and relation hierarchies to such an
expressive decidable fragment of hybrid logic does not endanger decidability.
A further result proved in this work is that the logic extending the
considered fragment with the addition of graded modalities (the modal
counterpart of number restrictions of description logics) has an undecidable
satisfiability problem, unless further syntactical restrictions are placed on
the universal graded modality.
We study the satisfiability problem for two-variable first-order logic over structures with one transitive relation. We show that the problem is decidable in 2-NExpTime for the fragment consisting of formulas where existential quantifiers are guarded by transitive atoms. As this fragment enjoys neither the finite model property nor the tree model property, to show decidability we introduce a novel model construction technique based on the infinite Ramsey theorem.
We also point out why the technique is not sufficient to obtain decidability for the full two-variable logic with one transitive relation; hence, contrary to our previous claim, [FO$^2$ with one transitive relation is decidable, STACS 2013: 317-328], the status of the latter problem remains open.
We give a uniform presentation of representation and decidability results related to the Kripke-style semantics of several non-classical logics. We show that a general representation theorem (which has as particular instances the representation theorems as algebras of sets for Boolean algebras, distributive lattices and semilattices) extends in a natural way to several classes of operators and allows to establish a relationship between algebraic and Kripke-style models. We illustrate the ideas on several examples. We conclude by showing how the Kripke-style models thus obtained can be used (if first-order axiomatizable) for automated theorem proving by resolution for some non-classical logics.
We study the problem of answering a union of Boolean conjunctive queries q against a database Δ, and a logical theory ϕ which falls in the guarded fragment with transitive guards (GF + TG). We trace the frontier between decidability and undecidability of the problem under consideration. Surprisingly, we show that query answering under GF2 + TG, i.e., the two-variable fragment of GF + TG, is already undecidable (even without equality), whereas its monadic fragment is decidable; in fact, it is 2exptime-complete in combined complexity and coNP-complete in data complexity. We also show that for a restricted class of queries, query answering under GF+TG is decidable.
We consider the satisfiability and finite satisfiability problems for extensions of the two-variable fragment of first-order logic in which an equivalence closure operator can be applied to a fixed number of binary predicates. We show that the satisfiability problem for two-variable, first-order logic with equivalence closure applied to two binary predicates is in 2NEXPTIME, and we obtain a matching lower bound by showing that the satisfiability problem for two-variable first-order logic in the presence of two equivalence relations is 2NEXPTIME-hard. The logics in question lack the finite model property; however, we show that the same complexity bounds hold for the corresponding finite satisfiability problems. We further show that the satisfiability (=finite satisfiability) problem for the two-variable fragment of first-order logic with equivalence closure applied to a single binary predicate is NEXPTIME-complete.
Definition des fragments modaux de la logique des predicats a partir de formules du premier ordre qui sont des traductions des proprietes poly-modales elementaires. Distinguant les fragments variables et finis des fragments lies a un quantificateur, l'A. developpe une version semantique des fragments gardes en remplacant les liens syntaxiques par des restrictions sur les types d'attribution dans les modeles generalises. Se referant a l'algebre cylindrique, l'A. indique les nouvelles directions que peuvent prendre les theoremes de Tarski dans un environnement mathematique: celle des contraintes structurelles speciales, des extensions infinies, de la logique modale etendue et d'une semantique dynamique
Consider a partially ordered set (T, <, =), where T is non-empty, < is an irreflexive and transitive relation on T and = is equality. We can regard T as a set of moments of time and < as the earlier-later relation. We refer to any fixed (T, <, =) as a flow of time.
This is the second volume of a systematic two-volume presentation of the various areas of research in the field of structural complexity. The mathematical theory of computation has developed into a broad and rich discipline within which the theory of algorithmic complexity can be approached from several points of view. This volume is addressed to graduate students and researchers and assumes knowledge of the topics treated in the first volume but is otherwise nearly self-contained. Topics covered include vector machines, parallel computation, alternation, uniform circuit complexity, isomorphism, biimmunity and complexity cores, relativization and positive relativization, the low and high hierarchies, Kolmogorov complexity and probability classes. Numerous exercises and references are given.
It is a classical result of Mortimer that , first-order logic with two variables, is decidable for satisfiability. We show that going beyond by adding any one of the following leads to an undecidable logic:– very weak forms of recursion, viz.¶(i) transitive closure
operations¶(ii) (restricted) monadic fixed-point operations¶– weak access to cardinalities, through the Härtig (or equicardinality)
quantifier¶– a choice construct known as Hilbert's -operator.
In fact all these extensions of prove to be undecidable both for satisfiability, and for satisfiability in finite structures. Moreover most of them are hard
for , the first level of the analytical hierachy, and thus have a much higher degree of undecidability than first-order logic.
A computer program capable of acting intelligently in the world must have a general representation of the world in terms of which its inputs are interpreted. Designing such a program requires commitments about what knowledge is and how it is obtained. Thus, some of the major traditional problems of philosophy arise in artificial intelligence.