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This paper shows that {weakly}-non-overlapping, non-collapsing andshallow term rewriting systems are confluent, which is a newsufficient condition on confluence for non-left-linear systems.
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Context 2
... Merge is shown in Figure 4. If a TRS R is weakly non-overlapping, the output Figure 1 A. and the graph G 2 in Figure 3 B., Merge R1 (G 1 , G 2 ) produces G 2 in Figure 1 B. The steps M1 and M2 are demonstrated in Examples 9 and 7, respectively. ...
Context 3
... Merge is shown in Figure 4. If a TRS R is weakly non-overlapping, the output Figure 1 A. and the graph G 2 in Figure 3 B., Merge R1 (G 1 , G 2 ) produces G 2 in Figure 1 B. The steps M1 and M2 are demonstrated in Examples 9 and 7, respectively. ...
Citations
... simple-right-linearity [TO94,OOT95], an assumption that for any rewrite rule, the right-hand side is linear and no variables occurring more than once in the left-hand side occur in the right-hand side; strongly depthpreservation [GOO96,GOO98], a characteristic that for any rewrite rule and any variable appearing in its both sides, the minimal depth of the variable occurrences in the left-hand side is greater than or equal to the maximal depth of the variable occurrences in the right-hand side; and noncollapsingness [SO10,SOO15], meaning that right-hand sides of no rewrite rules are variables. However, no general criterion based on critical pair computations is known when rewrite relations do not terminate. ...
... It is layered but its second rule is rank increasing since d(x, f (x)) has rank 2 while f (x) has rank 1. This system is non-confluent, since f (f (c)) → d(f (c), f (f (c))) → d(f (f (c)), f (f (c))) → 0 while f (f (c)) → f (d(c, f (c))) → f (d(f (c), f (c))) → f (0) which generates the regular tree language {S → d(0, S), S → f (0)} not containing 0. Note that replacing the second rule by the right-linear rule f (x) → d(x, f (c)) yields a confluent system [SO10]. ...
... Layered systems are a decidable class relating to overlay systems [DOS87], for which overlaps computed with plain unification can only take place at the root of terms -hence their name -, and generalizes strongly non-overlapping systems [SO10] which admit no linearized overlaps at all. All these classes are Turing-complete since they contain a complete class [Klo93]. ...
This thesis is devoted to the confluence of rewrite systems in the absence of termination, for applications in first-order functional languages like MAUDE or higher-order languages with dependent types, as Dedukti. In the first case, the computations on infinite data structures do not terminate, while in the second case, untyped computations do not terminate because of beta-reduction. In the case where the computations terminate, confluence is reduced to that of critical peaks, the "minimal diverging computations", made of a minimal middle term called "overlap" which computes in two different ways, resulting in a so-called "critical pair". In the case of non-terminating computations, a main result is that left-linear rewrite rules that have no critical pairs are always confluent. This suggests that the notion of critical pairs plays a key role there too, but a general understanding of the confluence of non-terminating computations in terms of critical pairs is still missing.Our investigation of confluence is based on the decreasing diagrams method due to van Oostrom, which generalizes the techniques used previously for both terminating and non-terminating computations. The method is abstract in the sense that it applies to arbitrary relations on an abstract set. It equips each step of computations with a label taken from a well-founded set. A diverging computation, called peak, has a decreasing diagram if its extremities can be joined by steps whose direction and labels satisfy some constraints with respect to the peak's rewrites and labels. The strength of this technique is its completeness, that is, any confluent relation can be equipped with a well-founded set of labels such that all peaks have decreasing diagrams. The proof of completeness is based on Klop's notion of cofinal derivations, which is an infinite derivation playing the role of a normal form when computations do not terminate.In the first part, we revise the results of van Oostrom, and propose an alternative proof that extends the method to the "modulo" case, in which computations mix rewrite steps and equational steps. The completeness result is extended as well, via a generalization of cofinal derivations and the notion of strong coherence due to Jouannaud and Kirchner.The second part of the thesis applies the decreasing diagrams method and its generalization to concrete systems rewriting terms, as well as to several open problems. The recent application to the problems of higher-order computations in dependent type theory is not part of the thesis.
... Overlaps at different positions along a path from the root to a leaf of l are forbidden. Layered systems is a decidable class that relates to overlay systems [6], for which overlaps computed with plain unification can only take place at the root of terms -hence their name-, and generalizes strongly non-overlapping systems [24] which admit no linearized overlaps at all. All these classes are Turing-complete since they contain a complete class [16]. ...
... , c → f (c)}, which is layered but whose second rule is rank increasing since d(x 1 , x 2 ) unifies with d(y, f (y)). This system is non- c, f c)) → f 0 which generates the regular tree language {S → d(0, S), S → f 0} not containing 0. Note that replacing the second rule by the right linear rule f (x) → d(x, f (c)) yields a confluent system [24]. ...
We investigate the new, Turing-complete class of layered systems, whose
lefthand sides of rules can only be overlapped at a multiset of disjoint or
equal positions. Layered systems define a natural notion of rank for terms: the
maximal number of non-overlapping redexes along a path from the root to a leaf.
Overlappings are allowed in finite or infinite trees. Rules may be
non-terminating, non-left-linear, or non-right-linear. Using a novel
unification technique, cyclic unification, and the so-alled subrewriting
relation, we show that rank non-increasing layered systems are confluent
provided their cyclic critical pairs have cyclic-joinable decreasing diagrams.
... Overlaps at different positions along a path from the root to a leaf of l are forbidden. Layered systems is a decidable class that relates to overlay systems [6], for which overlaps computed with plain unification can only take place at the root of terms -hence their name-, and generalizes strongly non-overlapping systems [24] which admit no linearized overlaps at all. All these classes are Turing-complete since they contain a complete class [16]. ...
... Example 37. Consider the critical pair free system R = {d(x, x) → 0, f (x) → d(x, f (x)), c → f (c)}, which is layered but whose second rule is rank increasing since d(x 1 , x 2 ) unifies with d(y, f (y)). This system is non-confluent, since f (f c, f c)) → f 0 which generates the regular tree language {S → d(0, S), S → f 0} not containing 0. Note that replacing the second rule by the right linear rule f (x) → d(x, f (c)) yields a confluent system [24]. ...
We investigate a new, Turing-complete class of layered systems, whose linearized lefthand sides of rules can only be overlapped at the root position. Layered systems define a natural notion of rank for terms: the maximal number of redexes along a path from the root to a leaf. Overlappings are allowed in finite or infinite trees. Rules may be non-terminating, non-left-linear, or non-right- linear. Using a novel unification technique, cyclic unification, we show that rank non-increasing layered systems are confluent provided their cyclic critical pairs have cyclic-joinable decreasing diagrams.
A term is weakly shallow if each defined function symbol occurs either at the root or in the ground subterms, and a term rewriting system is weakly shallow if both sides of a rewrite rule are weakly shallow. This paper proves that non-E-overlapping, weakly-shallow, and non-collapsing term rewriting systems are confluent by extending reduction graph techniques in our previous work [19] with towers of expansions.
