Plaisted, D. (1985), Semantic confluence tests and completion methods, Inform. and Control 65, 182--215.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....correctness of the algorithm, it should be clear that one can allow left sides of depth d 3 for rules such that the normal form of the right side r has no path of length less than d 1. Checking normal forms of all instances is not a syntactic condition, but testing for ground reducibility is [Plaisted, 1985]. So we can replace Conditions (A) and (B) with the following: For each rule l r, we have jjljj 2, where is the length of the shortest path from the root of r to a ground reducible position. Since the ground normal form of any term is of depth at least 1, this condition ....

Plaisted, D. A.: Semantic confluence tests and completion methods. Information and Computation 65:182--215 (1985).

....this implies est(t)oe toe. While in the original estimation method for functional programs [Gie95d] functions had to be completely defined, here we have to extend the estimation method to incompletely defined functions. This allows to prove termination of CSs that are not sufficiently complete [Pla85], too. 9 All subterms g(u )oe in toe are E smaller than f(r ) If g is a defined symbol (g = f is possible) then IN f f must contain IN g g and by the induction hypothesis IN g g implies g(u )oe. Hence, we have est(t)oe toe and (as f (s ) est(t) is in IN f f and ....

D. A. Plaisted. Semantic confluence tests and completion methods. Inform. and Control, 65(2/3):182-215, 1985. 28

....0 , then: R = R 0 [ fl rg convergent ) l = r is an inductive theorem of R 0 . A term was defined to be inductively reducible w.r.t. a rewrite system R if all its ground instances were reducible. Thus, the concept of ground reducibility was introduced to the field. Plaisted had already shown in [Pla85] that the problem was decidable for arbitrary rewrite systems, but the decision procedure he used was cripplingly expensive (a tower of 7 exponentials) Jouannaud and Kounalis introduced the idea of determining ground reducibility by using a test set S of irreducible substitutions such that if ....

David A. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:182, 1985.

....procedure, it is known that =R 0 [E 0 is always equivalentto= E . The lemma follows immediately. ut Definition 11. A term t is ground reducible by a set R of rewrite rules, if toe is reducible by R for anyground substitution oe. Ground reducibilityhasbeenshown to be decidable byPlaisted [13]. Since our process requires a convergent set R of rewrite rules in order to be applicable, we shall in the following write R in steadofE to denote the initial set of equations. The proofby consistency procedure forconstructivetheories with final algebra semantics is based on the completion of ....

D. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:182--215, 1985.

....reducible with respect to a rewrite system R iff all instances toe 2 T (F) of t are reducible by R. This definition extends to ordered rewriting, replacing R with E; when E is a finite set of (unconstrained) equations. Ground reducibility is decidable for arbitrary finite term rewriting systems [Pla85]. We show here that it is undecidable for finite sets of equations: Theorem 3. The problem: Input: A finite set of (unconstrained) equations E, a term t, a lexicographic path ordering. Question: Is t ground reducible with respect to E; is undecidable. Proof: We reduce the halting problem ....

David A. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65(2/3):182--215, May/June 1985.

....ground reducible w.r.t. a rewrite system R iff all instances toe 2 T (F ) of t are reducible by R. This definition extends to ordered rewriting, replacing R with E; when E is a finite set of (unconstrained) equations. Ground reducibility is decidable for arbitrary finite term rewriting systems [19]. It is undecidable for finite sets of equations [9] We show it is decidable in the special case where all symbols occcurring in the set of equations E have arity 0 or 1, in that case we say that the equations are unary) and the ordering is rpo with a total precedence. In fact, we can state a ....

D. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:182--215, 1985.

....all cases have been covered by R and that t will be reducible for any inputs. Many papers have been devoted to decision of ground reducibility. Let us report a brief history of the milestones, starting only in 1985 with the general case. Ground reducibility was first shown decidable by D. Plaisted [13]. The algorithm is however quite complex: a tower of 9 exponentials though there is no explicit complexity analysis in the paper. D. Kapur et al. gave another decidability proof [11] which is conceptually simpler, though still very complicated, and whose complexity is a tower of 7 exponentials in ....

D. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:182--215, 1985.

.... : s = t ) Red(s) For every terms s; t such that s t 8 x: Red(x) P (x) N(x) P (0) N(0) P (s(x) P (x) N(p(x) N(x) 23 Red is a recursive predicate on ground terms. This result is however technically difficult to prove and there are several proofs in the literature (from [Pla85] to [CJ97] This implies the decidability of the consistency of an equation with A. Lemma 14 When E is a finite ground convergent term rewriting system, the above set A is a strongly normal I axiomatization. Proof: sketch) A is infinite, but recursive, as the ordering on terms is assumed to be ....

David Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:182--215, 1985.

....domains. But it should be noted that these kinds of difficulties and drawbacks are not specific to the rewriting and completion based approach but are in general problematic for any kind of ITP approach. Interesting variations of proof by consistency techniques have further been explored in [Pla85] and [KNZ86] Within the framework of equational reasoning and rewriting techniques the classical approach to ITP using induction schemas has been discussed e.g. in [GG88] Keeping close to rewriting and completion techniques the positive approach as sketched in chapter 2 and its relations to ....

D. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:182--215, 1985.

....of the quantifier prefix and restrictedness of the class of rewrite systems) It remains open whether positive quantified theories of one step rewriting are decidable. Note in this respect that ground reducibility expressed by a positive 8 9 sentence is decidable for the usual rewrite systems (Plaisted 1985), but is undecidable for conditional systems, both in the weak sense (Kaplan Choquer 1986) and in the strong sense (Vorobyov 1998) 9 . Another problem worth investigating is the non uniform decidability of theories of one step rewriting with finite prefixes. Given any finite term rewriting ....

Plaisted, D. (1985), `Semantic confluence tests and completion methods', Information and Control 65, 182--215.

....is, on the one hand, transparent and, on the other hand, very powerful. We briefly comment on related work. Note that we do not consider syntactic confluence criteria here (see [Wi95] for a recent overview) There is only little special work on proving ground confluence in the literature. In [Pl85] one can find a semantic method which is in principle very powerful, which however indicates no way to automatize the ground confluence test. Fr86, Ga87] use Knuth Bendix like completion. This method has the disadvantage of producing often too many new consequences which are not needed for a ....

D. Plaisted, Semantic confluence tests and completion methods, Inform. and Control 65(2/3) 1985, pp. 182-215.

....rule systems whose reduction relations are not (locally) confluent 46 (but necessarily non noetherian then 47 ) Therefore, syntactic confluence criteria for nonnoetherian conditional rule systems must be very difficult to develop. Semantic confluence criteria (in the style of Plaisted[24]) seem to require noetherian (or at least normalizing) 46 cf. 12] Example A, p. 36 47 cf. 12] Theorem 4, p. 39 23 reduction relations because they rely on the irreducible reducts of the terms; furthermore irreducibility is not (semi ) decidable. Thus, for our confluence criteria we require ....

David A. Plaisted. Semantic Confluence Tests and Completion Methods. Information and Control 65, 1985, pp. 182-215.

....termination of the whole rewrite system. More syntactic criteria for confluence of Gamma R C can be found in Sect. 15 of [13] Sometimes, however, confluence of Gamma R C can only be shown using the semantic knowledge of the specifier: either with semantic confluence criteria in the style of [11] (see Theorem 6.5 of [15] or in a way that only works for the specific set of constructor rules at hand. Before we can formally present our syntactic confluence criterion, we have to introduce some more notions. Let t 1 ; t 2 2 T (sig; V ) We call a constructor substitution oe a unifier of t 1 ....

D. A. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:182--215, 1985.

....systems whose reduction relations are not (locally) confluent. 50 These rule systems are necessarily non noetherian. 51 Therefore, syntactic confluence criteria for nonnoetherian conditional rule systems must be very difficult to develop. Semantic confluence criteria (in the style of Plaisted[31]) seem to require noetherian (or at least normalising) reduction relations because they rely on the irreducible reducts of the terms; furthermore irreducibility is not (semi ) decidable. Thus, for our confluence criteria we require = R to be noetherian . Even then the situation is not very ....

David A. Plaisted. Semantic Confluence Tests and Completion Methods. Information and Control 65, 1985, pp. 182-215.

....divided into constructors and defined functions. An equation between constructor terms signals an inconsistency. Jouannaud and Kounalis [13] admit arbitrary convergent rewrite system for presenting a theory. They introduce the notion of inductive reducibility to detect inconsistencies. Plaisted [18], among others, has shown that inductive reducibility is decidable for finite unconditional term rewriting systems. Fribourg [9] is the first to notice that not all critical pairs need to be computed for inductive completion. It suffices to consider only linear inferences for selected complete ....

David A. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:182--215, 1985.

....disunification. Ground reducibility Proofs by consistency and sufficient completeness are closely related to the ground reducibility problem: a term is ground reducible if all its ground instances are reducible. This problem has been proved decidable for ordinary term rewriting systems [782, 542]. There are two main approaches for testing ground reducibility: the first one constructs a finite automaton such that the term is ground reducible if and only if the language accepted by the automaton is empty [121, 229] The second one is based on test sets: the term is ground reducible if and ....

D. A. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65(2&3):182--215, 1985.

....detected if during completion a rule l r is generated such that l r and l is not inductively reducible by D. Then some ground instance loe is irreducible and greater than roe; thus, loe roe is not a consequence of D. The method requires a ground convergent rewrite system R for D. Plaisted [Pla85] proves the decidability of inductive reducibility for finitely many unconditional equations. ffl Bachmair [Bac88] refines the method by extending inductive reducibility to equations. An equation u v is inductively reducible if for any nontrivial ground instance uoe voe one of uoe or voe is ....

D. A. Plaisted. Semantic Confluence Tests and Completion Methods. Information and Control, 65:182--215, 1985.

....consistency of an infinite set of clauses without equality. Finally, let us point out that, under Jouannaud and Kounalis hypotheses, Red is a recursive predicate on non ground terms (i.e. it is possible to decide whether S:Red j= 9 x: Red(s) There are several proofs in the literature (from [Pla85] to [CJ97] This implies the decidability of the consistency of an equation with A. The dedicated proofs of ground reducibility are certainly more efficient than the general method (inconsistency proof) we are currently designing. But, of course, we have more freedom, both in the design of e ....

David Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:182--215, 1985.

.... minus(0; succ(0) 5 While in the original estimation method for functional programs [Gie95d] functions had to be completely defined, here we have to extend the estimation method to incompletely defined functions. This allows to prove termination of CSs that are not sufficiently complete [Pla85], too. Therefore we additionally have to demand that irreducible ground terms with a defined root symbol are minimal, i.e. we also demand the constraints MIN = ft f(r )jf 2 D; t; r are ground terms; f(r ) is E normal formg: If MIN is also satisfied, then irreducible terms like ....

D. A. Plaisted. Semantic confluence tests and completion methods. Inform. and Control, 65(2/3):182-215, 1985.

....of t are reducible by R. This property is used in automatic proof by induction in equational theories [10, 1] in proving properties of equational specifications [11] and in constraint solving in quotient algebras [9] Ground reducibility has been shown decidable in the general case by D. Plaisted [15], D. Kapur et al. 11] E. Kounalis [12] Recently, some of us proved a more general result : the first order theory of encompassment is decidable [3] More precisely, a term s encompasses t if some instance of t is a subterm of s which we write encomp t (s) We showed that the full first order ....

....on a pumping property of the tree languages: if there is a tree in the language, then there should be a tree of bounded depth, where the bound only depends on the characteristics of the automaton. That is the way we follow. We show a pumping lemma, using a technique inspired by Plaisted s proof [15]. Following [4] we can then solve symbolic constraints involving encompassment formulas and membership formulas. As a corollary, we get for example that ground reducibility is decidable for sorted rewrite systems when the sorts are defined by means of term declarations t : S, provided of course ....

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D. Plaisted. Semantic confluence tests and completion method. Information and Control, 65:182-- 215, 1985.

....ground reducible w.r.t. a rewrite system R iff all instances toe 2 T (F) of t are reducible by R. This definition extends to ordered rewriting, replacing R with E; when E is a finite set of (unconstrained) equations. Ground reducibility is decidable for arbitrary finite term rewriting systems [Pla85]. We show here that it is undecidable for finite sets of equations: Theorem 3 The problem: Input: A finite set of (unconstrained) equations E, a term t, a lexicographic path ordering. Question: Is t ground reducible w.r.t. E; is undecidable. Proof: We reduce the halting problem for a ....

David A. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65(2/3):182--215, May/June 1985.

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David A. Plaisted. Semantic confluence tests and completion methods. Information and Control, 65:122--157, 1985.

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Plaisted, D. (1985), Semantic confluence tests and completion methods, Inform. and Control 65, 182--215.

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Plaisted, D., "Semantic confluence tests and completion methods," Information and Control, 65 1985, 182-215. 31

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David A. Plaisted (1985). Semantic Confluence Tests and Completion Methods. Information and Control 65, pp. 182-215.

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