Completeness of a prover for dense linear orders

Hines, Larry

doi:10.1007/bf00263449

Cited by 8 publications

(8 citation statements)

References 6 publications

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“…This implies that for sets of clauses in which equality occurs only implicitly, as part of non-strict inequalities u Յ v, variable chaining is not needed. Corresponding results have been established for non-ordered chaining calculi; see Richter [1984] and Hines [1992].…”

Section: Variable Eliminationmentioning

confidence: 73%

“…Naturally, subterm chaining methods for general clauses [Manna and Waldinger 1986;1992] and completionlike procedures for unit clauses [Levy and Agustí 1993] have been proposed for such relations, but completeness requires chaining into variables of the functional-reflexive axioms (cf. the "variable instance pairs" in Bachmair et al [1986]), which is impractical in general.…”

Section: Partial Congruences a Partial Equivalence ϳ Is A Symmetric mentioning

confidence: 99%

“…Chaining essentially encodes certain resolution steps with the transitivity axiom. For example, if S is a transitive relation, chaining allows one to derive uSv from uSs and tSv, where is a most general unifier of s and t. Corresponding calculi have been described by Bledsoe, Kunen, and Shostak [1985] and Hines [1992].…”

Section: Introductionmentioning

confidence: 98%

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Ordered chaining calculi for first-order theories of transitive relations

Bachmair

Ganzinger

1998

J. ACM

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We propose inference systems for binary relations that satisfy composition laws such as transitivity. Our inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods as they are used in the context of resolution-type theorem proving. We establish the refutational completeness of these calculi and prove that our methods are compatible with the usual simplification techniques employed in refutational theorem provers, such as subsumption or tautology deletion. Various optimizations of the basic chaining calculus will be discussed for theories with equality and for total orderings. A key to the practicality of chaining methods is the extent to which so-called variable chaining can be avoided. We demonstrate that rewrite techniques considerably restrict variable chaining and that further restrictions are possible if the transitive relation under consideration satisfies additional properties, such as symmetry. But we also show that variable chaining cannot be completely avoided in general.

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Section: Variable Eliminationmentioning

confidence: 73%

Section: Partial Congruences a Partial Equivalence ϳ Is A Symmetric mentioning

confidence: 99%

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Ordered chaining calculi for first-order theories of transitive relations

Bachmair

Ganzinger

1998

J. ACM

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“…For dense total orderings without endpoints, Bledsoe and Hines (1980) proposed techniques for eliminating certain occurrences of variables from formulas. Bledsoe et al (1985) and Hines (1992) gave completeness results for these restricted chaining systems. Monotonicity or anti-monotonicity of functions with respect to special (transitive) relations led Manna and Waldinger (1986) to propose subterm chaining methods for general clauses but the proposed calculus was shown to be incomplete (Manna and Waldinger, 1992).…”

Section: An Example: Towards a Second-order Completion Proceduresmentioning

confidence: 97%

Bi-rewrite Systems

Levy

Agustí

1996

Journal of Symbolic Computation

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In this article we propose an extension of term rewriting techniques to automate the deduction in monotone pre-order theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a bi-rewrite system is allowed to be a subset of the corresponding inclusion relation ⊆ or ⊇ defined by the theory of I. In order to assure the decidability and completeness of such proof procedure we study the termination and commutation of − −− →The proof of the commutation property is based on a critical pair lemma, using an extended definition of critical pair. We also extend the existing techniques of rewriting modulo equalities to bi-rewriting modulo a set of inclusions. Although we center our attention on the completion processà la Knuth-Bendix, the same notion of extended critical pairs is suitable to be applied to the so-called unfailing completion procedures. The completion process is illustrated by means of an example corresponding to the theory of the union operator. We show that confluence of extended critical pairs may be ensured adding rule schemes. Such rule schemes contain variables denoting schemes of expressions, instead of expressions. We propose the use of the linear second-order typed λ-calculus to codify these expression schemes. Although the general second-order unification problem is only semi-decidable, the second-order unification problems we need to solve during the completion process are decidable.

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“…In their inference system, no chaining through variables is performed and no explicit inferences with transitivity, density, totality and the "no endpoints" axioms are computed. Completeness results for particular such systems of restricted chaining are proved by Bledsoe, Kunen and Shostak (1985) and Hines (1992). Theorem provers developed from these theoretical investigations have performed successfully in proving theorems such as the continuity of the sum of two continuous functions or the intermediate value theorem; see Bledsoe and Hines (1980), Hines (1988) or Hines (1990).…”

Section: Introductionmentioning

confidence: 99%

Ordered chaining for total orderings

Bachmair

Ganzinger

1994

Automated Deduction — CADE-12

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We design new inference systems for total orderings by applying rewrite techniques to chaining calculi. Equality relations may either be specified axiomatically or built into the deductive calculus via paramodulation or superposition. We demonstrate that our inference systems are compatible with a concept of (global) redundancy for clauses and inferences that covers such widely used simplification techniques as tautology deletion, subsumption, and demodulation. A key to the practicality of chaining techniques is the extent to which so-called variable chainings can be restricted. Syntactic ordering restrictions on terms and the rewrite techniques which account for their completeness considerably restrict variable chaining. We show that variable elimination is an admissible simplification techniques within our redundancy framework, and that consequently for dense total orderings without endpoints no variable chaining is needed at all.

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Completeness of a prover for dense linear orders

Cited by 8 publications

References 6 publications

Ordered chaining calculi for first-order theories of transitive relations

Ordered chaining calculi for first-order theories of transitive relations

Bi-rewrite Systems

Ordered chaining for total orderings

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