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47
Theorem Proving with Ordering and Equality Constrained Clauses
 Journal of Symbolic Computation
, 1995
"... constraint strategies and saturation Given a signature F , below we denote by S the set of all clauses built over F , and similarly by C the set of all constraints, and by EC the set of all equality constraints (which is a subset of C). Definition 3.1. An inference rule IR is a mapping of ntuples ..."
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Cited by 75 (20 self)
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constraint strategies and saturation Given a signature F , below we denote by S the set of all clauses built over F , and similarly by C the set of all constraints, and by EC the set of all equality constraints (which is a subset of C). Definition 3.1. An inference rule IR is a mapping of ntuples of clauses to sets of triples containing a clause, a constraint and an equality constraint: IR : S n \Gamma! P(hS; C; ECi) An inference system is a set of inference rules. Definition 3.2. A constraint inheritance strategy is a function mapping a clause, two constraints and an equality constraint to a clause and a constraint: H : S \Theta C \Theta C \Theta EC \Gamma! S \Theta C Inference systems and constraint inheritance strategies are combined to produce inferences in the usual sense: given constrained clauses C 1 [[T 1 ]]; : : : ; Cn [[T n ]], we obtain a conclusion C [[T ]] as follows. Applying an inference rule to C 1 ; : : : ; Cn we obtain a triple hD; OT;ET i. Then the constraint...
Basic Paramodulation
 Information and Computation
, 1995
"... We introduce a class of restrictions for the ordered paramodulation and superposition calculi (inspired by the basic strategy for narrowing), in which paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. In addition we introduce restrictions bas ..."
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Cited by 70 (12 self)
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We introduce a class of restrictions for the ordered paramodulation and superposition calculi (inspired by the basic strategy for narrowing), in which paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. In addition we introduce restrictions based on term selection rules and redex orderings, which are general criteria for delimiting the terms which are available for inferences. These refinements are compatible with standard ordering restrictions and are complete without paramodulation into variables or using functional reflexivity axioms. We prove refutational completeness in the context of deletion rules, such as simplification by rewriting (demodulation) and subsumption, and of techniques for eliminating redundant inferences.
Equational Reasoning In SaturationBased Theorem Proving
, 1998
"... INTRODUCTION Equational reasoning is fundamental in mathematics, logics, and many applications of formal methods in computer science. In this chapter we describe the theoretical concepts and results that form the basis of stateoftheart automated theorem provers for firstorder clause logic with ..."
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Cited by 38 (2 self)
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INTRODUCTION Equational reasoning is fundamental in mathematics, logics, and many applications of formal methods in computer science. In this chapter we describe the theoretical concepts and results that form the basis of stateoftheart automated theorem provers for firstorder clause logic with equality. We mainly concentrate on refinements of paramodulation, such as the superposition calculus, that have yielded the most promising results to date in automated equational reasoning. We begin with some preliminary material in section 2 and then explain, in section 3, why resolution with the congruence axioms is an impractical theorem proving method for equational logic. In section 4 we outline the main results about paramodulationa more direct equational inference rule. This section also contains a description of the modification method, which can be used to demonstrate that the functional reflexivity axioms are redundant in the context of paramodulation. The modification
ACsuperposition with constraints: No ACunifiers needed
 Proceedings 12th International Conference on Automated Deduction
, 1990
"... We prove the completeness of (basic) deduction strategies with constrained clauses modulo associativity and commutativity (AC). Here each inference generates one single conclusion with an additional equality s = AC t in its constraint (instead of one conclusion for each minimal ACunifier, i.e. expo ..."
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Cited by 29 (5 self)
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We prove the completeness of (basic) deduction strategies with constrained clauses modulo associativity and commutativity (AC). Here each inference generates one single conclusion with an additional equality s = AC t in its constraint (instead of one conclusion for each minimal ACunifier, i.e. exponentially many). Furthermore, computing ACunifiers is not needed at all. A clause C [[ T ]] is redundant if the constraint T is not ACunifiable. If C is the empty clause this has to be decided to know whether C [[ T ]] denotes an inconsistency. In all other cases any sound method to detect unsatisfiable constraints can be used. 1 Introduction Some fundamental ideas on applying symbolic constraints to theorem proving were given in [KKR90], where a constrained clause is a shorthand for its (infinite) set of ground instances satisfying the constraint. In a constrained equation f(x) ' a [[ x = g(y) ]], the equality `=' of the constraint is usually interpreted in T (F) (syntactic equality), ...
33 Basic Test Problems: A Practical Evaluation of Some Paramodulation Strategies
, 1996
"... Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort &qu ..."
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Cited by 23 (5 self)
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Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort "our rule is complete and it heavily prunes the search space; therefore it is efficient". 2 These positions are highly questionable and indicate that the authors have little or no experience with the practical use of automated inference systems. Restrictive rules (1) can block short, easytofind proofs, (2) can block proofs involving simple clauses, the type of clause on which many practical searches focus, (3) can require weakening of redundancy control such as subsumption and demodulation, and (4) can require the use of complex checks in deciding whether such rules should be applied. The only way to determ
Ordering Constraints on Trees
, 1994
"... We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, wellfounded and compatible with the tree constructors. The paper includes some new results, in par ..."
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Cited by 21 (1 self)
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We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, wellfounded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a nonunary signature.
Saturation of FirstOrder (Constrained) Clauses With The Saturate System
 REWRITING TECHNIQUES AND APPLICATIONS, 5TH INTERNATIONAL CONFERENCE, RTA93, VOLUME 690 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1993
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AssociativeCommutative Deduction with Constraints
, 1993
"... Associativecommutative equational reasoning is known to be highly complex for theorem proving. Hence, it is very important to focus deduction by adding constraints, such as unification and ordering, and to define efficient strategies, such as the basic requirements `a la Hullot. Constraints are f ..."
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Cited by 18 (3 self)
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Associativecommutative equational reasoning is known to be highly complex for theorem proving. Hence, it is very important to focus deduction by adding constraints, such as unification and ordering, and to define efficient strategies, such as the basic requirements `a la Hullot. Constraints are formulas used for pruning the set of ground instances of clauses deduced by a theorem prover. We propose here an extension of ACparamodulation and ACsuperposition with these constraint mechanisms ; we do not need to compute ACunifiers anymore. The method is proved to be refutationally complete, even with simplification. The power of this approach is exemplified by a very short proof of the equational version of SAM's Lemma using DATAC, our implementation of the strategy.
Prototyping completion with constraints using computational systems
"... We use computational systems to express a completion with constraints procedure that gives priority to simplifications. Computational systems are rewrite theories enriched by strategies. The implementation of completion in ELAN, an interpretor of computational systems, is especially convenient for e ..."
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Cited by 17 (9 self)
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We use computational systems to express a completion with constraints procedure that gives priority to simplifications. Computational systems are rewrite theories enriched by strategies. The implementation of completion in ELAN, an interpretor of computational systems, is especially convenient for experimenting with different simplification strategies, thanks to the powerful strategy language of ELAN.