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24
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), ...
Ordering Constraints on Trees
 Colloquium on Trees in Algebra and Programming
, 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 p ..."
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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. 1 Symbolic Constraints Constraints on trees are becoming popular in automated theorem proving, logic programming and in other fields thanks to their potential to represent large or even infinite sets of formulae in a nice and compact way. More precisely, a symbolic constraint system, also called a constraint system on trees, consists of a fragment of firstorder logic over a set of predicate symbols P and a set of function symbols F , together with a fixed interpretation of the predicate symbols in the algebra of finite trees T (F) (or sometimes the algebra of infinite trees I(F)) ov...
Normalised Rewriting and Normalised Completion
, 1994
"... We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algor ..."
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Cited by 19 (2 self)
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We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algorithm for normalised rewriting. It contains as an instance the usual AC completion algorithm, but also the wellknown Buchberger's algorithm for computing standard bases of polynomial ideals. We investigate the particular case of completion of ground equations, In this case we prove by a uniform method that completion modulo E terminates, for some interesting E. As a consequence, we obtain the decidability of the word problem for some classes of equational theories. We give implementation results which shows the efficiency of normalised completion with respect to completion modulo AC. 1 Introduction Equational axioms are very common in most sciences, including computer science. Equations can ...
A Total, Ground Path Ordering for Proving Termination of ACRewrite Systems
 Proc. 8th RTA, LNCS 1232
, 1997
"... . A new path ordering for showing termination of associativecommutative (AC) rewrite systems is defined. If the precedence relation on function symbols is total, the ordering is total on ground terms, but unlike the ordering proposed by Rubio and Nieuwenhuis, this ordering can orient the distrib ..."
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Cited by 16 (3 self)
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. A new path ordering for showing termination of associativecommutative (AC) rewrite systems is defined. If the precedence relation on function symbols is total, the ordering is total on ground terms, but unlike the ordering proposed by Rubio and Nieuwenhuis, this ordering can orient the distributivity property in the proper direction. The ordering is defined in a natural way using recursive path ordering with status as the underlying basis. This settles a longstanding problem in termination orderings for AC rewrite systems. The ordering can be used to define an ordering on nonground terms. 1 Introduction Rewriting techniques reduce the search space for finding proofs substantially because of the ability to orient equality, which is symmetric, into a terminating directed rewrite relation (!), which is antisymmetric, using well founded orderings. Rules are used for "simplifying" expressions by repeatedly replacing instances of lefthand sides by the corresponding righthand s...
Integrating linear arithmetic into superposition calculus
 In Computer Science Logic (CSL’07
, 2007
"... Abstract. We present a method of integrating linear rational arithmetic into superposition calculus for firstorder logic. One of our main results is completeness of the resulting calculus under some finiteness assumptions. 1 ..."
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Cited by 16 (3 self)
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Abstract. We present a method of integrating linear rational arithmetic into superposition calculus for firstorder logic. One of our main results is completeness of the resulting calculus under some finiteness assumptions. 1
Superposition Theorem Proving for Abelian Groups Represented as Integer Modules
 Theoretical Computer Science
, 1996
"... We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equation ..."
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Cited by 13 (4 self)
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We define a superposition calculus specialized for abelian groups represented as integer modules, and show its refutational completeness. This allows to substantially reduce the number of inferences compared to a standard superposition prover which applies the axioms directly. Specifically, equational literals are simplified, so that only the maximal term of the sums is on the lefthand side. Only certain minimal superpositions need to be considered; other superpositions which a standard prover would consider become redundant. This not only reduces the number of inferences, but also reduces the size of the ACunification problems which are generated. That is, ACunification is not necessary at the top of a term, only below some nonACsymbol. Further, we consider situations where the axioms give rise to variable overlaps and develop techniques to avoid these explosive cases where possible. 1 Introduction Historically, starting from plain resolution, more and more problematic axioms ha...
Termination of AssociativeCommutative Rewriting by Dependency Pairs
 9th International Conference on Rewriting Techniques and Applications, volume 1379 of Lecture
, 1998
"... A new criterion for termination of rewriting has been described by Arts and Giesl in 1997. We show how this criterion can be generalized to rewriting modulo associativity and commutativity. We also show how one can build weak ACcompatible reduction orderings which may be used in this criterion. ..."
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Cited by 13 (1 self)
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A new criterion for termination of rewriting has been described by Arts and Giesl in 1997. We show how this criterion can be generalized to rewriting modulo associativity and commutativity. We also show how one can build weak ACcompatible reduction orderings which may be used in this criterion.
A total ACcompatible ordering based on RPO
 Theoretical Computer Science
, 1995
"... We define a simplification ordering on terms which is ACcompatible and total on nonAC equivalent ground terms, without any restrictions on the signature like the number of ACsymbols or free symbols. Unlike previous work by Narendran and Rusinowitch [NR91], our ACRPO ordering is not based on poly ..."
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Cited by 12 (7 self)
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We define a simplification ordering on terms which is ACcompatible and total on nonAC equivalent ground terms, without any restrictions on the signature like the number of ACsymbols or free symbols. Unlike previous work by Narendran and Rusinowitch [NR91], our ACRPO ordering is not based on polynomial interpretations, but on a simple extension of the wellknown RPO ordering (with a total (arbitrary) precedence on the function symbols). This solves an open question posed e.g. by Bachmair [Bac92]. A second difference is that this ordering is also defined on terms with variables, which makes it applicable in practice for complete theorem proving strategies with builtin ACunification and for orienting nonground rewrite systems. The ordering is defined in a simple way by means of rewrite rules, and can be easily implemented, since its main component is RPO. 1 Introduction Automated termination proofs are wellknown to be crucial for using rewritinglike methods in theorem proving an...
On Narrowing, Refutation Proofs and Constraints
 Rewriting Techniques and Applications, 6th International Conferenc e, RTA95, LNCS 914
, 1995
"... . We develop a proof technique for dealing with narrowing and refutational theorem proving in a uniform way, clarifying the exact relationship between the existing results in both fields and allowing us to obtain several new results. Refinements of narrowing (basic, LSE, etc.) are instances of the t ..."
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Cited by 9 (4 self)
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. We develop a proof technique for dealing with narrowing and refutational theorem proving in a uniform way, clarifying the exact relationship between the existing results in both fields and allowing us to obtain several new results. Refinements of narrowing (basic, LSE, etc.) are instances of the technique, but are also defined here for arbitrary (possibly ordering and/or equality constrained or not yet convergent or saturated) Horn clauses, and shown compatible with simplification and other redundancy notions. By narrowing modulo equational theories like AC, compact representations of solutions, expressed by ACequality constraints, can be obtained. Computing ACunifiers is only needed at the end if one wants to "uncompress" such a constraint into its (doubly exponentially many) concrete substitutions. 1 Introduction Answer computation in some logic is at the heart of many applications in computer science, such as (functional) logic programming, automated theorem proving /discoverin...