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Disunification: a Survey
 Computational Logic: Essays in Honor of Alan
, 1991
"... Solving an equation in an algebra of terms is known as unification. Solving more complex formulas combining equations and involving in particular negation is called disunification. With such a broad definition, many works fall into the scope of disunification. The goal of this paper is to survey the ..."
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Cited by 56 (9 self)
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Solving an equation in an algebra of terms is known as unification. Solving more complex formulas combining equations and involving in particular negation is called disunification. With such a broad definition, many works fall into the scope of disunification. The goal of this paper is to survey these works and bring them together in a same framework. R'esum'e On appelle habituellement (algorithme d') unification un algorithme de r'esolution d'une 'equation dans une alg`ebre de termes. La r'esolution de formules plus complexes, comportant en particulier des n'egations, est appel'ee ici disunification. Avec une d'efinition aussi 'etendue, de nombreux travaux peuvent etre consid'er'es comme portant sur la disunification. L'objet de cet article de synth`ese est de rassembler tous ces travaux dans un meme formalisme. Laboratoire de Recherche en Informatique, Bat. 490, Universit'e de ParisSud, 91405 ORSAY cedex, France. Email: comon@lri.lri.fr i Contents 1 Syntax 5 1.1 Basic Defini...
Combination Techniques and Decision Problems for Disunification
 Theoretical Computer Science
"... Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories E 1 ; : : : ; E n in order to obtain a unification algorithm for the union E 1 [ : : : [ E n of the theories. Here we want to show that variants of this method m ..."
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Cited by 21 (6 self)
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Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories E 1 ; : : : ; E n in order to obtain a unification algorithm for the union E 1 [ : : : [ E n of the theories. Here we want to show that variants of this method may be used to decide solvability and ground solvability of disunification problems in E 1 [ : : : [E n . Our first result says that solvability of disunification problems in the free algebra of the combined theory E 1 [ : : : [E n is decidable if solvability of disunification problems with linear constant restrictions in the free algebras of the theories E i (i = 1; : : : ; n) is decidable. In order to decide ground solvability (i.e., solvability in the initial algebra) of disunification problems in E 1 [ : : : [ E n we have to consider a new kind of subproblem for the particular theories E i , namely solvability (in the free algebra) of disunification problems with linear constant restricti...
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 ...
Combination of Constraint Solving Techniques: An Algebraic Point of View
 In Proceedings of the 6th International Conference on Rewriting Techniques and Applications, volume 914 of Lecture Notes in Computer Science
"... . In a previous paper we have introduced a method that allows one to combine decision procedures for unifiability in disjoint equational theories. Lately, it has turned out that the prerequisite for this method to applynamely that unification with socalled linear constant restrictions is dec ..."
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Cited by 16 (7 self)
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. In a previous paper we have introduced a method that allows one to combine decision procedures for unifiability in disjoint equational theories. Lately, it has turned out that the prerequisite for this method to applynamely that unification with socalled linear constant restrictions is decidable in the single theoriesis equivalent to requiring decidability of the positive fragment of the first order theory of the equational theories. Thus, the combination method can also be seen as a tool for combining decision procedures for positive theories of free algebras defined by equational theories. Complementing this logical point of view, the present paper isolates an abstract algebraic property of free algebras called combinabilitythat clarifies why our combination method applies to such algebras. We use this algebraic point of view to introduce a new proof method that depends on abstract notions and results from universal algebra, as opposed to technical manipul...
Termination and Completion modulo Associativity, Commutativity and Identity
 Theoretical Computer Science
, 1992
"... Rewriting with associativity, commutativity and identity has been an open problem for a long time. In 1989, Baird, Peterson and Wilkerson introduced the notion of constrained rewriting, to avoid the problem of nontermination inherent to the use of identities. We build up on this idea in two ways: b ..."
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Cited by 11 (3 self)
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Rewriting with associativity, commutativity and identity has been an open problem for a long time. In 1989, Baird, Peterson and Wilkerson introduced the notion of constrained rewriting, to avoid the problem of nontermination inherent to the use of identities. We build up on this idea in two ways: by giving a complete set of rules for completion modulo these axioms; by showing how to build appropriate orderings for proving termination of constrained rewriting modulo associativity, commutativity and identity. 1 Introduction Equations are ubiquitous in mathematics and the sciences. Among the most common equations are associativity, commutativity and identity (existence of a neutral element). Rewriting is an efficient way of reasoning with equations, introduced by Knuth and Bendix [12]. When rewriting, equations are used in one direction chosen once and for all. Unfortunately, orientation alone is not a complete inference rule: given a set of equational axioms E, there may be equal terms...
On The Use Of Constraints In Automated Deduction
, 1995
"... . This paper presents three approaches dealing with constraints in automated deduction. Each of them illustrates a different point. The expression of strategies using constraints is shown through the example of a completion process using ordered and basic strategies. The schematization of complex un ..."
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Cited by 7 (1 self)
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. This paper presents three approaches dealing with constraints in automated deduction. Each of them illustrates a different point. The expression of strategies using constraints is shown through the example of a completion process using ordered and basic strategies. The schematization of complex unification problems through constraints is illustrated by the example of an equational theorem prover with associativity and commutativity axioms. The incorporation of builtin theories in a deduction process is done for a narrowing process which solves queries in theories defined by rewrite rules with builtin constraints. Advantages of using constraints in automated deduction are emphasized and new challenging problems in this area are pointed out. 1 Motivations Constraints have been introduced in automated deduction since about 1990, although one could find similar ideas in theory resolution [32] and in higherorder resolution [16]. The idea is to distinguish two levels of deduction and t...
On the Complexity of Boolean Unification
 INFORMATION PROCESSING LETTERS
, 1997
"... Unification modulo the theory of Boolean algebras has been investigated by several autors. Nevertheless, the exact complexity of the decision problem for unification with constants and general unification was not known. In this research note, we show that the decision problem is \Pi p 2  complete ..."
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Cited by 7 (1 self)
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Unification modulo the theory of Boolean algebras has been investigated by several autors. Nevertheless, the exact complexity of the decision problem for unification with constants and general unification was not known. In this research note, we show that the decision problem is \Pi p 2  complete for unification with constants and PSPACEcomplete for general unification. In contrast, the decision problem for elementary unification (where the terms to be unified contain only symbols of the signature of Boolean algebras) is "only" NPcomplete.
General A and AXUnification via Optimized Combination Procedures
, 1991
"... In a recent paper [BS91] we introduced a new unification algorithm for the combination of disjoint equational theories. Among other consequences we mentioned (1) that the algorithm provides us with a decision procedure for the solvability of general A and AIunification problems and (2) that Kapur ..."
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Cited by 6 (3 self)
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In a recent paper [BS91] we introduced a new unification algorithm for the combination of disjoint equational theories. Among other consequences we mentioned (1) that the algorithm provides us with a decision procedure for the solvability of general A and AIunification problems and (2) that Kapur and Narendran's result about the NPdecidability of the solvability of general AC and ACIunification problems (see [KN91]) may be obtained from our results. In [BS91] we did not give detailled proofs for these two consequences. In the present paper we will treat these problems in more detail. Moreover, we will use the two examples of general A and AIunification for a case study of possible optimizations of the basic combination procedure.
Optimisation Techniques for Combining Constraint Solvers
 IN MAARTEN DE RIJKE AND
, 1998
"... In recent years, techniques that had been developed for the combination of unification algorithms for equational theories were extended to combining constraint solvers. These techniques inherited an old deficit that was already present in the combination of equational theories which makes them rathe ..."
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Cited by 5 (2 self)
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In recent years, techniques that had been developed for the combination of unification algorithms for equational theories were extended to combining constraint solvers. These techniques inherited an old deficit that was already present in the combination of equational theories which makes them rather unsuitable for practical use: The underlying combination algorithms are highly nondeterministic. This paper is concerned with the practical problem of how to optimise the combination method of Baader and Schulz. We present an optimisation method, called the deductive method, which uses specific algorithms for the components to reach certain decisions deterministically. We also give a strategy how to select an order of nondeterministic decisions. Run time tests of our implementation indicate that the optimised combination method yields combined decision procedures that are efficient enough to be used in practice.