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A New Method for Undecidability Proofs of First Order Theories
 Journal of Symbolic Computation
, 1992
"... this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction ..."
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Cited by 30 (7 self)
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this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction
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...
Combination Problems for Commutative/Monoidal Theories or How Algebra Can Help in Equational Unification
 J. Applicable Algebra in Engineering, Communication and Computing
, 1996
"... We study the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. It encompasses important examples like the theories of Abelian monoids, idempotent Abelian monoids, and Abelian groups. This class has been introduced by the au ..."
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Cited by 7 (7 self)
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We study the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. It encompasses important examples like the theories of Abelian monoids, idempotent Abelian monoids, and Abelian groups. This class has been introduced by the authors independently of each other as "commutative theories " (Baader) and "monoidal theories" (Nutt). We show that commutative theories and monoidal theories indeed define the same class (modulo a translation of the signature), and we prove that it is undecidable whether a given theory belongs to it. In the remainder of the paper we investigate combinations of commutative/monoidal theories with other theories. We show that finitary commutative/monoidal theories always satisfy the requirements for applying general methods developed for the combination of unification algorithms for disjoint equational theories. Then we study the adjunction of monoids of homomorphisms to commutative /monoidal t...
"Syntactic" ACUnification
, 1994
"... The rules for unification in a simple syntactic theory, using Kirchner's mutation [15, 16] do not terminate in the case of associativecommutative theories. We show that in the case of a linear equation, these rules terminate, yielding a complete set of solved forms, each variable introduced by the u ..."
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The rules for unification in a simple syntactic theory, using Kirchner's mutation [15, 16] do not terminate in the case of associativecommutative theories. We show that in the case of a linear equation, these rules terminate, yielding a complete set of solved forms, each variable introduced by the unifiers corresponding to a (trivial) minimal solution of the (trivial) Diophantine equation where all coefficients are 1. A nonlinear problem can be first treated as a linear one, that is considering two occurrences of a same variable as two different variables. After this step, one has to solve the equations between the different values that have been obtained for the different occurrences of a same variable. We show that one can restrict the search of the solutions of these latter equations to linear substitutions. This result is based on the analysis of how the minimal solutions of a linear Diophantine equation can be builtup using the solutions of the trivial Diophantine equation asso...