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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 57 (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...
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
On Equality Upto Constraints over Finite Trees, Context Unification, and OneStep Rewriting
"... We introduce equality upto constraints over finite trees and investigate their expressiveness. Equality upto constraints subsume equality constraints, subtree constraints, and onestep rewriting constraints. ..."
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Cited by 27 (7 self)
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We introduce equality upto constraints over finite trees and investigate their expressiveness. Equality upto constraints subsume equality constraints, subtree constraints, and onestep rewriting constraints.
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...
Complete Axiomatizations of some Quotient Term Algebras
 In Proc. 18th Int. Coll. on Automata, Languages and Programming, Madrid, LNCS 510
, 1993
"... We show that T (F )= =E can be completely axiomatized when =E is a quasifree theory. Quasifree theories are a wider class of theories than permutative theories of [Mal71] for which Mal'cev gave decision results. As an example of application, we show that the first order theory of T (F )= =E is de ..."
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Cited by 10 (3 self)
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We show that T (F )= =E can be completely axiomatized when =E is a quasifree theory. Quasifree theories are a wider class of theories than permutative theories of [Mal71] for which Mal'cev gave decision results. As an example of application, we show that the first order theory of T (F )= =E is decidable when E is a set of ground equations. Besides, we prove that the \Sigma 1 fragment of the theory of T (F )= =E is decidable when E is a compact set of axioms. In particular, the existential fragment of the theory of associativecommutative function symbols is decidable. Introduction Mal'cev studied in the early sixties classes of locally free algebras that can be completely axiomatized [Mal71]. He proved in particular that what is today known as Clark's equality theory is decidable. He also studied some classes of permutative algebras in which, roughly, the axiom f(s 1 ; : : : ; s n ) = f(t 1 ; : : : ; t n ) ) s 1 = t 1 : : : s n = t n is replaced with f(s 1 ; : : : ; s n ) = f(t ...
About the Theory of Tree Embedding
 Proc. Int. Joint Conf. on Theory and Practice of Software Development, Lecture Notes in Computer Science
, 1993
"... . We show that the positive existential fragment of the theory of tree embedding is decidable. 1 Introduction Symbolic Constraints, i.e. formulae interpreted in some term structure, have been revealed to be extremely useful in logic programming and theorem proving. Among such constraints, the order ..."
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Cited by 6 (3 self)
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. We show that the positive existential fragment of the theory of tree embedding is decidable. 1 Introduction Symbolic Constraints, i.e. formulae interpreted in some term structure, have been revealed to be extremely useful in logic programming and theorem proving. Among such constraints, the ordering constraints can be used in expressing ordered strategies at the formula level instead of the inference level. This allows to cut further the search space, while keeping the completeness of the strategy [7]. Solving ordering constraints also allows for a nice lifting of orderings from the ground level to the terms with variables: define s ? t by 8~x:s ? t where ~x, the variables of s; t, range over all ground terms. This provides with more powerful orderings for termination proofs in rewriting theory. Up to now, the satisfiability of ordering constraints has been studied for some orderings on terms: Venkataraman showed that the existential fragment of the theory of the subterm ordering i...
A Logical Framework for Convergent Infinite Computations 1
"... The need of studying infinite computations has been emphasized in recent years, e.g., see (Vardi and Wolper, 1994). By infinite computations, one means the computations done by some programs that create nonterminating processes or very long time running processes. For such programs, the computation ..."
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The need of studying infinite computations has been emphasized in recent years, e.g., see (Vardi and Wolper, 1994). By infinite computations, one means the computations done by some programs that create nonterminating processes or very long time running processes. For such programs, the computations done by them usually go through infinite sequences of running states (or configurations), unlike finite computations in which only finite sequences of running states are involved.