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A New Correctness Proof of the NelsonOppen Combination Procedure
 Frontiers of Combining Systems, volume 3 of Applied Logic Series
, 1996
"... The NelsonOppen combination procedure, which combines satisfiability procedures for a class of firstorder theories by propagation of equalities between variables, is one of the most general combination methods in the field of theory combination. We describe a new nondeterministic version of the p ..."
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Cited by 92 (8 self)
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The NelsonOppen combination procedure, which combines satisfiability procedures for a class of firstorder theories by propagation of equalities between variables, is one of the most general combination methods in the field of theory combination. We describe a new nondeterministic version of the procedure that has been used to extend the Constraint Logic Programming Scheme to unions of constraint theories. The correctness proof of the procedure that we give in this paper not only constitutes a novel and easier proof of Nelson and Oppen's original results, but also shows that equality sharing between the satisfiability procedures of the component theories, the main idea of the method, can be confined to a restricted set of variables.
Unions of NonDisjoint Theories and Combinations of Satisfiability Procedures
 THEORETICAL COMPUTER SCIENCE
, 2001
"... In this paper we outline a theoretical framework for the combination of decision procedures for constraint satisfiability. We describe a general combination method which, given a procedure that decides constraint satisfiability with respect to a constraint theory T1 and one that decides constraint s ..."
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Cited by 44 (7 self)
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In this paper we outline a theoretical framework for the combination of decision procedures for constraint satisfiability. We describe a general combination method which, given a procedure that decides constraint satisfiability with respect to a constraint theory T1 and one that decides constraint satisfiability with respect to a constraint theory T2, produces a procedure that (semi)decides constraint satisfiability with respect to the union of T1 and T2. We provide a number of modeltheoretic conditions on the constraint language and the component constraint theories for the method to be sound and complete, with special emphasis on the case in which the signatures of the component theories are nondisjoint. We also describe some general classes of theories to which our combination results apply, and relate our approach to some of the existing combination methods in the field.
Combining Symbolic Constraint Solvers on Algebraic Domains
 Journal of Symbolic Computation
, 1994
"... ion An atomic constraint p ? (t 1 ; : : : ; t m ) is decomposed into a conjunction of pure atomic constraints by introducing new equations of the form (x = ? t), where t is an alien subterm in the constraint and x is a variable that does not appear in p ? (t 1 ; : : : ; t m ). This is formalized tha ..."
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Cited by 28 (7 self)
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ion An atomic constraint p ? (t 1 ; : : : ; t m ) is decomposed into a conjunction of pure atomic constraints by introducing new equations of the form (x = ? t), where t is an alien subterm in the constraint and x is a variable that does not appear in p ? (t 1 ; : : : ; t m ). This is formalized thanks to the notion of abstraction. Definition 4.2. Let T be a set of terms such that 8t 2 T ; 8u 2 X [ SC; t 6= E1[E2 u: A variable abstraction of the set of terms T is a surjective mapping \Pi from T to a set of variables included in X such that 8s; t 2 T ; \Pi(s) = \Pi(t) if and only if s =E1[E2 t: \Pi \Gamma1 denotes any substitution (with possibly infinite domain) such that \Pi(\Pi \Gamma1 (x)) = x for any variable x in the range of \Pi. It is important to note that building a variable abstraction relies on the decidability of E 1 [ E 2 equality in order to abstract equal alien subterms by the same variable. Let T = fu #R j u 2 T (F [ X ) and u #R2 T (F [ X )n(X [ SC)g...
Connecting ManySorted Theories’, The
 Journal of Symbolic Logic
, 2007
"... Abstract. Basically, the connection of two manysorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of ..."
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Cited by 24 (7 self)
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Abstract. Basically, the connection of two manysorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of universal formulae in the component theories transfers to their connection. In addition, we consider variants of the basic connection scheme. Our results can be seen as a generalization of the socalled Econnection approach for combining modal logics to an algebraic setting. §1. Introduction. The combination of decision procedures for logical theories arises in many areas of logic in computer science, such as constraint solving, automated deduction, term rewriting, modal logics, and description logics. In general, one has two firstorder theories T1 and T2 over signatures Σ1 and Σ2, for which validity of a certain type of formulae (e.g., universal, existential positive,
A New Approach for Combining Decision Procedures for the Word Problem, and Its Connection to the NelsonOppen Combination Method
 Proceedings of the 14th International Conference on Automated Deduction
, 1997
"... The NelsonOppen combination method can be used to combine decision procedures for the validity of quantifierfree formulae in firstorder theories with disjoint signatures, provided that the theories to be combined are stably infinite. We show that, even though equational theories need not sati ..."
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Cited by 21 (10 self)
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The NelsonOppen combination method can be used to combine decision procedures for the validity of quantifierfree formulae in firstorder theories with disjoint signatures, provided that the theories to be combined are stably infinite. We show that, even though equational theories need not satisfy this property, Nelson and Oppen's method can be applied, after some minor modifications, to combine decision procedures for the validity of quantifierfree formulae in equational theories.
A New Combination Procedure for the Word Problem that Generalizes Fusion Decidability Results in Modal Logics
 In David A. Basin and Michaël Rusinowitch, editors, IJCAR ’04
, 2004
"... Previous results for combining decision procedures for the word problem in the nondisjoint case do not apply to equational theories induced by modal logicswhose combination is not disjoint since they share the theory of Boolean algebras. Conversely, decidability results for the fusion of mod ..."
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Cited by 18 (9 self)
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Previous results for combining decision procedures for the word problem in the nondisjoint case do not apply to equational theories induced by modal logicswhose combination is not disjoint since they share the theory of Boolean algebras. Conversely, decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand, and thus do not generalize to other equational theories.
EUnification by Means of Tree Tuple Synchronized Grammars
, 1996
"... : The goal of this paper is both to give a Eunification procedure that always terminates, and to decide unifiability. For this, we assume that the equational theory is specified by a confluent and constructorbased rewrite system, and that four additional restrictions are satisfied. We give a proce ..."
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Cited by 10 (3 self)
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: The goal of this paper is both to give a Eunification procedure that always terminates, and to decide unifiability. For this, we assume that the equational theory is specified by a confluent and constructorbased rewrite system, and that four additional restrictions are satisfied. We give a procedure that represents the (possibly infinite) set of solutions thanks to a tree tuple synchronized grammar, and that can decide unifiability thanks to an emptiness test. Moreover we show that if only three of the four additional restrictions are satisfied then unifiability is undecidable. 1 Introduction First order Eunification [29] is a tool that plays an important role in automated deduction, in particular in functional logic programming and for solving symbolic constraints (see [4] for an extensive survey of the area). It consists in finding instances to variables that make two terms equal modulo an equational theory given by a set of equalities, i.e. it amounts to solve an equation (ca...
Constraint Logic Programming over Unions of Constraint Theories
, 1998
"... In this paper, we present an extension of the Ja#arLassez constraint logic programming scheme that operates with unions of constraint theories with di#erent signatures and decides the satisfiability of mixed constraints by appropriately combining the constraint solvers of the component theories ..."
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Cited by 10 (1 self)
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In this paper, we present an extension of the Ja#arLassez constraint logic programming scheme that operates with unions of constraint theories with di#erent signatures and decides the satisfiability of mixed constraints by appropriately combining the constraint solvers of the component theories. We describe the extended scheme, and provide logical and operational semantics for it along the lines of those given for the original scheme. We then show how the main soundness and completeness results of constraint logic programming lift to our extension.