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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 43 (6 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.
Combination Techniques for NonDisjoint Equational Theories
 Proceedings 12th International Conference on Automated Deduction
, 1994
"... ion variables which are variables coming from an abstraction, either during preprocessing or during the algorithm itself. 3. Introduced variables which are variables introduced by the unification algorithms for each theory. We make the very natural assumption that the unification algorithm for each ..."
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Cited by 25 (5 self)
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ion variables which are variables coming from an abstraction, either during preprocessing or during the algorithm itself. 3. Introduced variables which are variables introduced by the unification algorithms for each theory. We make the very natural assumption that the unification algorithm for each theory may recognize initial, abstraction and introduced variables and never assigns an introduced variable to a nonintroduced one or an abstraction variable to an initial one. With this assumption, our combination algorithm will always make an introduced variable appear in at most one \Gamma i . We may thus also suppose that the domain of each solution does not contain an introduced variable. This does not compromise the soundness of our algorithm. The combination algorithm is described by the two rules given in figure 2. In the rule UnifSolve i , ae SF is obtained by abstracting aliens in the range of ae by fresh variables. ae F i is the substitution such that xae = xae SF ae F i for al...
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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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 17 (8 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.
Fast Matching in Combinations of Regular Equational Theories
, 1996
"... We consider the problem of efficient term matching, modulo combinations of regular equational theories. Our general approach to the problem consists of three phases: compilation, matching and subproblem solving. We describe a technique for dealing with nonlinear variables in a pattern and show how ..."
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Cited by 15 (3 self)
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We consider the problem of efficient term matching, modulo combinations of regular equational theories. Our general approach to the problem consists of three phases: compilation, matching and subproblem solving. We describe a technique for dealing with nonlinear variables in a pattern and show how this technique is specialized to several specific equational theories. For matching in an ordersorted setting we discuss an important optimization for theories involving the associativity equation. Finally we sketch a new method of combining matching algorithms for regular collapse theories and give examples that involve the identity and idempotence equations.
Connecting manysorted structures and theories through adjoint functions
 In Proc. 5th FroCoS
, 2005
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RuleBased Constraint Programming
 Fundamenta Informaticae
, 1998
"... In this paper we present a view of constraint programming based on the notion of rewriting controlled by strategies. We argue that this concept allows us to describe in a unified way the constraint solving mechanism as well as the metalanguage needed to manipulate the constraints. This has the a ..."
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Cited by 9 (2 self)
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In this paper we present a view of constraint programming based on the notion of rewriting controlled by strategies. We argue that this concept allows us to describe in a unified way the constraint solving mechanism as well as the metalanguage needed to manipulate the constraints. This has the advantage to provide descriptions that are very close to the proof theoretical setting used now to describe constraint manipulations like unification or numerical constraint solving. We examplify the approach by presenting examples of constraint solvers descriptions and combinations written in the ELAN language. 1
Termination Modulo Combinations of Equational Theories
"... Abstract. Rewriting with rules R modulo axioms E is a widely used technique in both rulebased programming languages and in automated deduction. Termination methods for rewriting systems modulo specific axioms E (e.g., associativitycommutativity) are known. However, much less seems to be known abou ..."
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Cited by 6 (5 self)
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Abstract. Rewriting with rules R modulo axioms E is a widely used technique in both rulebased programming languages and in automated deduction. Termination methods for rewriting systems modulo specific axioms E (e.g., associativitycommutativity) are known. However, much less seems to be known about termination methods that can be modular in the set E of axioms. In fact, current termination tools and proof methods cannot be applied to commonly occurring combinations of axioms that fall outside their scope. This work proposes a modular termination proof method based on semantics and terminationpreserving transformations that can reduce the proof of termination of rules R modulo E to an equivalent proof of termination of the transformed rules modulo a typically much simpler set B of axioms. Our method is based on the notion of variants of a term recently proposed by Comon and Delaune. We illustrate its practical usefulness by considering the very common case in which E is an arbitrary combination of associativity, commutativity, left and rightidentity axioms for various function symbols. 1