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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 35 (4 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...
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 24 (4 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...
Cooperation of Decision Procedures for the Satisfiability Problem
 Frontiers of Combining Systems: Proceedings of the 1st International Workshop, Munich (Germany), Applied Logic
, 1996
"... : Constraint programming is strongly based on the use of solvers which are able to check satisfiability of constraints. We show in this paper a rulebased algorithm for solving in a modular way the satisfiability problem w.r.t. a class of theories Th. The case where Th is the union of two disjoint t ..."
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Cited by 22 (4 self)
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: Constraint programming is strongly based on the use of solvers which are able to check satisfiability of constraints. We show in this paper a rulebased algorithm for solving in a modular way the satisfiability problem w.r.t. a class of theories Th. The case where Th is the union of two disjoint theories Th 1 and Th 2 is known for a long time but we study here different cases where function symbols are shared by Th 1 and Th 2 . The chosen approach leads to a highly nondeterministic decomposition algorithm but drastically simplifies the understanding of the combination problem. The obtained decomposition algorithm is illustrated by the combination of nondisjoint equational theories. Keywords: constraint programming, decision procedure, satisfiability, combination problem (R'esum'e : tsvp) INRIALorraine & CRIN, email: Christophe.Ringeissen@loria.fr Unit de recherche INRIA Lorraine Technpole de NancyBrabois, Campus scientifique, 615 rue de Jardin Botanique, BP 101, 54600 VILLE...
External Rewriting for Skeptical Proof Assistants
, 2002
"... This paper presents the design, the implementation and experiments of the integration of syntactic, conditional possibly associativecommutative term rewriting into proof assistants based on constructive type theory. Our approach is called external since it consists in performing term rewriting in a ..."
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Cited by 18 (3 self)
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This paper presents the design, the implementation and experiments of the integration of syntactic, conditional possibly associativecommutative term rewriting into proof assistants based on constructive type theory. Our approach is called external since it consists in performing term rewriting in a speci c and ecient environment and to check the computations later in a proof assistant.
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 (1 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
A constructive decision procedure for equalities modulo AC
"... this paper an optimised constructive decision procedure for AC equalities based on the syntacticness of AC theories. The original motivation for it comes from our work [5] to incorporate term rewriting into the Coq proof assistant [3] using ELAN [7]. The main idea is to perform term rewriting in ELA ..."
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this paper an optimised constructive decision procedure for AC equalities based on the syntacticness of AC theories. The original motivation for it comes from our work [5] to incorporate term rewriting into the Coq proof assistant [3] using ELAN [7]. The main idea is to perform term rewriting in ELAN and to only use Coq for checking purpose. When considering AC rewriting, proof checking requires an ecient method to prove AC equality in Coq using two axioms of associativity and commutativity or possibly a nite set of equalities derived from them