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16
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 76 (4 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 35 (3 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...
On the Combination of Symbolic Constraints, Solution Domains, and Constraint Solvers
 In Proceedings of the First International Conference on Principles and Practice of Constraint Programming
"... When combining languages for symbolic constraints, one is typically faced with the problem of how to treat "mixed" constraints. The two main problems are (1) how to define a combined solution structure over which these constraints are to be solved, and (2) how to combine the constraint solving metho ..."
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Cited by 26 (3 self)
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When combining languages for symbolic constraints, one is typically faced with the problem of how to treat "mixed" constraints. The two main problems are (1) how to define a combined solution structure over which these constraints are to be solved, and (2) how to combine the constraint solving methods for pure constraints into one for mixed constraints. The paper introduces the notion of a "free amalgamated product" as a possible solution to the first problem. Subsequently, we define socalled simplycombinable structures (SCstructures). For SCstructures over disjoint signatures, a canonical amalgamation construction exists, which for the subclass of strong SCstructures yields the free amalgamated product. The combination technique of [BS92, BaS94a] can be used to combine constraint solvers for (strong) SCstructures over disjoint signatures into a solver for their (free) amalgamated product. In addition to term algebras modulo equational theories, the class of SCstru...
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...
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...
Combination of Constraint Solving Techniques: An Algebraic Point of View
 In Proceedings of the 6th International Conference on Rewriting Techniques and Applications, volume 914 of Lecture Notes in Computer Science
"... . In a previous paper we have introduced a method that allows one to combine decision procedures for unifiability in disjoint equational theories. Lately, it has turned out that the prerequisite for this method to applynamely that unification with socalled linear constant restrictions is dec ..."
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Cited by 16 (7 self)
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. In a previous paper we have introduced a method that allows one to combine decision procedures for unifiability in disjoint equational theories. Lately, it has turned out that the prerequisite for this method to applynamely that unification with socalled linear constant restrictions is decidable in the single theoriesis equivalent to requiring decidability of the positive fragment of the first order theory of the equational theories. Thus, the combination method can also be seen as a tool for combining decision procedures for positive theories of free algebras defined by equational theories. Complementing this logical point of view, the present paper isolates an abstract algebraic property of free algebras called combinabilitythat clarifies why our combination method applies to such algebras. We use this algebraic point of view to introduce a new proof method that depends on abstract notions and results from universal algebra, as opposed to technical manipul...
G.: A uniform approach to constraintsolving for lists, multisets, compact lists, and sets
 ACM Trans. Comput. Log
, 2008
"... Lists, multisets, and sets are wellknown data structures whose usefulness is widely recognized in various areas of Computer Science. They have been analyzed from an axiomatic point of view with a parametric approach in [Dovier et al. 1998] where the relevant unification algorithms have been develop ..."
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Cited by 9 (5 self)
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Lists, multisets, and sets are wellknown data structures whose usefulness is widely recognized in various areas of Computer Science. They have been analyzed from an axiomatic point of view with a parametric approach in [Dovier et al. 1998] where the relevant unification algorithms have been developed. In this paper we extend these results considering more general constraints, namely equality and membership constraints and their negative counterparts.
Combining Constraint Solving
, 2001
"... this paper. On the one hand, dening a semantics for the combined system may depend on methods and results from formal logic and universal algebra. On the other hand, an ecient combination of the actual constraint solvers often requires the possibility of communication and cooperation between the sol ..."
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Cited by 5 (0 self)
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this paper. On the one hand, dening a semantics for the combined system may depend on methods and results from formal logic and universal algebra. On the other hand, an ecient combination of the actual constraint solvers often requires the possibility of communication and cooperation between the solvers.