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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 74 (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.
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...
Unification in a combination of equational theories with shared constants and its application to Primal Algebras
 In Proceedings of the 1st International Conference on Logic Programming and Automated Reasoning, St. Petersburg (Russia), volume 624 of Lecture Notes in Artificial Intelligence
, 1992
"... . We extend the results on combination of disjoint equational theories to combination of equational theories where the only function symbols shared are constants. This is possible because there exist finitely many proper shared terms (the constants) which can be assumed irreducible in any equational ..."
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Cited by 15 (3 self)
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. We extend the results on combination of disjoint equational theories to combination of equational theories where the only function symbols shared are constants. This is possible because there exist finitely many proper shared terms (the constants) which can be assumed irreducible in any equational proof of the combined theory. We establish a connection between the equational combination framework and a more algebraic one. A unification algorithm provides a symbolic constraint solver in the combination of algebraic structures whose finite domains of values are non disjoint and correspond to constants. Primal algebras are particular finite algebras of practical relevance for manipulating hardware descriptions. 1 Introduction The combination problem for unification can be stated as follows: given two unification algorithms in two (consistent) equational theories E 1 on T (F 1 ; X) and E 2 on T (F 2 ; X), how to design a unification algorithm for E 1 [ E 2 on T (F 1 [ F 2 ; X)? The ge...
Constraint Logic Programming over Unions of Constraint Theories
 Journal of Functional and Logic Programming
, 1998
"... In this paper, we propose an extension of the JaffarLassez Constraint Logic Programming scheme that operates with unions of constraint theories with different signatures and decides the satisabi]Jty of mixed constraints by appropriately combining the constraint solvers of the component theories ..."
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Cited by 9 (1 self)
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In this paper, we propose an extension of the JaffarLassez Constraint Logic Programming scheme that operates with unions of constraint theories with different signatures and decides the satisabi]Jty 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 lnes of those given for the original scheme. Then we show how the main soundness and completeness results of Constraint Logic Programming lft to our extension.
Combination of Matching Algorithms
 Proceedings 11th Annual Symposium on Theoretical Aspects of Computer Science, Caen (France), volume 775 of Lecture Notes in Computer Science
, 1994
"... . This paper addresses the problem of systematically building a matching algorithm for the union of two disjoint equational theories. The question is under which conditions matching algorithms in the single theories are sufficient to obtain a matching algorithm in the combination? In general, the bl ..."
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Cited by 5 (0 self)
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. This paper addresses the problem of systematically building a matching algorithm for the union of two disjoint equational theories. The question is under which conditions matching algorithms in the single theories are sufficient to obtain a matching algorithm in the combination? In general, the blind use of combination techniques introduces unification. Two different restrictions are considered in order to reduce this unification to matching. First, we show that combining matching algorithms (with linear constant restriction) is always sufficient for solving a pure fragment of combined matching problems. Second, we present a combined matching algorithm which is complete for the largest class of theories where unification is not needed, including collapsefree regular theories and linear theories. 1 Introduction The process of matching is crucial in term rewriting, from automated deduction involving simplification rules to the implementation of operational semantics for programming l...
Unification Algorithms Cannot be Combined in Polynomial Time
 in Proceedings of the 13th International Conference on Automated Deduction, M.A. McRobbie and J.K. Slaney (Eds.), Springer LNAI 1104
, 1996
"... . We establish that there is no polynomialtime general combination algorithm for unification in finitary equational theories, unless the complexity class #P of counting problems is contained in the class FP of function problems solvable in polynomialtime. The prevalent view in complexity theory is ..."
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Cited by 4 (0 self)
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. We establish that there is no polynomialtime general combination algorithm for unification in finitary equational theories, unless the complexity class #P of counting problems is contained in the class FP of function problems solvable in polynomialtime. The prevalent view in complexity theory is that such a collapse is extremely unlikely for a number of reasons, including the fact that the containment of #P in FP implies that P = NP. Our main result is obtained by establishing the intractrability of the counting problem for general AGunification, where AG is the equational theory of Abelian groups. Specifically, we show that computing the cardinality of a minimal complete set of unifiers for general AGunification is a #Phard problem. In contrast, AGunification with constants is solvable in polynomial time. Since an algorithm for general AGunification can be obtained as a combination of a polynomialtime algorithm for AGunification with constants and a polynomialtime algorithm...
Negation in Combining Constraint Systems
 Communications of the ACM
, 1998
"... In a recent paper, Baader and Schulz presented a general method for the combination of constraint systems for purely positive constraints. But negation plays an important role in constraint solving. E.g., it is vital for constraint entailment. Therefore it is of interest to extend their results to t ..."
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Cited by 3 (0 self)
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In a recent paper, Baader and Schulz presented a general method for the combination of constraint systems for purely positive constraints. But negation plays an important role in constraint solving. E.g., it is vital for constraint entailment. Therefore it is of interest to extend their results to the combination of constraint problems containing negative constraints. We show that the combined solution domain introduced by Baader and Schulz is a domain in which one can solve positive and negative "mixed" constraints by presenting an algorithm that reduces solvability of positive and negative "mixed" constraints to solvability of pure constraints in the components. The existential theory in the combined solution domain is decidable if solvability of literals with socalled linear constant restrictions is decidable in the components. We also give a criterion for ground solvability of mixed constraints in the combined solution domain. The handling of negative constraints can be signific...
Combining Unification and Disunification Algorithms  Tractable and Intractable Instances
, 1996
"... We consider the problem of combining procedures that decide solvability of (dis)unification problems over disjoint equational theories. Partial answers to the following questions are given: ffl Which properties of the component theories imply intractability in the sense that there cannot be a polyn ..."
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Cited by 2 (1 self)
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We consider the problem of combining procedures that decide solvability of (dis)unification problems over disjoint equational theories. Partial answers to the following questions are given: ffl Which properties of the component theories imply intractability in the sense that there cannot be a polynomial combination algorithm, assuming P 6= NP? ffl Which general properties of the component theories guarantee tractability of the combination problem in the sense that there exists a deterministic and polynomial combination algorithm? A criterion is given that characterizes a large class K of equational theories E where general Eunification is always NPhard. We show that all regular equational theories E that contain a commutative or an associative function symbol belong to K. Other examples of equational theories in K concern nonregular cases as well. The combination algorithm described in [BS92] can be used to reduce solvability of general Eunification algorithms to solvability of...
A criterion for intractability of Eunification with free function symbols and its relevance for combination of unification algorithms
"... . All applications of equational unification in the area of term rewriting and theorem proving require algorithms for general Eunifica tion, i.e., Eunification with free function symbols. On this background, the complexity of general Eunification algorithms has been investigated for a large num ..."
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Cited by 2 (0 self)
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. All applications of equational unification in the area of term rewriting and theorem proving require algorithms for general Eunifica tion, i.e., Eunification with free function symbols. On this background, the complexity of general Eunification algorithms has been investigated for a large number of equational theories. For most of the relevant cases, the problem of deciding solvability of general Eunification problems was found to be NPhard. We offer a partial explanation. A criterion is given that characterizes a large class K of equational theories E where general Eunification is always NPhard. We show that all regular equational theories E that contain a commutative or an associative function symbol belong to K. Other examples of equational theories in K concern nonregular cases as well. The combination algorithm described in [BS92] can be used to reduce solvability of general Eunification algorithms to solvability of E and free (Robinson) unification problems with lin...