Results 1  10
of
12
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 ..."
Abstract

Cited by 74 (4 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 28 (7 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 26 (3 self)
 Add to MetaCart
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 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 ..."
Abstract

Cited by 16 (7 self)
 Add to MetaCart
. 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...
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 ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
. 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...
A Combinatory Logic Approach to Higherorder Eunification
 in Proceedings of the Eleventh International Conference on Automated Deduction, SpringerVerlag LNAI 607
, 1992
"... Let E be a firstorder equational theory. A translation of typed higherorder Eunification problems into a typed combinatory logic framework is presented and justified. The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modifi ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
Let E be a firstorder equational theory. A translation of typed higherorder Eunification problems into a typed combinatory logic framework is presented and justified. The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modification of ordinary narrowing is shown to be a complete method for enumerating higherorder Eunifiers. In fact, we treat a more general problem, in which the types of terms contain type variables. 1 Introduction Investigation of the interaction between firstorder and higherorder equational reasoning has emerged as an active line of research. The collective import of a recent series of papers, originating with [Bre88] and including (among others) [Bar90], [BG91a], [BG91b], [Dou92], [JO91] and [Oka89], is that when various typed calculi are enriched by firstorder equational theories, the validity problem is wellbehaved, and furthermore that the respective computational approaches to ...
Orderings, ACTheories and Symbolic Constraint Solving (Extended Abstract)
 In Tenth Annual IEEE Symposium on Logic in Computer Science
, 1995
"... We design combination techniques for symbolic constraint solving in the presence of associative and commutative (AC) function symbols. This yields an algorithm for solving ACRPO constraints (where ACRPO is the ACcompatible total reduction ordering of [16]), which was a missing ingredient for autom ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
We design combination techniques for symbolic constraint solving in the presence of associative and commutative (AC) function symbols. This yields an algorithm for solving ACRPO constraints (where ACRPO is the ACcompatible total reduction ordering of [16]), which was a missing ingredient for automated deduction strategies with ACconstraint inheritance [15, 19]. As in the ACunification case (actually the ACunification algorithm of [9] is an instance of ours), for this purpose we first study the pure case, i.e. we show how to solve ACordering constraints built over a single AC function symbol and variables. Since ACRPO is an interpretationbased ordering, our algorithm also requires the combination of algorithms for solving interpreted constraints and noninterpreted constraints.
A Constraint Solver in Finite Algebras and Its Combination With Unification Algorithms
, 1992
"... In the context of constraint logic programming and theorem proving, the development of constraint solvers on algebraic domains and their combination is of prime interest. A constraint solver in finite algebras is presented for a constraint language including equations, disequations and inequations o ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
In the context of constraint logic programming and theorem proving, the development of constraint solvers on algebraic domains and their combination is of prime interest. A constraint solver in finite algebras is presented for a constraint language including equations, disequations and inequations on finite domains. The method takes advantage of the embedding of a finite algebra in a primal algebra that can be presented, up to an isomorphism, by an equational presentation. We also show how to combine this constraint solver in finite algebras with other unification algorithms, by extending the techniques used for the combination of unification. 1 Introduction Finite algebras provide valuable domains for constraint logic programming. Unification in this context has attracted considerable interest for its applications: it is of practical relevance for manipulating hardware descriptions and solving formulas of propositional calculus; its implementation in constraint logic programming lan...
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 ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
. 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...
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 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
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...