Unification in a combination of equational theories with shared constants and its application to primal algebras

Ringeissen, Christophe

doi:10.1007/bfb0013067

Cited by 15 publications

(6 citation statements)

References 6 publications

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“…In (Domenjoud et al, 1994), we extend the result due to Nipkow (1991) about combination of matching algorithms in regular theories to the case where shared symbols satisfy an appropriate notion of constructors. By applying techniques introduced in (Ringeissen, 1992;Kirchner and Ringeissen, 1994), we are also able to combine non-disjoint partially linear theories provided that shared symbols are only constants like for instance in the union E 1 _ E 2 where…”

Section: Resultsmentioning

confidence: 99%

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Combining Decision Algorithms for Matching in the Union of Disjoint Equational Theories

Ringeissen

1996

Information and Computation

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This paper addresses the problem of systematically building a matching algorithm for the union of two disjoint theories E 1 _ E 2 provided that matching algorithms are known in both theories E 1 and E 2. 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, the investigated method is complete for the largest class of theories where unification is not needed, including regular collapse-free theories and linear theories. Syntactic conditions are given to define this class of theories in which solving the combined matching problem is performed in a modular way.

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Section: Resultsmentioning

confidence: 99%

“…However, it is possible to decide if a given B-extended matching problem 1 is matching combinable. B-unification with constant restriction is unitary (Ringeissen, 1992) and it is sufficient to check, according to Lemma 2, that the most general unifier of 1! w.r.t.…”

Section: Matching Combinable Theoriesmentioning

confidence: 99%

Combining Decision Algorithms for Matching in the Union of Disjoint Equational Theories

Ringeissen

1996

Information and Computation

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“…In this paper which extends Ringeissen (1992), Kirchner & Ringeissen (1992), we generalize the uni cation technique to constraint solvers on algebraic domains possibly sharing some constant symbols. We provide for such combination the construction of a combined constraint language and a standard interpretation in which solving a pure constraint is equivalent to solving it in one component of the combination.…”

Section: Introductionmentioning

confidence: 95%

Combining Symbolic Constraint Solvers on Algebraic Domains

Kirchner

Ringeissen

1994

Journal of Symbolic Computation

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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. As an example, a constraint solver in nite algebras is presented for a constraint language including for instance equations, disequations and inequations. By extending techniques used for the combination of uni cation in disjoint equational theories, we show how to combine constraint solvers on di erent algebraic domains that may share some constant symbols. We illustrate this technique by combining the constraint solver in nite algebras with other uni cation algorithms, and with another constraint solver on a di erent nite algebra.

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“…Therefore, previous work has focused on identifying specific conditions and methods in which the problem is decidable. We continue the investigation in this paper, building on previous combination results, [2,4,7] and [6]. We are able to develop a novel approach to the non-disjoint combination problem.…”

Section: Introductionmentioning

confidence: 79%

“…As a final remark, it is important to note that our algorithm C does not work in a symmetric way with respect to E 1 and E 2 , contrary to combination algorithms where we assume that any (Σ 1 ∩ Σ 2 )-equality is valid in E 1 if and only if it is valid in E 2 [2,4,7]. Indeed, the algorithm A 1 partially solves the problem modulo E 1 ∪ E 2 (and not only modulo E 1 ).…”

Section: Satisfying Restrictionmentioning

confidence: 99%

Hierarchical Combination of Unication Algorithms (Extended Abstract)

Erbatur¹,

Kapur²,

Marshall³

et al.

EPiC Series in Computing

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A critical question in unification theory is how to obtaina unification algorithm for the combination of non-disjointequational theories when there exists unification algorithmsfor the constituent theories. The problem is known to bedifficult and can easily be seen to be undecidable in thegeneral case. Therefore, previous work has focused onidentifying specific conditions and methods in which theproblem is decidable.We continue the investigation in this paper, building onprevious combination results and our own work.We are able to develop a novel approach to the non-disjointcombination problem. The approach is based on a new set ofrestrictions and combination method such that if the restrictionsare satisfied the method produces an algorithm for the unificationproblem in the union of non-disjoint equational theories.

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Unification in a combination of equational theories with shared constants and its application to primal algebras

Cited by 15 publications

References 6 publications

Combining Decision Algorithms for Matching in the Union of Disjoint Equational Theories

Combining Decision Algorithms for Matching in the Union of Disjoint Equational Theories

Combining Symbolic Constraint Solvers on Algebraic Domains

Hierarchical Combination of Unication Algorithms (Extended Abstract)

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