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49
Specification and Proof in Membership Equational Logic
 Theoretical Computer Science
, 1996
"... : This paper is part of a longterm effort to increase expressiveness of algebraic specification languages while at the same time having a simple semantic basis on which efficient execution by rewriting and powerful theoremproving tools can be based. In particular, our rewriting techniques provide ..."
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Cited by 104 (49 self)
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: This paper is part of a longterm effort to increase expressiveness of algebraic specification languages while at the same time having a simple semantic basis on which efficient execution by rewriting and powerful theoremproving tools can be based. In particular, our rewriting techniques provide semantic foundations for Maude's functional sublanguage, where they have been efficiently implemented. Membership equational logic is quite simple, and yet quite powerful. Its atomic formulae are equations and sort membership assertions, and its sentences are Horn clauses. It extends in a conservative way both ordersorted equational logic and partial algebra approaches, while Horn logic can be very easily encoded. After introducing the basic concepts of the logic, we give conditions and proof rules with which efficient equational deduction by rewriting can be achieved. We also give completion techniques to transform a specification into one meeting these conditions. We address the important ...
On the Undecidability of SecondOrder Unification
 INFORMATION AND COMPUTATION
, 2000
"... ... this paper, and it is the starting point for proving some novel results about the undecidability of secondorder unification presented in the rest of the paper. We prove that secondorder unification is undecidable in the following three cases: (1) each secondorder variable occurs at most t ..."
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Cited by 33 (16 self)
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... this paper, and it is the starting point for proving some novel results about the undecidability of secondorder unification presented in the rest of the paper. We prove that secondorder unification is undecidable in the following three cases: (1) each secondorder variable occurs at most twice and there are only two secondorder variables; (2) there is only one secondorder variable and it is unary; (3) the following conditions (i)#(iv) hold for some fixed integer n: (i) the arguments of all secondorder variables are ground terms of size <n, (ii) the arity of all secondorder variables is <n, (iii) the number of occurrences of secondorder variables is #5, (iv) there is either a single secondorder variable or there are two secondorder variables and no firstorder variables.
Birewrite systems
, 1996
"... In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations ..."
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Cited by 29 (9 self)
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In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations −−− → R ⊆ and −−− → R ⊇ , and seek a common term c such that a −−−→ R ⊆ c and b −−−→
A Uniform Approach to Underspecification and Parallelism
 In Proceedings ACL'97
, 1997
"... We propose a unified flamework in which to treat semantic underspecification and parallelism phenomena in discourse. The framework employs a constraint language that can express equality and subtree relations between finite trees. In addition, our constraint language can express the equality upto r ..."
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Cited by 26 (9 self)
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We propose a unified flamework in which to treat semantic underspecification and parallelism phenomena in discourse. The framework employs a constraint language that can express equality and subtree relations between finite trees. In addition, our constraint language can express the equality upto relation over trees which captures parallelism between them. The constraints are solved by context unification. We demonstrate the use of our framework at the examples of quantifier scope, ellipsis, and their interaction. 1 I
Solvability of context equations with two context variables is decidable
 THE JOURNAL OF SYMBOLIC COMPUTATION
, 1999
"... Context unification is a natural variant of second order unification that represents a generalization of word unification at the same time. While second order unification is wellknown to be undecidable and word unification is decidable it is currently open if solvability of context equations is deci ..."
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Cited by 25 (2 self)
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Context unification is a natural variant of second order unification that represents a generalization of word unification at the same time. While second order unification is wellknown to be undecidable and word unification is decidable it is currently open if solvability of context equations is decidable. We show that solvability of systems of context equations with two context variables is decidable. The context variables may have an arbitrary number of occurrences, and the equations may contain an arbitrary number of individual variables as well. The result holds under the assumption that the first order background signature is finite.
On Unification of Terms with Integer Exponents
 Mathematical Systems Theory
, 1995
"... this paper, we got informed of the work of G. Salzer, which has been done independently from ours [6]. He uses a quite different formalism, but he shows essentially the same results as ours, except that his formalism is slightly more powerful since he can express for example the set of all complete ..."
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Cited by 24 (2 self)
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this paper, we got informed of the work of G. Salzer, which has been done independently from ours [6]. He uses a quite different formalism, but he shows essentially the same results as ours, except that his formalism is slightly more powerful since he can express for example the set of all complete finite binary trees with internal nodes labeled with f . Indeed, his syntax allows for multilple holes in the terms, in which case, the semantics is different from what we considered in section 7. In order to express it shortly, you can assume that the holes
Unification of Infinite Sets of Terms Schematized by Primal Grammars
 Theoretical Computer Science
, 1996
"... Infinite sets of terms appear frequently at different places in computer science. On the other hand, several practically oriented parts of logic and computer science require the manipulated objects to be finite or finitely representable. Schematizations present a suitable formalism to manipulate fin ..."
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Cited by 23 (3 self)
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Infinite sets of terms appear frequently at different places in computer science. On the other hand, several practically oriented parts of logic and computer science require the manipulated objects to be finite or finitely representable. Schematizations present a suitable formalism to manipulate finitely infinite sets of terms. Since schematizations provide a different approach to solve the same kind of problems as constraints do, they can be viewed as a new type of constraints. The paper presents a new recurrent schematization called primal grammars. The main idea behind the primal grammars is to use primitive recursion as the generating engine of infinite sets. The evaluation of primal grammars is based on substitution and rewriting, hence no particular semantics for them is necessary. This fact allows also a natural integration of primal grammars into Prolog, into functional languages or into other rewritebased applications. Primal grammars have a decidable unification problem and ...
A decision algorithm for stratified context unification
 FACHBEREICH INFORMATIK, J.W. GOETHEUNIVERSITAT
, 1999
"... Context unification is a variant of second order unification and also a generalization of string unification. Currently it is not known whether context unification is decidable. A specialization of context unification is stratified context unification. Recently, it turned out that stratified context ..."
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Cited by 17 (1 self)
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Context unification is a variant of second order unification and also a generalization of string unification. Currently it is not known whether context unification is decidable. A specialization of context unification is stratified context unification. Recently, it turned out that stratified context unification and onestep rewrite constraints are equivalent. This paper contains a description of a decision algorithm SCU for stratified context unification, which shows decidability of stratified context unification as well as of satisfiability of onestep rewrite constraints.
On the Complexity of Linear and Stratified Context Matching Problems
, 2001
"... We investigate the complexity landscape of context matching with respect to the number of occurrences of variables (i.e. linearity vs. varity 2) and various restrictions of stratification. We show that stratified context matching (SCM) and varity 2 context matching are NPcomplete, but that stratifi ..."
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Cited by 15 (1 self)
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We investigate the complexity landscape of context matching with respect to the number of occurrences of variables (i.e. linearity vs. varity 2) and various restrictions of stratification. We show that stratified context matching (SCM) and varity 2 context matching are NPcomplete, but that stratified simultaneous monadic context matching (SSMCM) is in P. SSMCM is equivalent to stratified simultaneous word matching (SSWM). We also show that the linear and the Comonrestricted case are in P and of time complexity O(n³). We give an algorithm for context matching and discuss how the performance of the general case can be improved through the use of information derived from polynomial approximations of the problem.
Decidable and undecidable secondorder unification problems
 In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications (RTA’98), volume 1379 of LNCS
, 1998
"... Abstract. There is a close relationship between word unification and secondorder unification. This similarity has been exploited for instance for proving decidability of monadic secondorder unification. Word unification can be easily decided by transformation rules (similar to the ones applied in ..."
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Cited by 15 (9 self)
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Abstract. There is a close relationship between word unification and secondorder unification. This similarity has been exploited for instance for proving decidability of monadic secondorder unification. Word unification can be easily decided by transformation rules (similar to the ones applied in higherorder unification procedures) when variables are restricted to occur at most twice. Hence a wellknown open question was the decidability of secondorder unification under this same restriction. Here we answer this question negatively by reducing simultaneous rigid Eunification to secondorder unification. This reduction, together with an inverse reduction found by Degtyarev and Voronkov, states an equivalence relationship between both unification problems. Our reduction is in some sense reversible, providing decidability results for cases when simultaneous rigid Eunification is decidable. This happens, for example, for onevariable problems where the variable occurs at most twice (because rigid Eunification is decidable for just one equation). We also prove decidability when no variable occurs more than once, hence significantly narrowing the gap between decidable and undecidable secondorder unification problems with variable occurrence restrictions. 1