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DynamicallyTyped Computations for OrderSorted Equational Presentations (Extended Abstract)
 Proc. 21st International Colloquium on Automata, Languages, and Programming, volume 820 of Lecture Notes in Computer Science
, 1994
"... Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in ordersorted algebras, with very few restrictions on the allowed presentations. We adopt an algebraic framework w ..."
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Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in ordersorted algebras, with very few restrictions on the allowed presentations. We adopt an algebraic framework where equational, membership and existence formulas can be expressed. A complete deduction calculus is provided to incorporate the interaction between all these formulas. The notion of decorated terms is proposed to memorize local sort information, dynamically changed by a rewriting process. A completion procedure for equational presentations with ordered sorts computes a set of rewrite rules with which not only equational theorems of the form (t = t 0 ), but also typing theorems of the for...
How to Transform Canonical Decreasing CTRSs into Equivalent Canonical TRSs
 In Proceedings of the 4th International Workshop on Conditional Term Rewriting Systems
, 1994
"... We prove constructively that the class of groundconfluent and decreasing conditional term rewriting systems (CTRSs) (without extra variables) coincides with the class of orthogonal and terminating, unconditional term rewriting systems (TRSs). TRSs being included in CTRSs, this result follows from a ..."
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Cited by 9 (0 self)
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We prove constructively that the class of groundconfluent and decreasing conditional term rewriting systems (CTRSs) (without extra variables) coincides with the class of orthogonal and terminating, unconditional term rewriting systems (TRSs). TRSs being included in CTRSs, this result follows from a transformation from any groundconfluent and decreasing CTRS specifying a computable function f into a TRS with the mentioned properties for f . The generated TRS is ordersorted, but we outline a similar transformation yielding an unsorted TRS.
Transforming Conditional Rewrite Systems with Extra Variables into Unconditional Systems
 In: Proc. LPAR '99, Tblisi
, 1999
"... . Deterministic conditional rewrite systems are interesting because they permit extra variables on the righthand sides of the rules. If such a system is quasireductive, then it is terminating and has a computable rewrite relation. It will be shown that every deterministic CTRS R can be transformed ..."
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Cited by 8 (2 self)
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. Deterministic conditional rewrite systems are interesting because they permit extra variables on the righthand sides of the rules. If such a system is quasireductive, then it is terminating and has a computable rewrite relation. It will be shown that every deterministic CTRS R can be transformed into an unconditional TRS U(R) such that termination of U(R) implies quasireductivity of R. The main theorem states that quasireductivity of R implies innermost termination of U(R). These results have interesting applications in two different areas: modularity in term rewriting and termination proofs of wellmoded logic programs. 1 Introduction Conditional term rewriting systems (CTRSs) are the basis of functional logic programming; see [Han94] for an overview of this field. In CTRSs variables on the righthand side of a rewrite rule which do not occur on the lefthand side are often forbidden. This is because it is in general not clear how to instantiate them. On the other hand, a rest...
Termination Checker and KnuthBendix Completion Tools for Maude Equational Specifications
, 2000
"... This document explains the design and use of a termination checker tool and of a KnuthBendix completion tool. The termination checker tool checks whether an equational specication terminates, and the KnuthBendix completion tool tries to complete an equational speci cation. These tools can be used ..."
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This document explains the design and use of a termination checker tool and of a KnuthBendix completion tool. The termination checker tool checks whether an equational specication terminates, and the KnuthBendix completion tool tries to complete an equational speci cation. These tools can be used to prove the termination or to complete ordersorted equational specications in Maude [7, 6, 4]. The tools have been written entirely in Maude and are in fact executable specications in rewriting logic [17] of the formal inference system that they implement. The fact that rewriting logic is reective [8, 3], and that Maude eciently supports reective rewriting logic computations [5, 4] is systematically exploited in the design of the tools. Contents 1
On Deterministic Conditional Rewriting
 MIT LABORATORY FOR COMPUTER SCIENCE
, 1997
"... The class of Deterministic Conditional Term Rewriting Systems (DCTRSs) is of utmost importance for the tight relationships exhibited with functional programming, logic programming and inductive reasoning. However, its analysis is extremely difficult, and to date there are only very few works on the ..."
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The class of Deterministic Conditional Term Rewriting Systems (DCTRSs) is of utmost importance for the tight relationships exhibited with functional programming, logic programming and inductive reasoning. However, its analysis is extremely difficult, and to date there are only very few works on the subject, each analyzing a particular aspect of DCTRSs. In this paper, we perform a thorough analysis of DCTRSs, ranging from the study of termination criteria, to new verification methods for the major properties of DCTRSs like termination and confluence, to the identification of subclasses of DTCRSs that exhibit a particularly nice behaviour. Moreover, we also address the study of modularity of DCTRSs, providing a number of new powerful results. This is particularly important, since to the best of our knowledge there is so far not a single result on the modularity of DCTRSs, and of 3CTRSs in general. Finally, most of the analysis of the paper is performed relying on the recent tool of unra...
New Completeness Results For Lazy Conditional Narrowing
 In Proceedings of 6h International Workshop on Unification (UNIF 2002
, 2002
"... In this paper we consider the lazy conditional narrowing calculus LCNC of [4]. LCNC is the extension of lnc to conditional term rewrite systems (CTRSs for short). The extension is motivated by the observation that CTRSs are much more expressive than unconditional TRSs for describing interesting prob ..."
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In this paper we consider the lazy conditional narrowing calculus LCNC of [4]. LCNC is the extension of lnc to conditional term rewrite systems (CTRSs for short). The extension is motivated by the observation that CTRSs are much more expressive than unconditional TRSs for describing interesting problems in natural and concise way. However, the additional expressive power raises two problems: (1) confluence is insufficient to guarantee the completeness with respect to normalized solutions, and (2) the search space increases dramatically because the conditions of the applied rewrite rule are added to the current goal. In [4] several completeness results for lcnc are presented. The only result which does not assume some kind of termination assumption does not permit extra variables in the conditions and righthand sides of the rewrite rules. In this paper we show the completeness of LCNC with leftmost selection for the class of (confluent) deterministic oriented CTRSs. Determinism was introduced by Ganzinger [1] and has proved to be very useful for the study of the (unique) termination behavior of wellmoded Horn clause programs (cf. [5]).
OrderSorted Termination: the Unsorted Way
 In Proceedings from NIK'95: Norwegian Conference on Informatics, Gran (Hadeland
, 1996
"... We consider the problem of proving termination of ordersorted rewrite systems. The dominating method for proving termination of ordersorted systems has been to simply ignore sort information, and use the techniques developed for unsorted rewriting. The problem with this approach is that many order ..."
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We consider the problem of proving termination of ordersorted rewrite systems. The dominating method for proving termination of ordersorted systems has been to simply ignore sort information, and use the techniques developed for unsorted rewriting. The problem with this approach is that many ordersorted rewrite systems terminate because of the structure of the set of sorts. In these cases the corresponding unsorted system would not terminate. In this paper we approach the problem of ordersorted termination by mapping the ordersorted rewrite system into an unsorted one suchthat termination of the latter implies termination of the former. By encoding sort information into the unsorted mapping, we are able to use general purpose termination orderings to prove termination of ordersorted rewrite systems whose termination depend on the sort hierarchy. We present a sequence of gradually stronger methods, and show that a previously published method is contained in ours as a special case. 1
Cancellative Abelian Monoids in Refutational Theorem Proving
 PHD THESIS, INSTITUT FÜR INFORMATIK, UNIVERSITÄT DES SAARLANDES
, 1997
"... ..."
On QuasiReductive and QuasiSimplifying Deterministic Conditional Rewrite Systems
 Proceedings of the 4th International Symposium on Functional and Logic Programming ', Vol. 1722 of
"... . Deterministic conditional rewrite systems permit extra variables on the righthand sides of the rules. If such a system is quasireductive or quasisimplifying, then it is terminating and has a computable rewrite relation. This paper provides new criteria for showing quasireductivity and quasisim ..."
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. Deterministic conditional rewrite systems permit extra variables on the righthand sides of the rules. If such a system is quasireductive or quasisimplifying, then it is terminating and has a computable rewrite relation. This paper provides new criteria for showing quasireductivity and quasisimplifyingness. In this context, another criterion from [ALS94] will be rectified and a claim in [Mar96] will be refuted. Moreover, we will investigate under which conditions the properties exhibit a modular behavior. 1 Introduction Conditional term rewriting systems (CTRSs) are the basis for the integration of the functional and logic programming paradigms; see [Han94] for an overview of this field. In these systems variables on the righthand side of a rewrite rule which do not occur on the lefthand side are problematic because it is in general not clear how to instantiate them. On the other hand, a restricted use of these extra variables enables a more natural and efficient way of writing...
Polymorphically ordersorted types in OBJ3
, 1997
"... . OBJ3 [GWM + 93] is a functional programming language with firstorder function types. OBJ3 has two special features: overloading of function symbols and the possibility to order the sorts. This ordering is induced by set inclusion on the carrier sets. We call the feature to be able to order ..."
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. OBJ3 [GWM + 93] is a functional programming language with firstorder function types. OBJ3 has two special features: overloading of function symbols and the possibility to order the sorts. This ordering is induced by set inclusion on the carrier sets. We call the feature to be able to order the sorts inclusion set subtyping. The algebraic semantics of OBJ3 is based on the theory of ordersorted algebras [GM89]. Furthermore, OBJ3 allows parameterized programming [Gog90]. However, the concepts of higherorder functions and parametric polymorphism are only emulated by parameters of OBJ3 modules. In this paper we show how to extend OBJ3 by parametric polymorphism in an elegant way. We call this extended language OBJP. In the second part of the paper we describe the operational semantics of OBJP. The operational semantics is a translation of OBJP programs into programs without overloading and subtypes. Here, we improve the approaches of Goguen, Jouannaud, and Mesegu...