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Confluence of Curried TermRewriting Systems
 Journal of Symbolic Computation
, 1995
"... Reduction Systems Definition 2.2. An Abstract Reduction System (short: ARS) consists of a set A and a sequence ! i of binary relations on A, labelled by some set I. We often drop the label if I is a singleton. We write A j= P if the ARS A = (A; ! i ; : : : ); i 2 I has the property P . Further we ..."
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Reduction Systems Definition 2.2. An Abstract Reduction System (short: ARS) consists of a set A and a sequence ! i of binary relations on A, labelled by some set I. We often drop the label if I is a singleton. We write A j= P if the ARS A = (A; ! i ; : : : ); i 2 I has the property P . Further we write A j= P Q iff A j= P and A j= Q. An ARS A = (A; !) has the diamond property , A j= \Sigma, iff /;! ` !;/. It has the ChurchRosser property (is confluent), A j= CR, iff (A; !!) j= \Sigma. Given an ARS A = (A; !), we write CR(t) as shorthand for (fu j t !! ug; !) j= CR. 4 Stefan Kahrs Under most circumstances, confluence is a useful property of ARSs, mainly because: if (A; !) j= CR, and if two elements x; y 2 A are equivalent w.r.t. the smallest equivalence containing !, then there is a z 2 A such that x!! z //y. Roughly: the ARS decides the equivalence. An ARS A = (A; ! a ; ! b ) commutes directly , A j= CD, iff / a ; ! b ` ! b ; / a . To prove confluence of an ARS, it is sometimes...
On the Modularity of Confluence of ConstructorSharing Term Rewriting Systems
 In Proceedings of the 19th colloquium on Trees in Algebra and Programming, LNCS 787
, 1994
"... . Toyama's Theorem states that confluence is a modular property of disjoint term rewriting systems. This theorem does not generalize to combined systems with shared constructors. Thus the question arises naturally whether there are sufficient conditions which ensure the modularity of confluence in t ..."
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. Toyama's Theorem states that confluence is a modular property of disjoint term rewriting systems. This theorem does not generalize to combined systems with shared constructors. Thus the question arises naturally whether there are sufficient conditions which ensure the modularity of confluence in the presence of shared constructors. In particular, Kurihara and Krishna Rao posed the problem whether there are interesting sufficient conditions independent of termination. This question appeared as Problem 59 in the list of open problems in the theory of rewriting published recently [DJK93]. The present paper gives an affirmative answer to that question. Among other sufficient criteria, it is shown that confluence is preserved under the combination of constructorsharing systems if the systems are also normalizing. This in conjunction with the fact that normalization is modular for those systems implies the modularity of semicompleteness. 1 Introduction It is wellknown from software engi...
Compositional Term Rewriting: An Algebraic Proof of Toyama's Theorem
 Rewriting Techniques and Applications, 7th International Conference, number 1103 in Lecture Notes in Computer Science
, 1996
"... This article proposes a compositional semantics for term rewriting systems, i.e. a semantics preserving structuring operations such as the disjoint union. The semantics is based on the categorical construct of a monad, adapting the treatment of universal algebra in category theory to term rewriting ..."
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This article proposes a compositional semantics for term rewriting systems, i.e. a semantics preserving structuring operations such as the disjoint union. The semantics is based on the categorical construct of a monad, adapting the treatment of universal algebra in category theory to term rewriting systems. As an example, the preservation of confluence under the disjoint union of two term rewriting systems is shown, obtaining an algebraic proof of Toyama's theorem, generalised slightly to term rewriting systems introducing variables on the righthand side of the rules.
Modularity in Manysorted Term Rewriting Systems
, 1993
"... Manysorted term rewriting systems (MTRS) are an extension of the formalism of term rewriting systems (TRS). The direct sum of TRS's is generalized to the direct sum of MTRS's. Some equivalence between "taking the direct sum" and "eliminating the sorts" is established: Component closed reduction pro ..."
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Manysorted term rewriting systems (MTRS) are an extension of the formalism of term rewriting systems (TRS). The direct sum of TRS's is generalized to the direct sum of MTRS's. Some equivalence between "taking the direct sum" and "eliminating the sorts" is established: Component closed reduction properties which are resistant against disjoint union (modularity), are also resistant against sort elimination (persistence). The reverse has been proved earlier. 1 Introduction 1.1 Related Work Term rewriting systems (TRS's) have been developed in the forties as a model for computation. They were used in those days for the study on the calculus (Church). Another issue formed the study on completion procedures on equational theories, started by Knuth and Bendix. In this setting, the main goal of term rewriting systems is to give an implementation of a theory of (undirected) equations. The completion procedure delivers a TRS, in which the rules have an unique direction. The applications o...
Abstract Modularity
, 2005
"... Modular rewriting seeks criteria under which rewrite systems inherit properties from their smaller subsystems. This divide and conquer methodology is particularly useful for reasoning about large systems where other techniques fail to scale adequately. Research has typically focused on reasoning a ..."
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Modular rewriting seeks criteria under which rewrite systems inherit properties from their smaller subsystems. This divide and conquer methodology is particularly useful for reasoning about large systems where other techniques fail to scale adequately. Research has typically focused on reasoning about the modularity of specific properties for specific ways of combining specific forms of rewriting. This paper is, we believe, the first to ask a much more general question. Namely, what can be said about modularity independently of the specific form of rewriting, combination and property at hand. A priori there is no reason to believe that anything can actually be said about modularity without reference to the specifics of the particular systems etc. However, this paper shows that, quite surprisingly, much can indeed be said.