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Equations and rewrite rules: a survey
 In Formal Language Theory: Perspectives and Open Problems
, 1980
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A 2Categorical Presentation of Term Graph Rewriting
 CATEGORY THEORY AND COMPUTER SCIENCE, VOLUME 1290 OF LNCS
, 1997
"... It is wellknown that a term rewriting system can be faithfully described by a cartesian 2category, where horizontal arrows represent terms, and cells represent rewriting sequences. In this paper we propose a similar, original 2categorical presentation for term graph rewriting. Building on a re ..."
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Cited by 34 (17 self)
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It is wellknown that a term rewriting system can be faithfully described by a cartesian 2category, where horizontal arrows represent terms, and cells represent rewriting sequences. In this paper we propose a similar, original 2categorical presentation for term graph rewriting. Building on a result presented in [8], which shows that term graphs over a given signature are in onetoone correspondence with arrows of a gsmonoidal category freely generated from the signature, we associate with a term graph rewriting system a gsmonoidal 2category, and show that cells faithfully represent its rewriting sequences. We exploit the categorical framework to relate term graph rewriting and term rewriting, since gsmonoidal (2)categories can be regarded as "weak" cartesian (2)categories, where certain (2)naturality axioms have been dropped.
Lazy rewriting on eager machinery
 ACM Transactions on Programming Languages and Systems
, 2000
"... The article introduces a novel notion of lazy rewriting. By annotating argument positions as lazy, redundant rewrite steps are avoided, and the termination behaviour of a term rewriting system can be improved. Some transformations of rewrite rules enable an implementation using the same primitives a ..."
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Cited by 23 (1 self)
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The article introduces a novel notion of lazy rewriting. By annotating argument positions as lazy, redundant rewrite steps are avoided, and the termination behaviour of a term rewriting system can be improved. Some transformations of rewrite rules enable an implementation using the same primitives as an implementation of eager rewriting. 1
(Cyclic) Term Graph Rewriting is adequate for Rational Parallel Term Rewriting
 CGH
, 1997
"... Acyclic Term Graphs are able to represent terms with sharing, and the relationship between Term Graph Rewriting (TGR) and Term Rewrtiting (TR) is now well understood [BvEG + 87, HP91]. During the last years, some researchers considered the extension of TGR to possibly cyclic term graphs, which ..."
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Cited by 20 (6 self)
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Acyclic Term Graphs are able to represent terms with sharing, and the relationship between Term Graph Rewriting (TGR) and Term Rewrtiting (TR) is now well understood [BvEG + 87, HP91]. During the last years, some researchers considered the extension of TGR to possibly cyclic term graphs, which can represent possibly infinite, rational terms. In [KKSdV94] the authors formalize the classical relationship between TGR and TR as an "adequate mapping" between rewriting systems, and extend it by proving that unraveling is an adequate mapping from cyclic TGR to rational, infinitary term rewriting: In fact, a single graph reduction may correspond to an infinite sequence of term reductions. Using the same notions, we propose a different adequacy result, showing that unraveling is an adequate mapping from cyclic TGR to rational parallel term rewriting, where at each reduction infinitely many rules can be applied in parallel. We also argue that our adequacy result is more natural...
Single Assignment C  Entwurf und Implementierung einer CVariante mit spezieller Unterstützung shapeinvarianter ArrayOperationen
, 1996
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Rewriting On Cyclic Structures: Equivalence Between The Operational And The Categorical Description
, 1999
"... . We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2theories. We show that this presentation is equivalent to the wellaccepted operational definition proposed by Barendregt et aliibut for the case of circular redexes, fo ..."
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Cited by 12 (6 self)
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. We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2theories. We show that this presentation is equivalent to the wellaccepted operational definition proposed by Barendregt et aliibut for the case of circular redexes, for which we propose (and justify formally) a different treatment. The categorical framework allows us to model in a concise way also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures, and to relate term graph rewriting to other rewriting formalisms. R'esum'e. Nous pr'esentons une formulation cat'egorique de la r'e'ecriture des graphes cycliques des termes, bas'ee sur une variante de 2theorie alg'ebrique. Nous prouvons que cette pr'esentation est 'equivalente `a la d'efinition op'erationnelle propos'ee par Barendregt et d'autres auteurs, mais pas dons le cas des radicaux circulaires, pour lesquels nous proposons (et justifions formellem...
Bisimilarity in Term Graph Rewriting
 Information and Computation
, 1998
"... We present a survey of confluence properties of (acyclic) term graph rewriting. Results and counterexamples are given for different kinds of term graph rewriting: besides plain applications of rewrite rules, extensions with the operations of collapsing and copying, and with both operations togethe ..."
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Cited by 2 (1 self)
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We present a survey of confluence properties of (acyclic) term graph rewriting. Results and counterexamples are given for different kinds of term graph rewriting: besides plain applications of rewrite rules, extensions with the operations of collapsing and copying, and with both operations together are considered. Collapsing and copying together constitute bisimilarity of term graphs. We establish sufficient conditions forand counterexamples toconfluence, confluence modulo bisimilarity and the ChurchRosser property modulo bisimilarity. Moreover, we address rewriting modulo bisimilarity, that is, rewriting of bisimilarity classes of term graphs. 1991 Computing Reviews Classification System: F.1.1, F.4.1, F.4.2 Keywords and Phrases: term graph rewriting, bisimilarity, confluence, ChurchRosser property Note: Work carried out under project SEN2.2, Data Manipulation. Part of the research of the third author was performed while he was on leave at CWI by a grant of the HCM ne...
Theory and Applications
"... Infinitary rewriting generalises usual finitary rewriting by providing infinite reduction sequences with a notion of convergence. The idea of – at least conceptually – assigning a meaning to infinite derivations is wellknown, for example, from lazy functional programming or from process calculi. In ..."
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Infinitary rewriting generalises usual finitary rewriting by providing infinite reduction sequences with a notion of convergence. The idea of – at least conceptually – assigning a meaning to infinite derivations is wellknown, for example, from lazy functional programming or from process calculi. Infinitary rewriting makes it possible to apply rewriting in order to obtain a formal model for such infinite derivations. The goal of this thesis is to comprehensively survey the field of infinitary term rewriting, to point out its shortcomings, and to try to overcome some of these shortcomings. The most significant problem that arises in infinitary rewriting is the inherent difficulty to finitely represent and, hence, to implement it. To this end, we consider term graph rewriting, which is able to finitely represent restricted forms of infinitary term rewriting. Moreover, we study different models that are used to formalise infinite reduction sequences: The wellestablished metric approach as well as an alternative approach using partial orders. Both methods together with the consequent infinitary versions of confluence and termination properties are analysed on an abstract level. Based on this, we argue that the partial order model has more advantageous properties and represents