## (Cyclic) Term Graph Rewriting is adequate for Rational Parallel Term Rewriting (1997)

Venue: | CGH |

Citations: | 20 - 6 self |

### BibTeX

@TECHREPORT{Corradini97(cyclic)term,

author = {Andrea Corradini and Frank Drewes},

title = {(Cyclic) Term Graph Rewriting is adequate for Rational Parallel Term Rewriting},

institution = {CGH},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

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...

### Citations

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(Show Context)
Citation Context ... general definitions to the category of term graphs defined in Section 3). The basic categorical construction in the algebraic definition of graph rewriting is that of pushout. Definition 12 (pushout =-=[ML71]-=- and pushout complement [Ehr87]). Given a category C and two arrows b : K ! B, d : K ! D of C, a triple hH; h : B ! H; c : D ! Hi as in Figure 3 (a) is called a pushout of hb; di if [Commutativity] h ... |

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(Show Context)
Citation Context ...lent in the other approach, can be reduced to the same term, ?, by a single parallel reduction. The paper is organized as follows. In Section 2 we summarize the basic definitions about infinite terms =-=[GTWW77]-=-, orthogonal term rewriting [HL91], and parallel term rewriting [Cor93]. In Section 3 we introduce (possibly cyclic) term graphs, and we make precise their relationship with (sets of) rational terms, ... |

84 |
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(Show Context)
Citation Context ...educed to the same term, ?, by a single parallel reduction. The paper is organized as follows. In Section 2 we summarize the basic definitions about infinite terms [GTWW77], orthogonal term rewriting =-=[HL91]-=-, and parallel term rewriting [Cor93]. In Section 3 we introduce (possibly cyclic) term graphs, and we make precise their relationship with (sets of) rational terms, via the unraveling function. Algeb... |

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(Show Context)
Citation Context ...-phase construction, which only makes sense for graphs that are intended to represent terms. On the contrary, the algebraic approach used in [HP91, CR93] uses the so-called "double-pushout approa=-=ch" ([Ehr87]-=-), which is a general graph rewriting formalism, and applies it to a category of term graphs. 3 During the last years, some authors considered the extension of term graph rewriting to the cyclic case,... |

38 |
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(Show Context)
Citation Context ...he notion of convergence is concerned, the wellknown topological structure of (possibly infinite) terms is used, which, equipped with a suitable notion of distance, form a complete ultra-metric space =-=[AN80]-=-. However, it is worth stressing that this adequacy result treats circular redexes in an ad hoc way, because the term reduction sequence corresponding to the reduction of a circular redex is not stron... |

19 |
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(Show Context)
Citation Context ...: K ! D and r always exists, and therefore in the above hypotheses there exists a direct derivation G ) p;g H. ut The proof of the last proposition is reported (for the equivalent category Jungle) in =-=[CR93]-=-, where also general conditions for the existence of pushouts are presented. 8 It is worth stressing that in the hypotheses of the last proposition, the pushout complement object D has the same nodes ... |

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(Show Context)
Citation Context ...mber of redexes in f ! : in a single step all the occurrences of f in f ! are replaced by g. 3 A variation of the algebraic approach making use of a single pushout construction is used for example in =-=[Ken91]-=-. Other differences among the various proposal for TGR concern the way terms are represented as graphs, but these are a mere syntactical issue (see [CMR + 91]). 2 f ffl @ABCED oo R // gffl @ABCED oo H... |

9 |
Term rewriting in CT \Sigma
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(Show Context)
Citation Context ...y of transfinite term rewriting is no more helpful, because it cannot justify this result. We need instead the notion of "infinite parallel rewriting", that has been introduced by the first =-=author in [Cor93]-=-. Such an extension of term rewriting has been defined by exploiting the complete partial ordered structure of CT \Sigma , the collection of possibly infinite, possibly partial terms over signature \S... |

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6 |
Is parallelism already concurrency? Part 1: Derivations in graph grammars
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- 1987
(Show Context)
Citation Context ...tegory of graphs, and it can be applied to term graphs as a specific instance. Thus general theorems proposed in the literature and concerning parallel reductions, equivalences of reduction sequences =-=[Kre87]-=-, and truly concurrent semantics in terms of event structures [CEL + 96] can be applied to TGR as well. Let Graph be a fixed category of graphs (below we will apply the general definitions to the cate... |

4 |
Implementing Term Rewriting by Jungle Evaluation. Informatique th'eorique et
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- 1991
(Show Context)
Citation Context ...ral many non-isomorphic graphs satisfying this last properties (i.e., graphs G such that U [G] = T ): The representation proposed above is the most compact one, 10 and it is called fully-collapsed in =-=[HP91]-=-. It is worth stressing that the results in Section 5 are independent of the chosen representation. The above definitions made clear the relationship between objects of category TGraph \Sigma and term... |

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2 |
Vries. On the Adequacy of Graph Rrewriting for Simulating Term Rewriting
- Kennaway, Klop, et al.
- 1994
(Show Context)
Citation Context ...ow 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 autho-=-rs 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... |