## Compositional Term Rewriting: An Algebraic Proof of Toyama's Theorem (1996)

Venue: | Rewriting Techniques and Applications, 7th International Conference, number 1103 in Lecture Notes in Computer Science |

Citations: | 9 - 2 self |

### BibTeX

@INPROCEEDINGS{Lüth96compositionalterm,

author = {Christoph Lüth},

title = {Compositional Term Rewriting: An Algebraic Proof of Toyama's Theorem},

booktitle = {Rewriting Techniques and Applications, 7th International Conference, number 1103 in Lecture Notes in Computer Science},

year = {1996},

pages = {261--275},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

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 right-hand side of the rules.

### Citations

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Citation Context ...e former is trivial, the latter requires another easy induction). Then the monad laws follow from Def. 1, and T \Theta is a Pre-enriched monad. 2.4 Compositionality As observed by Goguen and Burstall =-=[GB92]-=-, most structuring operations for algebraic specifications are colimits, either in the category of syntactic presentations (here, term rewriting systems), or in the category of semantic representation... |

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Citation Context ...s, no knowledge of them is either assumed or even necessary for a basic understanding of what follows; 2 a gentler introduction into enriched category theory than the somewhat demanding standard text =-=[Kel82]-=- is [Bor94, Chapter 6]. This article is an extract of the author's forthcoming thesis [Lu96]. Without referring to it explicitly in the following, the thesis will present the material of the present a... |

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Citation Context ...oup while the author was affiliated with Edinburgh University. 1.1 Preliminaries We assume a working knowledge of term rewriting systems, and category theory as gained from the first five chapters of =-=[Mac71]-=- (the notation and terminology of which will be used here, and to which we will often refer). Although this work involves enriched categories, no knowledge of them is either assumed or even necessary ... |

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Citation Context ... here. The morphisms of the latter are natural transformations respecting the unit and multiplication (monad morphisms, see [BW85, Section 3.6]). It is actually a Cat-enriched category (or 2-category =-=[KS74]-=-), so the colimit has to have an additional colimiting property on 2-cells: given two pairs of monad morphisms ff; ff 0 : T1 ! S, fi; fi 0 : T1 ! S and two modifications (see [KS74]) fl : ff ! ff 0 ; ... |

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Citation Context ...ding and their units are monomorphisms. Call such monads regular, then we have the main theorem: Theorem 15. The coproduct of two confluent regular monads is confluent. The original theorem by Toyama =-=[Toy87]-=- of course does not consider categories (corresponding to named reductions); it corresponds to Theorem 15 for two confluent, regular monads on Pre. It is obtained as a corollary from that theorem by o... |

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Citation Context ...substituting a term into a variable, and a variable for itself, yields the identity (in an informal notation, x[t=x] = t and t[x=x] = t). It turns out this is all one needs for universal algebra (see =-=[Man76]-=-). By adding a reduction structure, we will extend this to term rewriting systems, but let us first see how to treat signatures. 2 At some points, footnotes will point out technical details and side i... |

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Citation Context ...re under context and substitution (given by the monad) and the choice of the mathematical structure used to model the reduction (preorders or categories; other possibilities include Sesqui-categories =-=[Ste94]-=-, or graphs or binary relations to model the one-step reduction). It can handle some forms of non-standard rewriting as well. For example, combining the treatment of equational presentations and term ... |

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2 |
Compositional Categorical Term Rewriting in Structured Algebraic Specifications
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Citation Context ... follows; 2 a gentler introduction into enriched category theory than the somewhat demanding standard text [Kel82] is [Bor94, Chapter 6]. This article is an extract of the author's forthcoming thesis =-=[Lu96]-=-. Without referring to it explicitly in the following, the thesis will present the material of the present article, some of which can only be adumbrated due to length limitations, in far more detail. ... |