## CPO models for infinite term rewriting (1995)

Venue: | in Proc. AMAST'95, LNCS 936 |

Citations: | 10 - 7 self |

### BibTeX

@INPROCEEDINGS{Corradini95cpomodels,

author = {Andrea Corradini and Fabio Gadducci},

title = {CPO models for infinite term rewriting},

booktitle = {in Proc. AMAST'95, LNCS 936},

year = {1995},

pages = {368--384},

publisher = {Springer Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

. Infinite terms in universal algebras are a well-known topic since the seminal work of the ADJ group [1]. The recent interest in the field of term rewriting (tr) for infinite terms is due to the use of term graph rewriting to implement tr, where terms are represented by graphs: so, a cyclic graph is a finitary description of a possibly infinite term. In this paper we introduce infinite rewriting logic, working on the framework of rewriting logic proposed by Jos'e Meseguer [13, 14]. We provide a simple algebraic presentation of infinite computations, recovering the infinite parallel term rewriting, originally presented by one of the authors ([6]) to extend the classical, set-theoretical approach to tr with infinite terms. Moreover, we put all the formalism on firm theoretical bases, providing (for the first time, to the best of our knowledge, for infinitary rewriting systems) a clean algebraic semantics by means of (internal) 2-categories. 1 Introduction Term rewriting sy...

### Citations

779 | Rewrite systems - Dershowitz, Jouannaud - 1990 |

507 |
Conditional rewriting logic as a unified model of concurrency
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- 1992
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Citation Context ...t 0 . ut It is worth noting here that although the parallel rewriting logic of Definition 18 is slightly different from that of Meseguer (in the unconditional case), they are equivalent. Actually, in =-=[14] the -=-sequents appearing in the premises of the congruence and replacement rules are not restricted to be one-step, but they can be arbitrary. As a consequence, unlike in our definition, the operator "... |

418 |
Category theory for computer science
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- 1999
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Citation Context ...rections for future works. 2 Background We present here most of the technical notions we use along the paper. For some basic categorical definitions (as functor, product, etc.) we refer the reader to =-=[3]-=-. 2.1 Algebras and continuous algebras In this section we recall the definitions of the categories of algebras and of continuous algebras, introducing them in two equivalent ways: first in the traditi... |

194 |
Initial algebra semantics and continuous algebras
- Goguen, Thatcher, et al.
- 1977
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Citation Context ...imento di Informatica, Corso Italia 40, 56125 Pisa, Italy (fandrea,gadduccig@di.unipi.it) Abstract. Infinite terms in universal algebras are a well-known topic since the seminal work of the ADJ group =-=[1]-=-. The recent interest in the field of term rewriting (tr) for infinite terms is due to the use of term graph rewriting to implement tr, where terms are represented by graphs: so, a cyclic graph is a f... |

177 |
Functorial semantics of algebraic theories
- Lawvere
- 1963
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Citation Context ...parating in a better way the "\Sigma -structure" from the additional algebraic structure that the carrier can enjoy. We start defining the Lawvere theory Th(\Sigma) associated with a signatu=-=re \Sigma [12]-=-. This is a cartesian category having natural numbers as objects, generated in a free way from the operators of \Sigma . The relevant property (on which we will come back later) is that arrows from m ... |

31 |
Review of the elements of
- Kelly, Street
- 1974
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Citation Context ...ure, obtained via a suitable closure operation, turns out to be a 2-category called the Lawvere 2-theory of R. First of all, let us introduce the notion of 2-category: for further details we refer to =-=[11]-=-. Definition29 (2-categories). A 2-category C consists of a collection fa; b; : : :g of objects, or 0-cells, a collection ff; g; : : :g of arrows, or 1-cells, and a collection fff; fi; : : :g of trans... |

22 | Sesqui-pushout rewriting
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Citation Context ... proposed by Jos'e Meseguer [13, 14]. We provide a simple algebraic presentation of infinite computations, recovering the infinite parallel term rewriting, originally presented by one of the authors (=-=[6]-=-) to extend the classical, set-theoretical approach to tr with infinite terms. Moreover, we put all the formalism on firm theoretical bases, providing (for the first time, to the best of our knowledge... |

21 | Cyclic) term graph rewriting is adequate for rational parallel term rewriting
- Corradini, Drewes
- 1997
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Citation Context ... obtained by applying the redexes in any order. This is easily achieved by allowing only for the parallel application of non-overlapping, left-linear redexes: the definitions below summarize those in =-=[6, 5]-=- (see also [2]). Intuitively, finite parallel rewriting can be defined easily by exploiting the confluence of orthogonal term rewriting. In fact, the parallel reduction of a finite number of redexes i... |

20 |
Foundations of equational deduction: A categorical treatment of equational proofs and unification algorithms
- Rydeheard, Stell
- 1987
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Citation Context ...all the possible computations the system can perform, subject to a suitable equivalence. Alternatively, more abstract models for trs's have been recently proposed, relying on the use of 2-categories (=-=[4, 15, 16, 17]-=-) and functor categories ([13]). A functor category has functors as objects and natural transformations as arrows. A cat-enriched structure as a 2-category is given by a category such that also each h... |

18 | Relating two categorical models of term rewriting
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- 1995
(Show Context)
Citation Context ...all the possible computations the system can perform, subject to a suitable equivalence. Alternatively, more abstract models for trs's have been recently proposed, relying on the use of 2-categories (=-=[4, 15, 16, 17]-=-) and functor categories ([13]). A functor category has functors as objects and natural transformations as arrows. A cat-enriched structure as a 2-category is given by a category such that also each h... |

16 |
An abstract formulation for rewrite systems
- Power
- 1989
(Show Context)
Citation Context ...all the possible computations the system can perform, subject to a suitable equivalence. Alternatively, more abstract models for trs's have been recently proposed, relying on the use of 2-categories (=-=[4, 15, 16, 17]-=-) and functor categories ([13]). A functor category has functors as objects and natural transformations as arrows. A cat-enriched structure as a 2-category is given by a category such that also each h... |

15 |
de Vries. The adequacy of term graph rewriting for simulating term rewriting
- Kennaway, Klop, et al.
- 1993
(Show Context)
Citation Context ...to model rewrites. In fact, in term graph rewriting, a finite, cyclic graph may represent an infinite term, and a single rewriting step can be equivalent to an infinite sequence of rewrites (see e.g. =-=[6, 8, 10]-=-). In this paper we introduce a new approach to infinitary term rewriting. Our starting point has been the seminal work of Jos'e Meseguer about rewriting logic (rl). The idea underlying rl is to take ... |

13 |
Modelling term rewriting systems by sesqui-categories
- Stell
- 1994
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Citation Context |

1 |
Computational Semantics of Term Rewriting Systems, chapter 8 of Algebraic Methods
- Boudol
- 1985
(Show Context)
Citation Context ...ying the redexes in any order. This is easily achieved by allowing only for the parallel application of non-overlapping, left-linear redexes: the definitions below summarize those in [6, 5] (see also =-=[2]-=-). Intuitively, finite parallel rewriting can be defined easily by exploiting the confluence of orthogonal term rewriting. In fact, the parallel reduction of a finite number of redexes is defined simp... |

1 |
Functorial Semantics of Rewrite Systems, appendix of Rewriting as a Unified Model of Concurrency, SRI
- Meseguer
- 1990
(Show Context)
Citation Context ...: so, a cyclic graph is a finitary description of a possibly infinite term. In this paper we introduce infinite rewriting logic, working on the framework of rewriting logic proposed by Jos'e Meseguer =-=[13, 14]-=-. We provide a simple algebraic presentation of infinite computations, recovering the infinite parallel term rewriting, originally presented by one of the authors ([6]) to extend the classical, set-th... |