## An Algebraic Presentation of Term Graphs, via GS-Monoidal Categories (1999)

Venue: | Applied Categorical Structures |

Citations: | 38 - 25 self |

### BibTeX

@ARTICLE{Corradini99analgebraic,

author = {Andrea Corradini and Fabio Gadducci},

title = {An Algebraic Presentation of Term Graphs, via GS-Monoidal Categories},

journal = {Applied Categorical Structures},

year = {1999},

volume = {7},

pages = {7--299}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the well-known characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are one-to-one with the arrows of the free gs-monoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r), providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gs-monoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...

### Citations

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Citation Context ...e topic or intuition. As far as our work is concerned, let us summarise the motivations that led us to consider a categorical presentation of term graphs. When introducing gs-monoidal categories (see =-=[21, 22]-=-: We called them smonoidal, since we were not yet aware of Pfender's work cited below), definitivo.tex; 30/11/1999; 16:52; p.33 34 our intention was to enrich symmetric monoidal categories in order to... |

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Citation Context ... need of our definitivo.tex; 30/11/1999; 16:52; p.11 12 main theorem. 3 For a full account on (symmetric) monoidal categories we refer the reader to [44]; gs-monoidal theories have been introduced in =-=[21]-=-, under the name of s-monoidal theories. DEFINITION 10. (Symmetric monoidal categories). A monoidal categorysC is a triple C = hC 0 ;\Omega ; ei where C 0 is a category, e 2 C 0 is a distinguished obj... |

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Citation Context ...yclic term graph unravels to a rational term, i.e., a possibly infinite term with a finite number of distinct sub-terms. The relationship between rational terms and cyclic term graphs is addressed in =-=[9, 38]-=-, while in [11] rational terms are related to -terms (which are essentially a subclass of term graphs): None of these papers, however, phrases this relationship categorically as a functor between suit... |

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19 |
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18 | Implementing term rewriting by jungle evaluation
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Citation Context ...mmon agreement on the way term graphs should be defined. In fact, they have been represented as directed graphs satisfying a number of constraints [4], as suitable labeled hyper-graphs called jungles =-=[32]-=-, or as sets of recursive equations [1], among others. This paper proposes a first contribution towards a categorical theory of term graph rewriting, which parallels the categorical description of ter... |

17 |
Natural associativity and commutativity
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- 1963
(Show Context)
Citation Context .... Intuitively, symmetric monoidal categories formalise in categorical terms the basic notions of pairing and permutation, as shown by the many coherence results for these categories (originating from =-=[45]-=-). The following well-known fact provides a simple characterisation of symmetries that will be of use in the next section. LEMMA 12. (Symmetries are permutations). For any n 2 IlN, the set of symmetri... |

16 |
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14 |
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14 |
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13 |
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- 1997
(Show Context)
Citation Context ...iding often a more syntactical presentation of the latter. Various papers in the literature discuss term graph rewriting using a logical presentation in terms of inference rules and axioms, including =-=[26, 27, 49]-=-. As far as the topic of the present paper is concerned, we just want to recall the inference rule that defines term graphs as an extension of standard first order terms. Roughly, the usual syntax is ... |

12 |
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11 |
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10 |
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8 |
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- 1995
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Citation Context ...l-behaved' category of partial maps. To this aim, they introduce p-categories, which were very close to ours if we required r to be natural. Also related to ours are the copy-categories introduced in =-=[28]-=-. Walters et alii investigate the algebraic structure of relations, regarded as arrows of suitable categories of spans [36, 37]. They have on the operational side an interpretation in terms of process... |

7 |
On partial algebras
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Citation Context ...ons of) standard terms are obtained. The analysis of the categorical structure of classes of algebras of partial maps is the main topic of the work by Hoenke and other former East Germany algebraists =-=[8, 30, 31]-=-. They introduce dht-symmetric monoidal categories, which can be regarded as gs-monoidal ones with a zero-object, and where transformation r is natural while ! in general is not. A functorial presenta... |