## Coinduction for recursive data types: partial orders, metric spaces and Omega-categories (2000)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Worrell00coinductionfor,

author = {James Worrell},

title = {Coinduction for recursive data types: partial orders, metric spaces and Omega-categories},

year = {2000}

}

### OpenURL

### Abstract

In this paper we prove coinduction theorems for nal coalgebras of endofunctors on categories of partial orders and (generalized) metric spaces. These results characterize the order, respectively the metric, on a nal coalgebra as maximum amongst all simulations. As suggested in [15], and motivated by the idea that partial orders and metric spaces are types of enriched category, the notion of simulation is based on the enriched categorical counterpart of relations, called bimodules. In fact, the results above arise as instances of a coinduction theorem, parametric in a quantale applying to nal coalgebras of endofunctors on the category of all (small) all) 50147 and 18636-3 Also, we give a condition under which the operational notion of simulation coincides with the denotational notion of nal semantics. 1 Introduction Coinduction is a principle for reasoning about potentially innite or circular elements of recursive data types, like streams, processes or exact reals [14,5,7]. Typ...

### Citations

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Citation Context ...s would seem to add an onerous burden in employing the above coinduction theorem. We go on to show how this problem can be circumvented by a technique resembling Milner 's use of `bisimulation up-to' =-=[12-=-]; this is demonstrated by an example in section 8. 13 Proposition 5.7 Given an -category X and a function f : X 0 ! , then ^ f(x) def = W fX(x 0 ; x) f(x 0 ) j x 0 2 X 0 g denes an -functor X ! . Pro... |

403 | Bisimulation through probabilistic testing
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Citation Context ... functors, dummy in thesrst variable. 7 Example 1: probabilistic bisimulation In [18], De Vink and Rutten give a coalgebraic analysis of probabilistic bisimulation (rst introduced by Larsen and Skou [=-=9]-=-) using the Aczel-Mendler notion of coalgebraic bisimulation. One striking aspect of that work is that they use a non-trivial result in graph theory { the max- ow min-cut theorem { to prove that their... |

377 | Basic concepts of enriched category theory, volume 64
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Citation Context ...) is contained in the identity relation on X: R X . 3 -categories In this section we recall the notions of 30972-5 , -functor and-bimodule. These are exactly as in enriched (or V-)category theory [8]=-=-=-, but specialized to the case in which V is a quantale (this makes things substantially simpler) . Choosing appropriately, we recover preorders and monotone maps; another choice of yields generalized ... |

239 | The lazy lambda calculus
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Citation Context ... { a kind of relation between two 5200-9176sA special case of this is the notion 2 of a metric relation between two generalized metric spaces X and Y { assigning to each pair x 2 X; y 2 Y; a value in =-=[0; 1]-=- measuring `how related they are'. In [15] Rutten considers coalgebras of several given endofunctors on the category Gms of generalized metric spaces and non-expansive maps; for each of the functors h... |

162 | A nal coalgebra theorem - Aczel, Mendler - 1989 |

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Citation Context ...n realised that it is fruitful to view categories of domains as being enriched (or V-)categories. In this paper we follow Wagner [19] and Rutten [16] (both of whom take their inspiration from Lawvere =-=[10]-=-) in adopting the view that domains do not just organize themselves in an enriched category: we take enriched categories as our domains. More precisely, a domain is an 50135-2 for some complete lattic... |

92 |
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Citation Context ...riant, re ective subcategory of C then asnal coalgebra of T viewed as an endofunctor on D remains asnal coalgebra for T viewed as an endofunctor on C (Freyd calls this the re ective subcategory lemma =-=[-=-6]). Thus, thesnal coalgebra of the restriction of F to Set, whose existence is established in [18], is in fact thesnal F -coalgebra. 16 Next, recall the following denitions from [9]. Denition 7.1 A d... |

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Citation Context ...that two elements are in relation R one must exhibit a relation containing those two elements and closed under unfolding. The name coinduction for such a principle seems to be due to Milner and Tofte =-=[11]-=-; it is similar to the way bisimulations are used in concurrency theory. The central notions in the category theoretic account of coinduction are: coalgebra, homomorphism of coalgebras and bisimulatio... |

74 | Rutten: Bisimulation for Probabilistic Transition Systems: A Coalgebraic Approach
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Citation Context ...ontrast, in the order theoretic counterpart, theorem ??, we require a continuity property on simulations. mixed-variant functors, dummy in thesrst variable. 7 Example 1: probabilistic bisimulation In =-=[18-=-], De Vink and Rutten give a coalgebraic analysis of probabilistic bisimulation (rst introduced by Larsen and Skou [9]) using the Aczel-Mendler notion of coalgebraic bisimulation. One striking aspect ... |

43 | A 2-categorical approach to change of base and geometric morphisms
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Citation Context ...he following is readily veried. Proposition 2.2 [15, Prop 1.2] In the situation of denition 2.1, R is an F - bisimulation i for all x 2 X; y 2 Y : hx; yi 2 R ) hx;syi 2 F 2 (F 1 ) : 2 Following [4,15]=-=-=-, who were interested in `lifting' endofunctors on regular categories to endofunctors on their category of relations, let us write Rel F for the graph homomorphism Rel ! Rel with (Rel F )X = FX and (R... |

37 | A coinduction principle for recursive data types based on bisimulation
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Citation Context ...ational notion ofsnal semantics. 1 Introduction Coinduction is a principle for reasoning about potentially innite or circular elements of recursive data types, like streams, processes or exact reals [=-=14,5,7]-=-. Typically, one takes data types to be objects of some category of `domains', and type constructors to be endofunctors on that category. Recursive data types can be modelled assxed points, or invaria... |

37 |
Elements of generalized ultrametric domain theory
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(Show Context)
Citation Context ...s a quantale (this makes things substantially simpler) . Choosing appropriately, we recover preorders and monotone maps; another choice of yields generalized metric spaces and non-expansive maps (cf. =-=[10,19,16-=-]). Denition 3.1 A quantale is a complete lattice with a binary, associative operation , such that for every element a, both asand a have right adjoints (equivalently preserves all joins). A quantale ... |

14 |
An axiomatics for bicategories of modules
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Citation Context ...ry morphism R in Rel can be tabulated by span of functions X 1sR 2 ! Y in the sense that R = 2 1 { just take 1 and 2 to be the projections associated to R. Correspondingly, we recall from [3] how 5013 dules can be tabulated by a cospan of 33888-16 Denition 4.1 ([3]) Suppose R : Y # X is a bimodule, then the collage of R, denoted Coll(R), is the -category with the disjoint union of Y 0 an... |

12 | Circular Coinduction
- Rosu, Goguen
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(Show Context)
Citation Context ...ational notion ofsnal semantics. 1 Introduction Coinduction is a principle for reasoning about potentially innite or circular elements of recursive data types, like streams, processes or exact reals [=-=14,5,7]-=-. Typically, one takes data types to be objects of some category of `domains', and type constructors to be endofunctors on that category. Recursive data types can be modelled assxed points, or invaria... |

2 |
A coinduction principle for recursively de domains
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(Show Context)
Citation Context ...ion, and, by relating it to our categorical notion of simulation, we show that the metric on Fix(F ) is the maximum metric applicative bisimulation. This is patterned after some earlier work of Pitts =-=[13-=-] and Abramsky [1] in an order theoretic setting. They consider the minimal solution of a domain equation for the untyped lazy -calculus in Cppo ? , and show that the order relation on their solution ... |

2 |
On the foundations of coalgebra semantics: non-wellfounded sets, partial orders, metric spaces
- Turi, Rutten
- 1998
(Show Context)
Citation Context ...Rel F is actually a lax functor. It follows from [4] that Rel F is a lax functor i it is a functor i F preserves weak pullbacks. The coinduction principle forsnal F -coalgebras in Set is as follows [17]. Coindution: any bisimulation R on asnal F -coalgebra (X; ) is contained in the identity relation on X: R X . 3 -categories In this section we recall the notions of 30972-5 , -functor and-bimodul... |

1 |
Universal Coalgebra: a theory of systems. To appear, Theoretical Computer Science. Also Technical report CS-R9652, Centrum voor Wiskunde en Informatica
- Rutten
- 1996
(Show Context)
Citation Context ...ational notion ofsnal semantics. 1 Introduction Coinduction is a principle for reasoning about potentially innite or circular elements of recursive data types, like streams, processes or exact reals [=-=14,5,7]-=-. Typically, one takes data types to be objects of some category of `domains', and type constructors to be endofunctors on that category. Recursive data types can be modelled assxed points, or invaria... |

1 |
Relators and metric bisimulations. Proceedings of 1st workshop in coalgebraic methods in computer science, elec. notes in theor. comp. sci., vol 11
- Rutten
- 1998
(Show Context)
Citation Context ... on categories of partial orders and (generalized) metric spaces. These results characterize the order, respectively the metric, on asnal coalgebra as maximum amongst all simulations. As suggested in =-=[15]-=-, and motivated by the idea that partial orders and metric spaces are types of enriched category, the notion of simulation is based on the enriched categorical counterpart of relations, called bimodul... |

1 |
Liminf convergence for 32879-103
- Wagner
- 1997
(Show Context)
Citation Context ...-coalgebra is maximum amongst ordered F -bisimulations. It has long been realised that it is fruitful to view categories of domains as being enriched (or V-)categories. In this paper we follow Wagner =-=[19]-=- and Rutten [16] (both of whom take their inspiration from Lawvere [10]) in adopting the view that domains do not just organize themselves in an enriched category: we take enriched categories as our d... |