## Precategories for Combining Probabilistic Automata (1999)

Venue: | Electronic Notes in Theoretical Computer Science |

Citations: | 10 - 6 self |

### BibTeX

@INPROCEEDINGS{Mateus99precategoriesfor,

author = {Paulo Mateus and Amílcar Sernadas and Cristina Sernadas},

title = {Precategories for Combining Probabilistic Automata},

booktitle = {Electronic Notes in Theoretical Computer Science},

year = {1999},

pages = {15--16}

}

### OpenURL

### Abstract

A relaxed notion of category is presented having in mind the categorical caracterization of the mechanisms for combining probabilistic automata, since the composition of the appropriate morphisms is not always defined. A detailed discussion of the required notion of morphism is provided. The partiality of composition of such morphisms is illustrated at the abstract level of countable probability spaces. The relevant fragment of the theory of the proposed precategories is developed, including (constrained) products and Cartesian liftings. Precategories are precisely placed in the universe of neocategories. Some classical results from category theory are shown to carry over to precategories. Other results are shown not to hold in general. As an application, the precategorical universal constructs are used for characterizing the basic mechanisms for combining probabilistic automata: aggregation, interconnection and state constraining. Mathematics Subject Classifications: 18A10 68Q75. Ke...

### Citations

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Citation Context ...ic automata, the notions of aggregation and interconnection of nonprobabilistic automata, and the basics of category theory. We also use Cartesian liftings that are presented from first principles in =-=[BW90]. 2 What m-=-orphism? The key problem in following the categorical imperative of [Gog91, WN95] when dealing with probabilistic automata is to find the "right" notion of morphism between two such automata... |

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Citation Context ...ransition systems. The requirement on the probability front is very natural given the results of the previous section. The concept of zig-zag morphism, used to characterize probabilistic bisimulation =-=[BDEP97]-=-, is a special case of the proposed morphism. They coincide when considering a fixed set of actions and a surjective map between the states. Of course, such surjective morphisms always compose but the... |

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Citation Context ...terial precategory S over R is the pair hF 0 ; ,!i. Therefore we look at materialness as the precategory counterpart of concreteness of categories. For details on concrete categories see for instance =-=[AHS90]-=-. We now state a lemma that will be used several times in the sequel. Lemma 4.4 Let F : S ! R be the prefunctor induced by a material precategorysS over R. Then: F (f) ffi R F (g) = id F (A) ) f ffi S... |

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Citation Context ...bilistic automaton, aggregation, interconnection. 1 Introduction Many relaxed notions of category have been studied with different motivations in mind, starting with Ehresmann's work on neocategories =-=[Ehr65]-=- which was continued in [Cop80]. More recently, Herrlich and his associates have been working on semicategories [HS98]. All these notions can be seen as special cases of compositional graphs (where co... |

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Citation Context ...h different motivations in mind, starting with Ehresmann's work on neocategories [Ehr65] which was continued in [Cop80]. More recently, Herrlich and his associates have been working on semicategories =-=[HS98]-=-. All these notions can be seen as special cases of compositional graphs (where composition of f : A ! B and g : B ! C is not always defined). The differences consist of different laws for the identit... |

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Citation Context ..., interconnection. 1 Introduction Many relaxed notions of category have been studied with different motivations in mind, starting with Ehresmann's work on neocategories [Ehr65] which was continued in =-=[Cop80]-=-. More recently, Herrlich and his associates have been working on semicategories [HS98]. All these notions can be seen as special cases of compositional graphs (where composition of f : A ! B and g : ... |

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