J. A. Goguen. Minimal realization of machines in closed categories. Bulletin of the AMS, 78:777--783, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....classically a category of functions; below we take observational equivalence classes of trees. By minimal realization we understand the functorial construction of a universal machine realizing any behaviour, that is we expect the minimal realization process to be locally right adjoint to behaviour [Gog72, RSW98]. The minimal realization is denoted N (for Nerode) As with minimization, behaviour carries the structure of a lax functor, and minimized machines and behaviours are (bi )equivalent. The situation we have been describing can be summed up in: MinMach Beh Mach # # # # ## # # # # # ## ....

J. A. Goguen. Minimal realization of machines in closed categories. Bulletin of the AMS, 78:777--783, 1972.

....classically a category of functions; belowwe take observational equivalence classes of trees. By minimal realization we understand the functorial construction of a universal machine realizing anybehaviour, that is we expect the minimal realization process to be locally right adjoint to behaviour [Gog72, RSW98]. The minimal realization is denoted # (for Nerode) As with minimization, behaviour carries the structure of a lax functor, and minimized machines and behaviours are (bi )equivalent. The situation we have been describing can be summed up in: ####### ### #### # # # # # # ## # # # # # ## # # # # # ....

J. A. Goguen. Minimal realization of machines in closed categories. Bulletin of the AMS, 78:777-783, 1972.

....Box 94079, 1090 GB Amsterdam (NL) Kruislaan 413, 1098 SJ Amsterdam (NL) Telephone 31 20 592 9333 Telefax 31 20 592 4199 Automata and Behaviours in Categories of Processes Bart Jacobs CWI P.O. Box 94079, 1090 GB Amsterdam, The Netherlands (bjacobs cwi.nl) Abstract An early result of Goguen [4, 5] describes the fundamental adjunction between categories of deterministic automata and their behaviours. Our first step is to redefine (morphisms in) these categories of automata and behaviours so that a restriction in Goguen s approach can be avoided. Subsequently we give a coalgebraic analysis ....

....define process combinators on this category of behaviour, in such a way that this functor B preserves combinators (i.e. commutes with suitable functors) This yields a form of compositionality: the behaviour of a complex automaton may be understood from the behaviour of its parts. Joseph Goguen [4, 5] in the early 1970s (and again in [6] defined categories of (deterministic) automata and behaviours and showed that under certain restrictions the behaviour functor B has a right adjoint R, for realization. This is a fundamental result, bringing a number of minimal re 2. Deterministic ....

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J.A. Goguen. Minimal realization of machines in closed categories. Bull. Amer. Math. Soc., 78(5):777--783, 1972.

....from categorical automata theory and from topos theory 1.1 Introduction Categorical automata theory as the area of theoretical computer science had been arisen at the beginning of seventys. In this area the first fundamental investigations were done by J.A.Goguen, M.A. Arbib and e.g.Manes (see [2] [8] These investigations had been concentrated mainly on the problem of minimal implementation of total reaction morphisms, i.e. the problem of construction the optimal (in a certain sence) automaton in a category using information about the automaton s external behaviour. Categorical ....

Goguen J.A.: Minimal realization of machines in closed categories, Bull. Am. Math. Soc., 78, (1972), p. 777-783.

....notion of randomly timed automaton should be as general as possible in order to be able to support di erent execution policies. Di erent policies are studied in [19,8] for the case of stochastic Petri nets, but their usefulness extends to randomly timed automata as well. Following the style of [16,13,5,1,15,6,2,29,26] proposed for classical automata, we adopt a categorial approach to the development of the theory of randomly timed automata. However, probabilistic structures raise some speci c problems to category theory [22,25] Fortunately, we are able to avoid working with precategories as advocated in these ....

J. Goguen. Minimal realization of machines in closed categories. Bull. Amer. Math. Soc., 78:777-783, 1972. 28

.... enjoying a wider interest, perhaps partly due to recent results linking it to coalgebraic semantics [2, 34, 5, 37, 25, 36, 23, 35] Minimal realisation of machines, a subject with a long history, was one of the earliest topics in computer science to be treated in an elegant categorical manner [14, 13, 4, 3]. The relationship between behavioural equivalence and bisimulation seems to have suddenly become known or at least suggested, through various different articles [25, 36, 23, 35] In this paper we touch briefly on this relationship, and show that the proof technique developed in [17, 27] for ....

Joseph Goguen. Minimal realization of machines in closed categories. Bulletin of the American Mathematical Society, 78(5):777--783, 1972.

....described by monads on the hom categories, and the algebras are minimal automata. With an appropriate definition of the behaviour of Mealy automata, we are able to prove a minimal realization theorem which extends Nerode s theorem [Ner] It provides a variant of Goguen s minimal realization theory [Gog] and we extend this to include serial composition. The local situation, i.e. in a single hom category, is summarized in the following diagram. The reachable automata from X to Y are denoted AR (X; Y ) the subcategory of minimized automata is A M R (X; Y ) and behaviours from X to Y are ....

J. A. Goguen. Minimal realization of machines in closed categories. Bulletin of the AMS, 78:777--783, 1972.

....in our hybrid setting. Such an adjunction captures the fundamental relation between machines which can perform certain behaviour, and behaviours which can be realized in a certain (universal) way. Such behaviour realization adjunctions are typical in mathematical system theory, following work [9, 10, 11] of Goguen. We adapt this approach in a minor way, by taking the morphisms between input sets in contravariant direction (like in [17] in order to avoid some unnecessary restrictions. We shall use our behaviour realization adjunction to provide a setting in which to discuss simulations and ....

J.A. Goguen. Minimal realization of machines in closed categories. Bull. Amer. Math. Soc., 78(5):777-- 783, 1972.

....that, by using a more general notion of sheaf as a functor on a category with a Grothendieck topology [24] we can obtain an adjunction between system specifications and sheaves of objects. Our definition of object class specification is similar to abstract definitions of automata in categories [1, 2, 10, 12]. Since our definition was motivated by Goguen and Diaconescu s independent sum construction in hidden sorted algebra, it would be interesting to examine the relationship between Nerode equivalence in automata theory and behavioural equivalence in hidden sorted algebra. Mazurkiewicz s process ....

Joseph Goguen. Minimal realization of machines in closed categories. Bulletin of the American Mathematical Society, 78(5):777--783, 1972.

....gives the minimal realisation of a behaviour [18] Because right adjoints are uniquely determined, this provides a convenient abstract characterisation of minimal realisation. Moreover, this characterisation extends to, and even suggests, more general minimal realisation situations, e.g. see [17]. 5.4 Syntax and Semantics. One of the more spectacular adjoints is that between syntax and semantics for algebraic theories, again due to Lawvere in his thesis; see [49] 5.5 Cartesian Closed Categories. A Cartesian closed category has binary products, and a right adjoint to each functor sending ....

Joseph Goguen. Minimal realization of machines in closed categories. Bulletin of the American Mathematical Society, 78(5):777--783, 1972.

.... I formulated the minimal realization of automata as an adjoint functor [38] this soon evolved into much more general results about the minimal realization of machines in categories, which gave a neat unification of system theory (in the sense of electrical engineering) with automaton theory [36]. I consider this a major vindication of the categorical approach to systems. 2.3 Abstract Data Types and Algebraic Semantics The history of programming languages, and to a large extent of software engineering as a whole, can be seen as a succession of ever more powerful abstraction mechanisms. ....

Joseph Goguen. Minimal realization of machines in closed categories. Bulletin of the American Mathematical Society, 78(5):777--783, 1972.

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