J. A. Goguen, A categorical manifesto, in Math. Struct. in Comp. Sci. 1(1) (1991) 49--67.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....introduction and the proofs of the theorems. 3.1 Category theory Category theory was developed in an attempt to unify simple abstract concepts that were applicable in many branches of mathematics. Excellent introductions to category theory and its application in Computer Science can be found in [Pier90, Gogu91, Pier91, Aspe91, Barr96]. 3.1.1 Basic definitions Definition 3.1. A category C is a collection of objects and a collection of arrows satisfying the CATEGORIES following properties: ffl For each arrow f there is a domain object dom(f) and a codomain object codom(f) and by writing f : x y it is indicated that x = ....

J. A. Goguen, "A Categorical Manifesto", Mathematical Structures in Computer Science, vol. 1, pp. 49--68, 1991.

.... even in the total setting are brought forward in [26] If, on the other hand, extensionality is for some reason required by the user, it may easily be enforced: spec Extensionality = forall a; b : Type; f ; g : a b (8x : a f (x ) g(x ) f = g From a categorical point of view (see [10] on why we think this is relevant) intensional models are attractive because they are in direct correspondence with the natural class of categorical models; see below for references and further details. Methodologically speaking, intensional models o er the advantage of allowing a more ....

J. Goguen, A categorical manifesto, Math. Struct. Comput. Sci. 1 (1991), 49-67.

....free constructions. Since free constructions preserve colimits the construction of the model of computations is also compositional w.r.t. all composition operations for programs that can be modeled by colimits. Recall that colimits are the general categorical construction to put things together [Gog91], thus the restriction is acceptable. 2. Applying the STS method to a suitable algebraic presentation of graphs we obtain an algebra of graph derivations with sequential and parallel composition operations for free, given by the model of computations delivered by the method. 3. The categorical ....

J.A. Goguen. A categorical manifesto. MSCS, 1, 1991.

....of initial models of signatures in a setting with partial functions (see [1] for a simple example) If, on the other hand, extensionality is for some reason required by the user, it may easily be enforced by means of a suitable polymorphic axiom; see below. From a categorical point of view (see [11] on why we think this is relevant) intensional models are attractive because they are in direct correspondence with the natural class of categorical models; see below for references and further details. Methodologically speaking, intensional models o er the advantage of allowing a more ....

J. Goguen, A categorical manifesto, Math. Struct. Comput. Sci. 1 (1991), 49-67.

....BCHG 97] Herein, we tackle the problem of aggregating, interconnecting and constraining probabilistic automata, having in mind future work in compositional verification of probabilistic systems. We adopt the viewpoint of category theory, following the so called categorial imperative of [Gog91, WN95] However, the categorial approach to probability raises many difficulties related to the fact that the most natural morphisms do not always compose. We start the paper by first presenting and justifying the choice of a measure preserving map as the candidate to the notion of morphism ....

....over the possible next states. Extending our results to fully non deterministic automata, as considered for example in [LSS94, BK97] does not raise any essential new problem, but it would make the notation much heavier. 2 What morphism The key problem in following the categorial imperative of [Gog91, WN95] when dealing with probabilistic automata is to find the right notion of morphism between two such automata. Here, as always, right means suitable for the goals in mind, and our goals are to establish aggregation and interconnection of probabilistic automata as categorial constructions. ....

Joseph A. Goguen. A categorical manifesto. Math. Structures Comput. Sci., 1(1):49--67, 1991.

.... practice that one can recognise in the literature on SA. First of all, one of the myths of Software Engineering, is that the use of diagrammatic notation is not formal, that it is at best semi formal. Well, CT is all about chasing diagrams In fact, one of the basic principles summarised in [13] is that complex systems can be usefully identified with diagrams, system components and connectors corresponding to nodes, and interconnections being established through the edges of the diagrams. The subtlety here is that the word diagram in CT has a formal meaning and, at the same time, ....

....not mathematical , making appeal more to the intuition than formality. All the results can be found in [9] We should again stress that much of the work that we shall present is inspired by the work of J. Goguen and can derived from the general principles enunciated in the Categorical Manifesto [13]. The only disadvantage of previous papers on the subject is that they are based on mathematical models of system behaviour, namely sheaf theoretic ones, which are seldom part of the day to day vocabulary of the software architect (or, for that matter, the software engineer) More recent ....

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J . Goguen, "A Categorical Manifesto", Mathematical Structures in Computer Science 1(1), 1991, 49-67.

....the reusability of software components as suggested in [Krueger 92] A glossary of the notation introduced throughout the paper is provided at the end. 2 The categorical imperative A general explanation of why and how Category Theory (CT) is useful in Computing has already been given in [Goguen 91] it would be difficult for us to improve on it. Nevertheless, it seems important to state the basic principles which make it useful for the purpose at hand (and one that has previously not been extensively explored) namely to give evidence of its applicability for handling multiple formalisms ....

J.Goguen, "A Categorical Manifesto", Mathematical Structures in Computer Science 1(1), 1991, 49-67.

.... in [13] is that morphisms can be used to express interaction between components, so that given a category of widgets, the operation of putting a system of widgets together to form some super widget corresponds to taking the colimit of the diagram of widgets that shows how to interconnect them [15]. As shown in [5,16] these categorical principles can be used to formalise process models for concurrent systems such as transition systems, synchronisation trees, event structures, etc. We shall illustrate the approach using a trace based model. A process alphabet is a finite set, and a process ....

....denoting the whole set of elements of sort a. 4 Concluding remarks In this paper, we proposed a formalisation for the property according to which a framework for system design supports the separation between computation and coordination. We used Goguen s categorical approach to systems design [13,15] as a platform for the formalisation. The perceived advantages of the proposed notion of coordination are the following. On the one hand, it provides us with a way of checking whether a given formalism supports the separation between computation and coordination, which we take as being a good ....

. J.Goguen, "A Categorical Manifesto", Mathematical Structures in Computer Science

.... to systems is that morphisms can be used to express interaction between components, so that given a category of widgets, the operation of putting a system of widgets together to form some super widget corresponds to taking the colimit of the diagram of widgets that shows how to interconnect them [5]. The specific application of this approach to coordination abstracts general principles from previous applications of categorical techniques to parallel program design centred on the notion of superposition [8] In this approach, we assume that a notion of system (be it system specifications, ....

J.Goguen, "A Categorical Manifesto", Mathematical Structures in Computer Science 1(1), 1991, 49-67.

....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 automata 2 alization constructions for ....

J.A. Goguen. A categorical manifesto. Math. Struct. Comp. Sci., 1(1):49--67, 1991.

....the construction that characterizes the algebraic approach we are presenting, and it is shown in Figure 5. Intuitively, regarding graphs as distributed states of a system, a pushout is a sort of generalized union that specifies how to merge together two states having a common substate [45]. For example, if the right square of Figure 5 is a pushout in Graph, then graph H is obtained by gluing together graphs R and D along the common subgraph K, and r and m are the resulting injections. Therefore the double pushout construction can be interpreted as follows. In order to apply ....

J.A. Goguen. A categorical manifesto. Math. Struc. Comput. Sci., 1, 1991.

....why it is the time of scientific declarations often called manifestos as, for instance, The Object Oriented Database System Manifesto by M. Atkinson et al. ( ABD 89] Third Generation Data Base System Manifesto by M. Stonebraker et al.( SRL 90] A Categorical Manifesto by J. Goguen ( Gog89] So, we have also called the declaration presented below a Manifesto. It is written by a specialist in database design (who has more than a decade experience in semantic modeling and designing very large databases, and about two year experience in applying category theory concepts to his ....

J.A. Goguen. A categorical manifesto. Technical report, SRI International, SRI-CSL-89-8, 1989.

....probabilistic automata, having in mind future work in compositional verification of probabilistic systems. The objective here is to provide a categorical characterization of these mechanisms for relating and combining probabilistic automata, following the so called categorical imperative of [Gog91, WN95]. In fact, we have to adopt the precategorical point of view given that precategories are the appropriate structures for this work. We start the paper by first presenting and justifying the choice of a measure preserving map as the candidate to the notion of morphism between probability spaces, ....

....probabilistic 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 morphism 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. We expect to obtain aggregation as a categorical product and interconnection as a Cartesian lifting, following the program of [WN95] originally developed for processes and classical ....

J. Goguen. A categorical manifesto. Math. Structures Comput. Sci., 1(1):49--67, 1991.

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J. A. Goguen, A categorical manifesto, in Math. Struct. in Comp. Sci. 1(1) (1991) 49--67.

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J.Goguen, "A Categorical Manifesto", Mathematical Structures in Computer Science 1(1), 1991, 49-67.

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J.A. Goguen. A categorical manifesto. Math. Struct. Comp. Sci., 1(1):49--67, 1991.

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J.A. Goguen "A categorical Manifesto" Mathematical Structures in Computer Science vol. 1, n.1 (1991)

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J.A. Goguen. A categorical manifesto. Math. Struct. Comp. Sci., 1(1):49--67, 1991.

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