Joseph A. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49--67, March 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and manipulation. As a result, researchers proposed several frameworks defining the model theoretic semantics of the object oriented paradigm. Category theory (CT) seems to be an useful device to express these concepts in an understandable and unified way. Diskin and Goguen in their manifests [4, 9] pointed out that CT is a formalism supporting a high level of abstraction that helps to simplify Supported by the grant of the Czech Grant Agency No. 102 96 0986 Object Oriented Database Model and by the grant of the Czech Ministry of Education No. 0630 Modeling of Inheritance in the ....

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

....second part of the de nition ensures that the D chosen to construct the pushout is the minimal such D amongst all the candidates D 0 . The generalisation of this operation to several objects and morphisms is called a colimit. A practical interpretation for the colimit is given by Goguen in [5]: Given a species of structure, say widgets, then the result of interconnecting a system of widgets to form a super widget corresponds to taking the colimit of the diagram of widgets in which the morphisms show how they are interconnected. De nition: pullback. A pullback of a pair of morphisms ....

....p 0 . C A g D B p q f D 0 w R p 0 H H H H H H H Hj q 0 A A A A A A A AU Figure 2: Pullback of two morphisms f and g The generalisation of this operation to several objects and morphisms is called limit. There is also an intuition for limit in [5]: A diagram D in a category C can be seen as a system of constraints, and then a limit of D represents all possible solutions of the system. 3 Modular speci cation of systems Category theory has been used for a number of years as a framework for composing formal speci cations based on early ....

J.A. Goguen. A Categorical Manifesto. Mathematical Structures in Computer Science, 1(1), March 1991.

....composition, while synchronization of nets can be modeled as a product (see [39, 26] Since left right adjoint functors preserve colimits limits, a 5 semantics de ned via an adjunction turns out to be compositional with respect to such operations. The reader is referred to Goguen s paper [18] for an extensive discussion on the usefulness of category theory in computer science. Relation with deterministic processes. The problem of providing a truly concurrent semantics for contextual nets based on (deterministic) processes has been faced by various authors (see, e.g. 28, 17, 19, 9, ....

....the unfolding of [37] is carried out in the case of safe nite c nets, the major di erence is the fact that our unfolding is de ned at categorical level and establishes a core ection between the categories of semi weighted and occurrence c nets. This result ensures the naturality of the semantics [18] (the unfolding is the unique right adjoint to the inclusion functor) as well as its compositionality with respect to operations on nets de ned in terms of limits (e.g. synchronization) We already mentioned that Winskel s construction has been generalized in [25] not only to the subclass of ....

J.A. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1, 1991. 44

....with each other in a suitable sense, then these local refinements induce a refinement of the global composed specifications. The clue to compositionality is the categorical approach 1 , in particular using (co)limits to put specifications together, as suggested in a very general setting in [Gog91,BG77]. The universal property of a categorical colimit is that every compatible family of morphisms (refinements) of the components uniquely induces a morphism (refinement) of the composition. This categorical approach has been introduced for labeled transition systems as models for concurrency (among ....

J. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49--67, 1991.

....in use which make a clear distinction between visible and hidden sorts, the equality being strict on the visible sorts and behavioral on the hidden sorts, that is, meant as indistinguishability under experiments . We generically call these logics behavioral logics. These include hidden algebra [6 9], coherent hidden algebra [4,5] observational logic [12,1] the equational version) and other recent generalizations of hidden algebra [17,16] In order to show that a logic does not admit a complete axiomatization it is necessary and sufficient to show that the satisfaction problem is not ....

....implies the incompleteness of the other more relaxed behavioral logics. We have tried to prove our incompleteness results for the weakest (most restrictive) logics, so as to make them as general as possible. 2. 1 Fixed Data Hidden algebra first appeared in [6] and was further investigated in [7 9] and many others. Here, we present an over simplified version of hidden algebra logic and refer to it as basic fixed data hidden algebra in the rest of the paper. Definition 2.1 A hidden signature is a fv; hg sorted signature Sigma, where v is the visible sort and h is the hidden sort, consisting ....

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Joseph Goguen and Grant Malcolm. Hidden coinduction. Mathematical Structures in Computer Science, to appear 1999.

....introductions to the subject the reader is referred to the books by Barr and Wells [8] and Pierce [114] which are targeted specifically for audiences in computer science. Shorter introductions, together with further pointers to the numerous applications of categories in computing, are provided in [115, 40, 119, 26]. The definitive text on category theory is MacLane s book [88] 5.1 Graphs A graph consists of a set of objects O (vertices) a set of arrows A (edges) and a pair of functions # 0 , # 1 : A # O, called the source and target functions respectively. It is customary to write a : u # w to denote ....

....two arrows f : C # A and g : C # B in a category C having common source C. The idea is that a pushout construction combines the objects A and B into a third object P by making as few identifications as possible (those specified by f and g) and by adding nothing which is essentially new [40]. The resulting object P comes equipped with arrows p 1 : A # P and p 2 : B # P showing how A and B are included in P . Definition 5.6. A commutative square C f # A y g p1 y B p2 # P in a category is called a pushout square if whenever there is object Q and ....

J. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49--67, 1991.

....subcategories M Mod( Sigma) and as arrows all inclusion functors M 1 M 2 . By definition of Spec( Sigma) and Sub( Sigma) we can formulate the usual categorical presentation of the Galois correspondence arising from any (satisfaction) relation as an adjunction th( Sigma) a mod( Sigma) In [4], the description of a Galois correspondence in the language of adjoint functors is given as a typical example for categorical overkill. However, such a formulation will turn out to be a suitable way to get a better insight into the structure of logical systems. Proposition 2.4 (Galois ....

J. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49--67, 1991.

....of a formal definition to a wide audience, and there is a strong preference for a highly conceptual formalization, then category theory will be the most ideal instrument. However, if a wider audience needs to be addressed, a more mundane formalization may be more appropriate. Goguen [Gog91] argued that category theory can provide help with dealing with abstraction and representation independence: in computing science, more abstract viewpoints are often more useful, because of the need to achieve independence from the overwhelmingly complex details of how things are represented or ....

J.A. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49--67, 1991.

....and hidden sorts, the equality being strict on the visible sorts and behavioral on 1 Also Fundamentals of Computing, Faculty of Mathematics, University of Bucharest, Romania. Buss and Rosu the hidden sorts, that is, meant as indistinguishability under experiments . These include hidden algebra [7 10], coherent hidden algebra [5,6] observational logic [13,1] the equational version) and other recent generalizations of hidden algebra [18,17] Since 1940, increasingly refined techniques for showing incompleteness have been developed. The most usual method is to show that the satisfaction ....

....satisfaction can be classified in two categories, depending on whether a fixed data algebra is assumed for all models or not. In this section we briefly discuss them pointing out some of their main properties. 2. 1 Fixed Data Hidden algebra first appeared in [7] and was further investigated in [8 10] and many others. Definition 2.1 A hidden signature is a triple ( Psi; D; Sigma) often written Sigma D or just Sigma, where ffl Psi is a V sorted signature, ffl D is a Psi algebra called the data algebra, ffl Sigma is a (V [ H) sorted signature extending Psi such that: S1) if w 2 ....

Joseph Goguen and Grant Malcolm. Hidden coinduction. Mathematical Structures in Computer Science, to appear 1999.

....machine instructions or even processes. When Shp = Cat and = Id : Cat Cat, a category of systems reduces to a category of diagrams in C. Such categories are sometimes used to define limits and colimits as functors. Goguen has long advocated diagrams as semantic objects in computing. See [25,26] for an overview and further references. According to Goguen a diagram represents a system in a broad sense. The exact computational interpretation depends on the underlying category, but for example, the objects in the diagram may represent processes and the morphisms the interconnections. ....

....a product which may serve different roles. By temporal we refer to the fact that interface is itself a system with transitions which evolves. The separation of internal and observable behaviour as described above is not new. It appears in Ferrari et al. 21,22] and has been advocated by Goguen [25,26] in his theory of systems. As here, Goguen defines the behaviour of a system to be the limit of a diagram. Another instance is due to Cockett and Spooner [14 16] They construct categories in which morphisms are typed processes. Processes are spans in a category of conventional transition ....

Joseph A. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49--67, 1991.

....respectively. 2.3 Composable signatures and amalgamation Given a species of structure, say widgets, then the result of interconnecting a system of widgets to form a super widget corresponds to taking the colimit of the diagram of widgets in which the morphisms show how they are interconnected. [Gog91] If we want to apply this to combine signatures and theories, we have to assume that signature categories are cocomplete. Moreover, this combination should be reflected by the semantics, that is, the model functor should be continuous. But there are very common institutions such as unsorted logics ....

J. A. Goguen. A categorical manifesto. Mathematical Structures in Computer Science 1, 49--67, 1991.

....an appropriate approach to the actual software engineering. The Internet facilitates distributed cooperative projects and implicitly distributed cooperative software systems whose behavior is difficult to catch using only strict equalities of states. At the author s knowledge, coinduction [11, 22, 18, 12, 13, 15] and context induction [16, 6, 1] are among the most frequent techniques to prove behavioral equivalences of states and they both require human intervention. Proving automatically behavioral equivalences is a useful feature of specification languages. CafeOBJ [4, 5] has implemented behavioral ....

....and multiplications with prime numbers, so an infinite basis. 2 Hidden Algebra Hidden algebra [7, 9] appeared to give algebraic semantics for the object paradigm. A comprehensive survey for the case in which operations are all behavioral and have at most one hidden argument can be found in [12, 13]. We remind the reader the basic definitions and results of hidden algebra in its slightly extended form: Definition 1. A hidden signature is a triple ( Psi ; D; Sigma ) often noted Sigma , where Psi is a V sorted signature and D is a Psi algebra, called the data algebra, Sigma is ....

Joseph Goguen and Grant Malcolm. Hidden coinduction. Mathematical Structures in Computer Science, to appear 1999.

....we shall not be worried with providing category theoretic characterizations of the constructions. The point is not that we are less convinced of the importance of categorial techniques in the study of combinations, of any nature, as shown by a long history of fruitful results (see, for instance, [25]) in the lines of the principles advocated in the past by [29] However, for the purpose of this paper, it seemed that the categorial apparatus would make it too long and, at the same time, somehow deviate the attention of the reader from the main purpose of relating synchronization, ....

J. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49--67, 1991.

....eighties category theory has also found its way into computer science. Applications of category theory can be found in such diverse fields as automata and systems theory, formal specifications and abstract data types, type theory, domain theory, and constructive algorithmics. As pointed out by [Gog91] category theory can provide help with at least the following: ffl Formulating definitions and theories. In computing science, it is often more difficult to formulate concepts and results than to give a proof. As stated by [AHS90] category theory provides a language with a convenient symbolism ....

J.A. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49--67, 1991.

....an appropriate approach to the actual software engineering. The Internet facilitates distributive cooperative projects and implicitly distributive cooperative software systems whose behavior is difficult to catch using only strict equalities of states. At the author s knowledge, coinduction [7, 8] and context induction [10] are among the most frequent techniques to prove behavioral equivalences of states and they both require human intervention. Proving automatically behavioral equivalences is a useful feature of specification languages. CafeOBJ [1, 2] has implemented behavioral rewriting ....

Joseph Goguen and Grant Malcolm. Hidden coinduction. Mathematical Structures in Computer Science, to appear 1999.

....of single transformations, that yield closure operations on transformation systems and extended morphisms. The purpose of this section is to define compositions of whole transformation systems, that model how components of open systems can be put together. Following the categorical imperative (see [Gog91]) putting components together means to construct their colimit. First the interconnection of the components have to be specified by morphisms that define the correspondences. In most applications the correspondence between components is relational. Therefore they are not immediately related by a ....

J. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49--67, 1991.

.... a formal definition of auxiliary structure we assume the existence of the categories of configurations and effects (e.g. states in S and actions in A of the associated transition systems are arrows of categories) The advantages of using category theory in computer science are well summarized in [36]. We just remark here the following aspects: a) suitable classes of (structure preserving) functors between categories (representing transition systems) offer an immediate definition of simulation morphism between the underlying systems; b) considering categories in the small (i.e. objects ....

J.A. Goguen. A Categorical Manifesto. Mathematical Structures in Computer Science 1. 1991. pp. 49--67.

....: A B and g : B C, we let f ; g denote their composition, a morphism A C; also, we let 1 A denote the identity morphism at an object A. Sections 4.1 and 7 use colimits, and Section 7 also uses universal constructions. A general discussion of category theory for Computing Science is given in [14]. 3 Hidden Sorted Algebra This section gives a brief summary of basic concepts from (overloaded many sorted) hidden sorted algebra, with some motivation, following the lines of [15] The basic intuition is that an object has a state, which is hidden, i.e. only observed through the e ect of ....

Joseph Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49{ 67, March 1991.

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Joseph A. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49--67, March 1991.

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J. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1:49--67, 1991.

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J.A. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49--67, 1991.

No context found.

Joseph Goguen and Grant Malcolm. Hidden coinduction. Mathematical Structures in Computer Science, to appear 1999.

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J. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1:49-67, 1991.

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J. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1:49--67, 1991.

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

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