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A Categorical Manifesto
 Mathematical Structures in Computer Science
, 1991
"... : This paper tries to explain why and how category theory is useful in computing science, by giving guidelines for applying seven basic categorical concepts: category, functor, natural transformation, limit, adjoint, colimit and comma category. Some examples, intuition, and references are given for ..."
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Cited by 100 (5 self)
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: This paper tries to explain why and how category theory is useful in computing science, by giving guidelines for applying seven basic categorical concepts: category, functor, natural transformation, limit, adjoint, colimit and comma category. Some examples, intuition, and references are given for each concept, but completeness is not attempted. Some additional categorical concepts and some suggestions for further research are also mentioned. The paper concludes with some philosophical discussion. 0 Introduction This paper tries to explain why category theory is useful in computing science. The basic answer is that computing science is a young field that is growing rapidly, is poorly organised, and needs all the help it can get, and that 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. The seven guidelines of this paper can h...
Little Theories
 Automated DeductionCADE11, volume 607 of Lecture Notes in Computer Science
, 1992
"... In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to forma ..."
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Cited by 49 (16 self)
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In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to formalize mathematics, and we describe how imps, an Interactive Mathematical Proof System, supports it.
On the Structure of Highlevel Nets
 Helsinki University of Technology
, 1995
"... : The structure of Highlevel nets is studied from an algebraic and a logical point of view using Algebraic nets as an example. First the category of Algebraic nets is defined and the semantics given through an unfolding construction. Other kinds of Highlevel net formalisms are then presented. It is ..."
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Cited by 10 (0 self)
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: The structure of Highlevel nets is studied from an algebraic and a logical point of view using Algebraic nets as an example. First the category of Algebraic nets is defined and the semantics given through an unfolding construction. Other kinds of Highlevel net formalisms are then presented. It is shown that nets given in these formalisms can be transformed into equivalent Algebraic nets. Then the semantics of nets in terms of universal constructions is discussed. A definition of Algebraic nets in terms of structured transition systems is proposed. The semantics of the Algebraic net is then given as a free completion of this structured transition system to a category. As an alternative also a sheaf semantics of nets is examined. Here the semantics of the net arises as a limit of a diagram of sheaves. Next Algebraic nets are characterized as encodings of special morphisms called foldings. Each algebraic net gives rise to a surjective morphism between Petri nets and conversely each sur...
Semantics of architectural specifications in Casl
 Proc. 4th Intl. Conf. Fundamental Approaches to Software Engineering FASE’01, Springer LNCS 2029
, 2001
"... Abstract. We present a semantics for architectural specifications in Casl, including an extended static analysis compatible with modeltheoretic requirements. The main obstacle here is the lack of amalgamation for Casl models. To circumvent this problem, we extend the Casl logic by introducing enrich ..."
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Cited by 8 (6 self)
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Abstract. We present a semantics for architectural specifications in Casl, including an extended static analysis compatible with modeltheoretic requirements. The main obstacle here is the lack of amalgamation for Casl models. To circumvent this problem, we extend the Casl logic by introducing enriched signatures, where subsort embeddings form a category rather than just a preorder. The extended model functor has amalgamation, which makes it possible to express the amalgamability conditions in the semantic rules in static terms. Using these concepts, we develop the semantics at various levels in an institutionindependent fashion.
Presheaves as Configured Specifications
"... The paper addresses a notion of configuring systems, constructing them from specified component parts with specified sharing. This notion is independent of any underlying specification language and has been abstractly identified with the taking of colimits in category theory. Mathematically it is kn ..."
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Cited by 2 (0 self)
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The paper addresses a notion of configuring systems, constructing them from specified component parts with specified sharing. This notion is independent of any underlying specification language and has been abstractly identified with the taking of colimits in category theory. Mathematically it is known that these can be expressed by presheaves and the present paper applies this idea to configuration. We interpret the category theory informally as follows. Suppose C is a category whose objects are interpreted as specifications, and for which each morphism u : X ! Y is interpreted as contravariant "instance reduction", reducing instances of specification Y to instances of X . Then a presheaf P : Set C op represents a collection of instances that is closed under reduction. We develop an algebraic account of presheaves in which we present configurations by generators (for components) and relations (for shared reducts), and we outline a proposed configuration language based on the techniques. Oriat uses diagrams to express colimits of specifications, and we show that Oriat's category Diag(C) of finite diagrams is equivalent to the category of finitely presented presheaves over C.