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A Compositional Approach to Structuring and Refinement of Typed Graph Grammars
 Proc. of SEGRAGRA'95 "Graph Rewriting and Computation", Electronic Notes of TCS
, 1995
"... Based on a categorical semantics that has been developed for typed graph grammars we uses colimits (pushouts) to model composition and (reverse) graph grammar morphisms to describe refinements of typed graph grammars. Composition of graph grammars w.r.t. common subgrammars is shown to be compatible ..."
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Cited by 11 (2 self)
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Based on a categorical semantics that has been developed for typed graph grammars we uses colimits (pushouts) to model composition and (reverse) graph grammar morphisms to describe refinements of typed graph grammars. Composition of graph grammars w.r.t. common subgrammars is shown to be compatible with the semantics, i.e. the semantics of the composed grammar is obtained as the composition of the semantics of the component grammars. Moreover, the structure of a composed grammar is preserved during a refinement step in the sense that compatible refinements of the components induce a refinement of the composition. The concepts and results are illustrated by an example. 1 Introduction This contribution addresses the structuring and refinement of typed graph grammars defined according to the algebraic double pushout approach [5]. Typed graph grammars are introduced in [2] for the double pushout approach (cf. [6] for a corresponding notion in the singlepushout setting). They generalize ...
Enriched Categories as Models of Computation
 in Proc. Fifth Italian Conference on Theoretical Computer Science, ICTCS'95 , World Scientific
, 1996
"... . In this paper we discuss a general methodology to provide a categorical semantics for a wide class of computational systems, whose behaviour can be described by a suitable set of transition steps. We open our survey presenting some results on the semantics of Petri Nets. Starting from this, we ela ..."
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Cited by 11 (4 self)
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. In this paper we discuss a general methodology to provide a categorical semantics for a wide class of computational systems, whose behaviour can be described by a suitable set of transition steps. We open our survey presenting some results on the semantics of Petri Nets. Starting from this, we elaborate a twosteps procedure allowing for the description of all the sequences of transitions performed by a given system, and equipping them with a suitable equivalence relation. This relation provides the sistem under analisys with a concurrent semantics: equivalence classes denote families of "computationally equivalent" behaviours, corresponding to the execution of the same set of (causally) independent transition steps. 1 Introduction The latest years have seen a wide amount of different approaches to the semantics of computional sistems: a variety that, if only for the comparison between the various formalisms, calls for a unified framework. In this paper we aim to show that enriched ...
Duality in knowledge sharing
 IN 7TH INTERNATIONAL SYMPOSIUM ON ARTIFICIAL INTELLIGENCE AND MATHEMATICS, FT
, 2002
"... I propose a formalisation of knowledge sharing scenarios that aims at capturing the crucial role played by an existing duality between ontological theories one wants to merge and particular situations that need to be linked. I use diagrams in the Chu category and colimits over these diagrams to acco ..."
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Cited by 11 (9 self)
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I propose a formalisation of knowledge sharing scenarios that aims at capturing the crucial role played by an existing duality between ontological theories one wants to merge and particular situations that need to be linked. I use diagrams in the Chu category and colimits over these diagrams to account for the reliability and optimality of knowledge sharing systems. Furthermore, I show how we may obtain a deeper understanding of a system that shares knowledge between a probabilistic logic program and Bayesian belief networks by reanalysing the scenario in terms of the present approach.
Precategories for Combining Probabilistic Automata
 Electronic Notes in Theoretical Computer Science
, 1999
"... 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 partial ..."
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Cited by 10 (6 self)
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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...
Management of Evolving Specifications Using Category Theory
, 1998
"... Structure is important in large specifications for understanding, testing and managing change. Category theory has been explored as framework for providing this structure, and has been successfully used to compose specifications. This work has typically adopted a "correct by construction" approach: ..."
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Cited by 10 (0 self)
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Structure is important in large specifications for understanding, testing and managing change. Category theory has been explored as framework for providing this structure, and has been successfully used to compose specifications. This work has typically adopted a "correct by construction" approach: components are specified, proved correct and then composed together in such a way to preserve their properties. However, in a large project, it is desirable to be able to mix specification and composition steps such that at any particular moment in the process, we may have established only some of the properties of the components, and some of the composition relations. In this paper we propose adaptations to the categorical framework in order to manage evolving specifications. We demonstrate the utility of the framework on the analysis of a part of a software change request for the Space Shuttle.
Concurrent Computing: from Petri Nets to Graph Grammars
 Electronic Notes in Theoretical Computer Science
, 1995
"... Petri nets are widely accepted as a specification formalism for concurrent and distributed systems. One of the reasons of their success is the fact that they are equipped with a rich theory, including wellunderstood concurrent semantics; they also provide an interesting benchmark for tools and tech ..."
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Cited by 9 (0 self)
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Petri nets are widely accepted as a specification formalism for concurrent and distributed systems. One of the reasons of their success is the fact that they are equipped with a rich theory, including wellunderstood concurrent semantics; they also provide an interesting benchmark for tools and techniques for the description of concurrent systems. Graph grammars can be regarded as a proper generalization of Petri nets, where the current state of a system is described by a graph instead as by a collection of tokens. In this tutorial paper I will review some basic definitions and constructions concerning the concurrent semantics of nets, and I will show to what extent corresponding notions have been developed for graph grammars. Most of such results come out from a joint research by the Berlin and Pisa COMPUGRAPH groups. 1 Introduction The nets which owe their name to Carl Adam Petri [28,29] have been the first formal tool proposed for the specification of the behaviour of systems which...
Conservativity in Structured Ontologies
"... Using category theoretic notions, in particular diagrams and their colimits, we provide a common semantic backbone for various notions of modularity in structured ontologies, and outline a general approach for representing (heterogeneous) combinations of ontologies through interfaces of various kind ..."
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Cited by 9 (4 self)
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Using category theoretic notions, in particular diagrams and their colimits, we provide a common semantic backbone for various notions of modularity in structured ontologies, and outline a general approach for representing (heterogeneous) combinations of ontologies through interfaces of various kinds, based on the theory of institutions. This covers theory interpretations, (definitional) language extensions, symbol identifications, and conservative extensions. In particular, we study the problem of inheriting conservativity between subtheories in a diagram to its colimit ontology, and apply this to the problem of localisation of reasoning in ‘modular ontology languages’ such as DDLs or Econnections.
Mathematics of generic specifications for model management
 Encyclopedia of Database Technologies and Applications
, 2005
"... This article (further referred to as MathI), and the next one (further referred to as MathII, see p. 359), form a mathematical companion to the article in this encyclopedia on Generic Model Management (further referred to as GenMMt, see p.258). Articles MathI and II present the basics of the arro ..."
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Cited by 9 (7 self)
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This article (further referred to as MathI), and the next one (further referred to as MathII, see p. 359), form a mathematical companion to the article in this encyclopedia on Generic Model Management (further referred to as GenMMt, see p.258). Articles MathI and II present the basics of the arrow diagram machinery that provides model management with truly generic specifications. Particularly, it allows us to build a generic pattern for heterogeneous data and schema transformation, which is presented in MathII for the first time in the literature.
On the Role of Category Theory in the Area of Algebraic Specifications
 In LNCS , Proc. WADT11
, 1996
"... . The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic specifications. In detail we discuss different approaches to an abstract theory of specification logics. Further we present a uniform framework for developing pa ..."
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Cited by 9 (2 self)
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. The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic specifications. In detail we discuss different approaches to an abstract theory of specification logics. Further we present a uniform framework for developing particular specification logics. We make use of `classifying categories', to present categories of algebras as functor categories and to obtain necessary basic results for particular specification logics in a uniform manner. The specification logics considered are: equational logic for total algebras, conditional equational logic for partial algebras, and rewrite logic for concurrent systems. 1 Category Theory and Applications in Computer Science Category theory has been developed as a mathematical theory over 50 years and has influenced not only almost all branches of structural mathematics but also the development of several areas of computer science. It is the aim of this paper to review t...
Interconnection of Object Specifications
 Formal Methods and Object Technology
, 1996
"... ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, and ap ..."
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Cited by 8 (2 self)
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ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, and applying no methods corresponds to the identity of the monoid. A monoid is a set M with an associative binary operation ffl M : M \ThetaM ! M , usually referred to as `multiplication', which has an identity element e M 2 M . If M = (M; ffl M ; e M ) is a monoid, we often write just M for M, and e for e M ; moreover for m;m 0 2 M , we usually write mm 0 instead of m ffl M m 0 . For example, A , the set of lists containing elements of A, together with concatenation ++ : A \ThetaA ! A and the empty list [ ] 2 A , is a monoid. This example is especially important for the material in later sections. A monoid homomorphism is a structure preserving map between the carriers of ...