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A Categorical Programming Language
, 1987
"... A theory of data types and a programming language based on category theory are presented. Data types play a crucial role in programming. They enable us to write programs easily and elegantly. Various programming languages have been developed, each of which may use different kinds of data types. Ther ..."
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A theory of data types and a programming language based on category theory are presented. Data types play a crucial role in programming. They enable us to write programs easily and elegantly. Various programming languages have been developed, each of which may use different kinds of data types. Therefore, it becomes important to organize data types systematically so that we can understand the relationship between one data type and another and investigate future directions which lead us to discover exciting new data types. There have been several approaches to systematically organize data types: algebraic specification methods using algebras, domain theory using complete partially ordered sets and type theory using the connection between logics and data types. Here, we use category theory. Category theory has proved to be remarkably good at revealing the nature of mathematical objects, and we use it to understand the true nature of data types in programming.
Algebraic logic, varieties of algebras, and algebraic varieties
, 1995
"... Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could ..."
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Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.
A Comparison of Two ToposTheoretic Approaches to quantum theory,” arXiv:1010.2031 [mathph
"... iv ..."
The Definition of Extended ML
, 1994
"... This document formally defines the syntax and semantics of the Extended ML language. It is based directly on the published semantics of Standard ML in an attempt to ensure compatibility between the two languages. LFCS, Department of Computer Science, University of Edinburgh, Edinburgh, Scotland. ..."
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This document formally defines the syntax and semantics of the Extended ML language. It is based directly on the published semantics of Standard ML in an attempt to ensure compatibility between the two languages. LFCS, Department of Computer Science, University of Edinburgh, Edinburgh, Scotland. y Institute of Informatics, Warsaw University, and Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland. ii CONTENTS Contents 1 Introduction 1 1.1 Behavioural equivalence : : : : : : : : : : : : : : : : : : : : : : : : 3 1.2 Metalanguage : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Syntax of the Core 8 2.1 Reserved Words : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 2.2 Special constants : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 2.3 Comments : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.4 Identifiers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.5 Lexical analysis : : : :...
Covers for Monoids
"... . In this contribution to the structure theory of semigroups, we propose a unified generalisation of a string of results on group extensions, originating on the one hand in the seminal structure and covering theorems of McAlister and on the other, in Ash's celebrated solution of the Rhodes conj ..."
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. In this contribution to the structure theory of semigroups, we propose a unified generalisation of a string of results on group extensions, originating on the one hand in the seminal structure and covering theorems of McAlister and on the other, in Ash's celebrated solution of the Rhodes conjecture in finite semigroup theory. McAlister proved that each inverse monoid admits an Eunitary cover, and gave a structure theorem for Eunitary inverse monoids. Subsequent generalisations extended one or both results to orthodox monoids (McAlister, Szendrei, Takizawa), regular monoids (Trotter), Edense semigroups in which the idempotents form a semilattice (Margolis and Pin, Fountain), and Edense semigroups in which the idempotents form a subsemigroup (Almeida, Pin and Weil, Zhonghao Jiang). We show that any Edense monoid admits a Dunitary Edense cover and we provide a structure theorem for Dunitary Edense monoids, in terms of groups acting on a category. Here D(M) is the least weakly s...
Relational Graph Rewritings
 Theoretical Computer Science, Vol
, 1995
"... This note presents a new formalization of graph rewritings which generalizes traditional graph rewritings. Relational notions of graphs and their rewritings are introduced and several properties about graph rewritings are discussed using relational calculus (theory of binary relations). Single pusho ..."
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This note presents a new formalization of graph rewritings which generalizes traditional graph rewritings. Relational notions of graphs and their rewritings are introduced and several properties about graph rewritings are discussed using relational calculus (theory of binary relations). Single pushout approaches to graph rewritings proposed by Raoult and Kennaway are compared with our rewritings of relational (labeled) graph. Moreover a more general sufficient condition for two rewritings to commute and a theorem concerning critical pairs useful to demonstrate the confluency of graph rewriting systems are also given. 1 Introduction There are many researches [17,9,13,14,1618,2022] on graph grammars and graph rewritings which have a lot of applications including software specification, data bases, analysis of concurrent systems, developmental biology and many others. In these one of the advantages of categorical graph rewritings is to produce a universal reduction which eases theoret...
Categorical Framework for the Transformation of ObjectOriented Systems: Models and Data. Technical report 02010/01, Fachhochschule für die Wirtschaft Hannover, 2010. Submitted for publication in
 Journal of Symbolic Computation. http://fhdwdev.ha.bib.de/docmgr/index.php?module=fileview&objectId=525
"... Abstract: Refactoring of information systems is hard, for two reasons. On the one hand, large databases exist which have to be adjusted. On the other hand, many programs access that data. These programs all have to be migrated in a consistent manner such that their semantics does not change. It cann ..."
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Abstract: Refactoring of information systems is hard, for two reasons. On the one hand, large databases exist which have to be adjusted. On the other hand, many programs access that data. These programs all have to be migrated in a consistent manner such that their semantics does not change. It cannot be relied upon, however, that no running processes exist during such a migration. Consequently, a refactoring of an information system needs to take care of the migration of data, programs, and processes. This paper extends the model described in [SLK10] by operations, messages, and methods, which allows to model complete objectoriented systems. 1 Methods are expressed by special doublepushout graph transformations. Homomorphisms are used for the typing of the instance level as well as for the description of refactorings which specify the addition, folding, and unfolding of schema elements. Finally, a categorical framework is presented which allows to derive instance migrations from schema transformations in such a way that programs and processes to the old schema are correctly migrated into programs and processes to the new schema.
A Physical Quantum Model in a Smooth Topos
, 2008
"... We strengthen the case that the new logical perspective afforded by topos theory is suitable to the task of describing the physical world around us. In exploring some of the aspects of construction of a simple quantummechanical system in a mathematical universe different from that represented by se ..."
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We strengthen the case that the new logical perspective afforded by topos theory is suitable to the task of describing the physical world around us. In exploring some of the aspects of construction of a simple quantummechanical system in a mathematical universe different from that represented by set theory, we show that more thought and a better appreciation of the assumptions going into any mathematical model of the physical world are needed. We reflect on some of the mathematical consequences of this wider perspective of physics, explaining one interpretation of probabilistic values and numerical calculations in two mathematical universes governed by the less restrictive intuitionistic logic, and known to support the theory of synthetic differential geometry.
Refactoring ObjectOriented Systems
"... Abstract. Refactoring of information systems is hard, for two reasons. On the one hand, large databases exist which have to be adjusted. On the other hand, many programs access that data. These programs all have to be migrated in a consistent manner such that their semantics does not change. It cann ..."
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Abstract. Refactoring of information systems is hard, for two reasons. On the one hand, large databases exist which have to be adjusted. On the other hand, many programs access that data. These programs all have to be migrated in a consistent manner such that their semantics does not change. It cannot be relied upon, however, that no running processes exist during such a migration. Consequently, a refactoring of an information system needs to take care of the migration of data, programs, and processes. This paper introduces a model for complete objectoriented systems, describing the schema level with classes, associations, operations, and inheritance as well as the instance level with objects, links, methods, and messages. Methods are expressed by special doublepushout graph transformations. Homomorphisms are used for the typing of the instance level as well as for the description of refactorings which specify the addition, folding, and unfolding of schema elements. Finally, a categorial framework is presented which allows to derive instance migrations from schema transformations in such a way that programs and processes to the old schema are correctly migrated into programs and processes to the new schema. 1