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.

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.

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 : : : :...

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos T (A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra A. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum �(A) in T (A), which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on �, and self-adjoint elements of A define continuous functions (more precisely, locale maps) from � to Scott’s interval domain. Noting that open subsets of �(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T (A). These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.

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 [1-7,9,13,14,16-18,20-22] 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...

. 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 E-unitary cover, and gave a structure theorem for E-unitary inverse monoids. Subsequent generalisations extended one or both results to orthodox monoids (McAlister, Szendrei, Takizawa), regular monoids (Trotter), E-dense semigroups in which the idempotents form a semilattice (Margolis and Pin, Fountain), and E-dense semigroups in which the idempotents form a subsemigroup (Almeida, Pin and Weil, Zhonghao Jiang). We show that any E-dense monoid admits a D-unitary E-dense cover and we provide a structure theorem for D-unitary E-dense monoids, in terms of groups acting on a category. Here D(M) is the least weakly s...

Syntax. The current syntax allows multiple concrete syntactic representations of the same graph structure. Investigate an abstract syntax (canonical form) for ER Graphs. The mapping from the (various) concrete syntaxes to the abstract shall be defined. . Formalization. The current specification of ER Graphs representations, and its key concepts are not formalized. Formalize the definitions of these concepts using logical or other mathematical tools, such as those used in the AI/KR community. . Subgraph Relations. The current structure does not allow specification of subgraphs, or relationships between graphs, between graphs and subgraphs, or intersection and union of graphs. The only method of nesting graph definitions is currently reification. . Relation Properties. What properties are needed for defining relationships in a relationship graphs? Examples include total orders, symmetry, and other mathematical properties of relations. If such properties are needed, the rules need to b...

Introduction The very first sentence of Mac Lane and Moerdijk [5] says: A topos can be considered both as a "generalized space" and as a "generalized universe of sets". The "generalized universe of sets" aspect of toposes is relatively easy to understand and is well documented in the literature: start with Goldblatt [1] and proceed via Mac Lane and Moerdijk [5], or MacLarty [4], to Johnstone [2]. The basic trick is to use categorical properties to characterize set-theoretic constructions in the category of sets, and thence to transfer them to other categories that are sufficiently similar. The generalized spaces, on the other hand, though present in ideas of toposes right from their introduction by Grothendieck, are somewhat mysterious. Much of this is because the generalized universes of sets are not direct expressions of the spatial idea but represent it by a mathematical duality. My aim here is at least to present a clear picture of how intuitions of generalized spaces

This paper studies operations for creating new variants of a program that relate, in a well-defined way, to existing variants of the program. We formalize a program modification as a (special kind of) function from programs to programs, and study the algebra of these program modifications. We make use of the algebraic structure to formalize several intuitive concepts, such as that of "compatibility among program modifications", and establish several new results concerning the problems of program merging and separating consecutive edits. We also identify a category in which the objects are programs and the morphisms are program modifications, and show how program integration relates to the pushout in this category

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 object-oriented 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 double-pushout 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