Much attention has been given recently to the mechanism of fibring of logics, allowing free mixing of the connectives and using proof rules from both logics. Fibring seems to be a rather useful and general form of combination of logics that deserves detailed study. It is now well understood at the proof-theoretic level. However, the semantics of fibring is still insufficiently understood. Herein we provide a categorial definition of both proof-theoretic and model-theoretic fibring for logics without terms. To this end, we introduce the categories of Hilbert calculi, interpretation systems and logic system presentations. By choosing appropriate notions of morphism it is possible to obtain pure fibring as a coproduct. Fibring with shared symbols is then easily obtained by cocartesian lifting from the category of signatures. Soundness is shown to be preserved by these constructions. We illustrate the constructions within propositional modal logic.

: This paper uses concepts from sheaf theory to explicate phenomena in concurrent systems, including object, inheritance, deadlock, and non-interference, as used in computer security. The approach is very general, and applies not only to concurrent object oriented systems, but also to systems of differential equations, electrical circuits, hardware description languges, and much more. Time can be discrete or continuous, linear or branching, and distribution is allowed over space as well as time. Concepts from category theory help to achieve this generality: objects are modeled by sheaves; inheritance by sheaf morphisms; systems by diagrams; and interconnections by diagrams of diagrams. In addition, behaviour is given by limit, and the result of interconnection by colimit. The approach is illustrated with many examples, including a semantics for a simple concurrent object-based programming language. 1 Introduction Many popular formalisms for concurrent systems are syntactic (or "formal...

A completeness theorem is established for logics with congruence endowed with general semantics (in the style of general frames). As a corollary, completeness is shown to be preserved by bring logics with congruence provided that congruence is retained in the resulting logic. The class of logics with equivalence is shown to be closed under bring and to be included in the class of logics with congruence. Thus, completeness is shown to be preserved by bring logics with equivalence and general semantics. An example is provided showing that completeness is not always preserved by bring logics endowed with standard (non general) semantics. A categorial characterization of bring is provided using coproducts and cocartesian liftings. 1 Introduction Much attention has been recently given to the problems of combining logics and obtaining transference results. Besides leading to very interesting applications whenever it is necessary to work with dierent logics at the same time, ...

We give a direct type-theoretic characterization of the basic mechanisms of object-oriented programming, including objects, methods, message passing, and subtyping, by introducing an explicit constructor for object types and suitable introduction, elimination, and equality rules. The resulting abstract framework provides a basis for justifying and comparing previous encodings of objects based on recursive record types (Cardelli, 1984; Cardelli, 1992; Bruce, 1994; Cook et al., 1990; Mitchell, 1990a) and encodings based on existential types (Pierce & Turner, 1994).

ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe is written Set instead of U . The El-operator is omitted. For example the \Pi-type is described by the following constant and equality declarations (understood in every valid context): ` \Pi : (oe: Set; : (oe)Set)Set ` App : (oe: Set; : (oe)Set; m: \Pi(oe; ); n: oe) (m) ` : (oe: Set; : (oe)Set; m: (x: oe) (x))\Pi(oe; ) oe: Set; : (oe)Set; m: (x: oe) (x); n: oe ` App(oe; ; (oe; ; m); n) = m(n) Notice, how terms with free variables are represented as framework abstractions (in the type of ) and how substitution is represented as framework application (in the type of App and in the equation). In this way the burden of dealing correctly with variables, substitution, and binding is s...

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Although Entity-Relationship (ER) modelling techniques are commonly used for information modelling, Object-Role Modelling (ORM) techniques are becoming increasingly popular, partly because they include detailed design procedures providing guidelines for the modeller. As with the ER approach, a number of different ORM techniques exist. In this paper, we propose an integration of two theoretically well founded ORM techniques: FORM and PSM. Our main focus is on a common terminological framework, and on the notion of subtyping. Subtyping has long been an important feature of semantic approaches to conceptual schema design. It is also the concept in which FORM and PSM differ the most in their formalization. The subtyping issue is discussed from three different viewpoints covering syntactical, identification, and population issues. Finally, a wider comparison of approaches to subtyping is made, which encompasses other ER-based and ORM-based information modelling techniques, and highlights how formal subtype definitions facilitate a comprehensive specification of subtype constraints.

Denotational semantics [Sch86] is a powerful framework for describing programming languages; however, its descriptions lack modularity: conceptually independent language features influence each others' semantics. We address this problem by presenting a theory of modular denotational semantics. Following Mosses [Mos92], we divide a semantics into two parts, a computation ADT and a language ADT (abstract data type). The computation ADT represents the basic semantic structure of the language. The language ADT represents the actual language constructs, as described by a grammar. We define the language ADT using the computation ADT; in fact, language constructs are polymorphic over many different computation ADTs. Following Moggi [Mog89a], we build the computation ADT from composable parts, using monads and monad transformers. These techniques allow us to build many different computation ADTs, and, since our language constructs are polymorphic, many different language semantics. We autom...

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It is well-known that a term rewriting system can be faithfully described by a cartesian 2-category, where horizontal arrows represent terms, and cells represent rewriting sequences. In this paper we propose a similar, original 2-categorical presentation for term graph rewriting. Building on a result presented in [8], which shows that term graphs over a given signature are in one-to-one correspondence with arrows of a gs-monoidal category freely generated from the signature, we associate with a term graph rewriting system a gs-monoidal 2-category, and show that cells faithfully represent its rewriting sequences. We exploit the categorical framework to relate term graph rewriting and term rewriting, since gs-monoidal (2-)categories can be regarded as "weak" cartesian (2-)categories, where certain (2-)naturality axioms have been dropped.