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316
Fibring of logics as a categorial construction
 Journal of Logic and Computation
, 1999
"... 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 p ..."
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Cited by 51 (31 self)
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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 prooftheoretic level. However, the semantics of fibring is still insufficiently understood. Herein we provide a categorial definition of both prooftheoretic and modeltheoretic 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.
Computations in a Free Lie Algebra
, 1998
"... Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the BakerCampbellHausdorff formula and the recently developed Lie group methods for integration of differential equations on manifolds. This paper is concerned with complexity ..."
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Cited by 51 (15 self)
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Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the BakerCampbellHausdorff formula and the recently developed Lie group methods for integration of differential equations on manifolds. This paper is concerned with complexity and optimization of such computations in the general case where the Lie algebra is free, i.e. no specific assumptions are made about its structure. It is shown how transformations applied to the original variables of a problem yield elements of a graded free Lie algebra whose homogeneous subspaces are of much smaller dimension than the original ungraded one. This can lead to substantial reduction of the number of commutator computations. Witts formula for counting commutators in a free Lie algebra is generalized to the case of a general grading. This provides accurate bounds on the complexity. The interplay between symbolic and numerical computations is also discussed, exemplified by the new...
Sheaf Semantics for Concurrent Interacting Objects
 Mathematical Structures in Computer Science
, 1992
"... : This paper uses concepts from sheaf theory to explicate phenomena in concurrent systems, including object, inheritance, deadlock, and noninterference, as used in computer security. The approach is very general, and applies not only to concurrent object oriented systems, but also to systems of dif ..."
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Cited by 45 (5 self)
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: This paper uses concepts from sheaf theory to explicate phenomena in concurrent systems, including object, inheritance, deadlock, and noninterference, 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 objectbased programming language. 1 Introduction Many popular formalisms for concurrent systems are syntactic (or "formal...
Fibring: Completeness Preservation
 Journal of Symbolic Logic
, 2000
"... 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 ..."
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Cited by 45 (23 self)
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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, ...
Syntax and Semantics of Dependent Types
 Semantics and Logics of Computation
, 1997
"... 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 ..."
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Cited by 40 (4 self)
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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 Eloperator is omitted. For example the \Pitype 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...
On the Interpretation of Type Theory in Locally Cartesian Closed Categories
 Proceedings of Computer Science Logic, Lecture Notes in Computer Science
, 1994
"... . We show how to construct a model of dependent type theory (category with attributes) from a locally cartesian closed category (lccc). This allows to define a semantic function interpreting the syntax of type theory in an lccc. We sketch an application which gives rise to an interpretation of exten ..."
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Cited by 38 (1 self)
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. We show how to construct a model of dependent type theory (category with attributes) from a locally cartesian closed category (lccc). This allows to define a semantic function interpreting the syntax of type theory in an lccc. We sketch an application which gives rise to an interpretation of extensional type theory in intensional type theory. 1 Introduction and Motivation Interpreting dependent type theory in locally cartesian closed categories (lcccs) and more generally in (non split) fibrational models like the ones described in [7] is an intricate problem. The reason is that in order to interpret terms associated with substitution like pairing for \Sigma types or application for \Pitypes one needs a semantical equivalent to syntactic substitution. To clarify the issue let us have a look at the "naive" approach described in Seely's seminal paper [14] which contains a subtle inaccuracy. Assume some dependently typed calculus like the one defined in [10] and an lccc C (a category ...
A Unifying TypeTheoretic Framework for Objects
, 1993
"... We give a direct typetheoretic characterization of the basic mechanisms of objectoriented 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 abst ..."
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Cited by 37 (9 self)
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We give a direct typetheoretic characterization of the basic mechanisms of objectoriented 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).
Semantic Lego
, 1995
"... 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. Follo ..."
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Cited by 35 (0 self)
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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...
Subtyping and Polymorphism in ObjectRole Modelling
, 1995
"... Although EntityRelationship (ER) modelling techniques are commonly used for information modelling, ObjectRole 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 nu ..."
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Cited by 35 (22 self)
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Although EntityRelationship (ER) modelling techniques are commonly used for information modelling, ObjectRole 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 ERbased and ORMbased information modelling techniques, and highlights how formal subtype definitions facilitate a comprehensive specification of subtype constraints.