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Types, Abstraction, and Parametric Polymorphism, Part 2
, 1991
"... The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and P ..."
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Cited by 53 (1 self)
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The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and PLcategory models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.
Categories and groupoids
, 1971
"... In 1968, when this book was written, categories had been around for 20 years and groupoids for twice as long. Category theory had by then become widely accepted as an essential tool in many parts of mathematics and a number of books on the subject had appeared, or were about to appear (e.g. [13, 22, ..."
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Cited by 40 (2 self)
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In 1968, when this book was written, categories had been around for 20 years and groupoids for twice as long. Category theory had by then become widely accepted as an essential tool in many parts of mathematics and a number of books on the subject had appeared, or were about to appear (e.g. [13, 22, 37, 58, 65] 1). By contrast, the use of groupoids was confined to a small number of pioneering articles, notably by Ehresmann [12] and Mackey [57], which were largely ignored by the mathematical community. Indeed groupoids were generally considered at that time not to be a subject for serious study. It was argued by several wellknown mathematicians that group theory sufficed for all situations where groupoids might be used, since a connected groupoid could be reduced to a group and a set. Curiously, this argument, which makes no appeal to elegance, was not applied to vector spaces: it was well known that the analogous reduction in this case is not canonical, and so is not available, when there is extra structure, even such simple structure as an endomorphism. Recently, Corfield in [41] has discussed methodological issues in mathematics with this topic, the resistance to the notion of groupoids, as a prime example. My book was intended chiefly as an attempt to reverse this general assessment of the time by presenting applications of groupoids to group theory
Process and Term Tile Logic
, 1998
"... In a similar way as 2categories can be regarded as a special case of double categories, rewriting logic (in the unconditional case) can be embedded into the more general tile logic, where also sideeffects and rewriting synchronization are considered. Since rewriting logic is the semantic basis o ..."
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Cited by 33 (25 self)
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In a similar way as 2categories can be regarded as a special case of double categories, rewriting logic (in the unconditional case) can be embedded into the more general tile logic, where also sideeffects and rewriting synchronization are considered. Since rewriting logic is the semantic basis of several language implementation efforts, it is useful to map tile logic back into rewriting logic in a conservative way, to obtain executable specifications of tile systems. We extend the results of earlier work by two of the authors, focusing on some interesting cases where the mathematical structures representing configurations (i.e., states) and effects (i.e., observable actions) are very similar, in the sense that they have in common some auxiliary structure (e.g., for tupling, projecting, etc.). In particular, we give in full detail the descriptions of two such cases where (net) processlike and usual term structures are employed. Corresponding to these two cases, we introduce two ca...
Mapping Tile Logic into Rewriting Logic
, 1998
"... . 1 Introduction Mapping Tile Logic into Rewriting Logic meseguer@csl.sri.com ugo@di.unipi.it Jos'e Meseguer and Ugo Montanari Rewriting logic [27, 28, 31] extends to concurrent systems with state changes the body of theory developed within the algebraic semantics approach. It can also be Rewriti ..."
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Cited by 32 (23 self)
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. 1 Introduction Mapping Tile Logic into Rewriting Logic meseguer@csl.sri.com ugo@di.unipi.it Jos'e Meseguer and Ugo Montanari Rewriting logic [27, 28, 31] extends to concurrent systems with state changes the body of theory developed within the algebraic semantics approach. It can also be Rewriting logic Tile logic membership equational logic 2 double 2VHcategories internal strategies uniform Metodi e Strumenti per la Progettazione e la Verifica di Sistemi Eterogenei Connessi mediante Reti di Comunicazione CONFER2 COORDINA Computer Science Laboratory, SRI International, Menlo Park, Dipartimento di Informatica, Universit`a di Pisa, extends to concurrent systems with state changes the body of theory developed within the algebraic semantics approach. It is both a foundational tool and the kernel language of several implementation efforts (Cafe, ELAN, Maude). extends (unconditional) rewriting logic since it takes into account state changes with side effects and synchronization. It is ...
Saturated semantics for reactive systems
 LOGIC IN COMPUTER SCIENCE
, 2006
"... The semantics of process calculi has traditionally been specified by labelled transition systems (LTS), but with the development of name calculi it turned out that reaction rules (i.e., unlabelled transition rules) are often more natural. This leads to the question of how behavioural equivalences (b ..."
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Cited by 27 (15 self)
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The semantics of process calculi has traditionally been specified by labelled transition systems (LTS), but with the development of name calculi it turned out that reaction rules (i.e., unlabelled transition rules) are often more natural. This leads to the question of how behavioural equivalences (bisimilarity, trace equivalence, etc.) defined for LTS can be transferred to unlabelled transition systems. Recently, in order to answer this question, several proposals have been made with the aim of automatically deriving an LTS from reaction rules in such a way that the resulting equivalences are congruences. Furthermore these equivalences should agree with the intended semantics, whenever one exists. In this paper we propose saturated semantics, based on a weaker notion of observation and orthogonal to all the previous proposals, and we demonstrate the appropriateness of our semantics by means of two examples: logic programming and a subset of the open πcalculus. Indeed, we prove that our equivalences are congruences and that they coincide with logical equivalence and open bisimilarity respectively, while equivalences studied in previous works are strictly finer.
Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
 Mem. Amer. Math. Soc
"... The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjun ..."
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Cited by 18 (8 self)
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The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this
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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Cited by 13 (5 self)
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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.
Executable Tile Specifications for Process Calculi
, 1999
"... . Tile logic extends rewriting logic by taking into account sideeffects and rewriting synchronization. These aspects are very important when we model process calculi, because they allow us to express the dynamic interaction between processes and "the rest of the world". Since rewriting logic is the ..."
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Cited by 13 (10 self)
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. Tile logic extends rewriting logic by taking into account sideeffects and rewriting synchronization. These aspects are very important when we model process calculi, because they allow us to express the dynamic interaction between processes and "the rest of the world". Since rewriting logic is the semantic basis of several language implementation efforts, an executable specification of tile systems can be obtained by mapping tile logic back into rewriting logic, in a conservative way. However, a correct rewriting implementation of tile logic requires the development of a metalayer to control rewritings, i.e., to discard computations that do not correspond to any deduction in tile logic. We show how such methodology can be applied to term tile systems that cover and extend a wideclass of SOS formats for the specification of process calculi. The wellknown casestudy of full CCS, where the term tile format is needed to deal with recursion (in the form of the replicator operator), is di...
The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
, 2007
"... Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained ..."
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Cited by 12 (0 self)
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Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. This development has formed a major part of Gordon Plotkin’s mature work, and we study its history here, in particular asking why Lawvere theories were eclipsed by monads in the 1960’s, and how the renewed interest in them in a computer science setting might develop in future.
Tile Transition Systems as Structured Coalgebras
 Fundamentals of Computation Theory, volume 1684 of LNCS
, 1999
"... . The aim of this paper is to investigate the relation between two models of concurrent systems: tile rewrite systems and coalgebras. Tiles are rewrite rules with side effects which are endowed with operations of parallel and sequential composition and synchronization. Their models can be described ..."
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Cited by 4 (2 self)
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. The aim of this paper is to investigate the relation between two models of concurrent systems: tile rewrite systems and coalgebras. Tiles are rewrite rules with side effects which are endowed with operations of parallel and sequential composition and synchronization. Their models can be described as monoidal double categories. Coalgebras can be considered, in a suitable mathematical setting, as dual to algebras. They can be used as models of dynamical systems with hidden states in order to study concepts of observational equivalence and bisimilarity in a more general setting. In order to capture in the coalgebraic presentation the algebraic structure given by the composition operations on tiles, coalgebras have to be endowed with an algebraic structure as well. This leads to the concept of structured coalgebras, i.e., coalgebras for an endofunctor on a category of algebras. However, structured coalgebras are more restrictive than tile models. Those models which can be presented as st...