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Categorical Term Rewriting: Monads and Modularity
 University of Edinburgh
, 1998
"... Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting syste ..."
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Cited by 11 (6 self)
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Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting systems from smaller ones. Of particular interest is whether key properties are modular, thatis,ifthe components of a structured term rewriting system satisfy a property, then does the term rewriting system as a whole? A body of literature addresses this problem, but most of the results and proofs depend on strong syntactic conditions and do not easily generalize. Although many specific modularity results are known, a coherent framework which explains the underlying principles behind these results is lacking. This thesis posits that part of the problem is the usual, concrete and syntaxoriented semantics of term rewriting systems, and that a semantics is needed which on the one hand elides unnecessary syntactic details but on the other hand still possesses enough expressive power to model the key concepts arising from
From SOS specifications to structured coalgebras: How to make bisimulation a congruence
 ENTCS, 19(0):118 – 141
, 1999
"... In this paper we address the issue of providing a structured coalgebra presentation of transition systems with algebraic structure on states determined by an equational specification Γ. More precisely, we aim at representing such systems as coalgebras for an endofunctor on the category of Γalgebras ..."
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Cited by 10 (2 self)
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In this paper we address the issue of providing a structured coalgebra presentation of transition systems with algebraic structure on states determined by an equational specification Γ. More precisely, we aim at representing such systems as coalgebras for an endofunctor on the category of Γalgebras. The systems we consider are specified by using a quite general format of SOS rules, the algebraic format, which in general does not guarantee that bisimilarity is a congruence. We first show that the structured coalgebra representation works only for systems where transitions out of complex states can be derived from transitions out of corresponding component states. This decomposition property of transitions indeed ensures that bisimilarity is a congruence. For a system not satisfying this requirement, next we propose a closure construction which adds context transitions, i.e., transitions that spontaneously embed a state into a bigger context or viceversa. The notion of bisimulation for the enriched system coincides with the notion of dynamic bisimilarity for the original one, that is, with the coarsest bisimulation which is a congruence. This is sufficient to ensure that the structured coalgebra representation works for the systems obtained as result of the closure construction. 1
Sketches: Outline with references
, 1993
"... This package contains the original article, written in December, 1993, and this addendum, ..."
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Cited by 3 (0 self)
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This package contains the original article, written in December, 1993, and this addendum,
An Algebra of Graph Derivations Using Finite (co) Limit Double Theories
"... Graph transformation systems have been introduced for the formal specication of software systems. States are thereby modeled as graphs, and computations as graph derivations according to the rules of the specication. Operations on graph derivations provide means to reason about the distribution ..."
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Cited by 2 (1 self)
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Graph transformation systems have been introduced for the formal specication of software systems. States are thereby modeled as graphs, and computations as graph derivations according to the rules of the specication. Operations on graph derivations provide means to reason about the distribution and composition of computations. In this paper we discuss the development of an algebra of graph derivations as a descriptive model of graph transformation systems. For that purpose we use a categorical three level approach for the construction of models of computations based on structured transition systems. Categorically the algebra of graph derivations can then be characterized as a free double category with nite horizontal colimits.
Generalised Sketches as an algebraic graphbased framework for semantic modeling and database design
, 1997
"... A graphbased specification language and the corresponding machinery are described as stating a basic framework for semantic modeling and database design. It is shown that a few challenging theoretical questions in the area, and some of hot practical problems as well, can be successfully approached ..."
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Cited by 1 (0 self)
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A graphbased specification language and the corresponding machinery are described as stating a basic framework for semantic modeling and database design. It is shown that a few challenging theoretical questions in the area, and some of hot practical problems as well, can be successfully approached in the framework. The machinery has its origin in the classical sketches invented by Ehresmann and is close to their generalization recently proposed by Makkai. There are two essential distinctions from Makkai's sketches. One consists in a different  more direct  formalization of sketches that categorists (and database designers) usually draw. The second distinction is more fundamental and consists in introducing operational sketches specifying complex diagram operations over ordinary (predicate) sketches, correspondingly, models of operational sketches are diagram algebras. Together with the notion of parsing operational sketches, this is the main mathematical contribution of the pape...
Sketches, Views and PatternBased Reasoning
"... Abstract—The mathematical theory of sketches provides a graphical framework for describing and relating knowledge representations and their models. Maps between sketches can extract domainspecific context from a sketch, express knowledge dynamics and be used to manage representations created for di ..."
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Abstract—The mathematical theory of sketches provides a graphical framework for describing and relating knowledge representations and their models. Maps between sketches can extract domainspecific context from a sketch, express knowledge dynamics and be used to manage representations created for distinct applications or by different analysts. There are precise connections between classes of sketches and fragments of firstorder, infinitary predicate logic. EA sketches are a particular class that is related to entityattributerelation diagrams and can be implemented using features available in many relational database systems. In this paper we illustrate sketch theory through development of a simple human terrain model. We apply the theory to an example of aligning sketchbased knowledge representations and compare the approach to one using OWL/RDF. We describe the computational infrastructure that is available for working with sketches and outline research challenges. I.
ON CATEGORIES OF COHESIVE, ACTIVE SETS AND OTHER DYNAMIC SYSTEMS BY
"... This work is intended to contribute to a program of developing general concepts and methods for applying category theory to the modeling and solution o f scientic problems. It was motivated by my experiences as an engineering student studying continuum and kinetic models of
uid
ows. The language o ..."
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This work is intended to contribute to a program of developing general concepts and methods for applying category theory to the modeling and solution o f scientic problems. It was motivated by my experiences as an engineering student studying continuum and kinetic models of
uid
ows. The language of category theory is employed because it facilitates precise comparisons between diverse types of structures. Using this language to investigate relationships between
uid
ow models requires categorical specications of constitutive relations and of idioms occurring across classications of dynamic systems. It is shown that a categorytheoretic denition of chaotic system applies not only to the Smale horseshoe, a standard chaotic system, but also to Conway's \Game of Life " automaton. Symbolic dynamics of the \Dining Philosophers " relational system is computed. A category composed of stochastic matrices is dened and some of its elementary properties are developed. A categorical variant of symbolic dynamics is applied to a nite stochastic process. Using pointwise Kan extension formulas, conditions ensuring existence of certain representations among categories of dynamic systems are proved. iii For my parents iv Acknowledgments Thanks, Mom and Dad. You are the greatest. I am nally done! Thanks to my wife Susan for being supportive, generous, and trusting while I've worked on this and while we lived far apart. Thanks to my grandparents Byron Oliver, Wesley Wojtowicz, and Josephine Oliver. You are greatly missed. Aloha and thank you Janet Wojtowicz for the many things you still do for our family. Thanks Michele and Shane for the wellwishes.
Specifying, Programming and Verifying with Equational Logic
"... 1 Introduction Programming is difficult, as shown by the fact that debugging a program usually takes more time than creating it; moreover, the difficulty of debugging increases nonlinearly with program size. One reason for such phenomena is the astonishing complexity and subtlety of the semantics o ..."
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1 Introduction Programming is difficult, as shown by the fact that debugging a program usually takes more time than creating it; moreover, the difficulty of debugging increases nonlinearly with program size. One reason for such phenomena is the astonishing complexity and subtlety of the semantics of most widely used programming languages, due mainly to the desire for high efficiency on conventional processors. But rapid increases in the power and flexibility of hardware, and in the need for greater reliability and security in applications, suggest that it may be valuable to consider alternative approaches, based on higher level languages with much simpler semantics, despite the undoubted inertia of tradition, and the difficulty of learning new languages and new paradigms. This paper focuses on the OBJ family of languages, which have semantics based on various extensions of (first order) equational logic. The OBJ languages are logical programming languages, in which programs are theories, and computation is deduction, which makes it possible to do specification, programming and verification in a unified framework. This paper is mainly intended to introduce and motivate the material that it covers, rather than to provide a thorough mathematical exposition. Consequently, there are many references and several examples, but all proofs and many technical details are omitted.