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14
A Tutorial on (Co)Algebras and (Co)Induction
 EATCS Bulletin
, 1997
"... . Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition pr ..."
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Cited by 228 (34 self)
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. Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition principle, and as a proof principle for such structures. But there are also important dual "coalgebraic" structures, which do not come equipped with constructor operations but with what are sometimes called "destructor" operations (also called observers, accessors, transition maps, or mutators). Spaces of infinite data (including, for example, infinite lists, and nonwellfounded sets) are generally of this kind. In general, dynamical systems with a hidden, blackbox state space, to which a user only has limited access via specified (observer or mutator) operations, are coalgebras of various kinds. Such coalgebraic systems are common in computer science. And "coinduction" is the appropriate te...
Relational Reasoning about Functions and Nondeterminism
, 1998
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Dissertation Series publications. Copies may be obtained by contacting: BRICS ..."
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Cited by 31 (4 self)
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Dissertation Series publications. Copies may be obtained by contacting: BRICS
Normalization and the Yoneda Embedding
"... this paper we describe a new, categorical approach to normalization in typed  ..."
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Cited by 22 (3 self)
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this paper we describe a new, categorical approach to normalization in typed 
Relating Operational and Denotational Semantics for Input/Output Effects
, 1999
"... We study the longstanding problem of semantics for input/output (I/O) expressed using sideeffects. Our vehicle is a small higherorder imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational se ..."
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Cited by 7 (3 self)
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We study the longstanding problem of semantics for input/output (I/O) expressed using sideeffects. Our vehicle is a small higherorder imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational semantics for I/O effects. We use a novel labelled transition system that uniformly expresses both applicative and imperative computation. We make a standard definition of bisimilarity. We prove bisimilarity is a congruence using Howe's method. Next, we define a metalanguage M in which we may give a denotational semantics to O. M generalises Crole and Pitts' FIXlogic by adding in a parameterised recursive datatype, which is used to model I/O. M comes equipped both with an operational semantics and a domaintheoretic semantics in the category CPPO of cppos (bottompointed posets with joins of !chains) and Scott continuous functions. We use the CPPO semantics to prove that M is computationally...
A Theory of Program Refinement
, 1998
"... We give a canonical program refinement calculus based on the lambda calculus and classical firstorder predicate logic, and study its proof theory and semantics. The intention is to construct a metalanguage for refinement in which basic principles of program development can be studied. The idea is t ..."
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Cited by 6 (1 self)
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We give a canonical program refinement calculus based on the lambda calculus and classical firstorder predicate logic, and study its proof theory and semantics. The intention is to construct a metalanguage for refinement in which basic principles of program development can be studied. The idea is that it should be possible to induce a refinement calculus in a generic manner from a programming language and a program logic. For concreteness, we adopt the simplytyped lambda calculus augmented with primitive recursion as a paradigmatic typed functional programming language, and use classical firstorder logic as a simple program logic. A key feature is the construction of the refinement calculus in a modular fashion, as the combination of two orthogonal extensions to the underlying programming language (in this case, the simplytyped lambda calculus). The crucial observation is that a refinement calculus is given by extending a programming language to allow indeterminate expressions (or ‘stubs’) involving the construction ‘some program x such that P ’. Factoring this into ‘some x...’
Innocent Game Models of Untyped λCalculus
 Theoretical Computer Science
, 2000
"... We present a new denotational model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [9] and Nickau [17], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models D and ..."
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Cited by 3 (1 self)
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We present a new denotational model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [9] and Nickau [17], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models D and DREC as global sections of a reexive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that these are sensible algebras but are neither extensional nor universal. We then introduce a new representation of innocent strategies in an economical form. We show a strong connexion between the economical form of the denotation of a term in the game models and a variablefree form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call eectively almosteverywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a model D...
Innocent Game Models of Untyped λCalculus
, 2000
"... We present a new denotation model for the untyped λcalculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between "question" and "answer" moves is removed. We first construct mod ..."
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Cited by 3 (1 self)
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We present a new denotation model for the untyped λcalculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between "question" and "answer" moves is removed. We first construct models D and DREC as global sections of a reflexive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that these are sensible algebras but are neither extensional nor universal. We then introduce a new representation of innocent strategies in an economical form. We show a stong connexion between the economical form of the denotation of a term in the game models and a variablefree form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call effectively almosteverywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a ...
Innocent Game Models of Untyped
, 2000
"... We present a new denotation model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models ..."
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Cited by 1 (0 self)
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We present a new denotation model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96], but the traditional distinction between \question" and \answer" moves is removed. We rst construct models D and DREC as global sections of a reexive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that these are sensible algebras but are neither extensional nor universal. We then introduce a new representation of innocent strategies in an economical form. We show a stong connexion between the economical form of the denotation of a term in the game models and a variablefree form of the Nakajima tree of the term. Using this we show that the denable elements of DREC are precisely what we call eectively almosteverywhere copycat (EAC) strategies. The category A EAC with these strategies as morphisms gives rise to a ...
Database Engineering from the Category Theory Viewpoint
"... Abstract. This paper gives an overview of XML formal models, summarizes database engineering practices, problems and their evolution. We focus on categorical aspects of XML formal models. Many formal models such as XML Data Model, XQuery Data Model or Algebra for XML can be described in terms of cat ..."
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Cited by 1 (0 self)
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Abstract. This paper gives an overview of XML formal models, summarizes database engineering practices, problems and their evolution. We focus on categorical aspects of XML formal models. Many formal models such as XML Data Model, XQuery Data Model or Algebra for XML can be described in terms of category theory. This kind of description allows to consider generic properties of these formalisms, e.g. expressive power, optimization, reduction or translation between them, among others. These properties are rather crucial to comparison of different XML formal models and to consequent decision which formal system should be used to solve a concrete problem. This work aim is to be the basis for further research in the area of XML formal models where category theory is applied. 1