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Models of Sharing Graphs: A Categorical Semantics of let and letrec
, 1997
"... To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sha ..."
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Cited by 62 (10 self)
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To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is firstorder acyclic sharing graphs represented by letsyntax, and others are extensions with higherorder constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced
Categorical Logic of Names and Abstraction in Action Calculi
, 1993
"... ion elimination Definition 3.1. A monoidal category where every object has a commutative comonoid structure is said to be semicartesian. An action category is a K\Omega category with a distinguished admissible commutative comonoid structure on every object. A semicartesian category is cartesi ..."
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Cited by 21 (9 self)
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ion elimination Definition 3.1. A monoidal category where every object has a commutative comonoid structure is said to be semicartesian. An action category is a K\Omega category with a distinguished admissible commutative comonoid structure on every object. A semicartesian category is cartesian if and only if each object carries a unique comonoid structure, and such structures form two natural families, \Delta and !. The naturality means that all morphisms of the category must be comonoid homomorphisms. In action categories, the property of semicartesianness is fixed as structure: on each object, a particular comonoid structure is chosen. This choice may be constrained by some given graphic operations, with respect to which the structures must be admissible. The proof of proposition 2.6 shows that such structures determine the abstraction operators, and are determined by them. This is the essence of the equivalence of action categories and action calculi. As the embodiment of 2...
Types and Models for HigherOrder Action Calculi
, 1997
"... . Milner introduced action calculi as a framework for representing models of interactive behaviour. He also introduced the higherorder action calculi, which add higherorder features to the basic setting. We present type theories for action calculi and higherorder action calculi, and give the categ ..."
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Cited by 6 (5 self)
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. Milner introduced action calculi as a framework for representing models of interactive behaviour. He also introduced the higherorder action calculi, which add higherorder features to the basic setting. We present type theories for action calculi and higherorder action calculi, and give the categorical models of the higherorder calculi. As applications, we give a semantic proof of the conservativity of higherorder action calculi over action calculi, and a precise connection with Moggi's computational lambda calculus and notions of computation. 1 Introduction Milner introduced action calculi as a framework for representing models of interactive behaviour [Mil96]. He also introduced two conservative extensions: higherorder action calculi [Mil94a], which add higherorder features to the basic setting, and reflexive action calculi [Mil94b], which give recursion in the presence of the higherorder features. Various examples, which explore the role of action calculi as a general frame...
Symmetric Action Calculi
 Theoretical Computer Science
, 1999
"... Many calculi for describing interactive behaviour involve names, nameabstraction and namerestriction. Milner's reflexive action calculi provide a framework for exploring such calculi. It is based on names and nameabstraction. We introduce an alternative framework, the symmetric action calculi, ba ..."
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Cited by 5 (1 self)
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Many calculi for describing interactive behaviour involve names, nameabstraction and namerestriction. Milner's reflexive action calculi provide a framework for exploring such calculi. It is based on names and nameabstraction. We introduce an alternative framework, the symmetric action calculi, based on names, conames and namerestriction (or hiding). Nameabstraction is intepreted as a derived operator. The symmetric framework conservatively extends the reflexive framework. It allows for a natural intepretation of a variety of calculi: we give interpretations for the calculus, the I calculus and a variant of the fusion calculus. We then give a combinatory version of the symmetric framework, in which namerestriction also is expressed as a derived operator. This combinatory account provides an intermediate step between our nonstandard use of names in graphs, and the more standard graphical structure arising from category theory. To conclude, we briey illustrate the connection ...
Specifications of Processes under Expanding Signatures
, 1998
"... We propose a specification logic based on a ¯calculus enriched with explicit signatures and certain connectives expressing component decomposition and name privacy. This logic is interpreted on certain variants of action calculi of Milner, called herein generative action structures, and that model ..."
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We propose a specification logic based on a ¯calculus enriched with explicit signatures and certain connectives expressing component decomposition and name privacy. This logic is interpreted on certain variants of action calculi of Milner, called herein generative action structures, and that model processes that interact in an environment of dynamically extensible signatures. To illustrate this basic framework, a specification logic for compositional reasoning about local and global properties of object systems with mobile features is presented. 1 Introduction Most of the proposed statebased specification logics for concurrent interacting modules or objects (for instance, [13, 5]) assume that computations are carried out over data defined with techniques from algebraic specification, and that objects provide a fixed set of services, both determined at specification time by a given signature. Nevertheless, many existing computational frameworks rely on the use of dynamically generate...
John Power Edinburgh, January 1995
"... control structures do not have an explicit set of names, and they fit more naturally into the usual use of category theory in computer science, particularly in denotational semantics. Here, we go one step further. We drop the one (mild) categorically unnatural condition on abstract control structure ..."
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control structures do not have an explicit set of names, and they fit more naturally into the usual use of category theory in computer science, particularly in denotational semantics. Here, we go one step further. We drop the one (mild) categorically unnatural condition on abstract control structures, and prove an equivalence with what we call elementary control structures. Elementary control structures fit immediately into the general semantic theory of "notions of computation" of [PR], for which the simplest situation studied has a base category B with finite products and an identity on objects strict symmetric monoidal functor into a symmetric monoidal category C. That paper includes a mild generalisation of that setting, in order to incorporate the start of a study of adding state to traditional denotational semantics. So, the equivalence we prove here proves that control structures fit naturally and simply into the broader study of denotational semantics. In Section 2, we review...
Periodic Progress Report
"... Machines, G'erard Boudol from INRIA SophiaAntipolis and Ugo Montanari from University of Pisa prepared the area report on Calculi, Rajagopal Nagarajan from Imperial College did the area report on Logics for Concurrency and the calculus, Benjamin Pierce from Edinburgh University (and Cambridge) and ..."
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Machines, G'erard Boudol from INRIA SophiaAntipolis and Ugo Montanari from University of Pisa prepared the area report on Calculi, Rajagopal Nagarajan from Imperial College did the area report on Logics for Concurrency and the calculus, Benjamin Pierce from Edinburgh University (and Cambridge) and Bent Thomsen from ECRC wrote the area report on Programming Languages. 3 4 CHAPTER 1. OVERVIEW Chapter 2 Executive summary The overall objective of the CONFER action is to create both the theoretical foundations and the technology for combining the expressive power of the functional and the concurrent computational models. The action is organised around four main areas: ffl Foundational Models and Abstract Machines ffl Calculi ffl Logics for Concurrency and the calculus ffl Programming Languages The objectives for the third year of CONFER put forth at the end of Year 2 of CONFER have been achieved at the end of Year 3. Significant results beyond these objectives have also been obt...