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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 63 (9 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
An Algebraic Presentation of Term Graphs, via GSMonoidal Categories
 Applied Categorical Structures
, 1999
"... . We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the wellknown characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particula ..."
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Cited by 38 (25 self)
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. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the wellknown characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are onetoone with the arrows of the free gsmonoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r), providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gsmonoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...
Comparing Logics for Rewriting: Rewriting logic, action calculi and tile logic
 Theoretical Computer Science
, 2002
"... The large diffusion of concurrent and distributed systems has spawned in recent years a variety of new formalisms, equipped with features for supporting an easy specification of such systems. The aim of our paper is to analyze three proposals, namely rewriting logic, action calculi and tile logic, c ..."
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Cited by 13 (3 self)
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The large diffusion of concurrent and distributed systems has spawned in recent years a variety of new formalisms, equipped with features for supporting an easy specification of such systems. The aim of our paper is to analyze three proposals, namely rewriting logic, action calculi and tile logic, chosen among those formalisms designed for the description of rulebased systems. For each of these logics we first try to understand their foundations, then we briefly sketch some applications. The overall goal of our work is to find out a common layout where these logics can be recast, thus allowing for a comparison and an evaluation of their specific features.
Logical Predicates for Intuitionistic Linear Type Theories
 In Typed Lambda Calculi and Applications (TLCA'99), Lecture Notes in Computer Science 1581
, 1999
"... We develop a notion of Kripkelike parameterized logical predicates for two fragments of intuitionistic linear logic (MILL and DILL) in terms of their categorytheoretic models. Such logical predicates are derived from the categorical glueing construction combined with the free symmetric monoidal co ..."
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Cited by 11 (4 self)
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We develop a notion of Kripkelike parameterized logical predicates for two fragments of intuitionistic linear logic (MILL and DILL) in terms of their categorytheoretic models. Such logical predicates are derived from the categorical glueing construction combined with the free symmetric monoidal cocompletion. As applications, we obtain full completeness results of translations between linear type theories.
A logic of events
, 2003
"... There is a wellestablished theory and practice for creating correctbyconstruction functional programs by extracting them from constructive proofs of assertions of the form ∀x: A.∃y: B.R(x, y). There have been several efforts to extend this methodology to concurrent programs, say by using linear l ..."
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Cited by 11 (7 self)
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There is a wellestablished theory and practice for creating correctbyconstruction functional programs by extracting them from constructive proofs of assertions of the form ∀x: A.∃y: B.R(x, y). There have been several efforts to extend this methodology to concurrent programs, say by using linear logic, but there is no practice and the results are limited. In this paper we define a logic of events that justifies the extraction of correct distributed processes from constructive proofs that system specifications are achievable, and we describe an implementation of an extraction process in the context of constructive type theory. We show that a class of message automata, similar to IO automata and to active objects, are realizers for this logic. We provide a relative consistency result for the logic. We show an example of protocol derivation in this logic, and show how to embed temporal logics such as T LA+ in the event logic. 1
Functorial boxes in string diagrams
, 2006
"... String diagrams were introduced by Roger Penrose as a handy notation to manipulate morphisms in a monoidal category. In principle, this graphical notation should encompass the various pictorial systems introduced in prooftheory (like JeanYves Girard’s proofnets) and in concurrency theory (like Ro ..."
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Cited by 9 (3 self)
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String diagrams were introduced by Roger Penrose as a handy notation to manipulate morphisms in a monoidal category. In principle, this graphical notation should encompass the various pictorial systems introduced in prooftheory (like JeanYves Girard’s proofnets) and in concurrency theory (like Robin Milner’s bigraphs). This is not the case however, at least because string diagrams do not accomodate boxes — a key ingredient in these pictorial systems. In this short tutorial, based on our accidental rediscovery of an idea by Robin Cockett and Robert Seely, we explain how string diagrams may be extended with a notion of functorial box to depict a functor separating an inside world (its source category) from an outside world (its target category). We expose two elementary applications of the notation: first, we characterize graphically when a faithful balanced monoidal functor F: C − → D transports a trace operator from the category D
Type Assignment and Termination of Interaction Nets
"... Interaction nets have proved to be a useful tool for the study of computational aspects of different formalisms (e.g. calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semisi ..."
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Cited by 7 (4 self)
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Interaction nets have proved to be a useful tool for the study of computational aspects of different formalisms (e.g. calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semisimple interaction nets as a programming language, and present a type assignment system using intersection types. First we show that interactions preserve types (i.e. the system enjoys subject reduction), and we compare this type assignment system with the intersection systems for calculus and term rewriting systems. Then we define a recursion scheme that ensures termination of all interaction sequences. By relaxing the scheme and using the type assignment system, we derive another sufficient condition for termination of interaction nets. Finally, we show that although the type system based on general intersection types is not decidable, its restriction to rank 2 types is, and we give an algo...
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...
Categorical Glueing and Logical Predicates for Models of Linear Logic
, 1999
"... We give a series of glueing constructions for categorical models of fragments of linear logic. Specifically, we consider the glueing of (i) symmetric monoidal closed categories (models of Multiplicative Intuitionistic Linear Logic), (ii) symmetric monoidal adjunctions (for interpreting the modality ..."
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Cited by 5 (3 self)
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We give a series of glueing constructions for categorical models of fragments of linear logic. Specifically, we consider the glueing of (i) symmetric monoidal closed categories (models of Multiplicative Intuitionistic Linear Logic), (ii) symmetric monoidal adjunctions (for interpreting the modality !) and (iii) autonomous categories (models of Multiplicative Linear Logic); the glueing construction for autonomous categories is a mild generalization of the double glueing construction due to Hyland and Tan. Each of the glueing techniques can be used for creating interesting models of linear logic. In particular, we use them, together with the free symmetric monoidal cocompletion, for deriving Kripkelike parameterized logical predicates (logical relations) for the fragments of linear logic. As an application, we show full completeness results for translations between linear type theories. Contents 1 Introduction 3 2 Preliminaries 4 2.1 Symmetric Monoidal Structures . . . . . . . ....