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44
Strong Stability and the Incompleteness of Stable Models for λCalculus
 ANNALS OF PURE AND APPLIED LOGIC
, 1999
"... We prove that the class of stable models is incomplete with respect to pure λcalculus. More precisely, we show that no stable model has the same theory as the strongly stable version of Park's model. This incompleteness proof can be adapted to the continuous case, giving an incompleteness proof for ..."
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We prove that the class of stable models is incomplete with respect to pure λcalculus. More precisely, we show that no stable model has the same theory as the strongly stable version of Park's model. This incompleteness proof can be adapted to the continuous case, giving an incompleteness proof for this case which is much simpler than the original proof by Honsell an Ronchi della Rocca. Moreover, we isolate a very simple finite set, F , of equations and inequations, which has neither a stable nor a continuous model, and which is included in Th(P fs ) and in T
Coherent Banach spaces: a continuous denotational semantics
 Theoretical Computer Science
, 1999
"... We present a denotational semantics based on Banach spaces; it is inspired from the familiar coherent semantics of linear logic, the role of coherence being played by the norm: coherence is rendered by a supremum, whereas incoherence is rendered by a sum, and cliques are rendered by vectors of norm ..."
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Cited by 18 (3 self)
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We present a denotational semantics based on Banach spaces; it is inspired from the familiar coherent semantics of linear logic, the role of coherence being played by the norm: coherence is rendered by a supremum, whereas incoherence is rendered by a sum, and cliques are rendered by vectors of norm at most 1. The basic constructs of linear (and therefore intuitionistic) logic are implemented in this framework: positive connectives yield ℓ 1like norms and negative connectives yield ℓ ∞like norms. The problem of nonreflexivity of Banach spaces is handled by specifying the dual in ¡ advance, whereas the exponential connectives (i.e. intuitionistic implication) are handled by means of analytical functions on the open unit ball. The fact that this ball is open (and not closed) explains the absence of a simple solution to the question of a topological cartesian closed
Games on graphs and sequentially realizable functionals
 In Logic in Computer Science 02
, 2002
"... We present a new category of games on graphs and derive from it a model for Intuitionistic Linear Logic. Our category has the computational flavour of concrete data structures but embeds fully and faithfully in an abstract games model. It differs markedly from the usual Intuitionistic Linear Logic s ..."
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Cited by 17 (2 self)
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We present a new category of games on graphs and derive from it a model for Intuitionistic Linear Logic. Our category has the computational flavour of concrete data structures but embeds fully and faithfully in an abstract games model. It differs markedly from the usual Intuitionistic Linear Logic setting for sequential algorithms. However, we show that with a natural exponential we obtain a model for PCF essentially equivalent to the sequential algorithms model. We briefly consider a more extensional setting and the prospects for a better understanding of the Longley Conjecture. 1
A Continuum of Theories of Lambda Calculus Without Semantics
 16TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS 2001), IEEE COMPUTER
, 2001
"... In this paper we give a topological proof of the following result: There exist 2 @0 lambda theories of the untyped lambda calculus without a model in any semantics based on Scott's view of models as partially ordered sets and of functions as monotonic functions. As a consequence of this result, we ..."
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Cited by 16 (11 self)
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In this paper we give a topological proof of the following result: There exist 2 @0 lambda theories of the untyped lambda calculus without a model in any semantics based on Scott's view of models as partially ordered sets and of functions as monotonic functions. As a consequence of this result, we positively solve the conjecture, stated by BastoneroGouy [6, 7] and by Berline [10], that the strongly stable semantics is incomplete. 1
Between logic and quantic: a tract
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2003
"... We present a quantum interpretation of the perfect part of linear logic, by means of quantum coherent spaces. In particular this yields a novel interpretation of the reduction of the wave packet as the expression of ηconversion, a.k.a, extensionality. ..."
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Cited by 16 (1 self)
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We present a quantum interpretation of the perfect part of linear logic, by means of quantum coherent spaces. In particular this yields a novel interpretation of the reduction of the wave packet as the expression of ηconversion, a.k.a, extensionality.
Sequential algorithms and strongly stable functions
 in the Linear Summer School, Azores
, 2003
"... ..."
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 12 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Bistructures, Bidomains and Linear Logic
 in Proc. 21st ICALP
, 1997
"... Bistructures are a generalisation of event structures which allow a representation of spaces of functions at higher types in an orderextensional setting. The partial order of causal dependency is replaced by two orders, one associated with input and the other with output in the behaviour of func ..."
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Cited by 9 (3 self)
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Bistructures are a generalisation of event structures which allow a representation of spaces of functions at higher types in an orderextensional setting. The partial order of causal dependency is replaced by two orders, one associated with input and the other with output in the behaviour of functions. Bistructures form a categorical model of Girard's classical linear logic in which the involution of linear logic is modelled, roughly speaking, by a reversal of the roles of input and output. The comonad of the model has an associated coKleisli category which is closely related to that of Berry's bidomains (both have equivalent nontrivial full subcartesian closed categories).
Type Theory via Exact Categories (Extended Abstract)
 In Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science LICS '98
, 1998
"... Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why ..."
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Cited by 7 (0 self)
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Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R ` N\ThetaN on th...
Category theory for linear logicians
 Linear Logic in Computer Science
, 2004
"... This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categori ..."
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Cited by 7 (1 self)
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This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoidal closed, linearly distributive and ∗autonomous categories and their relation to multiplicative linear logic. We examine nonsymmetric monoidal categories, and consider them as models of noncommutative linear logic. We introduce traced monoidal categories, and discuss their relation to the geometry of interaction. The necessary aspects of the theory of monads is introduced in order to describe the categorical modelling of the exponentials. We conclude by briefly describing the notion of full completeness, a strong form of categorical completeness, which originated in the categorical model theory of linear logic. No knowledge of category theory is assumed, but we do assume knowledge of linear logic sequent calculus and the standard models of linear logic, and modest familiarity with typed lambda calculus. 0