Results 1  10
of
18
Concurrent Games and Full Completeness
, 1998
"... A new concurrent form of game semantics is introduced. This overcomes the problems which had arisen with previous, sequential forms of game semantics in modelling Linear Logic. It also admits an elegant and robust formalization. A Full Completeness Theorem for MultiplicativeAdditive Linear Logic is ..."
Abstract

Cited by 50 (17 self)
 Add to MetaCart
A new concurrent form of game semantics is introduced. This overcomes the problems which had arisen with previous, sequential forms of game semantics in modelling Linear Logic. It also admits an elegant and robust formalization. A Full Completeness Theorem for MultiplicativeAdditive Linear Logic is proved for this semantics. 1 Introduction This paper contains two main contributions: ffl the introduction of a new form of game semantics, which we call concurrent games. ffl a proof of full completeness of this semantics for MultiplicativeAdditive Linear Logic. We explain the significance of each of these in turn. Concurrent games Traditional forms of game semantics which have appeared in logic and computer science have been sequential in format: a play of the game is formalized as a sequence of moves. The key feature of this sequential format is the existence of a global schedule (or polarization) : in each (finite) position, it is (exactly) one player's turn to move 1 . This seq...
Proofs nets for unitfree multiplicativeadditive linear logic
 18th IEEE Intl. Symp. Logic in Computer Science (LICS’03
, 2003
"... A cornerstone of the theory of proof nets for unitfree multiplicative linear logic (MLL) is the abstract representation of cutfree proofs modulo inessential commutations of rules. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cutfr ..."
Abstract

Cited by 40 (4 self)
 Add to MetaCart
A cornerstone of the theory of proof nets for unitfree multiplicative linear logic (MLL) is the abstract representation of cutfree proofs modulo inessential commutations of rules. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cutfree monomial proof nets can correspond to the same cutfree proof. Thus the problem of finding a satisfactory notion of proof net for unitfree multiplicativeadditive linear logic (MALL) has remained open since the inception of linear logic in 1986. We present a new definition of MALL proof net which remains faithful to the cornerstone of the MLL theory. 1
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed ..."
Abstract

Cited by 28 (10 self)
 Add to MetaCart
We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
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 ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
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
Feedback for Linearly Distributive Categories: Traces and Fixpoints
, 1999
"... In the present paper, we develop the notion of a trace operator on a linearly distributive category, which amounts to essentially working within a subcategory (the core) which has the same sort of "type degeneracy" as a compact closed category. We also explore the possibility that an object may have ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
In the present paper, we develop the notion of a trace operator on a linearly distributive category, which amounts to essentially working within a subcategory (the core) which has the same sort of "type degeneracy" as a compact closed category. We also explore the possibility that an object may have several trace structures, introducing a notion of compatibility in this case. We show that if we restrict to compatible classes of trace operators, an object may have at most one trace structure (for a given tensor structure). We give a linearly distributive version of the "geometry of interaction" construction, and verify that we obtain a linearly distributive category in which traces become canonical. We explore the relationship between our notions of trace and fixpoint operators, and show that an object admits a fixpoint combinator precisely when it admits a trace and is a cocommutative comonoid. This generalises an observation of Hyland and Hasegawa.
Sequentiality vs. Concurrency in Games and Logic
 Math. Structures Comput. Sci
, 2001
"... Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic. ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.
Chu Spaces as a Semantic Bridge Between Linear Logic and Mathematics
 Theoretical Computer Science
, 1998
"... The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the selfdual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly interp ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the selfdual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly interpreting linear logic while fully and concretely embedding a comprehensive range of concrete categories of mathematics. Our main goal is to treat each end of this bridge in expository detail. In addition we introduce the dialectic lambdacalculus, and show that dinaturality semantics is not fully complete for the Chu interpretation of linear logic. 1 Introduction Linear logic was introduced by J.Y. Girard as a "logic behind logic." It separates logical reasoning into a core linear part in which formulas are merely moved around, and an auxiliary nonlinear part in which formulas may be deleted and copied. The core, multiplicative linear logic (MLL), is a substructural logic whose basic connect...
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 ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
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.
Games in the Semantics of Programming Languages
 Dept. of Philosophy, University of Amsterdam
, 1997
"... ion for PCF Motivated by the full completeness results, it became of compelling interest to reexamine perhaps the bestknown "open problem" in the semantics of programming languages, namely the "Full Abstraction problem for PCF", using the new tools provided by game semantics. 2 PCF is a highero ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
ion for PCF Motivated by the full completeness results, it became of compelling interest to reexamine perhaps the bestknown "open problem" in the semantics of programming languages, namely the "Full Abstraction problem for PCF", using the new tools provided by game semantics. 2 PCF is a higherorder functional programming language; modulo issues of the parameterpassing strategies, it forms a fragment of any programming language with higherorder procedures (which includes any reasonably expressive objectoriented language). The aspect of the Full Abstraction problem I personally found most interesting was: to construct a syntaxindependent model in which every element is the denotation of some program (note the analogy with full completeness, whose definition had in turn been motivated in part by this aspect of full abstraction). This is not how the problem was originally formulated, but by "general abstract nonsense", given such a model one can always quotient it to get a fully ab...
Entropic hopf algebras and models of noncommutative linear logic
 THEORY AND APPLICATIONS OF CATEGORIES 10
, 2002
"... We give a definition of categorical model for the multiplicative fragment of noncommutative logic. We call such structures entropic categories. We demonstrate the soundness and completeness of our axiomatization with respect to cutelimination. We then focus on several methods of building entropic ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
We give a definition of categorical model for the multiplicative fragment of noncommutative logic. We call such structures entropic categories. We demonstrate the soundness and completeness of our axiomatization with respect to cutelimination. We then focus on several methods of building entropic categories. Our first models are constructed via the notion of a partial bimonoid acting on a cocomplete category. We also explore an entropic version of the Chu construction, and apply it in this setting. It has recently been demonstrated that Hopf algebras provide an excellent framework for modeling a number of variants of multiplicative linear logic, such as commutative, braided and cyclic. We extend these ideas to the entropic setting by developing a new type of Hopf algebra, which we call entropic Hopf algebras. We show that the category of modules over an entropic Hopf algebra is an entropic category, (possibly after application of the Chu construction). Several examples are discussed, based first on the notion of a bigroup. Finally the TannakaKrein reconstruction theorem is extended to the entropic setting.