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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 19 (3 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
Sequential algorithms and strongly stable functions
 in the Linear Summer School, Azores
, 2003
"... ..."
Comparing Hierarchies of Types in Models of Linear Logic
, 2003
"... We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F : C D : G and transformations Id C ) GF and Id D ) FG, and (2) their exponentials ! are related by distri ..."
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Cited by 6 (3 self)
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We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F : C D : G and transformations Id C ) GF and Id D ) FG, and (2) their exponentials ! are related by distributive laws % : ! : ! M G ) G ! N commuting to the promotion rule. The key ingredient of the proof is a notion of backandforth translation between the hierarchies of types induced by M and N. We apply this result to compare (1) the qualitative and the quantitative hierarchies induced by the coherence (or hypercoherence) space model, (2) several paradigms of games semantics: errorfree vs. erroraware, alternated vs. nonalternated, backtracking vs. repetitive, uniform vs. nonuniform.
Parallel and Serial Hypercoherences
 Theoretical Computer Science, NorthHolland
, 1995
"... It is known that the strongly stable functions which arise in the semantics of PCF can be realized by sequential algorithms, which can be considered as deterministic strategies in games associated to PCF types. Studying the connection between strongly stable functions and sequential algorithms, two ..."
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Cited by 4 (0 self)
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It is known that the strongly stable functions which arise in the semantics of PCF can be realized by sequential algorithms, which can be considered as deterministic strategies in games associated to PCF types. Studying the connection between strongly stable functions and sequential algorithms, two dual classes of hypercoherences naturally arise: the parallel and serial hypercoherences. The objects belonging to the intersection of these two classes are in bijective correspondence with the socalled "serialparallel" graphs, that can essentially be considered as games. We show how to associate to any hypercoherence a parallel hypercoherence together with a projection onto the given hypercoherence and present some properties of this construction. Intuitively, it makes explicit the computational time of a hypercoherence.
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"... Comparing hierarchies of types in models of linear logic PaulAndré Melliès 1 We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F: C ⇄ D: G and transformations IdC ..."
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Comparing hierarchies of types in models of linear logic PaulAndré Melliès 1 We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F: C ⇄ D: G and transformations IdC ⇒ GF and IdD ⇒ F G, and
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"... In [BE01], Bucciarelli and Ehrhard propose a general tool for building a wide class of models of linear logic where a formula is interpreted as a set (the web) together with a kind of phase valued “coherence relation”. These interpretations are nonuniform in the sense that the semantics of a proof m ..."
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In [BE01], Bucciarelli and Ehrhard propose a general tool for building a wide class of models of linear logic where a formula is interpreted as a set (the web) together with a kind of phase valued “coherence relation”. These interpretations are nonuniform in the sense that the semantics of a proof makes no assumption about the behaviour of its possible counterproofs, unlike e.g. in the usual stable semantics where the argument of a stable functional is always a stable function. However, until now, it was suspected that this nonuniformity necessarily induces a kind of nondeterminism, namely that a “clique ” and an “anticlique ” could have more than one point in common. We provide a new nonuniform semantics of linear logic where this property of determinism is preserved. This is done by constructing the cofree exponential in the “nonuniform coherence space ” framework described at the end of [BE01]. We discuss the issue of sequentiality in this new model. Notations. We use the notation [ ] for multisets while the notation { } is, as usual, for sets. The pairwise union of multisets is denoted by a + sign and following this notation the generalised union is denoted by a � sign. The neutral element for this operation, the empty multiset, is denoted by []. If k ∈ N, k[a] denotes the multiset � k 1 [a]. If [ai  i ∈ I] is a multiset, its support is the set {ai  i ∈ I}. The cardinality ♯[ai  i ∈ I] of a multiset [ai  i ∈ I] is the cardinality ♯I of the set I. If m is a multiset we denote by supp(m) its support. The disjoint sum operation on sets is defined by setting A + B = {1} × A ∪ {0} × B. The categorical composition is denoted by �.