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What is a Categorical Model of the Differential and the Resource λCalculi?
"... The differential λcalculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows to apply a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitab ..."
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The differential λcalculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows to apply a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitable primitives to handle explicitly the resources used by a program during its execution. The differential operator also allows to write the full Taylor expansion of a program. Through this expansion every program can be decomposed into an infinite sum (representing nondeterministic choice) of ‘simpler’ programs that are strictly linear. The aim of this paper is to develop an abstract ‘model theory ’ for the untyped differential λcalculus. In particular, we investigate what should be a general categorical definition of denotational model for this calculus. Starting from the work of Blute, Cockett and Seely on differential categories we provide the notion of Cartesian closed differential category and we prove that linear reflexive objects living in such categories constitute sound models of the untyped differential λcalculus. We also give sufficient conditions for Cartesian closed differential categories to model the Taylor expansion. This entails that every model living in such categories equates all programs having the same full Taylor expansion. We then
The Scott model of Linear Logic is the extensional collapse of its relational model
, 2011
"... We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus. ..."
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We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus.
On probabilistic coherence spaces
, 2008
"... We introduce a probabilistic version of coherence spaces and show that these objects provide a model of linear logic. We build a model of the pure lambdacalculus in this setting and show how to interpret a probabilistic version of the functional language PCF. We give a probabilistic interpretation ..."
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We introduce a probabilistic version of coherence spaces and show that these objects provide a model of linear logic. We build a model of the pure lambdacalculus in this setting and show how to interpret a probabilistic version of the functional language PCF. We give a probabilistic interpretation of the semantics of probabilistic PCF closed terms of ground type.
ProjectTeam Moscova Mobility, Security, Concurrency, Verification and Analysis
"... c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1 2. Overall Objectives........................................................................ 1 ..."
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c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1 2. Overall Objectives........................................................................ 1
Exponentials with in nite multiplicities
"... Abstract. Given a semiring with unit which satis es some algebraic conditions, we de ne an exponential functor on the category of sets and relations which allows to de ne a denotational model of di erential linear logic and of the lambdacalculus with resources. We show that, when the semiring has ..."
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Abstract. Given a semiring with unit which satis es some algebraic conditions, we de ne an exponential functor on the category of sets and relations which allows to de ne a denotational model of di erential linear logic and of the lambdacalculus with resources. We show that, when the semiring has an element which is in nite in the sense that it is equal to its successor, this model does not validate the Taylor formula and that it is possible to build, in the associated Kleisli cartesian closed category, a model of the pure lambdacalculus which is not sensible. This is a quantitative analogue of the standard graph model construction in the
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"... We recently introduced an extensional model of the pure λcalculus living in a canonical cartesian closed category of sets and relations [6]. In the present paper, we study the nondeterministic features of this model. Unlike most traditional approaches, our way of interpreting nondeterminism does ..."
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We recently introduced an extensional model of the pure λcalculus living in a canonical cartesian closed category of sets and relations [6]. In the present paper, we study the nondeterministic features of this model. Unlike most traditional approaches, our way of interpreting nondeterminism does not require any additional powerdomain construction. We show that our model provides a straightforward semantics of nondeterminism (may convergence) by means of unions of interpretations, as well as of parallelism (must convergence) by means of a binary, nonidempotent operation available on the model, which is related to the mix rule of Linear Logic. More precisely, we introduce a λcalculus extended with nondeterministic choice and parallel composition, and we define its operational semantics (based on the may and must intuitions underlying our two additional operations). We describe the interpretation of this calculus in our model and show that this interpretation is ‘sensible ’ with respect to our operational semantics: a term
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"... We recently introduced an extensional model of the pure λcalculus living in a canonical cartesian closed category of sets and relations [6]. In the present paper, we study the nondeterministic features of this model. Unlike most traditional approaches, our way of interpreting nondeterminism does ..."
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We recently introduced an extensional model of the pure λcalculus living in a canonical cartesian closed category of sets and relations [6]. In the present paper, we study the nondeterministic features of this model. Unlike most traditional approaches, our way of interpreting nondeterminism does not require any additional powerdomain construction. We show that our model provides a straightforward semantics of nondeterminism (may convergence) by means of unions of interpretations, as well as of parallelism (must convergence) by means of a binary, nonidempotent operation available on the model, which is related to the mix rule of Linear Logic. More precisely, we introduce a λcalculus extended with nondeterministic choice and parallel composition, and we define its operational semantics (based on the may and must intuitions underlying our two additional operations). We describe the interpretation of this calculus in our model and show that this interpretation is ‘sensible ’ with respect to our operational semantics: a term converges if, and only if, it has a nonempty interpretation.
The Resource Lambda Calculus Is ShortSighted in Its Relational Model
"... Abstract. Relational semantics is one of the simplest and categorically most natural semantics of Linear Logic. The coKleisli category MRel model of the untyped λcalculus. That particular object of MRel is also a model of the resource λcalculus, deriving from Ehrhard and Regnier’s differential ex ..."
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Abstract. Relational semantics is one of the simplest and categorically most natural semantics of Linear Logic. The coKleisli category MRel model of the untyped λcalculus. That particular object of MRel is also a model of the resource λcalculus, deriving from Ehrhard and Regnier’s differential extension of Linear Logic and related to Boudol’s λcalculus with multiplicities. Bucciarelli et al. conjectured that model to be fullyabstract also for the resource λcalculus. We give a counterexample to the conjecture. As a byproduct we achieve a context lemma for the resource λcalculus.
A NONUNIFORM FINITARY RELATIONAL SEMANTICS
 THEORETICAL INFORMATICS AND APPLICATIONS
, 1999
"... We study iteration and recursion operators in the denotational semantics of typed λcalculi derived from the multiset relational model of linear logic. Although these operators are defined as fixpoints of typed functionals, we prove them finitary in the sense of Ehrhard's finiteness spaces. ..."
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We study iteration and recursion operators in the denotational semantics of typed λcalculi derived from the multiset relational model of linear logic. Although these operators are defined as fixpoints of typed functionals, we prove them finitary in the sense of Ehrhard's finiteness spaces.