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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
A relational semantics for parallelism and nondeterminism in . . .
, 2011
"... 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
On the characterization of models of H∗
"... We give a characterization, with respect to a large class of models of untyped λcalculus, of those models that are fully abstract for headnormalization, i.e., whose equational theory is H∗. An extensional Kmodel D is fully abstract if and only if it is hyperimmune, i.e., nonwell founded chains o ..."
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We give a characterization, with respect to a large class of models of untyped λcalculus, of those models that are fully abstract for headnormalization, i.e., whose equational theory is H∗. An extensional Kmodel D is fully abstract if and only if it is hyperimmune, i.e., nonwell founded chains of elements of D cannot be captured by any recursive function.
On Geometry of Interaction for Polarized Linear Logic
, 2014
"... We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP, which is the multiplicative fragment (without structural rules) of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multipoints to various categorical models of GoI. Multi ..."
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We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP, which is the multiplicative fragment (without structural rules) of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multipoints to various categorical models of GoI. Multipoints are shown to play an essential role in semantically characterizing the dynamics of proof networks in polarized proof theory. For example, they permit us to characterize the key feature of polarization, focusing, as well as playing a fundamental role in allowing us to construct concrete polarized GoI models. Our approach to polarized GoI involves two independent studies, based on different categorical perspectives of GoI. (i) Inspired by the work of Abramsky, Haghverdi, and Scott, a polarized GoI situation is dened which considers multipoints added to a traced monoidal category with an appropriate re
exive object U. Categorical versions of Girard's Execution formula (taking into account the multipoints) are dened, as well as the GoI
Creative Commons Attribution License. On Linear Information Systems
"... Scott’s information systems provide a categorically equivalent, intensional description of Scott domains and continuous functions. Following a well established pattern in denotational semantics, we define a linear version of information systems, providing a model of intuitionistic linear logic (a n ..."
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Scott’s information systems provide a categorically equivalent, intensional description of Scott domains and continuous functions. Following a well established pattern in denotational semantics, we define a linear version of information systems, providing a model of intuitionistic linear logic (a newSeely category), with a “settheoretic ” interpretation of exponentials that recovers Scott continuous functions via the coKleisli construction. From a domain theoretic point of view, linear information systems are equivalent to prime algebraic Scott domains, which in turn generalize prime algebraic lattices, already known to provide a model of classical linear logic. 1
Extensional collapse situations I: nontermination and unrecoverable errors
"... Abstract. We consider a simple model of higher order, functional computations over the booleans. Then, we enrich the model in order to encompass nontermination and unrecoverable errors, taken separately or jointly. We show that the models so defined form a lattice when ordered by the extensional ..."
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Abstract. We consider a simple model of higher order, functional computations over the booleans. Then, we enrich the model in order to encompass nontermination and unrecoverable errors, taken separately or jointly. We show that the models so defined form a lattice when ordered by the extensional collapse situation relation, introduced in order to compare models with respect to the amount of “intensional information ” that they provide on computation. The proofs are carried out by exhibiting suitable applied λcalculi, and by exploiting the fundamental lemma of logical relations. 1