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17
The Differential LambdaCalculus
 Theoretical Computer Science
, 2001
"... We present an extension of the lambdacalculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual ..."
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Cited by 44 (9 self)
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We present an extension of the lambdacalculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambdacalculus.
Hybridizing a logical framework
 In International Workshop on Hybrid Logic 2006 (HyLo 2006), Electronic Notes in Computer Science
, 2006
"... The logical framework LF is a constructive type theory of dependent functions that can elegantly encode many other logical systems. Prior work has studied the benefits of extending it to the linear logical framework LLF, for the incorporation linear logic features into the type theory affords good r ..."
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Cited by 20 (1 self)
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The logical framework LF is a constructive type theory of dependent functions that can elegantly encode many other logical systems. Prior work has studied the benefits of extending it to the linear logical framework LLF, for the incorporation linear logic features into the type theory affords good representations of state change. We describe and argue for the usefulness of an extension of LF by features inspired by hybrid logic, which has several benefits. For one, it shows how linear logic features can be decomposed into primitive operations manipulating abstract resource labels. More importantly, it makes it possible to realize a metalogical framework capable of reasoning about stateful deductive systems encoded in the style familiar from prior work with LLF, taking advantage of familiar methodologies used for metatheoretic reasoning in LF.Acknowledgments From the very first computer science course I took at CMU, Frank Pfenning has been an exceptional teacher and mentor. For his patience, breadth of knowledge, and mathematical good taste I am extremely thankful. No less do I owe to the other two major contributors to my programming languages
Böhm trees, Krivine machine and the Taylor expansion of ordinary lambdaterms
, 2005
"... We show that, given an ordinary lambdaterm and a normal resource lambdaterm which appears in the normal form of its Taylor expansion, the unique resource term of the Taylor expansion of the ordinary lambdaterm reducing to this normal resource term can be obtained by running a version of the Krivi ..."
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Cited by 17 (5 self)
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We show that, given an ordinary lambdaterm and a normal resource lambdaterm which appears in the normal form of its Taylor expansion, the unique resource term of the Taylor expansion of the ordinary lambdaterm reducing to this normal resource term can be obtained by running a version of the Krivine abstract machine.
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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Cited by 5 (1 self)
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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
Implicit Polymorphic Type System for the Blue Calculus
, 1997
"... The Blue Calculus is a direct extension of both the lambda and the pi calculi. In a preliminary work from Gérard Boudol, a simple type system was given that incorporates Curry's type inference for the lambdacalculus. In the present paper we study an implicit polymorphic type system, adapted from th ..."
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Cited by 4 (2 self)
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The Blue Calculus is a direct extension of both the lambda and the pi calculi. In a preliminary work from Gérard Boudol, a simple type system was given that incorporates Curry's type inference for the lambdacalculus. In the present paper we study an implicit polymorphic type system, adapted from the ML typing discipline. Our typing system enjoys subject reduction and principal type properties and we give results on the complexity for the type inference problem. These are interesting results for the blue calculus as a programming notation for higherorder concurrency.
Functions as SessionTyped Processes
"... We study typedirected encodings of the simplytyped λcalculus in a sessiontyped πcalculus. The translations proceed in two steps: standard embeddings of simplytyped λcalculus in a linear λcalculus, followed by a standard translation of linear natural deduction to linear sequent calculus. We ..."
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Cited by 4 (3 self)
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We study typedirected encodings of the simplytyped λcalculus in a sessiontyped πcalculus. The translations proceed in two steps: standard embeddings of simplytyped λcalculus in a linear λcalculus, followed by a standard translation of linear natural deduction to linear sequent calculus. We have shown in prior work how to give a CurryHoward interpretation of the proofs in the linear sequent calculus as πcalculus processes subject to a session type discipline. We show that the resulting translations induce sharing and copying parallel evaluation strategies for the original λterms, thereby providing a new logically motivated explanation for these strategies.
A Note on Intersection Types
, 1995
"... : Following J.L. Krivine, we call D the type inference system introduced by M. Coppo and M. Dezani where types are propositional formulae written with conjunction and implication from propositional letters  there is no special constant !. We show here that the wellknown result on D, stating th ..."
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Cited by 3 (0 self)
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: Following J.L. Krivine, we call D the type inference system introduced by M. Coppo and M. Dezani where types are propositional formulae written with conjunction and implication from propositional letters  there is no special constant !. We show here that the wellknown result on D, stating that any term which possesses a type in D strongly normalises does not need a new reducibility argument, but is a mere consequence of strong normalization for natural deduction restricted to the conjunction and implication. The proof of strong normalization for natural deduction, and therefore our result, as opposed to reducibility arguments, can be carried out within primitive recursive arithmetic. On the other hand, this enlightens the relation between & and & that G. Pottinger has already wondered about, and can be applied to other situations, like the lambda calculus with multiplicities of G. Boudol. Keywords: Lambda calculus , intersection types , strong normalization. Logic, proof th...
A finiteness structure on resource terms
 IN LICS
, 2010
"... We study the Taylor expansion of lambdaterms in a nondeterministic or algebraic setting, where terms can be added. The target language is a resource lambda calculus based on a differential lambdacalculus we introduced recently. This operation is not possible in the general untyped case where redu ..."
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Cited by 3 (1 self)
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We study the Taylor expansion of lambdaterms in a nondeterministic or algebraic setting, where terms can be added. The target language is a resource lambda calculus based on a differential lambdacalculus we introduced recently. This operation is not possible in the general untyped case where reduction can produce unbounded coefficients. We endow resource terms with a finiteness structure (in the sense of our earlier work on finiteness spaces) and show that the Taylor expansions of terms typeable in Girard’s system F are finitary by a reducibility method.
The Discriminating Power of Multiplicities in the λCalculus
, 1996
"... The λcalculus with multiplicities is a refinement of the lazy λcalculus where the argument in an application comes with a multiplicity, which is an upper bound to the number of its uses. This introduces potential deadlocks in the evaluation. We study the discriminating power of this calculus over ..."
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Cited by 2 (0 self)
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The λcalculus with multiplicities is a refinement of the lazy λcalculus where the argument in an application comes with a multiplicity, which is an upper bound to the number of its uses. This introduces potential deadlocks in the evaluation. We study the discriminating power of this calculus over the usual λterms. We prove in particular that the observational equivalence induced by contexts with multiplicities coincides with the equality of LévyLongo trees associated with λterms. This is a consequence of the characterization we give of the corresponding observational precongruence, as an intensional preorder involving etaexpansion, namely Ong's lazy PlotkinScottEngeler preorder.
The Discriminating Power of Multiplicities in the LambdaCalculus
, 1996
"... The calculus with multiplicities is a refinement of the lazy calculus where the argument in an application comes with a multiplicity, which is an upper bound to the number of its uses. This introduces potential deadlocks in the evaluation. We study the discriminating power of this calculus over th ..."
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Cited by 1 (0 self)
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The calculus with multiplicities is a refinement of the lazy calculus where the argument in an application comes with a multiplicity, which is an upper bound to the number of its uses. This introduces potential deadlocks in the evaluation. We study the discriminating power of this calculus over the usual terms. We prove in particular that the observational equivalence induced by contexts with multiplicities coincides with the equality of L'evyLongo trees associated with terms. This is a consequence of the characterization we give of the corresponding observational precongruence, as an intensional preorder involving jexpansion, namely Ong's lazy PlotkinScottEngeler preorder. 1 Introduction The calculus with multiplicities was introduced in [5] for the purpose of studying the relationship between the calculus and Milner's ßcalculus [13]. It is a "resource conscious" refinement of the calculus, based on the following observation: in an application MN the argument N is infini...