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42
On linear combinations of λterms
"... Abstract. We define an extension of λcalculus with linear combinations, endowing the set of terms with a structure of Rmodule, where R is a fixed set of scalars. Terms are moreover subject to identities similar to usual pointwise definition of linear combinations of functions with values in a vect ..."
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Abstract. We define an extension of λcalculus with linear combinations, endowing the set of terms with a structure of Rmodule, where R is a fixed set of scalars. Terms are moreover subject to identities similar to usual pointwise definition of linear combinations of functions with values in a vector space. We then extend βreduction on those algebraic λterms as follows: at + u reduces to at ′ + u as soon as term t reduces to t ′ and a is a nonzero scalar. We prove that reduction is confluent. Under the assumption that the set R of scalars is positive (i.e. a sum of scalars is zero iff all of them are zero), we show that this algebraic λcalculus is a conservative extension of ordinary λcalculus. On the other hand, we show that if R admits negative elements, then every term reduces to every other term. We investigate the causes of that collapse, and discuss some possible fixes. Preliminary definitions and notations. Recall that a rig (also known as “semiring with zero and unit”) is the same as a ring, without the condition that every element admits an opposite for addition. Let R be a rig. We write R • for R \ {0}. We denote by letters a, b, c the elements of R, and say that R is positive if, for all a, b ∈ R, a + b = 0 implies a = 0 and b = 0. An example of positive rig is N, the set of natural numbers, with usual addition and multiplication. If i, j ∈ N, we write [i; j] for the set {k ∈ N; i ≤ k ≤ j}. Also, we write application of λterms à la Krivine: (s)t denotes the application of term s to term t. 1
Algebraic λcalculus
"... We introduce an extension of λcalculus by endowing the set of terms with a structure of vector space (or more generally of module), over a fixed set of scalars. Terms are moreover subject to identities similar to usual pointwise definition of linear combinations of functions with values in a vector ..."
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We introduce an extension of λcalculus by endowing the set of terms with a structure of vector space (or more generally of module), over a fixed set of scalars. Terms are moreover subject to identities similar to usual pointwise definition of linear combinations of functions with values in a vector space. We then study a natural extension of βreduction in this setting: we prove it is confluent, then discuss consistency and conservativity over ordinary λcalculus. We also provide normalization results for a simple type system.
Convolution ¯ λµcalculus
 of Lecture Notes in Computer Science
, 2007
"... We define an extension of Herbelin’s ¯ λµcalculus, introducing a product operation on contexts (in the sense of lists of arguments, or stacks in environment machines), similar to the convolution product of distributions. This is the computational couterpart of some new semantical constructions, ext ..."
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We define an extension of Herbelin’s ¯ λµcalculus, introducing a product operation on contexts (in the sense of lists of arguments, or stacks in environment machines), similar to the convolution product of distributions. This is the computational couterpart of some new semantical constructions, extending models of EhrhardRegnier’s differential interaction nets, along the lines of Laurent’s polarization of linear logic. We demonstrate this correspondence by providing this calculus with a denotational semantics inside a lambdamodel in the category of sets and relations. 1
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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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.
Constructing differential categories and deconstructing categories of games
 In Luca Aceto, Monika Henzinger, and Jiri Sgall, editors, ICALP (2), volume 6756 of Lecture Notes in Computer Science
, 2011
"... Abstract. We present an abstract construction for building differential categories useful to model resource sensitive calculi, and we apply it to categories of games. In one instance, we recover a category previously used to give a fully abstract model of a nondeterministic imperative language. The ..."
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Abstract. We present an abstract construction for building differential categories useful to model resource sensitive calculi, and we apply it to categories of games. In one instance, we recover a category previously used to give a fully abstract model of a nondeterministic imperative language. The construction exposes the differential structure already present in this model. A second instance corresponds to a new Cartesian differential category of games. We give a model of a Resource PCF in this category and show that it enjoys the finite definability property. Comparison with a relational semantics reveals that the latter also possesses this property and is fully abstract. 1
Complexity of strongly normalising λterms via nonidempotent intersection types
"... We present a typing system for the λcalculus, with nonidempotent intersection types. As it is the case in (some) systems with idempotent intersections, a λterm is typable if and only if it is strongly normalising. Nonidempotency brings some further information into typing trees, such as a bound o ..."
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We present a typing system for the λcalculus, with nonidempotent intersection types. As it is the case in (some) systems with idempotent intersections, a λterm is typable if and only if it is strongly normalising. Nonidempotency brings some further information into typing trees, such as a bound on the longest βreduction sequence reducing a term to its normal form. We actually present these results in Klop’s extension of λcalculus, where the bound that is read in the typing tree of a term is refined into an exact measure of the longest reduction sequence. This complexity result is, for longest reduction sequences, the counterpart of de Carvalho’s result for linear headreduction sequences.
Full Abstraction for Resource Calculus with Tests
 In CSL, Lecture Notes in Computer Science
, 2011
"... We study the semantics of a resource sensitive extension of the λcalculus in a canonical reflexive object of a category of sets and relations, a relational version of the original Scott D ∞ model of the pure λcalculus. This calculus is related to Boudol’s resource calculus and is derived from Ehrh ..."
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We study the semantics of a resource sensitive extension of the λcalculus in a canonical reflexive object of a category of sets and relations, a relational version of the original Scott D ∞ model of the pure λcalculus. This calculus is related to Boudol’s resource calculus and is derived from Ehrhard and Regnier’s differential extension of Linear Logic and of the λcalculus. We extend it with new constructions, to be understood as implementing a very simple exception mechanism, and with a “must ” parallel composition. These new operations allow to associate a context of this calculus with any point of the model and to prove full abstraction for the finite subcalculus where ordinary λcalculus application is not allowed. The result is then extended to the full calculus by means of a Taylor Expansion formula. 1998 ACM Subject Classification F.4.1 Lambda calculus and related systems
Resource combinatory algebras
"... Abstract. We initiate a purely algebraic study of Ehrhard and Regnier’s resource λcalculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambdaalgebras and resource lambdaabstraction algebras. We establish the relations between them, laying down fou ..."
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Abstract. We initiate a purely algebraic study of Ehrhard and Regnier’s resource λcalculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambdaalgebras and resource lambdaabstraction algebras. We establish the relations between them, laying down foundations for a model theory of resource λcalculus. We also show that the ideal completion of a resource combinatory (resp. lambda, lambdaabstraction) algebra induces a “classical ” combinatory (resp. lambda, lambdaabstraction) algebra, and that any model of the classical λcalculus raising from a resource lambdaalgebra determines a λtheory which equates all terms having the same Böhm tree. 1
Head Linear Reduction
, 2004
"... This paper defines head linear reduction, a reduction strategy of #terms that performs the minimal number of substitutions for reaching a head normal form. The definition relies on an extended notion of redex, and head linear reduction is therefore not a strategy in the exact usual sense. Krivine ..."
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This paper defines head linear reduction, a reduction strategy of #terms that performs the minimal number of substitutions for reaching a head normal form. The definition relies on an extended notion of redex, and head linear reduction is therefore not a strategy in the exact usual sense. Krivine 's Abstract Machine is proved to be sound by relating it both to head linear reduction and to usual head reduction. The first proof suggests a variant machine, the Pointer Abstract Machine, which is also proved to be sound with respect to head linear reduction.
DIFFERENTIAL RESTRICTION CATEGORIES
"... Abstract. We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of Rn in a way that is completely algebraic. We also give other models for the resulting structure ..."
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Abstract. We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of Rn in a way that is completely algebraic. We also give other models for the resulting structure, discuss what it means for a partial map to be additive or linear, and show that differential restriction structure can be lifted through various completion operations.