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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.
A system F accounting for scalars
, 2010
"... The algebraic λcalculus [40] and the linearalgebraic λcalculus [3] extend the λcalculus with the possibility of making arbitrary linear combinations of λcalculus terms (preserving ∑ αi.ti). In this paper we provide a finegrained, System Flike type system for the linearalgebraic λcalculus (L ..."
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The algebraic λcalculus [40] and the linearalgebraic λcalculus [3] extend the λcalculus with the possibility of making arbitrary linear combinations of λcalculus terms (preserving ∑ αi.ti). In this paper we provide a finegrained, System Flike type system for the linearalgebraic λcalculus (Lineal). We show that this scalar type system enjoys both the subjectreduction property and the strongnormalisationproperty, which constitute our main technical results. The latter yields a significant simplification of the linearalgebraic λcalculus itself, by removing the need for some restrictions in its reduction rules – and thus leaving it more intuitive. But the more important, original feature of this scalar type system is that it keeps track of ‘the amount of a type’ that this present in each term. As an example, we show how to use this type system in order to guarantee the welldefiniteness of probabilistic functions ( ∑ αi = 1) – thereby specializing Lineal into a probabilistic, higherorder λcalculus. Finally we begin to investigate the logic induced by the scalar type system, and prove a nocloning theorem expressed solely in terms of the possible proof methods in this logic. We discuss the potential connections with Linear Logic and Quantum Computation.
Lineal: A linearalgebraic λcalculus
, 2010
"... We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higherorder computation and linear algebra. This language extends the λcalculus with the possibility to make arbitrary linear combinations of te ..."
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We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higherorder computation and linear algebra. This language extends the λcalculus with the possibility to make arbitrary linear combinations of terms α.t + β.u. We describe how to “execute” this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higherorder. We mention the perspectives of this work in the field of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus.
Lineal: A linearalgebraic λcalculus
"... We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higherorder computation and linear algebra. This language extends the λcalculus with the possibility to make arbitrary linear combinations of te ..."
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We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higherorder computation and linear algebra. This language extends the λcalculus with the possibility to make arbitrary linear combinations of terms α.t + β.u. We describe how to “execute ” this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higherorder. We mention the perspectives of this work in the field of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus. 1. Motivations Knuth and Bendix have described a method to transform an equational theory into a rewrite system [35]. In this paper, we show that this can be achieved for the theory of vector spaces. This yields a computational definition of the no
c © A. Assaf and S.Perdrix This work is licensed under the Creative Commons Attribution License. Completeness of algebraic CPS simulations
, 2011
"... The algebraic lambda calculus (λalg) and the linear algebraic lambda calculus (λlin) are two extensions of the classical lambda calculus with linear combinations of terms. They arise independently in distinct contexts: the former is a fragment of the differential lambda calculus, the latter is a ca ..."
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The algebraic lambda calculus (λalg) and the linear algebraic lambda calculus (λlin) are two extensions of the classical lambda calculus with linear combinations of terms. They arise independently in distinct contexts: the former is a fragment of the differential lambda calculus, the latter is a candidate lambda calculus for quantum computation. They differ in the handling of application arguments and algebraic rules. The two languages can simulate each other using an algebraic extension of the wellknown callbyvalue and callbyname CPS translations. These simulations are sound, in that they preserve reductions. In this paper, we prove that the simulations are actually complete, strengthening the connection between the two languages. 1