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Leibniz international proceedings in informatics

Linear Lambda-Calculus is Linear

Alejandro Díaz-Caro, Gilles Dowek B

Published Jan 1, 2022• 0 citations
Publisher

Abstract

We prove a linearity theorem for an extension of linear logic with addition and multiplication by a scalar: the proofs of some propositions in this logic are linear in the algebraic sense. This work is part of a wider research program that aims at defining a logic whose proof language is a quantum programming language.
DOI
10.4230/lipics.fscd.2022.21

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Abstract We introduce the L!S{{\mathcal L}_!^{\mathcal S}}L!S​-calculus, a linear lambda-calculus extended with scalar multiplication and term addition, that acts as a proof language for intuitionistic linear logic. These algebraic operations enable the direct expression of linearity at the syntactic level, a property not typically available in standard proof-term calculi. Building upon previous work, we develop the L!S{{\mathcal L}_{!}^{\mathcal S}}L!S​-calculus as an extension of the LS{\mathcal L}^{\mathcal S}LS-calculus with the ! modality. We prove key meta-theoretical properties—subject reduction, confluence, strong normalization and an introduction property—as well as preserve the expressiveness of the original LS{\mathcal L}^{\mathcal S}LS-calculus, including the encoding of vectors and matrices, and the correspondence between proof-terms and linear functions. A denotational semantics is provided in the framework of linear categories with biproducts, ensuring a sound and adequate interpretation of the calculus. This work is part of a broader programme aiming to build a measurement-free quantum programming language grounded in linear logic.
The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine and continues to be of enormous benefit in the classical theory of computation. We propose that quantum computation, like its classical counterpart, may benefit from a version of the lambda calculus suitable for expressing and reasoning about quantum algorithms. In this paper we develop a quantum lambda calculus as an alternative model of quantum computation, which combines some of the benefits of both the quantum Turing machine and the quantum circuit models. The calculus turns out to be closely related to the linear lambda calculi used in the study of linear logic. We set up a computational model and an equational proof system for this calculus, and we argue that it is equivalent to the quantum Turing machine.
We introduce a minimal language combining both higher-order computation and linear algebra. Roughly, this is nothing else than the Lambda-calculus together with the possibility to make linear combinations of terms a.t+b.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 higher-order. We mention the perspectives of this work in field of quantum computation, which we show can be easily encoded in the calculus, as well as in other domains such as the interpretation of linear logic. Finally we prove the confluence of the calculus, this is our main result. Keywords: quantum programming language, quantum control, quantum logic, probabilistic and quantitative analysis, rewriting techniques.
With a view towards models of quantum computation, we define a functional language where all functions are linear operators by construction. A small step operational semantic (and hence an interpreter/simulator) is provided for this language in the form of a term rewrite systems. The linear-algebraic -calculus hereby constructed is linear in a different (yet related) sense to that, say, of the linear -calculus. These various notions of linearity are discussed in the context of quantum programming languages .