Scalar System F for Linear-Algebraic λ-Calculus: Towards a Quantum Physical Logic

Abstract

Session: Quantum CalculiInternational audienceThe Linear-Algebraic λ-Calculus extends the λ-calculus with the possibility of making arbitrary linear combinations of terms α.t+β.u. Since one can express fixed points over sums in this calculus, one has a notion of infinities arising, and hence indefinite forms. As a consequence, in order to guarantee the confluence, t−t does not always reduce to 0 - only if t is closed normal. In this paper we provide a System F like type system for the Linear-Algebraic λ-Calculus, which guarantees normalisation and hence no need for such restrictions, t−t always reduces to 0. Moreover this type system keeps track of 'the amount of a type'. As such it can be seen as probabilistic type system, guaranteeing that terms define correct probabilistic functions. It can also be seen as a step along the quest toward a quantum physical logic through the Curry-Howard isomorphism

Confluence via strong normalisation in an algebraic λ-calculus with rewriting

“…Table 1 shows the abstract syntax of types and terms of the calculus, where the terms are based on those of λ lin [3]. Our choice of explicit System F instead of a Curry style presentation [1,6] stems from the fact that, as shown in [2], the "factorisation" reduction rules (cf. Group F in Table 2) in a Curry style setting introduce some imprecisions.…”

“…Notice that we cannot reuse the proofs of previous typed versions of λ lin (e.g. [1,6]) since in [1] only terms of the same type can be added together, and in [6] the calculus under consideration is a fragment of λ CA . Therefore, none of them have the same set of terms as λ CA .…”

“…The rewrite system has been proved locally confluent [2], so by ensuring strong normalisation we obtain confluence [17]. This approach has been followed in other works [1,2,6] which discuss similar type systems with strong normalisation. While these type systems give us some information about the terms, they also impose some undesirable restrictions:…”

A Type System for the Vectorial Aspect of the Linear-Algebraic Lambda-Calculus

“…These languages are based on an interpretation of intuitionistic logic by linear logic. A second line of work [2,3,6] considers linear combinations of terms as a sort of "quantum superposition". This paper stems from this second approach.…”

“…The central question this paper is concerned with is the nature of the type system to be used. The solution we are proposing is an extension of two languages designed in [2] and [6].…”