Abstract. The algebraic lambda-calculus and the linear-algebraic lambda-calculus extend the lambda-calculus with the possibility of making arbitrary linear combinations of terms. In this paper we provide a fine-grained, System F -like type system for the linear-algebraic lambda-calculus. We show that this "Scalar " type system enjoys both the subject-reduction property and the strong-normalisation property, our main technical results. The latter yields a significant simplification of the linear-algebraic lambda-calculus itself, by removing the need for some restrictions in its reduction rules. But the more important, original feature of the Scalar type system is that it keeps track of 'the amount of a type' that is present in each term. As an example of its use, we show that it can serve as a guarantee that the normal form of a term is barycentric, i.e that its scalars are summing to one.

Abstract. We introduce a quantum-like classical computational concept, called affine computation, as a generalization of probabilistic computation. After giving the basics of affine computation, we define affine finite automata (AfA) and compare it with quantum and probabilistic finite automata (QFA and PFA, respectively) with respect to three basic language recognition modes. We show that, in the cases of bounded and unbounded error, AfAs are more powerful than QFAs and PFAs, and, in the case of nondeterministic computation, AfAs are more powerful than PFAs but equivalent to QFAs. Moreover, we show that exclusive affine languages form a superset of exclusive quantum and stochastic languages.

We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We show that the resulting typed lambda-calculus is strongly normalizing and features a weak subject-reduction

In this paper we present a semantics for a linear algebraic lambda-calculus based on realizability. This semantics characterizes a notion of unitarity in the system, answering a long standing issue. We derive from the semantics a set of typing rules for a simply-typed linear algebraic lambda-calculus, and show how it extends both to classical and quantum lambda-calculi.

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