Superdevelopments for Weak Reduction

We study superdevelopments in the weak lambda calculus of Cagman and Hindley, a confluent variant of the standard weak lambda calculus in which reduction below lambdas is forbidden. In contrast to developments, a superdevelopment from a term M allows not only residuals of redexes in M to be reduced but also some newly created ones. In the lambda calculus there are three ways new redexes may be created; in the weak lambda calculus a new form of redex creation is possible. We present labeled and simultaneous reduction formulations of superdevelopments for the weak lambda calculus and prove them equivalent

Completeness of algebraic CPS simulations

The algebraic lambda calculus and the linear algebraic lambda calculus 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 well-known call-by-value and call-by-name 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.Comment: In Proceedings DCM 2011, arXiv:1207.682

Linear-algebraic lambda-calculus

With a view towards models of quantum computation and/or the interpretation of linear logic, 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 system. The linear-algebraic lambda-calculus hereby constructed is linear in a different (yet related) sense to that, say, of the linear lambda-calculus. These various notions of linearity are discussed in the context of quantum programming languages. KEYWORDS: quantum lambda-calculus, linear lambda-calculus, $\lambda$-calculus, quantum logics.Comment: LaTeX, 23 pages, 10 figures and the LINEAL language interpreter/simulator file (see "other formats"). See the more recent arXiv:quant-ph/061219

A System F accounting for scalars

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 this 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 shown 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

Trees from Functions as Processes

Levy-Longo Trees and Bohm Trees are the best known tree structures on the {\lambda}-calculus. We give general conditions under which an encoding of the {\lambda}-calculus into the {\pi}-calculus is sound and complete with respect to such trees. We apply these conditions to various encodings of the call-by-name {\lambda}-calculus, showing how the two kinds of tree can be obtained by varying the behavioural equivalence adopted in the {\pi}-calculus and/or the encoding

GLC actors, artificial chemical connectomes, topological issues and knots

Based on graphic lambda calculus, we propose a program for a new model of asynchronous distributed computing, inspired from Hewitt Actor Model, as well as several investigation paths, concerning how one may graft lambda calculus and knot diagrammatics