λ-definition of function(al)s by normal forms

Abstract

Lambda-calculus is extended in order to represent a rather large class of recursive equation systems, implicitly characterizing function(al)s or mappings of some algebraic domain into arbitrary sets. Algebraic equality will then be represented by A~g-convertibility (or even reducibility). It is then proved, under very weak assumptions on the structure of the equations, that there always exist solutions in normal form (Interpretation theorem). Some features of the solutions, like the use of parametric representations of the algebraic constructors, higherorder solutions by currification, definability of functions on unions of algebras, etc., have been easily checked by a first implementation of the mentioned theorem, the CuCh machine. * This work has been pal'tially supported by grahts from ESPRIT BRA 7232 working group "Gentzen" and from MURST 40% (Italy).

Encoding the Factorisation Calculus

“…We have a signature containing one function symbol add, one nullary constructor zero, and one unary constructor succ; moreover in this simple case, the equation system is already complete. In fact, every partial recursive function (on natural numbers) can be defined by a canonical system of equations [5,6]. Given an equational system E over a signature Σ, we can also take it to define a term rewriting system on Ter(Σ) by reading each equation as a rewrite rule, i.e.…”

Böhm's Theorem

“…The notion of X-separability has interesting relationships with invertibility of λ-terms. The Böhm-out technique is at the basis of the implementation, presented in [Böhm et al, 1994], of the CuCh-machine, a λ-calculus interpreter introduced by Böhm and Gross in [Böhm and Gross, 1966].…”

The Impact of the Lambda Calculus in Logic and Computer Science

“…[This was the trick Kleene found at the dentist.] Now we will present the method of [24] and [19] to represent data types. Again we consider the example of labeled trees.…”

“…As a result, a much more efficient representation of algorithms on these data types can be given, than when these types were represented via numbers. This methodology was perfected in two different ways in [22] and [24] or [19]. The first paper does the representation in a way that can be typed; the other papers in an essentially stronger way, but one that cannot be typed.…”