Gödelization in the Untyped lambda-Calculus

It is well-known that one cannot inside the pure untyped lambda calculus determine equivalence. I.e., one cannot determine if two terms are beta-equivalent, even if they both have normal forms. This implies that it is impossible in the pure untyped lambda calculus to do Godelisation, i.e. to write a function that can convert a term to a representation of (the normal form of) that term, as equivalence of normal-form terms is decidable given their representation. If the lambda calculus is seen as a programming language, this means that you can't from the value of a function nd its text. Things are di erent for simply typed lambda calculus: Berger and Schwichtenberg showed that, given its type, it is possible to convert a function into a representation of its normal form. This was termed \an inverse to the evaluation function", as it turns values into representations. However, the main purpose was for normalising terms. Similarly, Goldberg has shown that for a subset (proper combinators) of the pure untyped lambda calculus, G odelisation is possible. However, the Godeliser itself is not a proper combinator, though it (as all closed lambda terms) can be written by combining proper combinators. In this paper, we investigate Godelisation for the full untyped lambda calculus. To overcome the theoretical impossibility of this, we extend the lambda calculus with a feature that allows limited manipulation of extensional aspects: A nite set of labels on lambda terms and a predicate for comparing these. Within this extended lambda calculus, we can convert terms in the subset corresponding to normal form terms in the classical lambda calculus into their representation. The extension of the lambda calculus (we conjecture) retains the Church-Rosser property. This implies that G odelisation must yield identical results for beta-equivalent terms. We show only that terms in normal form G odelise to their representation, but the implication is that any term that has a normal form will Godelise to a representation of its normal form. Hence, G odelisation can be used as a tool for normalising lambda terms.