A self-interpreter of lambda calculus having a normal form

Abstract

We formalize a technique introduced by Böhm and Piperno to solve systems of recursive equations in lambda calculus without the use of the fixed point combinator and using only normal forms. To this aim we introduce the notion of a canonical algebraic term rewriting system, and we show that any such system can be interpreted in the lambda calculus by the Böhm — Piperno technique in such a way that strong normalization is preserved. This allows us to improve some recent results of Mogensen concerning efficient gödelizations ⌈⌉:Λ→Λ of lambda calculus. In particular we prove that under a suitable gödelization there exist two lambda terms E (self-interpreter) and R (reductor), both having a normal form, such that for every (closed or open) lambda term M E⌈M⌉→M and if M has a normal form N, then R⌈M⌉→⌈N⌉.

Partially supported by MURST Research projects 40% “Modelli della computazione e dei linguaggi di programmazione”, and 60% “Specifiche, Concorrenza e Logica computazionale”, while the first author was holding a research position at the University of L'Aquila.

Partially supported by MURST Research projects 40% “Fondamenti dei linguaggi funzionali e logici”, and 60% “Progetto Ateneo”.