Applications of infinitary lambda calculus

We present an introduction to infinitary lambda calculus, highlighting its main properties. Subsequently we give three applications of infinitary lambda calculus. The first addresses the non-definability of Surjective Pairing, which was shown by the first author not to be definable in lambda calculus. We show how this result follows easily as an application of Berry’s Sequentiality Theorem, which itself can be proved in the setting of infinitary lambda calculus. The second pertains to the notion of relative recursiveness of number-theoretic functions. The third application concerns an explanation of counterexamples to confluence of lambda calculus extended with non-left-linear reduction rules: Adding non-left-linear reduction rules such as δxx→x or the reduction rules for Surjective Pairing to the lambda calculus yields non-confluence, as proved by the second author. We discuss how an extension to the infinitary lambda calculus, where Böhm trees can be directly manipulated as infinite terms, yields a more simple and intuitive explanation of the correctness of these Church-Rosser counterexamples.