Exploring the Computational Content of the Infinite Pigeonhole Principle

Abstract

The use of classical logic for some combinatorial proofs, as it is the case with Ramsey’s theorem, can be localized in the Infinite Pigeonhole (IPH) principle, stating that any infinite sequence which is finitely coloured has an infinite monochromatic subsequence. Since, in general, there is no computable functional producing such an infinite subsequence, we consider a Π20-corollary, proving the classical existence of a finite monochromatic subsequence of any given length. In order to obtain a program from this proof, we apply two methods for extraction: the refined A-Translation, as proposed by Berger et al. and Godel’s Dialectica interpretation. In this article, we compare the resulting programs with respect to their behaviour and complexity and indicate how they reflect the computational content of IPH.