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 colored has an infinite monochromatic subsequence. Since in general there is no computable functional producing such an infinite subsequence, we consider a Π 0 2-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 Gödel's Dialectica interpretation. In this paper, we compare the resulting programs with respect to their behavior and complexity and indicate how they reflect the computational content of IPH.

Expansion trees with cut

“…It is well known that cut-elimination and similar procedures in classical logic are typically nonconfluent; see, e.g., Urban (2000), Ratiu and Trifonov (2012), and Baaz et al (2005) for case studies and Baaz and Hetzl (2011) and Hetzl (2012) for asymptotic results. Neither the proof forests of Heijltjes (2010) nor the Herbrand nets of McKinley (2013) have a confluent reduction.…”

Herbrand-Confluence

“…From the point of view of applications this means that we have a choice among different programs that can be extracted. In [RT12] the authors show that two different extraction methods applied to the same proof produce two programs, one of polynomial and one of exponential average-case complexity. This phenomenon is further exemplified by case studies in [Urb00, BHL `05, BHL `08] as well as the asymptotic results [BH11,Het12b].…”

Proofs, Programs, Processes

“…Therefore, abstract mathematical objects do not need to be "constructivised", and it is possible to directly extract programs from abstract mathematical proofs. The uniform treatment of first-order quantifiers can be seen as a special case of the interpretations studied by Schwichtenberg [Sch08], Hernest and Oliva [HO08] and Ratiu and Trifonov [RT10], which allow for a fine control of the amount of computational information extracted from proofs. We illustrate our interpretation by extracting the average function on the real interval [−1, 1] from a proof.…”