The weak pigeonhole principle for function classes in S^1_2

TL;DR

This paper explores the provability of the surjective weak pigeonhole principle within the logical system S^1_2 for various function classes weaker than polynomial time, extending known results about the injective case.

Contribution

It investigates the provability of the surjective weak pigeonhole principle in S^1_2 for weaker function classes, complementing existing results on the injective case.

Findings

S^1_2 cannot prove the surjective weak pigeonhole principle for certain weaker function classes

The paper identifies specific classes where provability differs from the polynomial time case

Results suggest limitations of S^1_2 in formalizing surjective pigeonhole principles for these classes

Abstract

It is well known that S^1_2 cannot prove the injective weak pigeonhole principle for polynomial time functions unless RSA is insecure. In this note we investigate the provability of the surjective (dual) weak pigeonhole principle in S^1_2 for provably weaker function classes.