On some $\Sigma^{B}_{0}$-formulae generalizing counting principles over $V^{0}$

TL;DR

This paper formalizes and compares various counting principles over the bounded arithmetic theory $V^{0}$, exploring their independence and proof complexity, and establishing conditions for their provability.

Contribution

It introduces new formalizations of counting principles, conjectures their mutual independence, and provides proof complexity bounds within bounded arithmetic.

Findings

Mutual independence conjectures between counting principles and combinatorial theorems.

Sufficient conditions for proving the independence conjectures.

Lower bounds on proof evaluations for certain propositional proofs.

Abstract

We formalize various counting principles and compare their strengths over $V^{0}$. In particular, we conjecture the following mutual independence between: (1) a uniform version of modular counting principles and the pigeonhole principle for injections, (2) a version of the oddtown theorem and modular counting principles of modulus $p$, where $p$ is any natural number which is not a power of $2$, (3) and a version of Fisher's inequality and modular counting principles. Then, we give sufficient conditions to prove them. We give a variation of the notion of $PHP$-tree and $k$-evaluation to show that any Frege proof of the pigeonhole principle for injections admitting the uniform counting principle as an axiom scheme cannot have $o(n)$-evaluations. As for the remaining two, we utilize well-known notions of $p$-tree and $k$-evaluation and reduce the problems to the existence of certain families of polynomials witnessing violations of the corresponding combinatorial principles with low-degree Nullstellensatz proofs from the violation of the modular counting principle in concern.