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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2203.10237 |
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Table of Contents:
- 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.