## 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 class
✦ LIBER ✦
Dual weak pigeonhole principle, Boolean complexity, and derandomization
✍ Scribed by Emil Jeřábek
- Publisher
- Elsevier Science
- Year
- 2004
- Tongue
- English
- Weight
- 410 KB
- Volume
- 129
- Category
- Article
- ISSN
- 0168-0072
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We show that the known bounded-depth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a size-depth trade-o upper bound: there are proofs of size n O(d(log(n)) 2=d ) and depth O(d). This solves an open problem of Maciel et