Journal of Computer and System Sciences - 26th IEEE Conference on Foundations of Computer Science, October 21-23, 1985
On truth-table reducibility to SAT
Information and Computation
Exponential lower bounds for the pigeonhole principle
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Some consequences of cryptographical conjectures for S12 and EF
Information and Computation - Special issue: logic and computational complexity
A new proof of the weak Pigeonhole principle
Journal of Computer and System Sciences - Special issue on STOC 2000
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This paper considers the relational versions of the surjective, partial surjective, and multifunction weak pigeonhole principles for PV, @?"1^b, @?"1^b, and B(@?"1^b) formulas as well as relativizations of these formulas to higher levels of the bounded arithmetic hierarchy. We show that the partial surjective weak pigeonhole principle for @?"1^b formulas implies that for each k there is a string of length 2^2^n^^^k which is hard to block-recognize by circuits of size n^k. These principles in turn imply the partial surjective principle for @?"1^b formulas. We show that the surjective weak pigeonhole principle for B(@?"1^b) formulas in S"2^1 implies our hard-string principle which in turn implies the surjective weak pigeonhole principle for @?"1^b formulas. We introduce a class of predicates corresponding to poly-log length iterates of polynomial time computable predicates and show that over S"2^1, the multifunction weak pigeonhole principle for such predicates is equivalent to an ''iterative'' circuit block-recognition principle. A consequence of this is that if S"2^1 proves this principle then RSA is vulnerable to polynomial time attacks.