Computational limitations of small-depth circuits
Computational limitations of small-depth circuits
Exponential lower bounds for the pigeonhole principle
Computational Complexity
Bounded arithmetic, propositional logic, and complexity theory
Bounded arithmetic, propositional logic, and complexity theory
An exponential lower bound to the size of bounded depth Frege proofs of the Pigeonhole Principle
Random Structures & Algorithms
Feasibly constructive proofs and the propositional calculus (Preliminary Version)
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
Separating the polynomial-time hierarchy by oracles
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Improved witnessing and local improvement principles for second-order bounded arithmetic
ACM Transactions on Computational Logic (TOCL)
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We give a new characterization of the strict $$\forall {\Sigma^b_j}$$ sentences provable using $${\Sigma^b_k}$$ induction, for 1 驴 j 驴 k. As a small application we show that, in a certain sense, Buss's witnessing theorem for strict $${\Sigma^b_k}$$ formulas already holds over the relatively weak theory PV. We exhibit a combinatorial principle with the property that a lower bound for it in constant-depth Frege would imply that the narrow CNFs with short depth j Frege refutations form a strict hierarchy with j, and hence that the relativized bounded arithmetic hierarchy can be separated by a family of $$\forall {\Sigma^b_1}$$ sentences.