On the complexity of the parity argument and other inefficient proofs of existence
Journal of Computer and System Sciences - Special issue: 31st IEEE conference on foundations of computer science, Oct. 22–24, 1990
The independence of the modulo p counting principles
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Lower bounds to the size of constant-depth propositional proofs
Journal of Symbolic Logic
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
Proof complexity in algebraic systems and bounded depth Frege systems with modular counting
Computational Complexity
The relative complexity of NP search problems
Journal of Computer and System Sciences
On Interpolation and Automatization for Frege Systems
SIAM Journal on Computing
Linear gaps between degrees for the polynomial calculus modulo distinct primes
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Searching Constant Width Mazes Captures the AC0 Hierarchy
STACS '98 Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
The Complexity of Proving the Discrete Jordan Curve Theorem
ACM Transactions on Computational Logic (TOCL)
Time-space tradeoffs in resolution: superpolynomial lower bounds for superlinear space
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The st- connectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of the grid graph unless the paths cross somewhere.We prove that the propositional tautologies which encode the st-connectivity principle have polynomial-size Frege proofs and polynomial-size TC0-Frege proofs. For bounded-width grid graphs, the st-connectivity tautologies have polynomial-size resolution proofs. A key part of the proof is to show that the group with two generators, both of order two, has word problem in alternating logtime (Alogtime) and even in TC0.Conversely, we show that constant depth Frege proofs of the st-connectivity tautologies require near-exponential size. The proof uses a reduction from the pigeonhole principle, via tautologies that express a "directed single source" principle SINK, which is related to Papadimitriou's search classes PPAD and PPADS (or, PSK).The st-connectivity principle is related to Urquhart's propositional Hex tautologies, and we establish the same upper and lower bounds on proof complexity for the Hex tautologies. In addition, the Hex tautology is shown to be equivalent to the SINK tautologies and to the one-to-one onto pigeonhole principle.