Theoretical Computer Science
Journal of the ACM (JACM)
Resource-Bounded Kolmogorov Complexity Revisited
SIAM Journal on Computing
Nondeterministic Instance Complexity and Hard-to-Prove Tautologies
STACS '00 Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Some connections between nonuniform and uniform complexity classes
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Optimal proof systems imply complete sets for promise classes
Information and Computation
NP-Hard Sets Are Exponentially Dense Unless coNP C NP/poly
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Nondeterministic Instance Complexity and Proof Systems with Advice
LATA '09 Proceedings of the 3rd International Conference on Language and Automata Theory and Applications
Does Advice Help to Prove Propositional Tautologies?
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
Competing provers yield improved Karp-Lipton collapse results
Information and Computation
A tight Karp-Lipton collapse result in bounded arithmetic
ACM Transactions on Computational Logic (TOCL)
Verifying proofs in constant depth
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
On the amount of nonconstructivity in learning formal languages from positive data
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
Verifying proofs in constant depth
ACM Transactions on Computation Theory (TOCT)
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One of the starting points of propositional proof complexity is the seminal paper by Cook and Reckhow [J. Symbolic Logic, 1979], where they defined propositional proof systems as poly-time computable functions which have all propositional tautologies as their range. Motivated by provability consequences in bounded arithmetic, Cook and Kraji'cek [J. Symbolic Logic, 2007] have recently started the investigation of proof systems which are computed by poly-time functions using advice. In this paper we concentrate on three fundamental questions regarding this new model. First, we investigate whether a given language L admits a polynomially bounded proof system with advice. Depending on the complexity of the underlying language L and the amount and type of the advice used by the proof system, we obtain different characterizations for this problem. In particular, we show that this question is tightly linked with the question whether L has small nondeterministic instance complexity. The second question concerns the existence of optimal proof systems with advice. For propositional proof systems, Cook and Kraji'cek gave a surprising positive answer which we extend to all languages. These results show that providing proof systems with advice yields a more powerful model, but this model is also less directly applicable in practice. Our third question therefore asks whether the usage of advice in propositional proof systems can be simplified or even eliminated. While in principle, the advice can be very complex, we show that propositional proof systems with logarithmic advice are also computable in poly-time with access to a sparse NP-oracle. Employing a recent technique of Buhrman and Hitchcock [CCC, 2008] we also manage to transfer the advice from the proof to the proven formula, which leads to a more practical computational model.