Abstract
One of the starting points of propositional proof complexity is the seminal paper by Cook and Reckhow [6], 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 Krajíček [5] have recently started the investigation of proof systems which are computed by poly-time functions using advice. While this yields a more powerful model, it is also less directly applicable in practice.
In this note we investigate the question 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 proof systems with logarithmic advice are also computable in poly-time with access to a sparse NP-oracle. In addition, we show that if advice is ”not very helpful” for proving tautologies, then there exists an optimal propositional proof system without advice. In our main result, we prove that advice can be transferred from the proof to the formula, leading to an easier computational model. We obtain this result by employing a recent technique by Buhrman and Hitchcock [4].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Balcázar, J., Schöning, U.: Logarithmic advice classes. Theoretical Computer Science 99, 279–290 (1992)
Beyersdorff, O., Köbler, J., Müller, S.: Nondeterministic instance complexity and proof systems with advice. In: Proc. 3rd International Conference on Language and Automata Theory and Applications. LNCS, vol. 5457, pp. 164–175. Springer, Heidelberg (2009)
Beyersdorff, O., Müller, S.: A tight Karp-Lipton collapse result in bounded arithmetic. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 199–214. Springer, Heidelberg (2008)
Buhrman, H., Hitchcock, J.M.: NP-hard sets are exponentially dense unless coNP ⊆ NP/poly. In: Proc. 23rd Annual IEEE Conference on Computational Complexity, pp. 1–7 (2008)
Cook, S.A., Krajíček, J.: Consequences of the provability of NP ⊆ P/poly. The Journal of Symbolic Logic 72(4), 1353–1371 (2007)
Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. The Journal of Symbolic Logic 44(1), 36–50 (1979)
Köbler, J., Messner, J., Torán, J.: Optimal proof systems imply complete sets for promise classes. Information and Computation 184(1), 71–92 (2003)
Krajíček, J., Pudlák, P.: Propositional proof systems, the consistency of first order theories and the complexity of computations. The Journal of Symbolic Logic 54(3), 1063–1079 (1989)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Beyersdorff, O., Müller, S. (2009). Does Advice Help to Prove Propositional Tautologies?. In: Kullmann, O. (eds) Theory and Applications of Satisfiability Testing - SAT 2009. SAT 2009. Lecture Notes in Computer Science, vol 5584. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02777-2_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-02777-2_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02776-5
Online ISBN: 978-3-642-02777-2
eBook Packages: Computer ScienceComputer Science (R0)