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Hard Instances of Algorithms and Proof Systems

Published:01 May 2014Publication History
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Abstract

If the class TAUT of tautologies of propositional logic has no almost optimal algorithm, then every algorithm A deciding TAUT has a hard sequence, that is, a polynomial time computable sequence witnessing that A is not almost optimal. We show that this result extends to every Πt p-complete problem with t ≥1; however, assuming the Measure Hypothesis, there is a problem which has no almost optimal algorithm but is decided by an algorithm without hard sequences. For problems Q with an almost optimal algorithm, we analyze whether every algorithm deciding Q, which is not almost optimal, has a hard sequence.

References

  1. S. Ben-David and A. Gringauze. 1998. On the existence of propositional proof systems and oracle-relativized propositional logic. In Proceedings of the Electronic Colloquium on Computational Complexity (ECCC), TR98-021.Google ScholarGoogle Scholar
  2. L. Berman. 1976. On the structure of complete sets: Almost everywhere complexity and infinitely often speedup. In Proceedings of the 17th IEEE Symposium on Foundations of Computer Science (FOCS’76). 76--80. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. L. Berman and J. Hartmanis. 1977. On isomorphisms and density of NP and other complete sets. SIAM J. Comput. 6, 2, 305--322.Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. O. Beyersdorff and Z. Sadowski. 2011. Do there exist complete sets for promise classes? Math. Logic Quart. 57, 6, 535--550.Google ScholarGoogle ScholarCross RefCross Ref
  5. H. Buhrman, S. A. Fenner, and L. Fortnow. 1997. Results on resource-bounded measure. In Proceedings of the 24th International Colloquium on Automata, Languages and Programming (ICALP’97). Lecture Notes in Computer Science, vol. 1256, Springer, 188--194. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Y. Chen and J. Flum. 2010. On p-optimal proof systems and logics for PTIME. In Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP’10, Track B). Lecture Notes in Computer Science, 6199, vol. 6199, Springer, 321--332. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Y. Chen and J. Flum. 2011. Listings and logics. In Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science (LICS’11). IEEE Computer Society, 165--174. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Y. Chen and J. Flum. 2014. On optimal inverters. Bull. Symb. Logic. To appear.Google ScholarGoogle Scholar
  9. D. Gutfreund, R. Shaltiel, and A. Ta-Shma. 2007. If NP languages are hard on the worst-case, then it is easy to find their hard instances. Computat. Complex. 16, 4, 412--441. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. K-I Ko and D. J. Moore. 1981. Completeness, approximation and density. SIAM J. Comput. 10, 4, 787--796.Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. J. Köbler and J. Messner. 1998. Complete problems for promise classes by optimal proof systems for test sets. In Proceedings of the 13th Annual IEEE Conference on Computational Complexity (CCC’98). Springer, 132--140. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. J. Köbler, J. Messner, and J. Torán. 2003. Optimal proof systems imply complete sets for promise classes. Inf. Computat. 184, 1, 71--92. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. J. Krajíček. 1995. Bounded Arithmetic, Propositional Logic, and Complexity Theory. Cambridge University Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. J. Krajíček. 2012. A note on SAT algorithms and proof complexity. Inf. Process. Lett. 112, 12, 490--493. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. J. Krajíček. 2014. On the computational complexity of finding hard tautologies. Bull. Lond. Math. Soc. 46, 1, 111--125.Google ScholarGoogle ScholarCross RefCross Ref
  16. J. Krajíček and P. Pudlák. 1989. Propositional proof systems, the consistency of first order theories and the complexity of computations. J. Symb. Logic 54, 3, 1063--1079.Google ScholarGoogle ScholarCross RefCross Ref
  17. L. Levin. 1973. Universal sequential search problems. Prob. Inf. Transmiss. 9, 3, 265--266.Google ScholarGoogle Scholar
  18. J. H. Lutz. 1997a. Observations on measure and lowness for Δp2. Theory Comput. Syst. 30, 4, 429--442.Google ScholarGoogle ScholarCross RefCross Ref
  19. J. H. Lutz. 1997b. The quantitative structure of exponential time. Complexity Theory Retrospective II, 225--254. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. E. Mayordomo. 1994. Almost every set in exponential time is P-bi-Immune. Theoret. Comput. Sci. 136, 2, 487--506. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. J. Messner. 1999. On optimal algorithms and optimal proof systems. In Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS’99). Lecture Notes in Computer Science, vol. 1563, Springer, 541--550. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. J. Messner. 2000. On the simulation order of proof systems. Ph.D. Dissertation. University of Ulm.Google ScholarGoogle Scholar
  23. H. Monroe. 2011. Speedup for natural problems and noncomputability. Theoret. Comput. Sci. 412, 4--5, 478--481. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Z. Sadowski. 2002. On an optimal propositional proof system and the structure of easy subsets. Theoret. Comput. Sci. 288, 1, 181--193. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Z. Sadowski. 2007. Optimal proof systems, optimal acceptors and recursive presentability. Fund. Informat. 79, 1--2, 169--185. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. C. P. Schnorr. 1976. Optimal algorithms for self-reducible problems. In Proceedings of the 3rd International Colloquium on Automata, Languages and Programming (ICALP’76). Edinburgh University Press, 322--337.Google ScholarGoogle Scholar
  27. L. Stockmeyer. 1974. The complexity of decision problems in automata theory. Ph.D. Dissertation, MIT.Google ScholarGoogle Scholar
  28. O. V. Verbitsky. 1979. Optimal algorithms for coNP-sets and the problem EXP=NEXP. Matemat. zametki 50, 2, 37--46.Google ScholarGoogle Scholar

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