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Complexity of Propositional Proofs Under a Promise

  • Nachum Dershowitz
  • Iddo Tzameret
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

Abstract

We study – within the framework of propositional proof complexity – the problem of certifying unsatisfiability of CNF formulas under the promise that any satisfiable formula has many satisfying assignments, where “many” stands for an explicitly specified function Λ in the number of variables n. To this end, we develop propositional proof systems under different measures of promises (that is, different Λ) as extensions of resolution. This is done by augmenting resolution with axioms that, roughly, can eliminate sets of truth assignments defined by Boolean circuits. We then investigate the complexity of such systems, obtaining an exponential separation in the average-case between resolution under different size promises:

(i) Resolution has polynomial-size refutations for all unsatisfiable 3CNF formulas when the promise is ε.2 n , for any constant 0 < ε< 1.

(ii) There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is 2 δn (and the number of clauses is o(n 3/2)), for any constant 0 < δ< 1.

Keywords

proof complexity resolution random 3CNF promise problems 

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References

  1. 1.
    Beame, P., Karp, R., Pitassi, T., Saks, M.: The efficiency of resolution and Davis-Putnam procedures. SIAM J. Comput. 31(4), 1048–1075 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow—resolution made simple. J. ACM 48(2), 149–169 (2001)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. The Journal of Symbolic Logic 44(1), 36–50 (1979)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dershowitz, N., Tzameret, I.: Complexity of propositional proofs under a promise (full version) (2007), http://www.cs.tau.ac.il/~tzameret/PromiseProofs.pdf
  5. 5.
    Feige, U., Kim, J., Ofek, E.: Witnesses for non-satisfiability of dense random 3CNF formulas. In: Proc. 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 497–508 (October 2006)Google Scholar
  6. 6.
    Hirsch, E.: A fast deterministic algorithm for formulas that have many satisfying assignments. Logic Journal of the IGPL 6(1), 59–71 (1998)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Trevisan, L.: A note on approximate counting for k-DNF. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 417–426. Springer, Heidelberg (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Nachum Dershowitz
    • 1
    • 2
  • Iddo Tzameret
    • 1
  1. 1.School of Computer Science, Tel Aviv University, Ramat Aviv 69978Israel
  2. 2.Microsoft Research, Redmond, WA 98052USA

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