Complexity of Propositional Proofs Under a Promise
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.
Keywordsproof complexity resolution random 3CNF promise problems
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- 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.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
- 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