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.2n, for any constant 0 < ε< 1.
(ii) There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is 2δn (and the number of clauses is o(n 3/2)), for any constant 0 < δ< 1.
This work was carried out in partial fulfillment of the requirements for the Ph.D. degree of the second author and was supported in part by the Israel Science Foundation (grant no. 250/05).
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Dershowitz, N., Tzameret, I. (2007). Complexity of Propositional Proofs Under a Promise. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73420-8_27
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DOI: https://doi.org/10.1007/978-3-540-73420-8_27
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