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Complexity of propositional proofs under a promise

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 (i.e., 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: (1) Resolution has polynomial-size refutations for all unsatisfiable 3CNF formulas when the promise is &epsis;⋠2 n , for any constant 0<&epsis;<1. (2) There are no subexponential size resolution refutations for random 3CNF formulas, when the promise is 2 δ n , for any constant 0<δ<1 (and the number of clauses is O ( n 3/2−&epsis; ), for 0<&epsis;<1/2). “ Goods Satisfactory or Money Refunded ” —The Eaton Promise http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computational Logic (TOCL) Association for Computing Machinery

Complexity of propositional proofs under a promise

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Publisher
Association for Computing Machinery
Copyright
The ACM Portal is published by the Association for Computing Machinery. Copyright © 2010 ACM, Inc.
Subject
Proof theory
ISSN
1529-3785
DOI
10.1145/1740582.1740586
Publisher site
See Article on Publisher Site

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 (i.e., 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: (1) Resolution has polynomial-size refutations for all unsatisfiable 3CNF formulas when the promise is &epsis;⋠2 n , for any constant 0<&epsis;<1. (2) There are no subexponential size resolution refutations for random 3CNF formulas, when the promise is 2 δ n , for any constant 0<δ<1 (and the number of clauses is O ( n 3/2−&epsis; ), for 0<&epsis;<1/2). “ Goods Satisfactory or Money Refunded ” —The Eaton Promise

Journal

ACM Transactions on Computational Logic (TOCL) – Association for Computing Machinery

Published: May 1, 2010

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