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 eṡ2  n  , for any constant 0   (2) There are no subexponential size resolution refutations for random 3CNF formulas, when the promise is 2 Δ  n  , for any constant 0 O ( n  3/2−e ), for 0   “ Goods Satisfactory or Money Refunded ”   —The Eaton Promise
['Nachum Dershowitz', 'Iddo Tzameret']
Complexity of propositional proofs under a promise