Complexity of Propositional Proofs under a Promise

TL;DR

This paper explores the complexity of propositional proof systems under promises about the number of satisfying assignments, revealing exponential separations in proof sizes depending on the promise size and formula type.

Contribution

It introduces augmented resolution proof systems with axioms based on Boolean circuit-defined sets, analyzing their complexity under different promise measures.

Findings

Polynomial-size refutations for unsatisfiable 3CNF formulas with large promises.

No sub-exponential refutations for random 3CNF formulas with certain promise sizes.

Exponential separation in proof complexity depending on promise size and formula randomness.

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 $\Lam$ in the number of variables $n$. To this end, we develop propositional proof systems under different measures of promises (that is, different $\Lam$) 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 $\eps\cd2^n$, for any constant $0<\eps<1$. 2. There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is $2^{\delta n}$ (and the number of clauses is $o(n^{3/2})$), for any constant $0<\delta<1$.