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 \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 $0<\delta<1$.Comment: 32 pages; a preliminary version appeared in the Proceedings of ICALP'0

Cellular Automata are Generic

Any algorithm (in the sense of Gurevich's abstract-state-machine axiomatization of classical algorithms) operating over any arbitrary unordered domain can be simulated by a dynamic cellular automaton, that is, by a pattern-directed cellular automaton with unconstrained topology and with the power to create new cells. The advantage is that the latter is closer to physical reality. The overhead of our simulation is quadratic.Comment: In Proceedings DCM 2014, arXiv:1504.0192

Complexity of Propositional Proofs Under a Promise

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: (1) Resolution has polynomial-size refutations for all unsatisfiable 3CNF formulas when the promise is ε·2n, for any constant 0 &lt; ε &lt; 1. (2) There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is 2δn (and the number of clauses is o(n3/2)), for any constan