On the automatizability of resolution and related propositional proof systems

摘要

A propositional proof system is automatizable if there is an algorithm that, given a tautology, produces a proof in time polynomial in the size of its smallest proof. This notion can be weakened if we allow the algorithm to produce a proof in a stronger system within the same time bound. This new notion is called weak antomatizability. Among other characterizations, we prove that a system is weakly automatizable exactly when a weak form of the satisfiability problem is solvable in polynomial time. After studying the robustness of the definition, we prove the equivalence between: (i). Resolution is weakly automatizable, (ii) Res(k) is weakly automatizable, and (iii) Res(k) has feasible interpolation, when k > 1. In order to prove this result, we show that Res(2) has polynomial-size proofs of the reflection principle of Resolution, which is a version of consistency. We also show that Res(k), for every k > 1, proves its consistency in polynomial size, while Resolution does not. In fact, we show that Resolution proofs of its own consistency require almost exponential size. This gives a better lower bound for the monotone interpolation of Res(2) and a separation from Resolutionas a byproduct. Our techniques also give us a way to obtain a large class of examples that have small Resolution refutations but require relatively large width. This answers a question of Alekhnovich and Razborov related to whether Resolution is automatizable in quasipolynomial-time. (C) 2003 Elsevier Inc. All rights reserved. [References: 31]