On the automatizability of resolution and related propositional proof systems

  • Authors:

  • Albert Atserias;María Luisa Bonet

  • Affiliations:

  • Departament de Llenguatges i Sistemes Informátics, Universitat Politècnica de Catalunya, Barcelona, Spain;Departament de Llenguatges i Sistemes Informátics, Universitat Politècnica de Catalunya, Barcelona, Spain

  • Venue:
  • Information and Computation
  • Year:
  • 2004

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Abstract

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 automatizability. 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 Resolution as 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.