On Relativisation and Complexity Gap for Resolution-Based Proof Systems

Abstract

We study the proof complexity of Taut, the class of Second-Order Existential (SO∃) logical sentences which fail in all finite models. The Complexity-Gap theorem for Tree-like Resolution says that the shortest Tree-like Resolution refutation of any such sentence Φ is either fully exponential, $2^{\Omega \left(n\right)}$, or polynomial, $n^{O\left(1\right)}$, where n is the size of the finite model. Moreover, there is a very simple model-theoretics criteria which separates the two cases: the exponential lower bound holds if and only if Φ holds in some infinite model.

In the present paper we prove several generalisations and extensions of the Complexity-Gap theorem.

• 1 For a natural subclass of Taut, $Rel\left(Taut\right)$, there is a gap between polynomial Tree-like Resolution proofs and sub-exponential, $2^{\Omega \left(n^{\varepsilon }\right)}$, general (DAG-like) Resolution proofs, whilst the separating model-theoretic criteria is the same as before. $Rel\left(Taut\right)$ is the set of all sentences in Taut, relativised with respect to a unary predicate.

• 2 The gap for stronger systems, $\textrm{Res}^{*}\left(k\right)$, is between polynomial and $\exp \left(\Omega \left(\frac{\log k}{k}n\right)\right)$ for every k, 1≤ k≤ n. $\textrm{Res}^{*}\left(k\right)$ is an extension of Tree-like Resolution, in which literals are replaced by terms (i.e. conjunctions of literals) of size at most k. The lower bound is tight.

• 3 There is (as expected) no gap for any propositional proof system (including Tree-like Resolution) if we enrich the language of SO logic by a built-in order.