Effective Disjunction and Effective Interpolation in Suffciently Strong Proof Systems

TL;DR

This paper explores how certain properties of proof systems, like effective disjunction and interpolation, transfer from a strong system (Extended Frege) to weaker systems, with implications for complexity classes and algorithms.

Contribution

It establishes that effective disjunction and interpolation properties in Extended Frege systems imply similar properties in weaker proof systems and relates these to complexity class separations.

Findings

If EF has the uniform effective disjunction property, so do related proof systems.

EF's uniform effective interpolation property implies E equals NE intersect coNE.

An exponential time algorithm exists for separating NE-pairs under certain conditions.

Abstract

In this article, we deal with the uniform effective disjunction property and the uniform effective interpolation property, which are weaker versions of the classical effective disjunction property and the effective interpolation property.\\ The main result of the paper is as follows: Suppose the proof system $EF$ (Extended Frege) has the uniform effective disjunction property, then every sufficiently strong proof system $S$ that corresponds to a theory $T$, which is a theory in the same language as the theory $V_{1}^{1}$, also has the uniform effective disjunction property. Furthermore, if we assume that $EF$ has the uniform effective interpolation property, then the proof system $S$ also has the uniform effective interpolation property.\\ From this, it easily follows that if $EF$ has the uniform effective interpolation property, then for every disjoint $NE$-pair, there exists a set in $E$ that separates this pair. Thus, if $EF$ has the uniform effective interpolation property, it specifically holds that $NE \cap coNE = E$. Additionally, at the end of the article, the following is proven: Suppose the proof system $EF$ has the uniform effective interpolation property, and let $A_{1}$ and $A_{2}$ be a (not necessarily disjoint) NE-pair such that $A_{1} \cup A_{2} = \mathbb{N}$; then there exists an exponential time algorithm which for every input $n$ (of length $O(\log n)$) finds $i\in\{1,2\}$ such that $n\in A_{i}$.