Relativisation makes contradictions harder for Resolution

TL;DR

This paper investigates how relativisation affects the complexity of resolution proof systems, demonstrating that relativisation makes certain contradictions harder to resolve and establishing separations between different system variants.

Contribution

It introduces new separations between resolution systems with bounded conjunctions, showing how relativisation impacts their proof complexity and providing novel techniques for parameterized systems.

Findings

Relativisation increases the difficulty of refuting certain principles.

Separation results between Res(d) and Res(d+1) systems.

Parameterised systems Res(1) and Res(2) also shown to be separated.

Abstract

We provide a number of simplified and improved separations between pairs of Resolution-with-bounded-conjunction refutation systems, Res(d), as well as their tree-like versions, Res*(d). The contradictions we use are natural combinatorial principles: the Least number principle, LNP_n and an ordered variant thereof, the Induction principle, IP_n. LNP_n is known to be easy for Resolution. We prove that its relativisation is hard for Resolution, and more generally, the relativisation of LNP_n iterated d times provides a separation between Res(d) and Res(d+1). We prove the same result for the iterated relativisation of IP_n, where the tree-like variant Res*(d) is considered instead of Res(d). We go on to provide separations between the parameterized versions of Res(1) and Res(2). Here we are able again to use the relativisation of the LNP_n, but the classical proof breaks down and we are forced to use an alternative. Finally, we separate the parameterized versions of Res*(1) and Res*(2). Here, the relativisation of IP_n will not work as it is, and so we make a vectorising amendment to it in order to address this shortcoming