Small Depth Proof Systems

TL;DR

This paper investigates highly restricted proof systems with small-depth circuits, characterizing their power to represent regular languages and related problems, and exploring their implications for NP and graph reachability.

Contribution

It precisely characterizes the computational power needed for proof systems to capture all regular languages using small-depth circuits and explores their application to NP and graph problems.

Findings

All regular languages have polylogarithmic AC^0 proof systems.

Undirected Reachability and UnReachability have NC^0 proof systems.

MAJ has a polylog AC^0 proof system.

Abstract

A proof system for a language L is a function f such that Range(f) is exactly L. In this paper, we look at proofsystems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is: they can be computed by bounded fanin circuits of constant depth (NC^0), or of O(log log n) depth but with O(1) alternations (polylog AC^0). Each output bit depends on very few input bits; thus such proof systems correspond to a kind of local error-correction on a theorem-proof pair. We identify exactly how much power we need for proof systems to capture all regular languages. We show that all regular language have polylog AC^0 proof systems, and from a previous result (Beyersdorff et al, MFCS 2011, where NC^0 proof systems were first introduced), this is tight. Our technique also shows that MAJ has polylog AC^0 proof system. We explore the question of whether TAUT has NC^0 proof systems. Addressing this question about 2TAUT, and since 2TAUT is closely related to reachability in graphs, we ask the same question about Reachability. We show that both Undirected Reachability and Directed UnReachability have NC^0 proof systems, but Directed Reachability is still open. In the context of how much power is needed for proof systems for languages in NP, we observe that proof systems for a good fraction of languages in NP do not need the full power of AC^0; they have SAC^0 or coSAC^0 proof systems.