Learning algorithms versus automatability of Frege systems

TL;DR

This paper establishes an equivalence between provable learning of circuit classes and the automatability of propositional proof systems, linking proof complexity with learning theory under certain conditions.

Contribution

It introduces a framework connecting learning algorithms and proof system automatability, proving their equivalence for strong, well-behaved propositional proof systems.

Findings

Provable learning and automatability are equivalent under certain conditions.

The results apply to systems that simulate Jeřábek's WF system.

Existence of hard functions implies the equivalence for specific proof systems.

Abstract

We connect learning algorithms and algorithms automating proof search in propositional proof systems: for every sufficiently strong, well-behaved propositional proof system $P$, we prove that the following statements are equivalent, 1. Provable learning: $P$ proves efficiently that p-size circuits are learnable by subexponential-size circuits over the uniform distribution with membership queries. 2. Provable automatability: $P$ proves efficiently that $P$ is automatable by non-uniform circuits on propositional formulas expressing p-size circuit lower bounds. Here, $P$ is sufficiently strong and well-behaved if I.-III. holds: I. $P$ p-simulates Je\v{r}\'abek's system $WF$ (which strengthens the Extended Frege system $EF$ by a surjective weak pigeonhole principle); II. $P$ satisfies some basic properties of standard proof systems which p-simulate $WF$; III. $P$ proves efficiently for some Boolean function $h$ that $h$ is hard on average for circuits of subexponential size. For example, if III. holds for $P=WF$, then Items 1 and 2 are equivalent for $P=WF$. If there is a function $h\in NE\cap coNE$ which is hard on average for circuits of size $2^{n/4}$, for each sufficiently big $n$, then there is an explicit propositional proof system $P$ satisfying properties I.-III., i.e. the equivalence of Items 1 and 2 holds for $P$.