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 $P$.
翻译:我们把学习算法和算法连接起来,在标本证明系统中自动进行校对:对于每一个足够强大、良好、有条理的标本证明系统来说,我们证明以下声明是等效的,1美元。 可证实的学习:美元能有效地证明,在统一分布上,小电路的大小的电路可以通过成员问询来学习。 2. 可证实的自动兼容性:美元能有效地证明,美元能通过标本显示小电路下界的非统一电路的非统一电路进行自动分析。在这里,美元能足够强大,如果I.-III持有以下的报表:I.P.