Learning algorithms versus automatability of Frege systems

Conference item

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,
- Provable learning. P proves efficiently that p-size circuits are learnable by subexponential-size circuits over the uniform distribution with membership queries.
- 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 Jeřábek’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. The notion of automatability in Item 2 is slightly modified so that the automating algorithm outputs a proof of a given formula (expressing a p-size circuit lower bound) in p-time in the length of the shortest proof of a closely related but different formula (expressing an average-case subexponential-size circuit lower bound).
If there is a function h ∈ NE∩ 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.

Authors: Pich, J; Santhanam, R

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
• Host title: Leibniz International Proceedings in Informatics (LIPIcs)
• Journal: LIPIcs
• Volume: 229
• Pages: 101:1--101:20
• Place of publication: Dagstuhl, Germany
• Publication date: 2022-06-28
• Event title: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
• Event series: International Colloquium on Automata, Languages, and Programming
• DOI: 10.4230/LIPIcs.ICALP.2022.101
• ISSN: 1868-8969
• ISBN: 9783959772358
• Language: English
• Keywords: proof complexity; FFR; automatability; learning algorithms