Information in propositional proofs and algorithmic proof search

TL;DR

This paper explores the proof search problem in propositional logic, introducing an information-based measure to analyze proof complexity and establishing links between proof search efficiency and proof system optimality.

Contribution

It introduces a new information-efficiency measure for propositional proofs, linking proof search complexity with proof system optimality and providing insights into proof complexity bounds.

Findings

Existence of time-optimal proof search algorithms for fixed proof systems.

Characterization of proof search complexity via the information-efficiency function.

Connection between proof system optimality and information-efficiency optimality.

Abstract

We study from the proof complexity perspective the (informal) proof search problem: Is there an optimal way to search for propositional proofs? We note that for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal proof search algorithm exists without restricting proof systems iff a p-optimal proof system exists. To characterize precisely the time proof search algorithms need for individual formulas we introduce a new proof complexity measure based on algorithmic information concepts. In particular, to a proof system $P$ we attach {\bf information-efficiency function} $i_P(\tau)$ assigning to a tautology a natural number, and we show that: - $i_P(\tau)$ characterizes time any $P$-proof search algorithm has to use on $\tau$ and that for a fixed $P$ there is such an information-optimal algorithm, - a proof system is information-efficiency optimal iff it is p-optimal, - for non-automatizable systems $P$ there are formulas $\tau$ with short proofs but having large information measure $i_P(\tau)$. We isolate and motivate the problem to establish unconditional super-logarithmic lower bounds for $i_P(\tau)$ where no super-polynomial size lower bounds are known. We also point out connections of the new measure with some topics in proof complexity other than proof search.