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INFORMATION IN PROPOSITIONAL PROOFS AND ALGORITHMIC PROOF SEARCH

Part of: Proof theory and constructive mathematics Theory of computing

Published online by Cambridge University Press:  27 September 2021

JAN KRAJÍČEK Open the ORCID record for JAN KRAJÍČEK [Opens in a new window]
JAN KRAJÍČEK*
Affiliation:
FACULTY OF MATHEMATICS AND PHYSICS CHARLES UNIVERSITY SOKOLOVSKÁ 83, PRAGUE 186 75, THE CZECH REPUBLICE-mail:krajicek@karlin.mff.cuni.cz
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Abstract

We study from the proof complexity perspective the (informal) proof search problem (cf. [17, Sections 1.5 and 21.5]):

  • • Is there an optimal way to search for propositional proofs?

We note that, as a consequence of Levin’s universal search, 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 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 $ ,

  • • for a fixed P there is such an information-optimal algorithm (informally: it finds proofs of minimal information content),

  • • a proof system is information-efficiency optimal (its information-efficiency function is minimal up to a multiplicative constant) 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.

Keywords

propositional proof systemsalgorithmic informationproof search

MSC classification

Primary: 03F20: Complexity of proofs
Secondary: 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.)
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 2 , June 2022 , pp. 852 - 869
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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