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TR23-030 | 21st March 2023 09:25

A proof complexity conjecture and the Incompleteness theorem


Authors: Jan Krajicek
Publication: 21st March 2023 12:10
Downloads: 86


Given a sound first-order p-time theory $T$ capable of formalizing syntax of
first-order logic we define a p-time function $g_T$ that stretches all inputs by one
bit and we use its properties to show that $T$ must be incomplete. We leave it as an
open problem whether for some $T$ the range of $g_T$ intersects all infinite NP sets
(i.e. whether it is a proof complexity generator hard for all proof systems).

A propositional version of the construction shows that at least one of the following
three statements is true:

- there is no p-optimal propositional proof system (this is equivalent to the
non-existence of a time-optimal propositional proof search algorithm),

- $E \not\subseteq P/poly$,

- there exists function $h$ that stretches all inputs by one bit,
is computable in sub-exponential time and its range $Rng(h)$ intersects all infinite
N sets.

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