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INCOMPLETENESS IN THE FINITE DOMAIN

Published online by Cambridge University Press:  15 February 2018

PAVEL PUDLÁK
PAVEL PUDLÁK*
Affiliation:
INSTITUTE OF MATHEMATICS CZECH ACEDEMY OF SCIENCES PRAGUE, CZECH REPUBLICE-mail:pudlak@math.cas.cz
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Abstract

Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond NP ≠ coNP. These conjectures formally connect computational complexity with the difficulty of proving some sentences, which means that high computational complexity of a problem associated with a sentence implies that the sentence is not provable in a weak theory, or requires a long proof. Another reason for putting forward these conjectures is that some results in proof complexity seem to be special cases of such general statements and we want to formalize and fully understand these statements. Roughly speaking, we are trying to connect syntactic complexity, by which we mean the complexity of sentences and strengths of the theories in which they are provable, with the semantic concept of complexity of the computational problems represented by these sentences.

We have introduced the most fundamental conjectures in our earlier works [27, 33–35]. Our aim in this article is to present them in a more systematic way, along with several new conjectures, and prove new connections between them and some other statements studied before.

Keywords

03F2003D1568Q15finite consistencypropositional proof systemsdisjoint NP-setssearch problems
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 23 , Issue 4 , December 2017 , pp. 405 - 441
Copyright
Copyright © The Association for Symbolic Logic 2017 

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