k-Sat on groups and undecidability

Michael H. Freedman

Research output: Contribution to journalConference articlepeer-review

Abstract

The general Boolean formula can be quickly converted into a normal form, N3, the conjunction of triple disjunctions of literals, which is satisfiable if the original is. On the other hand, formulae in N2, the conjunction of 2-fold disjunctions of literals, can be checked for a satisfaction in polynomial time. Thus these two satisfaction problems, `2-sat' and `3-sat', have been considered as an interesting boundary point between P and NP. We define an infinite generalization of 2-sat and 3-sat which are respectively algorithmic and undecidable. As a corollary it is noted that the 3-colorability of doubly-periodic planar graphs is undecidable. It was suggested in [F] that a general approach to proving P≠NP would be to construct some infinitary limit of decision problems with the property that those admitting polynomial time algorithms would be decidable in this limit. The hope here is to exploit the strong connection between polynomial growth and finite dimensionality. Since logic has a method - self-reference - for establishing problems as undecidable, this technique applied to the limit could potentially show that the finite-decision problem lies outside of P. This paper supplies one way of extending k-sat to an infinite context in which decidability distinguishes 2-sat from 3-sat.

Original languageEnglish
Pages (from-to)572-576
Number of pages5
JournalConference Proceedings of the Annual ACM Symposium on Theory of Computing
DOIs
StatePublished - 1998
Externally publishedYes
EventProceedings of the 1998 30th Annual ACM Symposium on Theory of Computing - Dallas, TX, USA
Duration: 23 May 199826 May 1998

ASJC Scopus subject areas

  • Software

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