What makes a phase transition? Analysis of the random satisfiability problem

Katharina A. Zweig, Gergely Palla, T. Vicsek

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

In the last 30 years it was found that many combinatorial systems undergo phase transitions. One of the most important examples of these can be found among the random k-satisfiability problems (often referred to as k-SAT), asking whether there exists an assignment of Boolean values satisfying a Boolean formula composed of clauses with k random variables each. The random 3-SAT problem is reported to show various phase transitions at different critical values of the ratio of the number of clauses to the number of variables. The most famous of these occurs when the probability of finding a satisfiable instance suddenly drops from 1 to 0. This transition is associated with a rise in the hardness of the problem, but until now the correlation between any of the proposed phase transitions and the hardness is not totally clear. In this paper we will first show numerically that the number of solutions universally follows a lognormal distribution, thereby explaining the puzzling question of why the number of solutions is still exponential at the critical point. Moreover we provide evidence that the hardness of the closely related problem of counting the total number of solutions does not show any phase transition-like behavior. This raises the question of whether the probability of finding a satisfiable instance is really an order parameter of a phase transition or whether it is more likely to just show a simple sharp threshold phenomenon. More generally, this paper aims at starting a discussion where a simple sharp threshold phenomenon turns into a genuine phase transition.

Original languageEnglish
Pages (from-to)1501-1511
Number of pages11
JournalPhysica A: Statistical Mechanics and its Applications
Volume389
Issue number8
DOIs
Publication statusPublished - Apr 15 2010

Fingerprint

Satisfiability Problem
Phase Transition
Number of Solutions
Threshold Phenomena
Hardness
Sharp Threshold
hardness
Log Normal Distribution
thresholds
random variables
Order Parameter
Critical value
Counting
Critical point
Assignment
Random variable
Likely
critical point
counting

Keywords

  • Phase transition
  • Random satisfiability problem
  • Threshold phenomenon

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Statistics and Probability

Cite this

What makes a phase transition? Analysis of the random satisfiability problem. / Zweig, Katharina A.; Palla, Gergely; Vicsek, T.

In: Physica A: Statistical Mechanics and its Applications, Vol. 389, No. 8, 15.04.2010, p. 1501-1511.

Research output: Contribution to journalArticle

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