What makes a phase transition? Analysis of the random satisfiability problem
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 ksatisfiability problems (often referred to as kSAT), asking whether there exists an assignment of Boolean values satisfying a Boolean formula composed of clauses with k random variables each. The random 3SAT 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 transitionlike 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.
 Publication:

Physica A Statistical Mechanics and its Applications
 Pub Date:
 April 2010
 DOI:
 10.1016/j.physa.2009.12.051
 arXiv:
 arXiv:1002.0217
 Bibcode:
 2010PhyA..389.1501Z
 Keywords:

 Physics  Data Analysis;
 Statistics and Probability;
 Computer Science  Computational Complexity;
 Physics  Computational Physics
 EPrint:
 Physica A 389, 15011511 (2010)