### 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 language | English |
---|---|

Pages (from-to) | 1501-1511 |

Number of pages | 11 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 389 |

Issue number | 8 |

DOIs | |

Publication status | Published - Apr 15 2010 |

### Fingerprint

### Keywords

- Phase transition
- Random satisfiability problem
- Threshold phenomenon

### ASJC Scopus subject areas

- Condensed Matter Physics
- Statistics and Probability

### Cite this

*Physica A: Statistical Mechanics and its Applications*,

*389*(8), 1501-1511. https://doi.org/10.1016/j.physa.2009.12.051

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

Research output: Contribution to journal › Article

*Physica A: Statistical Mechanics and its Applications*, vol. 389, no. 8, pp. 1501-1511. https://doi.org/10.1016/j.physa.2009.12.051

}

TY - JOUR

T1 - What makes a phase transition? Analysis of the random satisfiability problem

AU - Zweig, Katharina A.

AU - Palla, Gergely

AU - Vicsek, T.

PY - 2010/4/15

Y1 - 2010/4/15

N2 - 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.

AB - 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.

KW - Phase transition

KW - Random satisfiability problem

KW - Threshold phenomenon

UR - http://www.scopus.com/inward/record.url?scp=74849100195&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=74849100195&partnerID=8YFLogxK

U2 - 10.1016/j.physa.2009.12.051

DO - 10.1016/j.physa.2009.12.051

M3 - Article

VL - 389

SP - 1501

EP - 1511

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

IS - 8

ER -