## Abstract

In recent years there has been significant interest in the study of random k-SAT formulae. For a given set of n Boolean variables, let B_{k} denote the set of all possible disjunctions of k distinct, non-complementary literals from its variables (k-clauses). A random k-SAT formula F_{k}(n,m) is formed by selecting uniformly and independently m clauses from B_{k} and taking their conjunction. Motivated by insights from statistical mechanics that suggest a possible relationship between the "order" of phase transitions and computational complexity, Monasson and Zecchina (Phys. Rev. E 56(2) (1997) 1357) proposed the random (2+p)-SAT model: for a given p∈[0,1], a random (2+p)-SAT formula, F_{2+p}(n,m), has m randomly chosen clauses over n variables, where pm clauses are chosen from B_{3} and (1-p)m from B_{2}. Using the heuristic "replica method" of statistical mechanics, Monasson and Zecchina gave a number of non-rigorous predictions on the behavior of random (2+p)-SAT formulae. In this paper we give the first rigorous results for random (2+p)-SAT, including the following surprising fact: for p≤2/5, with probability 1-o(1), a random (2+p)-SAT formula is satisfiable iff its 2-SAT subformula is satisfiable. That is, for p≤2/5, random (2+p)-SAT behaves like random 2-SAT.

Original language | English (US) |
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Pages (from-to) | 109-129 |

Number of pages | 21 |

Journal | Theoretical Computer Science |

Volume | 265 |

Issue number | 1-2 |

DOIs | |

State | Published - 2001 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science