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 Bk denote the set of all possible disjunctions of k distinct, non-complementary literals from its variables (k-clauses). A random k-SAT formula Fk(n,m) is formed by selecting uniformly and independently m clauses from Bk 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, F2+p(n,m), has m randomly chosen clauses over n variables, where pm clauses are chosen from B3 and (1-p)m from B2. 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.

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Journal Theoretical Computer Science
Achlioptas, D. (Dimitris), Kirousis, L.M. (Lefteris M.), Kranakis, E, & Krizanc, D. (Danny). (2001). Rigorous results for random (2+p)-SAT. Theoretical Computer Science, 265(1-2), 109–129. doi:10.1016/S0304-3975(01)00154-2