Phase transitions of PP-complete satisfiability problems

Bailey, Delbert D.; Dalmau, Víctor; Kolaitis, Phokion G.

doi:http://dx.doi.org/10.1016/j.dam.2006.09.014

Phase transitions of PP-complete satisfiability problems

Citation

Bailey DD, Dalmau V, Kolaitis PG. Phase transitions of PP-complete satisfiability problems. Discrete Appl Math. 2007 Jun 15;155(12):1627-39. DOI: 10.1016/j.dam.2006.09.014

Abstract

The complexity class PP consists of all decision problems solvable by polynomial-time probabilistic Turing machines. It is well known that PP is a highly intractable complexity class and that PP-complete problems are in all likelihood harder than NP-complete problems. We investigate the existence of phase transitions for a family of PP-complete Boolean satisfiability problems under the fixed clauses-to-variables ratio model. A typical member of this family is the decision problem # 3SAT: given a 3CNF-formula, is it satisfied by at least the square-root of the total number of possible truth assignments? We provide evidence to the effect that there is a critical ratio at which the asymptotic probability of # 3SAT undergoes a phase transition from 1 to 0. We obtain upper and lower bounds for by showing that . We also carry out a set of experiments on random instances of # 3SAT using a natural modification of the Davis–Putnam–Logemann–Loveland (DPLL) procedure. Our experimental results suggest that . Moreover, the average number of recursive calls of this modified DPLL procedure reaches a peak around 2.5 as well.