Phase transitions of PP-complete satisfiability problems

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(⩾2n/2)(⩾2n/2): 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 r3,2r3,2 at which the asymptotic probability of # 3SAT(⩾2n/2)(⩾2n/2) undergoes a phase transition from 1 to 0. We obtain upper and lower bounds for r3,2r3,2 by showing that 0.9227⩽r3,2⩽2.5950.9227⩽r3,2⩽2.595. We also carry out a set of experiments on random instances of # 3SAT(⩾2n/2)(⩾2n/2) using a natural modification of the Davis–Putnam–Logemann–Loveland (DPLL) procedure. Our experimental results suggest that r3,2≈2.5r3,2≈2.5. Moreover, the average number of recursive calls of this modified DPLL procedure reaches a peak around 2.5 as well.