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