Phase Transitions of PP-Complete Satisfiability Problems

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 PPcomplete Boolean satisfiability problems under the fixed clauses-to-variables ratio model. A typical member of this family is the decision problem #3S AT ( 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;2 at which the asymptotic probability of #3S AT ( 2n=2 ) undergoes a phase transition from 1 to 0. We obtain upper and lower bounds for r3;2 by showing that 0:9227 r3;2 2:595. We also carry out a set of experiments on random instances of #3S AT (2n=2 ) using a natural modification of the Davis-Putnam-Logemann-Loveland (DPLL) procedure. Our experimental results suggest that r3;2 2:5. Moreover, the average number of recursive calls of this modified DPLL procedure reaches a peak around 2:5 as well.