Experimental analyses on phase transitions in compiling satisfiability problems

In the past decade, a kind of well-known phenomena in many complex combinatorial problems such as satisfiability problem, called phase transition, have been widely studied. In this paper, the phase transition phenomena are investigated during compiling k -satisfiability problems into tractable languages with empirical methods. Ordered binary decision diagram and deterministic-decomposable negation normal form are selected as the tractable target languages for the compilation. Via intensive experiments, it can be concluded that an easy-hard-easy pattern exists during the compilations, which is only related to the ratio of the number of clauses to that of variables if we set k to a fixed value, rather than to the target languages. Moreover, it can be concluded that the space exhausted during the compilations grows exponentially with the number of variables growing, whereas there is also a phase transition separating the polynomial-increment region from the exponential-increment region. Additionally, it can be observed that there is a phase transition of prime implicants around peak points of the easy-hard-easy pattern and the ratios of random instances whose average lengths of prime implicants are larger than the threshold 0.5 change sharply. From these analyses, it can be concluded that prime implicant length and solution interchangeability are crucial impacts on sizes of compilation results.