Bose–Einstein condensation in satisfiability problems

This paper is concerned with the complex behavior arising in satisfiability problems. We present a new statistical physics-based characterization of the satisfiability problem. Specifically, we design an algorithm that is able to produce graphs starting from a k-SAT instance, in order to analyze them and show whether a Bose–Einstein condensation occurs. We observe that, analogously to complex networks, the networks of k-SAT instances follow Bose statistics and can undergo Bose–Einstein condensation. In particular, k-SAT instances move from a fit-get-rich network to a winner-takes-all network as the ratio of clauses to variables decreases, and the phase transition of k-SAT approximates the critical temperature for the Bose–Einstein condensation. Finally, we employ the fitness-based classification to enhance SAT solvers (e.g., ChainSAT) and obtain the consistently highest performing SAT solver for CNF formulas, and therefore a new class of efficient hardware and software verification tools.

► New BEC-driven algorithm to translate a k-SAT instance into a network. ► The network presents a BE phase transition as a function of α = clauses/variables. ► The k-SAT phase transition approximates the critical temperature for the BEC. ► We obtain the highest performing SAT solver by using a fitness-based classification. ► We connect k-SAT problem, complex networks and condensation phase transition.