Kramers-Wannier Duality of Statistical Mechanics Applied to the Boolean Satisfiability Problem of Computer Science

Abstract: We present a novel application of the Kramers-Wannier duality on one of the most important problems of computer science, the Boolean satisfiability problem (SAT). More specifically, we focus on sharp-SAT or equivalently #SAT - the problem of counting the number of solutions to a Boolean satisfaction formula. #SAT can be cast into a statistical-mechanical language, where it reduces to calculating the partition function of an Ising spin Hamiltonian with multi-spin interactions. We show that Kramers-Wannier duality can be generalized to apply to such multi-connected spin networks. We present an exact dual partner to #SAT and explicitly verify their equivalence with a few simple examples. It is shown that the NP-completeness of the original problem maps on the complexity of the dual problem of enumerating the number of non-negative solutions to a Diophantine system of equations. We discuss the implications of this duality and the prospects of similar dualities applied to computer science problems.