Tensor Network Contractions for #SAT

Abstract: The computational cost of counting the number of solutions satisfying a Boolean formula, which is a problem instance of #SAT, has proven subtle to quantify. Even when finding individual satisfying solutions is computationally easy (e.g. 2-SAT, which is in P), determining the number of solutions is #P-hard. Recently, computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics-inspired techniques. By these methods, we give an algorithm using an axiomatic tensor contraction language for n-variable #SAT instances with complexity $O((g+cd)^{O(1)} 2^c)$ where $c$ is the number of COPY-tensors, $g$ is the number of gates, and $d$ is the maximal degree of any COPY-tensor. Thus, counting problems can be solved efficiently when their tensor network expression has at most $O(\log c)$ COPY-tensors and polynomial fan-out. This framework also admits an intuitive proof of a variant of the Tovey conjecture (the r,1-SAT instance of the Dubois-Tovey theorem). This study increases the theory, expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science.