  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.
