Journal of Statistical Physics

, Volume 160, Issue 5, pp 1389–1404

Tensor Network Contractions for #SAT


DOI: 10.1007/s10955-015-1276-z

Cite this article as:
Biamonte, J.D., Morton, J. & Turner, J. J Stat Phys (2015) 160: 1389. doi:10.1007/s10955-015-1276-z


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 \(\mathsf{{P}}\)), determining the number of solutions can be #\(\mathsf{{P}}\)-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, n-variable counting problems can be solved efficiently when their tensor network expression has at most \(O(\log n)\) 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.


Statistical physics Complexity Computational physics Quantum physics 

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.ISI FoundationTorinoItaly
  2. 2.Department of MathematicsThe Pennsylvania State UniversityState CollegeUSA