Journal of Statistical Physics

, Volume 160, Issue 5, pp 1389–1404 | Cite as

Tensor Network Contractions for #SAT

  • Jacob D. Biamonte
  • Jason Morton
  • Jacob Turner


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 



We thank Tobias Fritz and Eduardo Mucciolo for providing feedback. JDB acknowledges financial support from the Fondazione Compagnia di San Paolo through the Q-ARACNE project and the Foundational Questions Institute (FQXi, under Grant FQXi-RFP3-1322). JM and JT acknowledges the NSF (under Grant NSF-1007808) for financial support.


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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

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