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- Title
Tensor Network Contractions for #SAT.
- Authors
Biamonte, Jacob; Morton, Jason; Turner, Jacob
- 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 $$\mathsf{{P}}$$ ), 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.
- Subjects
QUANTUM mechanics; QUANTUM theory; COMPUTER science; COGNITIVE science; BOOLEAN functions
- Publication
Journal of Statistical Physics, 2015, Vol 160, Issue 5, p1389
- ISSN
0022-4715
- Publication type
Article
- DOI
10.1007/s10955-015-1276-z