Mapping between Spin-Glass Three-Dimensional (3D) Ising Model and Boolean Satisfiability Problem

The common feature for a nontrivial hard problem is the existence of nontrivial topological structures, non-planarity graphs, nonlocalities, or long-range spin entanglements in a model system with randomness. For instance, the Boolean satisfiability (K-SAT) problems for K ≥ 3 MSATK≥3  are nontrivial...

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Vydáno v:Mathematics (Basel) Ročník 11; číslo 1; s. 237
Hlavní autor: Zhang, Zhidong
Médium: Journal Article
Jazyk:angličtina
Vydáno: Basel MDPI AG 01.01.2023
Témata:
Algebra
Algorithms
Artificial intelligence
Boolean
Boolean satisfiability
Complexity
computational complexity
Computer science
Exact solutions
Food science
Graphs
Ising model
Lower bounds
Machine learning
Magnetic fields
Mathematics
Phase transitions
Physics
Randomness
Spin glasses
spin-glass 3D Ising model
Three dimensional models
Topology
Transfer matrices
Two dimensional models
ISSN:2227-7390, 2227-7390
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Shrnutí:The common feature for a nontrivial hard problem is the existence of nontrivial topological structures, non-planarity graphs, nonlocalities, or long-range spin entanglements in a model system with randomness. For instance, the Boolean satisfiability (K-SAT) problems for K ≥ 3 MSATK≥3  are nontrivial, due to the existence of non-planarity graphs, nonlocalities, and the randomness. In this work, the relation between a spin-glass three-dimensional (3D) Ising model  MSGI3D  with the lattice size N = mnl and the K-SAT problems is investigated in detail. With the Clifford algebra representation, it is easy to reveal the existence of the long-range entanglements between Ising spins in the spin-glass 3D Ising lattice. The internal factors in the transfer matrices of the spin-glass 3D Ising model lead to the nontrivial topological structures and the nonlocalities. At first, we prove that the absolute minimum core (AMC) model MAMC3D exists in the spin-glass 3D Ising model, which is defined as a spin-glass 2D Ising model interacting with its nearest neighboring plane. Any algorithms, which use any approximations and/or break the long-range spin entanglements of the AMC model, cannot result in the exact solution of the spin-glass 3D Ising model. Second, we prove that the dual transformation between the spin-glass 3D Ising model and the spin-glass 3D Z2 lattice gauge model shows that it can be mapped to a K-SAT problem for K ≥ 4 also in the consideration of random interactions and frustrations. Third, we prove that the AMC model is equivalent to the K-SAT problem for K = 3. Because the lower bound of the computational complexity of the spin-glass 3D Ising model CLMSGI3D  is the computational complexity by brute force search of the AMC model CUMAMC3D, the lower bound of the computational complexity of the K-SAT problem for K ≥ 4 CLMSATK≥4  is the computational complexity by brute force search of the K-SAT problem for K = 3  CUMSATK=3. Namely, CLMSATK≥4=CLMSGI3D≥CUMAMC3D=CUMSATK=3. All of them are in subexponential and superpolynomial. Therefore, the computational complexity of the K-SAT problem for K ≥ 4 cannot be reduced to that of the K-SAT problem for K < 3.
Bibliografie:ObjectType-Article-1
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ISSN:2227-7390
2227-7390
DOI:10.3390/math11010237