Abstract
The statistical and algebraic properties of a family of random Boolean equations arestudied in this paper. Based on studying the mechanism of influence propagation fromsome variable fixed to 1, the giant strongly connected component and magnetization ofsolutions are investigated, which exhibit the splitting phenomenon of solution space and aferromagnetic transition. Furthermore, by analyzing the semi-group property of thesolution space and the influence of propagation from some variable fixed to 0, the scaleof generating elements is calculated, which undergoes linear, polynomial andexponential phases. Compared with the analysis by statistical mechanics, it is suggestedthat these different phases of algebraic complexity correspond to the structuralcomplexity of the solution space as replica symmetry, one-step replica symmetrybreaking and further-step replica symmetry breaking phases. It is supposed that thestructural complexity of solution space can be interpreted from the viewpoint ofalgebra.
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