Computer Science – Computational Complexity
Scientific paper
2007-12-19
New Journal of Physics 10 (2008) 023009
Computer Science
Computational Complexity
Submitted to New Journal of Physics
Scientific paper
10.1088/1367-2630/10/2/023009
We introduce a prime number generator in the form of a stochastic algorithm. The character of such algorithm gives rise to a continuous phase transition which distinguishes a phase where the algorithm is able to reduce the whole system of numbers into primes and a phase where the system reaches a frozen state with low prime density. In this paper we firstly pretend to give a broad characterization of this phase transition, both in terms of analytical and numerical analysis. Critical exponents are calculated, and data collapse is provided. Further on we redefine the model as a search problem, fitting it in the hallmark of computational complexity theory. We suggest that the system belongs to the class NP. The computational cost is maximal around the threshold, as common in many algorithmic phase transitions, revealing the presence of an easy-hard-easy pattern. We finally relate the nature of the phase transition to an average-case classification of the problem.
Lacasa Lucas
Luque Bartolo
Miramontes Octavio
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