随机素数发生器的相变和计算复杂性
Phase transition and computational complexity in a stochastic prime number generator
摘要：
我们引入一个随机算法的形式素数生成器。这样的算法的字符产生了一个连续的相变，区分一个阶段的算法是能够减少整个系统的数字到素数和一个阶段，系统达到冻结状态与低素数密度。在本文中，我们首先试图从解析和数值分析两个方面对这种相变给出一个广泛的表征。计算临界指数，并提供数据折叠。进一步，我们将模型重新定义为搜索问题，使其符合计算复杂性理论的特点。我们认为该系统属于NP类。计算成本在阈值附近是最大的，这在许多算法相变中很常见，揭示了易-难-易模式的存在。最后，我们将相变的性质与问题的平均情况分类联系起来。
英文摘要：
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.
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