是什么造成了相变？随机可满足性问题的分析
What makes a phase transition? Analysis of the random satisfiability problem
摘要：
在过去的30年里，人们发现许多组合系统都经历了相变。其中一个最重要的例子可以在随机k-可满足性问题（通常称为k-SAT）中找到，询问是否存在满足布尔公式的布尔值的赋值，该布尔公式由每个子句具有k个随机变量的子句组成。随机3-SAT问题在子句数与变量数之比的不同临界值下表现出不同的相变。其中最著名的是当找到可满足实例的概率突然从1下降到0时。这种转变与问题的硬度上升有关，但到目前为止，所提出的任何相变与硬度之间的相关性还不完全清楚。在本文中，我们将首先数值表明，解决方案的数量普遍遵循对数正态分布，从而解释了令人困惑的问题，为什么解决方案的数量仍然是指数在临界点。此外，我们提供的证据表明，硬度的密切相关的问题，计数的解决方案的总数并没有表现出任何相变样的行为。这就提出了一个问题，即找到一个可满足实例的概率是否真的是一个相变的序参量，或者它是否更有可能只是显示一个简单的尖锐阈值现象。更一般地说，本文旨在开始讨论一个简单的尖锐的阈值现象变成一个真正的相变。
英文摘要：
In the last 30 years it was found that many combinatorial systems undergo phase transitions. One of the most important examples of these can be found among the random k-satisfiability problems (often referred to as k-SAT), asking whether there exists an assignment of Boolean values satisfying a Boolean formula composed of clauses with k random variables each. The random 3-SAT problem is reported to show various phase transitions at different critical values of the ratio of the number of clauses to the number of variables. The most famous of these occurs when the probability of finding a satisfiable instance suddenly drops from 1 to 0. This transition is associated with a rise in the hardness of the problem, but until now the correlation between any of the proposed phase transitions and the hardness is not totally clear. In this paper we will first show numerically that the number of solutions universally follows a lognormal distribution, thereby explaining the puzzling question of why the number of solutions is still exponential at the critical point. Moreover we provide evidence that the hardness of the closely related problem of counting the total number of solutions does not show any phase transition-like behavior. This raises the question of whether the probability of finding a satisfiable instance is really an order parameter of a phase transition or whether it is more likely to just show a simple sharp threshold phenomenon. More generally, this paper aims at starting a discussion where a simple sharp threshold phenomenon turns into a genuine phase transition.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
