The probabilistic satisfiability of a logical expression is a fundamental concept known as the partition function in statistical physics and field theory, an evaluation of a related graph's Tutte polynomial in mathematics, and the Moore-Shannon network reliability of that graph in engineering. It is the crucial element for decision-making under uncertainty. Not surprisingly, it is provably hard to compute exactly or even to approximate. Many of these applications are concerned only with a subset of problems for which the solutions are monotonic functions. Here we extend the weak- and strong-coupling methods of statistical physics to heterogeneous satisfiability problems and introduce a novel approach to constructing lower and upper bounds on the approximation error for monotonic problems. These bounds combine information from both perturbative analyses to produce bounds that are tight in the sense that they are saturated by some problem instance that is compatible with all the information contained in either approximation.
翻译:逻辑表达法的概率相对性是一个基本概念,即统计物理学和实地理论中的分割函数,对相关图表数学中的图特多元数学的评估,以及该图工程中的摩尔-沙农网络可靠性。这是在不确定情况下决策的关键要素。毫不奇怪,精确或甚至近似很难计算。许多这些应用仅涉及一系列问题,其解决办法是单调函数。在这里,我们将统计物理的弱和强相联方法推广到差异性可调和问题,并引入一种新颖的方法,在单调问题近似误上构建一个下界和上界。这些边框结合了从两处的扰动性分析获得的信息,以产生在它们与任何近似都包含的所有信息相兼容的某个问题实例所饱和的界限。