We develop an analytical framework for Boolean Promise Constraint Satisfaction Problems (PCSPs) that studies polymorphisms through the notion of influence from Fourier analysis of Boolean functions. Extending the work of Brakensiek, Guruswami, and Sandeep [ICALP'21] on Ordered PCSPs, we identify two general phenomena in Boolean minions indicative of hardness or tractability: (1) preservation of coordinate influence under random 2-to-1 minors and (2) the presence of sharp thresholds. We demonstrate that these phenomena occur in broader settings than previously established, yielding new hardness/tractability results for minions consisting of unate or polynomial threshold functions.

翻译：我们为布尔承诺约束满足问题（PCSPs）建立了一个分析框架，该框架通过布尔函数傅里叶分析中的影响力概念来研究多态性。在Brakensiek、Guruswami和Sandeep [ICALP'21]关于有序PCSPs工作的基础上，我们识别出布尔小从属中指示困难性或可解性的两类一般现象：（1）在随机2对1小从属作用下坐标影响力的保持，以及（2）尖锐阈值的出现。我们证明这些现象出现在比先前已建立的更广泛场景中，从而为包含单值或多项式阈值函数的小从属提供了新的困难性/可解性结果。