随机与或树的广义商模型及其在可满足性中的应用
Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability
摘要：
本文的动机是以下满足性问题：随机一致地选择长度为n的布尔表达式，建立在一组k_n布尔变量上。这个表达式可满足的概率是多少？当n趋于无穷大时，本文所建立的随机布尔表达式模型是布尔Catalan树模型，它在文献中已经广泛地研究了常数序列（k_n）_{n \geq 1}。本文的根本突破是将已有的结果推广到任意（合理）整数序列（k_n）_{n \geq 1}上，特别是解决了上述可满足性问题.我们还分析了引入自然等价关系对布尔表达式集的影响。这个新的“商”模型恰好在k_n = n/ln n处表现出一个非常有趣的阈值（或饱和）现象。
英文摘要：
This article is motivated by the following satisfiability question: pick uniformly at random an and/or Boolean expression of length n, built on a set of k_n Boolean variables. What is the probability that this expression is satisfiable? asymptotically when n tends to infinity? The model of random Boolean expressions developed in the present paper is the model of Boolean Catalan trees, already extensively studied in the literature for a constant sequence (k_n)_{n\geq 1}. The fundamental breakthrough of this paper is to generalise the previous results to any (reasonable) sequence of integers (k_n)_{n\geq 1}, which enables us, in particular, to solve the above satisfiability question. We also analyse the effect of introducing a natural equivalence relation on the set of Boolean expressions. This new "quotient" model happens to exhibit a very interesting threshold (or saturation) phenomenon at k_n = n/ln n.
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。
