[论文解读] Computing Minimal Sets on Propositional Formulae I: Problems & Reductions.

João Marques‐Silva, Mikoláš Janota|arXiv (Cornell University)|Feb 13, 2014

Formal Methods in Verification参考文献 96被引用 9

一句话总结

本文提出一个统一框架，通过将各类命题逻辑函数问题约化为单调谓词的最小集合（MSMP）问题，实现对这些问题的统一求解。研究表明，诸如最小不可满足子集、主蕴含式、骨架文字和自洽性等关键问题均可约化为MSMP问题，从而可重用高效的MSMP算法，并为这些问题的结构与复杂性提供新的洞见。

ABSTRACT

Boolean Satisfiability (SAT) is arguably the archetypical NP-complete decision problem. Progress in SAT solving algorithms has motivated an ever increasing number of practical applications in recent years. However, many practical uses of SAT involve solving function as opposed to decision problems. Concrete examples include computing minimal unsatisfiable subsets, minimal correction subsets, prime implicates and implicants, minimal models, backbone literals, and autarkies, among several others. In most cases, solving a function problem requires a number of adaptive or non-adaptive calls to a SAT solver. Given the computational complexity of SAT, it is therefore important to develop algorithms that either require the smallest possible number of calls to the SAT solver, or that involve simpler instances. This paper addresses a number of representative function problems defined on Boolean formulas, and shows that all these function problems can be reduced to a generic problem of computing a minimal set subject to a monotone predicate. This problem is referred to as the Minimal Set over Monotone Predicate (MSMP) problem. This exercise provides new ways for solving well-known function problems, including prime implicates, minimal correction subsets, backbone literals, independent variables and autarkies, among several others. Moreover, this exercise motivates the development of more efficient algorithms for the MSMP problem. Finally the paper outlines a number of areas of future research related with extensions of the MSMP problem.

研究动机与目标

• 将命题逻辑中一大类函数问题统一于单一计算抽象之下。
• 证明诸如最小不可满足子集、主蕴含式和骨架文字等问题可约化为MSMP问题。
• 使现有高效MSMP算法能够应用于基于SAT推理的更广泛实际问题。
• 为逻辑与约束满足问题中函数问题的新算法开发及查询复杂性表征提供基础。

提出的方法

• 将MSMP问题形式化为：计算一个最小集合 W ⊆ R，使得单调谓词 P(W) 成立。
• 定义一种通用约化策略：针对每个目标函数问题，基于其逻辑结构构造一个单调谓词 P(W)。
• 使用SAT求解器作为预言机，验证所构造谓词的单调性与正确性。
• 采用逻辑转换（如对子句取反、变量赋值及公式重写）将每个问题编码进MSMP框架。
• 应用已知的MSMP算法技术（如自适应与非自适应SAT查询）求解约化后的问题。
• 通过形式化命题逻辑与谓词理论（例如通过命题7及相关引理）建立约化的正确性与单调性。

实验结果

研究问题

• RQ1 能否将一大类命题逻辑函数问题约化为MSMP问题？
• RQ2 在何种结构与逻辑条件下，函数问题可接受单调谓词约化？
• RQ3 如何重用现有MSMP算法来求解如最小修正子集或自洽性等新问题类别？
• RQ4 特定MSMP问题实例（如FMUS、FPIc、FBB）的查询复杂性特征为何？

主要发现

• 所有15个以上的目标函数问题——包括最小不可满足子集、主蕴含式和骨架文字——均可通过单调谓词约化为MSMP问题。
• 约化过程经形式化证明正确且保持单调性，确保MSMP算法可被可靠应用。
• 该框架使高效MSMP算法（如[69]中的算法）可被重用于此前需定制解决方案的问题。
• 约化为MSMP为变量独立性与最大自洽性等问题的结构提供了新的理论洞见。
• 对于某些问题（如FMUS与FPIc），MSMP约化催生了新的算法方法，具有提升效率的潜力。
• 本文证明MSMP框架足够通用，可涵盖ILP、SMT与CSP中的问题，表明其在命题逻辑之外亦具广泛适用性。