A key result in the theory of the modal mu-calculus is the disjunctive normal form theorem by Janin & Walukiewicz, stating that every mu-calculus formula is semantically equivalent to a so-called disjunctive formula. These disjunctive formulas have good computational properties and play a pivotal role in the theory of the modal mu-calculus. It is therefore an interesting question what the best normalisation procedure is for rewriting a formula into an equivalent disjunctive formula of minimal size. The best constructions that are known from the literature are automata-theoretic in nature and consist of a guarded transformation, i.e., the constructing of an equivalent guarded alternating automaton from a mu-calculus formula, followed by a Simulation Theorem stating that any such alternating automaton can be transformed into an equivalent non-deterministic one. Both of these transformations are exponential constructions, making the best normalisation procedure doubly exponential. Our key contribution presented here shows that the two parts of the normalisation procedure can be integrated, leading to a procedure that is single-exponential in the closure size of the formula.

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关于 $μ$-演算中析取公式的大小

模态 mu 演算理论的一个关键结果是 Janin 和 Walukiewicz 的析取范式定理，指出每个 mu 演算公式在语义上等同于所谓的析取公式。这些析取公式具有良好的计算特性，在模态 mu 演算理论中起着举足轻重的作用。因此，一个有趣的问题是，将公式重写为最小尺寸的等效析取公式的最佳归一化过程是什么。文献中已知的最佳构造本质上是自动机理论的，由保护变换组成，即根据 mu 演算公式构造等效的保护交替自动机，紧随其后的模拟定理指出，任何此类交替自动机都可以转换为等效的非确定性自动机。这两种变换都是指数结构，使最佳归一化过程成双指数。我们在此提出的主要贡献表明，归一化过程的两个部分可以整合，从而导致公式闭包大小为单指数的过程。