正则语言和含零半群的零一律
Zero-One Law for Regular Languages and Semigroups with Zero
摘要：
如果一个正则语言的渐近密度收敛于0或1，那么它就有0 - 1定律。我们证明了所有0 - 1语言的类在布尔运算和代数下是闭的。证明了正则语言有0 - 1律当且仅当它的句法幺半群有0元。我们的证明给出了正则语言的0 - 1律的代数和自动机刻画，并引出了以下两个推论：（i）存在一个O（nlog n）的算法来检验给定的正则语言是否具有0 - 1律。(ii)有限字上存在一阶逻辑的布尔闭包具有0 - 1律。
英文摘要：
A regular language has the zero-one law if its asymptotic density converges to either zero or one. We prove that the class of all zero-one languages is closed under Boolean operations and quotients. Moreover, we prove that a regular language has the zero-one law if and only if its syntactic monoid has a zero element. Our proof gives both algebraic and automata characterisation of the zero-one law for regular languages, and it leads the following two corollaries: (i) There is an O(n log n) algorithm for testing whether a given regular language has the zero-one law. (ii) The Boolean closure of existential first-order logic over finite words has the zero-one law.
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