基于自动鞅的形式语言的随机性
Randomness of formal languages via automatic martingales
摘要：
我们定义了一个概念的随机性的个人和集合的形式语言的基础上自动鞅作用于序列的话，从一些潜在的域。一个自动鞅赌输入的单词是否属于目标语言。然后将单个语言和语言集合的随机性定义为自动鞅根据输入的单词序列或文本在目标语言上下注以获得无限资本的失败。事实证明，形式语言的随机性在很大程度上依赖于文本。对于非常一般的文本类，任何非正则语言在单独考虑时都是随机的。至于语言的集合，非常一般的文本类只允许自动语言族的非随机性。另一方面，一个任意的可计算语言在特定的动态文本下被证明是非随机的。
英文摘要：
We define a notion of randomness for individual and collections of formal languages based on automatic martingales acting on sequences of words from some underlying domain. An automatic martingale bets if the incoming word belongs to the target language or not. Then randomness of both single languages and collections of languages is defined as a failure of automatic martingale to gain an unbounded capital by betting on the target language according to an incoming sequence of words, or a text. The randomness of formal languages turned out to be heavily dependent on the text. For very general classes of texts, any nonregular language happens to be random when considered individually. As for collections of languages, very general classes of texts permits nonrandomness of automatic families of languages only. On the other hand, an arbitrary computable language is be shown to be nonrandom under certain dynamic texts.
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