arXiv subjects
Publications and source records attributed to Birzhan Moldagaliyev.
This paper describes a construction of supermartingales realized as automatic functions. A capital of supermartingales is represented using automatic capital groups~(ACG). Properties of these automatic supermartingales are then studied. Automatic supermartingales induce a notion of random infinite binary sequence. We show that the class of random sequences coincide with that of disjunctive sequences.
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.
In this paper we define a notion of automatic randomness tests (ART) which capture measure theoretic typicalness of infinite binary sequences within the framework of automata theory. An individual ART is found to be equivalent to a deterministic Büchi automaton recognizing $ω$-language of (Lebesgue) measure zero. A collection of ART's induce a notion of automatic random sequence. We provide a purely combinatorial characterization of an automatic random sequence in the form of a disjunctive property for sequences. At last, we compare two kinds of automatic randomness tests presented in this paper.