Probabilistic automata

In mathematics and computer science, the probabilistic automaton (PA) is a generalization of the nondeterministic finite automaton; it includes the probability of a given transition into the transition function, turning it into a transition matrix or . Thus, the probabilistic automaton generalizes the concept of a Markov chain or subshift of finite type. The languages recognized by probabilistic automata are called stochastic languages; these include the regular languages as a subset. The number of stochastic languages is uncountable.

The concept was introduced by Michael O. Rabin in 1963; a certain special case is sometimes known as the Rabin automaton. In recent years, a variant has been formulated in terms of quantum probabilities, the quantum finite automaton.

The probabilistic automaton may be defined as an extension of a non-deterministic finite automaton ${\displaystyle (Q,\Sigma ,\delta ,q_{0},F)}$, together with two probabilities: the probability ${\displaystyle P}$ of a particular state transition taking place, and with the initial state ${\displaystyle q_{0}}$ replaced by a giving the probability of the automaton being in a given initial state.

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