# Finite State Machine

<mathematics, algorithm, theory> (FSM or "Finite State Automaton", "transducer") An abstract machine consisting of a set of states (including the initial state), a set of input events, a set of output events and a state transition function. The function takes the current state and an input event and returns the new set of output events and the next state. Some states may be designated as "terminal states". The state machine can also be viewed as a function which maps an ordered sequence of input events into a corresponding sequence of (sets of) output events.

A deterministic FSM is one where the next state is uniquely determined by a single input event. The next state of a non-deterministic FSM (NFA) depends not only on the current input event, but also on an arbitrary number of subsequent input events. Until these subsequent events occur it is not possible to determine which state the machine is in.

It is possible to automatically translate some (but not all) non-deterministic FSMs into deterministic ones which will produce the same output given the same input. [Is this true?]

In a probabilistic FSM [proper name?], there is a predetermined probability of each next state given the current state and input (compare Markov chain).

The terms "acceptor" and "transducer" are used particularly in language theory where automata are often considered as abstract machines capable of recognising a language (certain sequences of input events). An acceptor has a single Boolean output and accepts or rejects the input sequence by outputting true or false respectively, whereas a transducer translates the input into a sequence of output events.

FSMs are used in computability theory and in some practical applications such as regular expressions and digital logic design.

[J.H. Conway, "regular algebra and finite machines", 1971, Eds Chapman & Hall].

[S.C. Kleene, "Representation of events in nerve nets and finite automata", 1956, Automata Studies. Princeton].

[Hopcroft & Ullman, 1979, "Introduction to automata theory, languages and computations", Addison-Wesley].

[M. Crochemore "tranducters and repetitions", Theoritical. Comp. Sc. 46, 1986].

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Nearby terms: finite differencing « Finite State Automata « Finite State Automaton « Finite State Machine » first-order » first-order logic » first-order predicate logic