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 (DFA) is one where the next state is uniquely determinied by
a single input event. The next state of a nondeterministic 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 any nondeterministic FSM into a
deterministic one which will produce the same output given the same input. Each
state in the DFA represents the set of states the NFA might be in at a given
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
FSMs are used in computability theory and in some practical applications such as
regular expressions and digital logic design.
See also state transition diagram, Turing Machine.
[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
[M. Crochemore "tranducters and repetitions", Theoritical. Comp. Sc. 46, 1986].
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