<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 time.
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.
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