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# Finite state machine

finite state machine
A model of a computational system, consisting of a set of states (including a start state), an alphabet of symbols that serves as a set of possible inputs to the machine, and a transition function that maps each state to another state (or to itself) for any given input symbol. The machine operates by being fed a string of symbols, and moves through a series of states. The study of the computational power of finite state machines and other related machines is important in computer science and linguistics. The computational core of a Turing machine is a finite state machine. Also called finite state automaton.
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.
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 computations”, Addison-Wesley].
[M. Crochemore “tranducters and repetitions”, Theoritical. Comp. Sc. 46, 1986].
(2001-09-22)

Tagged:

• Finite-verb

noun 1. a verb form that distinguishes person, number, and tense, and also mood or aspect, as opens in She opens the door.

• Finitism

/ˈfaɪnaɪtˌɪzəm/ noun 1. (philosophy, logic) the view that only those entities may be admitted to mathematics that can be constructed in a finite number of steps, and only those propositions entertained whose truth can be proved in a finite number of steps Compare intuitionism

• Finito

[fi-nee-toh] /fɪˈni toʊ/ adjective, Informal. 1. finished; ended. noun phrase The putative mathematical constant by which a wrong answer is multiplied to get a right answer (1950s+)

• Finitude

[fin-i-tood, -tyood, fahy-ni-] /ˈfɪn ɪˌtud, -ˌtyud, ˈfaɪ nɪ-/ noun 1. a state or quality. n. 1640s, from finite + -ude.

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