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Finite-state machines are a class of automata studied in automata theory and the theory of computation. In computer science, finite-state machines are widely used in modeling of application behavior ( control theory ), design of hardware digital systems , software engineering , compilers , network protocols , and computational linguistics .
An automaton with a finite number of states is called a finite automaton (FA) or finite-state machine (FSM). The figure on the right illustrates a finite-state machine, which is a well-known type of automaton. This automaton consists of states (represented in the figure by circles) and transitions (represented by arrows).
The figure illustrates a deterministic finite automaton using a state diagram. In this example automaton, there are three states: S 0, S 1, and S 2 (denoted graphically by circles). The automaton takes a finite sequence of 0s and 1s as input. For each state, there is a transition arrow leading out to a next state for both 0 and 1.
Nondeterministic finite automaton with ε-moves (NFA-ε) is a further generalization to NFA. In this kind of automaton, the transition function is additionally defined on the empty string ε. A transition without consuming an input symbol is called an ε-transition and is represented in state diagrams by an arrow labeled "ε". ε-transitions ...
In the automata formulation, the Krohn–Rhodes theorem for finite automata states that given a finite automaton A with states Q and input set I, output alphabet U, then one can expand the states to Q' such that the new automaton A' embeds into a cascade of "simple", irreducible automata: In particular, A is emulated by a feed-forward cascade ...
Cellular automata have found application in various areas, including physics, theoretical biology and microstructure modeling. A cellular automaton consists of a regular grid of cells, each in one of a finite number of states, such as on and off (in contrast to a coupled map lattice). The grid can be in any finite number of dimensions.
Automata-based programming is a programming paradigm in which the program or part of it is thought of as a model of a finite-state machine (FSM) or any other (often more complicated) formal automaton (see automata theory). Sometimes a potentially infinite set of possible states is introduced, and such a set can have a complicated structure, not ...
A two-way deterministic finite automaton (2DFA) is an abstract machine, a generalized version of the deterministic finite automaton (DFA) which can revisit characters already processed. As in a DFA, there are a finite number of states with transitions between them based on the current character, but each transition is also labelled with a value ...