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A classic form of state diagram for a finite automaton (FA) is a directed graph with the following elements (Q, Σ, Z, δ, q 0, F): [2] [3] Vertices Q: a finite set of states, normally represented by circles and labeled with unique designator symbols or words written inside them; Input symbols Σ: a finite collection of input symbols or designators
A finite-state machine (FSM) or finite-state automaton (FSA, plural: automata), finite automaton, or simply a state machine, is a mathematical model of computation.It is an abstract machine that can be in exactly one of a finite number of states at any given time.
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.
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).
Let k be a positive integer, and let D = (Q, Σ k, δ, q 0, Δ, τ) be a deterministic finite automaton with output, where Q is the finite set of states; the input alphabet Σ k consists of the set {0,1,...,k-1} of possible digits in base-k notation; δ : Q × Σ k → Q is the transition function; q 0 ∈ Q is the initial state;
On each rightward move, the table can be updated using the old table values and the character that was in the previous cell. Since the original head-control had some fixed number of states, and there is a fixed number of states in the tape alphabet, the table has fixed size, and can therefore be computed by another finite state machine.
Such an automaton may be defined as a 5-tuple (Q, Σ, T, q 0, F), in which Q is the set of states, Σ is the set of input symbols, T is the transition function (mapping a state and an input symbol to a set of states), q 0 is the initial state, and F is the set of accepting states. The corresponding DFA has states corresponding to subsets of Q.
The set of states of automaton A, denoted states(A), need not be finite. This is a significant generalization of the usual notion of finite automata, as it enables modeling systems with unbounded data structures like counters and unbounded length queues. The set of start states (also known as initial states) is a non-empty subset of states.