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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.
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.
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.
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;
There can be multiple arrows for an input character if the finite-state machine is nondeterministic. Designate a state as the start state. The start state is given in the formal definition of a finite-state machine. Designate one or more states as accepting state. This is also given in the formal definition of a finite-state machine.
In particular, the intersection non-emptiness problem is defined as follows. Given a list of deterministic finite automata as input, the goal is to determine whether or not their associated regular languages have a non-empty intersection. In other, the goal is to determine if there exists a string that is accepted by all of the automata in the ...