Finite Automata

• F ⊆ Q is the set of accept or final states. The behavior of a DFA M on an input string x ∈ Σ∗ is as follows. It reads the symbols in x one at a time, from left to right. At each point in time, the automaton has a current state. Initially, the automaton is in state s. If the current state of the automaton is q, and the next symbol in x to be read is a, then the automaton on reading a transitions to the state δ(q, a), and continues processing the remaining symbols of x. After all the symbols in x have been read by the automaton, if the current state of the automaton is in F then the input is said to have been accepted (or the automaton returns the Boolean answer true on the input x), and if the current state is not in F then input is rejected (or the automaton returns false). The above informal description of computation can be convenient captured by the notion of a run. A run of M starting from state q on input x = a1a2 · · · an, where ai ∈ Σ, is a sequence of states q0, q1, . . . qn such that