Consider deterministic finite state automata , where is a finite set of states, is a finite set of characters, is a transition function, is a start state, and is a set of accept states. Also, denotes the empty string.
Let . For , returns an integer number represented by . For example, and . For , we define .
(1) Depict the state transition diagram of a deterministic finite state automaton that accepts .
(2) Depict the state transition diagram of a deterministic finite state automaton that accepts .
(3) Show , and of a deterministic finite state automaton that accepts for any . You may use mod to describe .
(4) Let , where returns the first character of string . Show , and of a deterministic finite state automaton that accepts for any . You may use mod to describe .