1. Finite Automata & Regular Languages
Sipser, Chap 1 Hopcroft, Motawi, Ullman, Chap 2

2. Deterministic Finite Automata
A DFA or deterministic finite automaton M is a 5-tuple, M = (Q, , , q0, F), where:
Q is a finite set of states of M
 is the finite input alphabet of M
: Q    Q is the state transition function
q0 is the start state of M
F  Q is the set of accepting states or final states of M

3. DFA Example State diagram Q = { q0, q1 }  = { 0, 1 } F = { q1 }  11 M
State diagram
Q = { q0, q1 }  = { 0, 1 } F = { q1 }  1 q0 q1
State Table

4. State table & state transition function
State transition function (q0, 0) = q0, (q0, 1) = q1 (q1, 0) = q1, (q1, 1) = q0  1 q0 q1

5. State transitions
If q, q’  Q, s  , and (q, s) = q’, then we say that q’ is an s-successor of q, or there is a transition from q to q’ on input s, and we write q s q’
Example: since (q0, 1) = q1, then there is a transition from q0 to q1 on input 1, and we write q0 1 q1.

6. State sequences
If a string of input symbols w = s0s1s2 … sk-1 takes M from initial state q0 to state qk, namely q0 s0 q1 s1 q2 s2 q3  … s[k-1] qk then we say that qk is a w-successor of q0, and write q0 w qk. Also q0q1q2 … qk is called an admissible state sequence for w.

7. Strings accepted by a DFA
Let M = (Q, , , q0, F) be a DFA, and w = s0s1s2 … sk-1  * be a string over alphabet . Then M accepts w if there exists an admissible state sequence q0q1q2 … qk for w, starting at initial state q0 and ending with state qk, where qk  F. That is, M accepts input string w if M ends up in one of the final states.

8. Language recognized by a DFA
The language L(M) that is recognized by a DFA, M = (Q, , , q0, F), is the set of all strings accepted by M. That is, L(M) = { w  * | M accepts w } = { w  * | q0 w qk, qk  F }.
Example: For the previous DFA, L(M) is the set of all strings of 0s and 1s with odd parity, that is, odd number of 1s.

9. DFA Example 2 Recognizer for 11*01* B 1 C A D 0,1 Trap
0,1
* means zero or more occurrences of the preceding symbol
Trap

10. DFA Example 2
M = (Q, , , q0, F), L(M) = 11*01* Q = { q0=A, B, C, D }  = { 0, 1 } F = { C }  1 A D B C

11. DFA Example 3
Modulo 3 counter B 1 2 A 1 2 2 C 1

12. DFA Example 3
M = (Q, , , q0, F) Q = { q0=A, B, C }  = { 0, 1, 2 } F = { A }  1 2 A B C

13. Regular Languages and DFAs
A language L  * is called regular if there exists a DFA M such that L(M)=L
Examples of regular languages
Binary strings with odd parity
Language described by 11*01*
Strings that form integers divisible by 3

14. Next
Variations on Finite Automata; e.g., Nondeterministic Finite Automata (NFAs)
Equivalences of these variations with DFAs
Regular expressions