1. CSCI 2670 Introduction to Theory of Computing
September 1, 2004

2. Agenda Last class Today Tomorrow Discussed non-determinism
Lots of excellent questions!
Today
Further exploration of NFA’s
Equivalence of DFA’s and NFA’s
Tomorrow
Closure of regular languages under regular operators

3. Announcement Quiz tomorrow (9/2)
DFA’s (informal description, formal definition, creation of, combining)
NFA’s (inform description, formal definition)

4. Non-deterministic finite automaton
A non-deterministic finite automaton is a
5-tuple (Q,,,q0,F), where
Q is a finite set of states
 is a (finite) alphabet
 : Q × ε  P(Q) is the transition function
 maps to sets of states
q0 is the start state, and
F  Q is the set of accept states

5. Example q1 q4 q3 q2 Q = {q1, q2, q3, q4} = {a, b, c} q0 = q1
a, c
b, c ε q1 q4 q3 q2
Q = {q1, q2, q3, q4}
= {a, b, c}
q0 = q1
F = {q2, q3, q4}

6. Example (cont.) a b c ε q1  {q2,q2,q3} q2 {q2} q3 {q3} q4 {q4} q1 q4
a, c
b, c ε q1 q4 q3 q2

7. Group project For each NFA Informally describe the behavior of the NFA
Try to construct a DFA accepting the same language

8. Group 1 1 1 ε ε 1 1
All strings containing an even number of 0’s or an even number of 1’s

9. Group 2 All prefixes of the string 0110
0,ε
0,ε
0,ε
0,ε 1 1
All prefixes of the string 0110
Proper prefixes may be followed by a 0

10. Group 3 All strings with a zero count divisible by 0 or 1 1 1 ε 1 ε 1ε 1 ε 1 1
All strings with a zero count divisible by 0 or 1

11. Group 4
0,1 1
0,1
0,1
All strings of length at least three whose third to last symbol is a 1

12. Group 5 All strings with alternating 0’s and 1’s that start with a 0 11
All strings with alternating 0’s and 1’s that start with a 0

13. Group 6 All strings containing either zero 1’s or three or more 1’s 11 ε 1 1
All strings containing either zero 1’s or three or more 1’s

14. Equivalence of DFAs and NFAs
Theorem: Every non-deterministic finite automaton has an equivalent deterministic finite automaton
Both FAs accept the same language
Proof method
Construction
Similar to method used for calculating strings
Follow all paths in parallel where states represent parallel paths

15. Proof idea
Given NFA M1={Q,,,q0,F} construct DFA M2={Q’,,’,q0’,F’} with L(M1)=L(M2)
Intuition
Recall :Q×εP(Q)
Our DFA’s transition function will generate paths within P(Q)
’: P(Q)×P(Q)