1. CSCI 2670 Introduction to Theory of Computing
September 2, 2004

2. Agenda Last class Today NFA examples
Presented theorem regarding equivalence of DFA’s and NFA’s
Today
Quiz
Prove four theorems
Equivalence of DFA’s and NFA’s
Union, concatenation, and Kleene star are regular operators

3. Next week
Section 1.3 and part of Section 1.4
Read pages

4. 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

5. 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)

6. Defining M2 Determine Q’, q0, and F’
Q’ = P(Q)
q0’ = {q0}
F’ = {R  Q’ | R  F  }
i.e., R contains at least one of M1’s accept states
Defining ’ (for now ignore ε jumps)

7. Example Q’={,{q1}, {q2}, {q3}, {q1,q2}, {q1,q3}, {q2,q3}, {q1,q2,q3}}q1 1 q3
Q’={,{q1}, {q2}, {q3}, {q1,q2}, {q1,q3}, {q2,q3}, {q1,q2,q3}}

8. Example 1 q2 q1 1 q3
q0’={q1}

9. Example 1 q2 q1 1 q3
F’={{q3}, {q1,q3}, {q2,q3}, {q1,q2,q3}}

10. Example 1 {q1} {q2,q3}  {q2} {q1,q3} {q3} {q1,q2} {q1,q2,q3} 1 1 q21
{q1}
{q2,q3} 
{q2}
{q1,q3}
{q3}
{q1,q2}
{q1,q2,q3} 1 q1 q2 q3 1 1

11. What about ε jumps? For each R  P(Q), define function E(R)
E(R) = {q | q can be reach by 0 or more ε jumps from some r  R}
Redefine ’(R,a) to include E(R)
’(R,a) = {q | q  E((r,a)) for some r  R}

12. Closure of NFA’s under regular operations
Remainder of class
Union
Concatenation
Kleene star

13. Union is a regular operation
Theorem: The class of regular languages is closed under the union operation
Proof approach: Assume A1 and A2 are both regular languages with A1=L(M1) and A2=L(M2) and create an NFA M such that L(M) = A1A2
Method: Proof by construction

14. Construct M from M1 and M2 M M1 ε ε M2

15. Concatenation is a regular operation
Theorem: The class of regular languages is closed under the concatenation operation
Proof approach: Assume A1 and A2 are both regular languages with A1=L(M1) and A2=L(M2) and create an NFA M such that L(M) = A1A2
Method: Proof by construction

16. Construct M from M1 and M2 M M1 ε ε M2

17. Kleene star is a regular operation
Theorem: The class of regular languages is closed under the Kleene operation
Proof approach: Assume A1 is a regular language with A1=L(M1) and create an NFA M such that L(M) = A1*
Method: Proof by construction

18. Construct M from M1 M ε ε M1

19. Have A Wonderful Labor Day!!!