CSCI 2670 Introduction to Theory of Computing

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

2. Agenda • Last class • 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 63 - 74

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 1 • Q’={,{q1}, {q2}, {q3}, {q1,q2}, {q1,q3}, {q2,q3}, {q1,q2,q3}} q2 0 q1 1 0 q3

8. Example 1 • q0’={q1} q2 0 q1 1 0 q3

9. Example 1 • F’={{q3}, {q1,q3}, {q2,q3}, {q1,q2,q3}} q2 0 q1 1 0 q3

10. 1 q2 0 q1 0 q3 Example 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. M1 M2 M Construct M from M1 and M2 ε ε

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

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. ε ε ε M1 M Construct M from M1

19. Have A Wonderful Labor Day!!!