Lecture 9 NFA Subset Construction Epsilon Transitions
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Lecture 9 NFA Subset Construction Epsilon Transitions

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Last Time: Readings 2.3Mutual Induction Proof revisited Languages denoted by regular expressionsExamplesRuby Regular ExpressionsTEST 1
Lecture 9 NFA Subset Construction Epsilon Transitions

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1. Lecture 9 NFA Subset Construction & Epsilon Transitions Topics: Thompson Construction Examples Thompson Construction Test 1

2. Last Time: Readings 2.3 Mutual Induction Proof revisited Languages denoted by regular expressions Examples Ruby Regular Expressions TEST 1 ? September 29th New: Readings section 2.2.3-2.4 Sample Test 1new Theorem 2.12 Proof Author?s Website Solutions Online e-NFA ? NFA e-NFA ? DFA subset construction Regular Expressions Relations Machines and Regular expressions Ruby: regular expressions Ruby Pickaxe Book Ruby: showmatch.rb

3. Regular Expressions & Ken Thompson http://en.wikipedia.org/wiki/Regular_expression http://en.wikipedia.org/wiki/Ken_Thompson

4. Grep Unix utility man grep man ?k regexp

5. Thompson Construction Based on recursive (inductive) definition of regular expressions We describe NFAs (with epsilon moves) that recognize the base cases. Then assuming we have NFAs for smaller expressions r and s we construct NFAs for r + s rs r*

6. Recall Recursive Definition of Reg Expr Definition of regular expressions over an alphabet S Base cases: if a e S then A is a regular expression and denotes L(a) = { a } e is a regular expression and denotes L(e) = { e } Recursive definition If r and s are regular expressions denoting the languages L(r) and L(s) then rs is a regular expression denoting L(r)L(s) r+s is a regular expression denoting L(r) L(s) r* is a regular expression denoting [L(r)]* =

7. Thompson Construction Base cases

8. Thompson Construction Recursive cases r + s rs

9. Thompson Construction Recursive cases r*

10. Thompson Construction Examples

13. Sample test 1 outline Proof Techniques Inductive proof mutual induction proof Given DFA Transition diagram Input ?abaa? L(M) Give DFA for L NFA NFA for L NFA ? DFA (Subset) eNFA eNFA for L e-closure (ECLOSE in text) eNFA ? DFA (Subset) Ruby regular expressions

14. Design DFA that accepts Language L Example For a DFA D how do you prove L(D) = L ?

15. Mutual Induction Proof Define three statements for a mutual induction proof that could help in proving that L(M) = L ={x e {0, 1}* | that x has a number of zeroes divisible by 3 }

18. HW solutions 2.5.1

19. 2.5.3 a,b The set of strings consisting of zero or more a?s followed by zero or more b?s followed by zero or more c?s The set of strings consisting of either 01 repeated one or more times or 010 repeated one of more times

20. 2.3.2 Subset NFA without e NFA DFA

21. 2.3.4 Done before The set of strings over {0,1, ? 9} such that the final digit has appeared before The set of strings over {0,1, ? 9} such that the final digit has not appeared before The set of strings of 0?s and 1?s such that there are two 0?s separated by a number of positions that is a multiple of 4. Note 0 is allowable multiple.

22. Tenth symbol from the right is a ?1?

23. Given L(M1) and L(M2) define DFA for intersection

25. Pop Quiz Induction proof

28. Homework: 2.5.1 2.5.3 a,b Extra Credit (20pts): Modify dfa1.rb to print all strings of length<8 that would be accepted by that DFA.


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