Lecture 6 NFA Subset Construction Epsilon Transitions

1. Lecture 6NFA Subset Construction & Epsilon Transitions Topics: Examples Subset Construction Author’s Website again Epsilon transitions Ruby - dfa1.rb

2. Last Time: Readings 2.3 HW Hints and Induction example Review Induction DFA for Union, revisit Example 2.4 Pop Quiz Uses of Finite automata NFA Delta-hat, a string accepted by an NFA, the language accepted Subset construction converting NFA ? equivalent DFA TEST 1 – September 29th New: Readings section 2.2.3-2.4 Examples Subset construction converting NFA? equivalent DFA Author’s Website Solutions Online

3. NFA example Figure 2.9 x What does the ?p mean? What does the *r mean? What is d(s, x) informally?

4. Subset Construction example Figure 2.9 Page 56 of text

5. Subset Construction example Figure 2.9 Page 56 of text

6. Subset Construction Significance Constructing an equivalent DFA from and NFA What does equivalent mean? Does equivalent mean have the same number of states? Equivalent means ? Why convert? What is better about an NFA? What is better about a DFA? We are interested in the power of these models? Can an NFA recognize a language that a DFA can’t? Can a DFA recognize a language that an NFA can’t?

7. Exercise 2.2.9 Solutions Online Author’s Website for Text http://infolab.stanford.edu/~ullman/ialc.html HW Solutions for starred (*) problems http://infolab.stanford.edu/~ullman/ialcsols/sol2.html 2.2.9 page 54 Prove If d(q0, a) = d(qf, a) for all a in S then for all w != e we have d(q0, w) = d(qf, w) by induction on the length of w. Basis Assume Then we need to show that Dr. Ullman’s (Jeff’s) slides from CS 154 http://infolab.stanford.edu/~ullman/ialc/jdu-slides.html

8. Homework and Test 1 HW 2 Extra Credit http://infolab.stanford.edu/~ullman/ialc/slides/slides1.pdf HW 3 4a 4b 5 6 HW 4 Pop Quiz

9. Subset Example from Author’s website slides2.pdf

10. Mutual Induction Proof Write up on back of Lecture Overview

11. Consider our old friend from HW 2.2.5b:L = {w e {0,1}* | the tenth symbol from the right end of w is a ‘1’ } If we convert an NFA with n states to a DFA using the subset construction what is the max number states in the DFA? Can we do better? Subset construction an example of “lazy evaluation” – i.e. consider only states we can get to from q0 DFA minimization is a topic for later

12. Ruby: Strings and DFAs (dfa1.rb) # DFA1.rb on Handouts page Now consider how to generate all strings in S* of length 6 # Idea: generate them from a list of the strings of length n-1 # by concatenating onto each string w of length n-1 each a e S (a recursive definition) # lists of strings of length n-1 and n strnm1 = Array.new(); strn = Array.new(); strnm1 = ["a", "b"] print "strnm1 = #{strnm1}\n“

13. [2,3,4,5, 6, 7, 8].each { |len| numstrings = 0 strnm1.each { |str| alphabet.each { |chr| x = chr + str strn[numstrings] = x numstrings = numstrings + 1 } } print "Strings of length #{len}:\n" strn.each { |str| print "#{str}\n" } strnm1 = strn strn = Array.new numstrings = 0 }

14. Theorem 2.11 For NFA there is Eq. DFA

15. Theorem 2.12 L is accepted by DFA if and only if L is accepted by NFA

16. Epsilon (e)-Transitions Keyword Searching Example : for, format, font

19. Epsilon Closure

20. Equivalent NFA (without e) for an NFA with e Convert NFA with e to an equivalent NFA without e Compute transitive closure of e arcs If p can reach state q by e arcs and d(r, a) contains p (there is a transition from r to q on input a) then add q to d(r, a) i.e. add a transition from r to q on input a xx

21. References and Homework Ruby pickaxe book Online http://whytheluckystiff.net/ruby/pickaxe/ Author’s Website for Text http://infolab.stanford.edu/~ullman/ialc.html Slides, HW, Exams