Lecture 6 NFA Subset Construction & Epsilon Transitions

1. Lecture 6NFA Subset Construction & Epsilon Transitions CSCE 355 Foundations of Computation Topics: • Examples Subset Construction • Author’s Website again • Epsilon transitions • Ruby - dfa1.rb Sept 15, 2008

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 – Monday 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 δ(s, x) informally? • Page 56 Transition Table (NFA) 0 1 p q r 0,1

4. Subset Construction example Figure 2.9 • Page 56 of text 0 1 p q r Equivalent DFA Table 0,1

5. Subset Construction example Figure 2.9 • Page 56 of text 0 1 p q r Equivalent DFA Table 0,1

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 δ(q0, a) = δ(qf, a) for all a in Σ then for all w != ε we have δ(q0, w) = δ(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

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

14. Mutual Induction Proof • Write up on back of Lecture Overview

15. Consider our old friend from HW 2.2.5b:L = {w ε{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 0,1 0,1 0,1 1 0,1 … 3 4 10 0 1 2 0,1

16. Ruby: Strings and DFAs (dfa1.rb) # DFA1.rb on Handouts page Now consider how to generate all strings in Σ* 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 εΣ (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“

17. [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 }

18. Theorem 2.11 For NFA there is Eq. DFA

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

20. Epsilon (ε)-Transitions • Keyword Searching Example: for, format, font

21. Epsilon Closure

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

23. 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