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

New: Readings section 2.2.3-2.4 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

NFA example Figure 2.9 Page 56 Transition Table (NFA) x What does the p mean? What does the *r mean? What is δ(s, x) informally? 1 p q r 0,1 State\Input 1 p { p, q } q *r

Subset Construction example Figure 2.9 Page 56 of text 1 p q r Equivalent DFA Table State\Input 1 { p } { p, q} { p } { p, r} *{ p, r} 0,1

Subset Construction example Figure 2.9 Page 56 of text 1 p q r Equivalent DFA Table State\Input 1 { p } { p, q} { p } { p, r} *{ p, r} 0,1

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?

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

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

Subset Example from Author’s website slides2.pdf

Mutual Induction Proof Write up on back of Lecture Overview

Consider our old friend from HW 2. 2. 5b: L = {w ε {0,1} 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’ } 0,1 0,1 0,1 1 0,1 3 4 … 10 1 2 0,1 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

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 ε Σ (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“

[2,3,4,5, 6, 7, 8]. each { |len| numstrings = 0 strnm1 [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

Theorem 2.11 For NFA there is Eq. DFA

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

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

Epsilon Closure

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

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