Lecture 6 NFA Subset Construction

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Lecture 6NFA Subset Construction Epsilon
Transitions
CSCE 355 Foundations of Computation

• Topics
• Examples Subset Construction
• Authors Website again
• Epsilon transitions
• Ruby - dfa1.rb

Sept 15, 2008
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• 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
• Authors Website Solutions Online

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NFA example Figure 2.9

• x

• What does the ?p mean?
• What does the r mean?
• What is d(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
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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 cant?
• Can a DFA recognize a language that an NFA cant?

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Exercise 2.2.9 Solutions Online

• Authors 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. Ullmans (Jeffs) slides from CS 154
• http//infolab.stanford.edu/ullman/ialc/jdu-slide
  s.html

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Homework and Test 1

• HW 2
• Extra Credit
• http//infolab.stanford.edu/ullman/ialc/slides/sl
  ides1.pdf
• HW 3
• 4a
• 4b
• 5
• 6
• HW 4
• Pop Quiz

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Subset Example from Authors website slides2.pdf
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Mutual Induction Proof

• Write up on back of Lecture Overview

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Consider our old friend from HW 2.2.5bL w e
0,1 the tenth symbol from the right end of w
is a 1
0,1
0,1
0,1
1
0,1
10
3
4

0
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

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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

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• 2,3,4,5, 6, 7, 8.each len
• numstrings 0
• strnm1.each str
• alphabet.each chr
• x chr str
• strnnumstrings x
• numstrings numstrings 1
• print "Strings of length len\n"
• strn.each str print "str\n"
• strnm1 strn
• strn Array.new
• numstrings 0

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Theorem 2.11 For NFA there is Eq. DFA
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Theorem 2.12 L is accepted by DFA if and only if
L is accepted by NFA
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Epsilon (e)-Transitions

• Keyword Searching Example for, format, font

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Epsilon Closure
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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

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References and Homework

• Ruby pickaxe book Online
• http//whytheluckystiff.net/ruby/pickaxe/
• Authors Website for Text
• http//infolab.stanford.edu/ullman/ialc.html
• Slides, HW, Exams