Lecture 9 NFA Subset Construction & Epsilon Transitions

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Lecture 9NFA Subset Construction Epsilon
Transitions
CSCE 355 Foundations of Computation

• Topics
• Thompson Construction
• Examples Thompson Construction
• Test 1

Sept 24, 2008
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• 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
• Authors 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

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Regular Expressions Ken Thompson

• http//en.wikipedia.org/wiki/Regular_expression
• http//en.wikipedia.org/wiki/Ken_Thompson

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Grep

• Unix utility
• man grep
• man k regexp

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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

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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)
• rs is a regular expression denoting L(r)
  L(s)
• r is a regular expression denoting
  L(r)

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Thompson Construction Base cases
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Thompson Construction Recursive cases

• r s

• rs

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Thompson Construction Recursive cases

• r

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Thompson Construction Examples
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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

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Design DFA that accepts Language L

• Example
• For a DFA D how do you prove L(D) L ?

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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

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HW solutions 2.5.1
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2.5.3 a,b

• The set of strings consisting of zero or more as
  followed by zero or more bs followed by zero or
  more cs
• The set of strings consisting of either 01
  repeated one or more times or 010 repeated one of
  more times

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2.3.2 Subset NFA without e

• NFA

• DFA

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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 0s and 1s such that there
  are two 0s separated by a number of positions
  that is a multiple of 4. Note 0 is allowable
  multiple.

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Tenth symbol from the right is a 1
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Given L(M1) and L(M2) define DFA for intersection
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Pop Quiz Induction proof
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Homework

• 2.5.1
• 2.5.3 a,b
• Extra Credit (20pts) Modify dfa1.rb to print all
  strings of lengthlt8 that would be accepted by
  that DFA.