Regular Expressions and Non-regular Languages

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Regular Expressions and Non-regular Languages

• http//cis.k.hosei.ac.jp/yukita/

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Expressions and their values
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Definition 1.26
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The values of atomic expressions
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Example 1.27
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Units for the binary operations
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Theorem 1.28

• A language is regular if and only if some regular
  expression describes it.
• We break down this theorem as follows.
• Lemma 1.29
• If a language is described by a regular
  expression, then it is regular.
• Lemma 1.32
• If a langulage is regular, then it is described
  by a regular expression.

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Proof of Lemma 1.29
a
Next three slides
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Case 4 Let N1, N2, and N correspond to R1, R2,
and R, respectively.
N
N1
e
e
N2
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Case 5 Let N1, N2, and N correspond to R1, R2,
and R, respectively.
N
e
e
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Case 6 Let N1 and N correspond to R1 and R,
respectively.
N
e
e
e
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Generalized Nondeterministic Finite Automaton

• is roughly a NFA in which the transition arrows
  may have regular expressions as labels.
• We assume the following standard form for
  convenience, which can always be attained with an
  easy modification.
• There is only one accept state and different from
  the start state.
• The start state has transition arrows going to
  every other state but no arrows coming in from
  any other state.
• There is only a single accept state, and it has
  arrows coming in from any other state but no
  arrows going to any other state.
• Except for the start and accept states, one arrow
  goes from every state to every other state and
  also from each state to itself.

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Standard Form of GNFA
...
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Standard Form of GNFA
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Equivalent GNFA with one fewer state
R4
qi
qj
qi
qj
R1
R3
qrip
R2
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Definition 1.33
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Computation with GNFA
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Converting GNFA
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Claim 1.34 For any GNFA G, Convert(G) is
equivalent to G.
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Proof continued
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Non-regularity
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Theorem 1.37 Pumping Lemma
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Proof of Th 1.37
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Example 1.38
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Example 1.39
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Alternative proof of 1.39
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Example 1.40
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Example 1.41 Unary Language
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Example 1.42 Pumping Down
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Problem 1.41 Differential Encoding
0
1
1
0xx1
0xx0
0
0
qstart
0
1xx0
1xx1
1
1
1
0