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Regular Expressions and Non-regular Languages

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A language is regular if and only if some regular expression describes it. We break down this theorem as ... Generalized Nondeterministic Finite Automaton ... – PowerPoint PPT presentation

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Title: Regular Expressions and Non-regular Languages


1
Regular Expressions and Non-regular Languages
  • http//cis.k.hosei.ac.jp/yukita/

2
Expressions and their values
3
Definition 1.26
4
The values of atomic expressions
5
Example 1.27
6
Units for the binary operations
7
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.

8
Proof of Lemma 1.29
a
Next three slides
9
Case 4 Let N1, N2, and N correspond to R1, R2,
and R, respectively.
N
N1
e
e
N2
10
Case 5 Let N1, N2, and N correspond to R1, R2,
and R, respectively.
N
e
e
11
Case 6 Let N1 and N correspond to R1 and R,
respectively.
N
e
e
e
12
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.

13
Standard Form of GNFA
...
14
Standard Form of GNFA
15
Equivalent GNFA with one fewer state
R4
qi
qj
qi
qj
R1
R3
qrip
R2
16
Definition 1.33
17
Computation with GNFA
18
Converting GNFA
19
Claim 1.34 For any GNFA G, Convert(G) is
equivalent to G.
20
Proof continued
21
Non-regularity
22
Theorem 1.37 Pumping Lemma
23
Proof of Th 1.37
24
Example 1.38
25
Example 1.39
26
Alternative proof of 1.39
27
Example 1.40
28
Example 1.41 Unary Language
29
Example 1.42 Pumping Down
30
Problem 1.41 Differential Encoding
0
1
1
0xx1
0xx0
0
0
qstart
0
1xx0
1xx1
1
1
1
0
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