COT 4210 Lecture Notes - 3

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COT 4210 Lecture Notes - 3

• By Njegos and Monika

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R is a regular expression if R is

1. a?S, a for some a in the alphabet S.
2. e
3. Ø
4. R1 U R2, where R1 and R2 are regular expression
5. R1o R2, where R1 and R2 are regular expressions
6. R1, where R1 is a regular expression

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• If A1 and A2 are regular then
• A1 U A2
• A1 o A2
• A1
• All are regular too.

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Regular Expression

• 010- zero or more 0s followed by one 1 and
  followed by any number of 0s or no 0
• (0 U e) 1- 0 followed by any number of 1s or
  any number of 1s
• 1 Ø - Take zero or more 1s and concatenated
  by empty set. 1 Ø Ø
• Ø- e Empty string

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N1 U N2
N
N1
e
e
N2
6
A1
e
e
e
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0 U 1
0
e
e
1
8
(0 U 1)
0
e
e
e
e
1
e
9
0 (0 U 1) 0
0
e
e
e
0
0
e
e
e
e
1
e
10
0 U 0 (0 U 1) 0
0
e
0
e
e
e
e
0
0
e
e
e
e
1
e
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Qj
R4
Qi
R3
R1
Qrip
R1 R2 R3 U R4
Qj
Qi
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GNFA

• GNFA stands for Generalized Nondeterministic
  Finite Automaton.
• GNFA are simply nondeterministic finite automaton
  wherein the transition arrows may have any
  regular expressions as labels, instead of only
  members of the alphabet.

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Restricted GNFA

• 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 every other state but no
  arrows going to any other state. Accept state is
  not the same as the start 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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EXAMPLE
0
1
0
1
e
Q2
Q3
0
Q1
S
1
e
F
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After Ripping Q3
0
1
e
Q2
Q1
S
1
010
e
F
16
101 U 010
R2
01
S
Q2
0
10
R3
R4
F
01(101 U 010)10 U 0
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• If we have n states in DFA then restricted GNFA
  is going to have (n2) states.