Difference between revisions of "Theoretical Aspects of Lexical Analysis/Exercise 4"

From Wiki**3

< Theoretical Aspects of Lexical Analysis
(→‎DFA)
(→‎DFA)
Line 173: Line 173:
  
  
−
{| width="100%"
+
Graphically, the DFA is represented as follows:
−
! style="text-align: left; font-weight:normal; vertical-align: top; width: 50%;" |Graphically, the DFA is represented as follows:
 
  
 
<graph>
 
<graph>
Line 206: Line 205:
 
</graph>
 
</graph>
  
−
Given the minimization tree to the right, the final minimal DFA is:
+
 
 +
The minimization tree is as follows.
 +
 
 +
<graph>
 +
digraph mintree {
 +
  node [shape=none,fixedsize=true,width=0.3,fontsize=10]
 +
  "{0, 1, 2, 3, 4, 5, 6}" -> "{0, 1, 2, 3}" [label="NF",fontsize=10]
 +
  "{0, 1, 2, 3, 4, 5, 6}" -> "{4, 5, 6}" [label="  F",fontsize=10]
 +
  //"{0, 1, 2, 3}" -> "{0, 1, 2, 3} " [label="  a",fontsize=10]
 +
  "{0, 1, 2, 3}" ->  "{0, 1, 2}"
 +
  "{0, 1, 2, 3}" -> "{3} " [label="  b",fontsize=10]
 +
  "{0, 1, 2}" -> "{0, 2} "
 +
  "{0, 1, 2}" -> "{1} " [label="  b",fontsize=10]
 +
  fontsize=10
 +
  //label="Minimization tree"
 +
}
 +
</graph>
 +
 
 +
The tree expansion for non-splitting sets has been omitted for simplicity ("a" transition for non-final states and the {0, 2} "a" and "b" transitions. The final states are all indistinguishable, regarding either "a" or "b" transitions.
 +
 
 +
Given the minimization tree above, the final minimal DFA is:
 
<graph>
 
<graph>
 
digraph dfamin {
 
digraph dfamin {
Line 226: Line 245:
 
}
 
}
 
</graph>
 
</graph>
−
 
−
! style="text-align: left; font-weight:normal; vertical-align: top; width: 50%;" | The minimization tree is as follows.
 
−
 
−
<graph>
 
−
digraph mintree {
 
−
  node [shape=none,fixedsize=true,width=0.3,fontsize=10]
 
−
  "{0, 1, 2, 3, 4, 5, 6}" -> "{0, 1, 2, 3}" [label="NF",fontsize=10]
 
−
  "{0, 1, 2, 3, 4, 5, 6}" -> "{4, 5, 6}" [label="  F",fontsize=10]
 
−
  //"{0, 1, 2, 3}" -> "{0, 1, 2, 3} " [label="  a",fontsize=10]
 
−
  "{0, 1, 2, 3}" ->  "{0, 1, 2}"
 
−
  "{0, 1, 2, 3}" -> "{3} " [label="  b",fontsize=10]
 
−
  "{0, 1, 2}" -> "{0, 2} "
 
−
  "{0, 1, 2}" -> "{1} " [label="  b",fontsize=10]
 
−
  fontsize=10
 
−
  //label="Minimization tree"
 
−
}
 
−
</graph>
 
−
 
−
The tree expansion for non-splitting sets has been omitted for simplicity ("a" transition for non-final states and the {0, 2} "a" and "b" transitions. The final states are all indistinguishable, regarding either "a" or "b" transitions.
 
−
|}
 
  
 
[[category:Teaching]]
 
[[category:Teaching]]
 
[[category:Compilers]]
 
[[category:Compilers]]
 
[[en:Theoretical Aspects of Lexical Analysis]]
 
[[en:Theoretical Aspects of Lexical Analysis]]

Revision as of 16:18, 25 March 2009

Use Thompson's algorithm to build the NFA for the following regular expression. Build the corresponding DFA and minimize it.

  • (a|b)*abb(a|b)*

NFA

The following is the result of applying Thompson's algorithm.


DFA

Determination table for the above NFA:

In α∈Σ move(In, α) ε-closure(move(In, α)) In+1 = ε-closure(move(In, α))
- - 0 0, 1, 2, 4, 7 0
0 a 3, 8 1, 2, 3, 4, 6, 7, 8 1
0 b 5 1, 2, 4, 5, 6, 7 2
1 a 3, 8 1, 2, 3, 4, 6, 7, 8 1
1 b 5, 9 1, 2, 4, 5, 6, 7, 9 3
2 a 3, 8 1, 2, 3, 4, 6, 7, 8 1
2 b 5 1, 2, 4, 5, 6, 7 2
3 a 3, 8 1, 2, 3, 4, 6, 7, 8 1
3 b 5, 10 1, 2, 4, 5, 6, 7, 10, 11, 12, 14, 17 4
4 a 3, 8, 13 1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16, 17 5
4 b 5, 15 1, 2, 4, 5, 6, 7, 11, 12, 14, 15, 16, 17 6
5 a 3, 8, 13 1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16, 17 5
5 b 5, 9, 15 1, 2, 4, 5, 6, 7, 9, 11, 12, 14, 15, 16, 17 7
6 a 3, 8, 13 1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16, 17 5
6 b 5, 15 1, 2, 4, 5, 6, 7, 11, 12, 14, 15, 16, 17 6
7 a 3, 8, 13 1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16, 17 5
7 b 5, 10, 15 1, 2, 4, 5, 6, 7, 10, 11, 12, 14, 15, 16, 17 8
8 a 3, 8, 13 1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16, 17 5
8 b 5, 15 1, 2, 4, 5, 6, 7, 11, 12, 14, 15, 16, 17 6


Graphically, the DFA is represented as follows: 



The minimization tree is as follows.


The tree expansion for non-splitting sets has been omitted for simplicity ("a" transition for non-final states and the {0, 2} "a" and "b" transitions. The final states are all indistinguishable, regarding either "a" or "b" transitions.

Given the minimization tree above, the final minimal DFA is:

Retrieved from "https://web.tecnico.ulisboa.pt/~david.matos/w/pt/index.php?title=Theoretical_Aspects_of_Lexical_Analysis/Exercise_4&oldid=3088"