Difference between revisions of "Theoretical Aspects of Lexical Analysis/Exercise 2"

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< Theoretical Aspects of Lexical Analysis
Line 4: Line 4:
  
 
== Solution ==
 
== Solution ==
 +
 +
=== NFA ===
 +
 +
The following is the result of applying Thompson's algorithm.
 +
 +
<graph>
 +
digraph nfa {
 +
    { node [shape=circle style=invis] start }
 +
  rankdir=LR; ratio=0.5
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  node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 11
 +
  node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
 +
  start -> 0
 +
  0 -> 1; 0 -> 11
 +
  1 -> 2; 1 -> 6
 +
  2 -> 3; 2 -> 5
 +
  3 -> 4 [label="a",fontsize=10]
 +
  4 -> 3; 4 -> 5
 +
  5 -> 10
 +
  6 -> 7; 6 -> 9
 +
  7 -> 8 [label="b",fontsize=10]
 +
  8 -> 7; 8 -> 9
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  9 -> 10
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  10 -> 1; 10 -> 11
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  fontsize=10
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  //label="NFA for (a*|b*)*"
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}
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</graph>
 +
 +
=== DFA ===
 +
 +
Determination table for the above NFA:
 +
 +
{| cellspacing="2"
 +
! style="padding-left: 20px; padding-right: 20px; background: wheat;" | I<sub>n</sub>
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! style="padding-left: 20px; padding-right: 20px; background: wheat;" | α∈Σ
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! style="padding-left: 20px; padding-right: 20px; background: wheat;" | move(I<sub>n</sub>, α)
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! style="padding-left: 20px; padding-right: 20px; background: wheat;" | ε-closure(move(I<sub>n</sub>, α))
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! style="padding-left: 20px; padding-right: 20px; background: wheat;" | I<sub>n+1</sub> = ε-closure(move(I<sub>n</sub>, α))
 +
|-
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! style="font-weight: normal; align: center; background: #ffffcc;" | -
 +
! style="font-weight: normal; align: center; background: #ffffcc;" | -
 +
! style="font-weight: normal; align: center; background: #ffffcc;" | 0
 +
! style="font-weight: normal; align: left;  background: #ffffcc;" | 0, 1, 2, 3, 5, 6, 7, 9, 10, '''11'''
 +
! style="font-weight: normal; align: center; background: #ffffcc;" | '''0'''
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|-
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! style="font-weight: normal; align: center; background: #e6e6e6;" | 0
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | a
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | 4
 +
! style="font-weight: normal; align: left;  background: #e6e6e6;" | 1, 2, 3, 4, 5, 6, 7, 9, 10, '''11'''
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | '''1'''
 +
|-
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | 0
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | b
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | 8
 +
! style="font-weight: normal; align: left;  background: #e6e6e6;" | 1, 2, 3, 5, 6, 7, 8, 9, 10, '''11'''
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | '''2'''
 +
|-
 +
! style="font-weight: normal; align: center; background: #ffffcc;" | 1
 +
! style="font-weight: normal; align: center; background: #ffffcc;" | a
 +
! style="font-weight: normal; align: center; background: #ffffcc;" | 4
 +
! style="font-weight: normal; align: left;  background: #ffffcc;" | 1, 2, 3, 4, 5, 6, 7, 9, 10, '''11'''
 +
! style="font-weight: normal; align: center; background: #ffffcc;" | '''1'''
 +
|-
 +
! style="font-weight: normal; align: center; background: #ffffcc;" | 1
 +
! style="font-weight: normal; align: center; background: #ffffcc;" | b
 +
! style="font-weight: normal; align: center; background: #ffffcc;" | 8
 +
! style="font-weight: normal; align: left;  background: #ffffcc;" | 1, 2, 3, 5, 6, 7, 8, 9, 10, '''11'''
 +
! style="font-weight: normal; align: center; background: #ffffcc;" | '''2'''
 +
|-
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! style="font-weight: normal; align: center; background: #e6e6e6;" | 2
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | a
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | 4
 +
! style="font-weight: normal; align: left;  background: #e6e6e6;" | 1, 2, 3, 4, 5, 6, 7, 9, 10, '''11'''
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | '''1'''
 +
|-
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | 2
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | b
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | 8
 +
! style="font-weight: normal; align: left;  background: #e6e6e6;" | 1, 2, 3, 5, 6, 7, 8, 9, 10, '''11'''
 +
! style="font-weight: normal; align: center; background: #e6e6e6;" | '''2'''
 +
|}
 +
 +
 +
{| width="100%"
 +
! style="text-align: left; font-weight:normal; vertical-align: top; width: 50%;" |Graphically, the DFA is represented as follows:
 +
 +
<graph>
 +
digraph dfa {
 +
    { node [shape=circle style=invis] start }
 +
  rankdir=LR; ratio=0.5
 +
  node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 0 1 2
 +
  node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
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  start -> 0
 +
  0 -> 1 [label="a"]
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  0 -> 2 [label="b"]
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  1 -> 1  [label="a"]
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  1 -> 2  [label="b"]
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  2 -> 1 [label="a"]
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  2 -> 2 [label="b"]
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  fontsize=10
 +
  //label="DFA for (a|b)*"
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}
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</graph>
 +
 +
Given the minimization tree to the right, the final minimal DFA is:
 +
<graph>
 +
digraph dfamin {
 +
    { node [shape=circle style=invis] start }
 +
  rankdir=LR; ratio=0.5
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  node [shape=doublecircle,fixedsize=true,width=0.4,fontsize=10]; 012
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  node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
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  start -> 012
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  012 -> 012 [label="a"]
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  012 -> 012 [label="b"]
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  fontsize=10
 +
  //label="DFA for (a|b)*"
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}
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</graph>
 +
 +
! style="text-align: left; font-weight:normal; vertical-align: top; width: 50%;" | The minimization tree is as follows. As can be seen, the states are indistinguishable.
 +
 +
<graph>
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digraph mintree {
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  node [shape=none,fixedsize=true,width=0.2,fontsize=10]
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  " {0, 1, 2}" -> "{}" [label="NF",fontsize=10]
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  " {0, 1, 2}" -> "{0, 1, 2}" [label="F",fontsize=10]
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  "{0, 1, 2}" -> "{0, 1, 2} " [label="a,b",fontsize=10]
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  fontsize=10
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  //label="Minimization tree"
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}
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</graph>
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|}
  
 
[[category:Teaching]]
 
[[category:Teaching]]
 
[[category:Compilers]]
 
[[category:Compilers]]
 
[[en:Theoretical Aspects of Lexical Analysis]]
 
[[en:Theoretical Aspects of Lexical Analysis]]

Revision as of 02:08, 22 March 2009

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

  • (a*|b*)*

Solution

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, 3, 5, 6, 7, 9, 10, 11 0
0 a 4 1, 2, 3, 4, 5, 6, 7, 9, 10, 11 1
0 b 8 1, 2, 3, 5, 6, 7, 8, 9, 10, 11 2
1 a 4 1, 2, 3, 4, 5, 6, 7, 9, 10, 11 1
1 b 8 1, 2, 3, 5, 6, 7, 8, 9, 10, 11 2
2 a 4 1, 2, 3, 4, 5, 6, 7, 9, 10, 11 1
2 b 8 1, 2, 3, 5, 6, 7, 8, 9, 10, 11 2


Graphically, the DFA is represented as follows:


Given the minimization tree to the right, the final minimal DFA is:


The minimization tree is as follows. As can be seen, the states are indistinguishable.


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