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

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< Theoretical Aspects of Lexical Analysis
(New page: Use Thompson's algorithm to build the NFA for the following regular expression. Build the corresponding DFA and minimize it. * <nowiki>(a|b)*abb(a|b)*</nowiki> == Solution == [[category...)
 
 
(24 intermediate revisions by the same user not shown)
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−
Use Thompson's algorithm to build the NFA for the following regular expression. Build the corresponding DFA and minimize it.  
+
{{TOCright}}
−
* <nowiki>(a|b)*abb(a|b)*</nowiki>
+
 
 +
==Problem ==
 +
Use Thompson's algorithm to build the NFA for the following regular expression. Build the corresponding DFA and minimize it.
 +
 
 +
* '''<nowiki>(a|b)*abb(a|b)*</nowiki>'''
  
 
== Solution ==
 
== Solution ==
  
−
[[category:Teaching]]
+
The non-deterministic finite automaton (NFA), built by applying Thompson's algorithm to the regular expression '''<nowiki>(a|b)*abb(a|b)*</nowiki>''' is the following:
−
[[category:Compilers]]
+
{{CollapsedCode|NFA|
 +
<dot-hack>
 +
digraph nfa {
 +
    { node [shape=circle style=invis] s }
 +
  rankdir=LR; ratio=0.5
 +
  node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 17
 +
  node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
 +
 
 +
  s -> 0
 +
  0 -> 1
 +
  1 -> 2
 +
  1 -> 4
 +
  2 -> 3 [label="a",fontsize=10]
 +
  4 -> 5 [label="b",fontsize=10]
 +
  3 -> 6
 +
  5 -> 6
 +
  6 -> 1
 +
  6 -> 7
 +
  0 -> 7
 +
 
 +
  7 -> 8 [label="a",fontsize=10]
 +
  8 -> 9 [label="b",fontsize=10]
 +
  9 -> 10 [label="b",fontsize=10]
 +
 
 +
  10 -> 11
 +
  11 -> 12
 +
  11 -> 14
 +
  12 -> 13 [label="a",fontsize=10]
 +
  14 -> 15 [label="b",fontsize=10]
 +
  13 -> 16
 +
  15 -> 16
 +
  16 -> 11
 +
  16 -> 17
 +
  10 -> 17
 +
 
 +
  fontsize=10
 +
}
 +
</dot-hack>
 +
}}
 +
 
 +
Applying the determination algorithm to the above NFA, the following determination table is obtained:
 +
 
 +
{{CollapsedCode|Determination table|
 +
<runphp>
 +
echo<<<___EOF___
 +
<table border="1" cellspacing="0"><colgroup span="3" width="84"></colgroup> <colgroup width="237"></colgroup> <colgroup width="84"></colgroup>
 +
<tbody>
 +
<tr>
 +
<td align="center" bgcolor="#FFCC99" height="44"><strong><span style="font-family: Arial;">In</span></strong></td>
 +
<td align="center" bgcolor="#FFCC99"><strong><span style="font-family: Arial;">&alpha;&isin;&Sigma;</span></strong></td>
 +
<td align="center" bgcolor="#FFCC99"><strong><span style="font-family: Arial;">move(In, &alpha;)</span></strong></td>
 +
<td align="center" bgcolor="#FFCC99"><strong><span style="font-family: Arial;">&epsilon;-closure(move(In, &alpha;))</span></strong></td>
 +
<td align="center" bgcolor="#FFCC99"><strong><span style="font-family: Arial;">In+1&nbsp;= &epsilon;-closure(move(In, &alpha;))</span></strong></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#FFFFCC" height="17"><strong><span style="font-family: Arial;">-</span></strong></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">-</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">0</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">0, 1, 2, 4, 7</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">0</span></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#F5F5F5" height="17"><span style="font-family: Arial;">0</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">a</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">3, 8</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1, 2, 3, 4, 6, 7, 8</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1</span></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#F5F5F5" height="17"><span style="font-family: Arial;">0</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">b</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">5</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1, 2, 4, 5, 6, 7</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">2</span></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#FFFFCC" height="17"><span style="font-family: Arial;">1</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">a</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">3, 8</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">1, 2, 3, 4, 6, 7, 8</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">1</span></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#FFFFCC" height="17"><span style="font-family: Arial;">1</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">b</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">5, 9</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">1, 2, 4, 5, 6, 7, 9</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">3</span></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#F5F5F5" height="17"><span style="font-family: Arial;">2</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">a</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">3, 8</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1, 2, 3, 4, 6, 7, 8</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1</span></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#F5F5F5" height="17"><span style="font-family: Arial;">2</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">b</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">5</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1, 2, 4, 5, 6, 7</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">2</span></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#FFFFCC" height="17"><span style="font-family: Arial;">3</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">a</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">3, 8</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">1, 2, 3, 4, 6, 7, 8</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">1</span></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#FFFFCC" height="17"><span style="font-family: Arial;">3</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">b</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">5, 10</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">1, 2, 4, 5, 6, 7, 10, 11, 12, 14,&nbsp;17</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><strong><span style="font-family: Arial;">4</span></strong></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#F5F5F5" height="17"><strong><span style="font-family: Arial;">4</span></strong></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">a</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">3, 8, 13</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16,&nbsp;17</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><strong><span style="font-family: Arial;">5</span></strong></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#F5F5F5" height="17"><strong><span style="font-family: Arial;">4</span></strong></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">b</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">5, 15</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1, 2, 4, 5, 6, 7, 11, 12, 14, 15, 16,&nbsp;17</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><strong><span style="font-family: Arial;">6</span></strong></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#FFFFCC" height="17"><strong><span style="font-family: Arial;">5</span></strong></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">a</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">3, 8, 13</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16,&nbsp;17</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><strong><span style="font-family: Arial;">5</span></strong></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#FFFFCC" height="17"><strong><span style="font-family: Arial;">5</span></strong></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">b</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">5, 9, 15</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">1, 2, 4, 5, 6, 7, 9, 11, 12, 14, 15, 16,&nbsp;17</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><strong><span style="font-family: Arial;">7</span></strong></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#F5F5F5" height="17"><strong><span style="font-family: Arial;">6</span></strong></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">a</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">3, 8, 13</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16,&nbsp;17</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><strong><span style="font-family: Arial;">5</span></strong></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#F5F5F5" height="17"><strong><span style="font-family: Arial;">6</span></strong></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">b</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">5, 15</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1, 2, 4, 5, 6, 7, 11, 12, 14, 15, 16,&nbsp;17</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><strong><span style="font-family: Arial;">6</span></strong></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#FFFFCC" height="17"><strong><span style="font-family: Arial;">7</span></strong></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">a</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">3, 8, 13</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16,&nbsp;17</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><strong><span style="font-family: Arial;">5</span></strong></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#FFFFCC" height="17"><strong><span style="font-family: Arial;">7</span></strong></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">b</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">5, 10, 15</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><span style="font-family: Arial;">1, 2, 4, 5, 6, 7, 10, 11, 12, 14, 15, 16,&nbsp;17</span></td>
 +
<td align="center" bgcolor="#FFFFCC"><strong><span style="font-family: Arial;">8</span></strong></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#F5F5F5" height="17"><strong><span style="font-family: Arial;">8</span></strong></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">a</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">3, 8, 13</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16,&nbsp;17</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><strong><span style="font-family: Arial;">5</span></strong></td>
 +
</tr>
 +
<tr>
 +
<td align="center" bgcolor="#F5F5F5" height="17"><strong><span style="font-family: Arial;">8</span></strong></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">b</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">5, 15</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><span style="font-family: Arial;">1, 2, 4, 5, 6, 7, 11, 12, 14, 15, 16,&nbsp;17</span></td>
 +
<td align="center" bgcolor="#F5F5F5"><strong><span style="font-family: Arial;">6</span></strong></td>
 +
</tr>
 +
</tbody>
 +
</table>
 +
___EOF___;
 +
</runphp>
 +
}}
 +
 
 +
Graphically, the DFA is represented as follows:
 +
 
 +
{{CollapsedCode|DFA|
 +
<dot-hack>
 +
digraph dfa {
 +
    { node [shape=circle style=invis] s }
 +
  rankdir=LR; ratio=0.5
 +
  node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 4 5 6 7 8
 +
  node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
 +
  s -> 0
 +
  0 -> 1 [label="a",fontsize=10]
 +
  0 -> 2 [label="b",fontsize=10]
 +
  1 -> 1  [label="a",fontsize=10]
 +
  1 -> 3  [label="b",fontsize=10]
 +
  2 -> 1 [label="a",fontsize=10]
 +
  2 -> 2 [label="b",fontsize=10]
 +
  3 -> 1 [label="a",fontsize=10]
 +
  3 -> 4 [label="b",fontsize=10]
 +
  4 -> 5 [label="a",fontsize=10]
 +
  4 -> 6 [label="b",fontsize=10]
 +
  5 -> 5 [label="a",fontsize=10]
 +
  5 -> 7 [label="b",fontsize=10]
 +
  6 -> 5 [label="a",fontsize=10]
 +
  6 -> 6 [label="b",fontsize=10]
 +
  7 -> 5 [label="a",fontsize=10]
 +
  7 -> 8 [label="b",fontsize=10]
 +
  8 -> 5 [label="a",fontsize=10]
 +
  8 -> 6 [label="b",fontsize=10]
 +
  fontsize=10
 +
}
 +
</dot-hack>
 +
}}
 +
The minimization tree is as follows:
 +
 
 +
{{CollapsedCode|Minimization tree|
 +
[[image:aula3p4mintree.jpg|350px]]
 +
}}
 +
 
 +
<!--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 both "a" or "b" transitions (remember that, at this stage, we assume that individual states -- i.e., the final states -- are all indistinguishable).
 +
-->
 +
Given the minimization tree above, the final minimal DFA is as follows:
 +
 
 +
{{CollapsedCode|Minimal DFA|
 +
<dot-hack>
 +
digraph dfamin {
 +
    { node [shape=circle style=invis] s }
 +
  rankdir=LR; ratio=0.5
 +
  node [shape=doublecircle,fixedsize=true,width=0.4,fontsize=10]; 45678
 +
  node [shape=circle,fixedsize=true,width=0.3,fontsize=10];
 +
  s -> 02
 +
  02 -> 1 [label="a",fontsize=10]
 +
  02 -> 02 [label="b",fontsize=10]
 +
  1 -> 1  [label="a",fontsize=10]
 +
  1 -> 3  [label="b",fontsize=10]
 +
  3 -> 1 [label="a",fontsize=10]
 +
  3 -> 45678 [label="b",fontsize=10]
 +
  45678 -> 45678 [label="a",fontsize=10]
 +
  45678 -> 45678 [label="b",fontsize=10]
 +
  fontsize=10
 +
}
 +
</dot-hack>
 +
}}
 +
 
 +
[[category:Compiladores]]
 +
[[category:Ensino]]
 +
 
 
[[en:Theoretical Aspects of Lexical Analysis]]
 
[[en:Theoretical Aspects of Lexical Analysis]]

Latest revision as of 22:44, 11 February 2019

Problem

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)*

Solution

The non-deterministic finite automaton (NFA), built by applying Thompson's algorithm to the regular expression (a|b)*abb(a|b)* is the following:

NFA

Applying the determination algorithm to the above NFA, the following determination table is obtained:

Determination table
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:

DFA

The minimization tree is as follows:

Minimization tree

Aula3p4mintree.jpg

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

Minimal DFA
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