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− | + | ==Problem == | |
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Use Thompson's algorithm to build the NFA for the following regular expression. Build the corresponding DFA and minimize it. | 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>''' | * '''<nowiki>(a|b)*abb(a|b)*</nowiki>''' | ||
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− | + | == Solution == | |
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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: | 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: | ||
− | < | + | {{CollapsedCode|NFA| |
+ | <dot-hack> | ||
digraph nfa { | digraph nfa { | ||
{ node [shape=circle style=invis] s } | { node [shape=circle style=invis] s } | ||
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fontsize=10 | fontsize=10 | ||
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} | } | ||
− | </ | + | </dot-hack> |
+ | }} | ||
Applying the determination algorithm to the above NFA, the following determination table is obtained: | Applying the determination algorithm to the above NFA, the following determination table is obtained: | ||
− | {| cellspacing=" | + | {{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;">α∈Σ</span></strong></td> | |
− | + | <td align="center" bgcolor="#FFCC99"><strong><span style="font-family: Arial;">move(In, α)</span></strong></td> | |
− | + | <td align="center" bgcolor="#FFCC99"><strong><span style="font-family: Arial;">ε-closure(move(In, α))</span></strong></td> | |
− | + | <td align="center" bgcolor="#FFCC99"><strong><span style="font-family: Arial;">In+1 = ε-closure(move(In, α))</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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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: | Graphically, the DFA is represented as follows: | ||
− | < | + | {{CollapsedCode|DFA| |
+ | <dot-hack> | ||
digraph dfa { | digraph dfa { | ||
{ node [shape=circle style=invis] s } | { node [shape=circle style=invis] s } | ||
Line 207: | Line 231: | ||
8 -> 6 [label="b",fontsize=10] | 8 -> 6 [label="b",fontsize=10] | ||
fontsize=10 | fontsize=10 | ||
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} | } | ||
− | </ | + | </dot-hack> |
− | + | }} | |
The minimization tree is as follows: | The minimization tree is as follows: | ||
− | + | {{CollapsedCode|Minimization tree| | |
− | + | [[image:aula3p4mintree.jpg|350px]] | |
− | + | }} | |
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− | } | ||
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+ | <!--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: | Given the minimization tree above, the final minimal DFA is as follows: | ||
− | < | + | {{CollapsedCode|Minimal DFA| |
+ | <dot-hack> | ||
digraph dfamin { | digraph dfamin { | ||
{ node [shape=circle style=invis] s } | { node [shape=circle style=invis] s } | ||
Line 247: | Line 261: | ||
45678 -> 45678 [label="b",fontsize=10] | 45678 -> 45678 [label="b",fontsize=10] | ||
fontsize=10 | fontsize=10 | ||
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} | } | ||
− | </ | + | </dot-hack> |
− | + | }} | |
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[[category:Compiladores]] | [[category:Compiladores]] |
Contents |
Use Thompson's algorithm to build the NFA for the following regular expression. Build the corresponding DFA and minimize it.
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 |
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Applying the determination algorithm to the above NFA, the following determination table is obtained:
Determination table | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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Graphically, the DFA is represented as follows:
DFA |
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The minimization tree is as follows:
Minimization tree |
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Given the minimization tree above, the final minimal DFA is as follows:
Minimal DFA |
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