Theoretical Aspects of Lexical Analysis/Exercise 1: Difference between revisions
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__NOTOC__ | __NOTOC__ | ||
<div class="section-container auto" data-section> | <div class="section-container auto" data-section> | ||
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The non-deterministic finite automaton (NFA), built by applying Thompson's algorithm to the regular expression '''<nowiki>(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)*</nowiki>''' is the following: | ||
{{CollapsedCode|NFA for <nowiki>(a|b)*</nowiki>| | {{CollapsedCode|NFA for <nowiki>(a|b)*</nowiki>| | ||
< | <kroki lang="graphviz"> | ||
digraph nfa { | digraph nfa { | ||
{ node [shape=circle style=invis] start } | { node [shape=circle style=invis] start } | ||
| Line 35: | Line 36: | ||
0 -> 7 | 0 -> 7 | ||
fontsize=10 | fontsize=10 | ||
} | } | ||
</ | </kroki> | ||
}} | }} | ||
| Line 93: | Line 93: | ||
{| width="100%" | {| width="100%" | ||
! style="text-align: left; font-weight:normal; vertical-align: top; width: 50%;" |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: | ||
< | <kroki lang="graphviz"> | ||
digraph dfa { | digraph dfa { | ||
{ node [shape=circle style=invis] start } | { node [shape=circle style=invis] start } | ||
| Line 107: | Line 107: | ||
2 -> 2 [label="b"] | 2 -> 2 [label="b"] | ||
fontsize=10 | fontsize=10 | ||
} | } | ||
</ | </kroki> | ||
Given the minimization tree to the right, the final minimal DFA is: | Given the minimization tree to the right, the final minimal DFA is: | ||
< | <kroki lang="graphviz"> | ||
digraph dfamin { | digraph dfamin { | ||
{ node [shape=circle style=invis] start } | { node [shape=circle style=invis] start } | ||
| Line 122: | Line 121: | ||
012 -> 012 [label="b"] | 012 -> 012 [label="b"] | ||
fontsize=10 | fontsize=10 | ||
/ | /*label="DFA for (a|b)*"*/ | ||
} | } | ||
</ | </kroki> | ||
! 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. | ! 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. | ||
< | <kroki lang="graphviz"> | ||
digraph mintree { | digraph mintree { | ||
node [shape=none,fixedsize=true,width=0.2,fontsize=10] | node [shape=none,fixedsize=true,width=0.2,fontsize=10] | ||
| Line 135: | Line 134: | ||
"{0, 1, 2}" -> "{0, 1, 2} " [label="a,b",fontsize=10] | "{0, 1, 2}" -> "{0, 1, 2} " [label="a,b",fontsize=10] | ||
fontsize=10 | fontsize=10 | ||
/ | /*label="Minimization tree"*/ | ||
} | } | ||
</ | </kroki> | ||
|} | |} | ||
<!-- ====================== END OF SOLUTION ====================== --> | <!-- ====================== END OF SOLUTION ====================== --> | ||
Latest revision as of 18:22, 26 April 2026
Problem
Use Thompson's algorithm to build the NFA for the following regular expression. Build the corresponding DFA and minimize it.
- (a|b)*
Solution
The non-deterministic finite automaton (NFA), built by applying Thompson's algorithm to the regular expression (a|b)* is the following:
| NFA for (a|b)* |
|---|
HTTP-Response:
error code: 1015
Diagram-Code:
digraph nfa {
{ node [shape=circle style=invis] start }
rankdir=LR; ratio=0.5
node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 7
node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
start -> 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
fontsize=10
}
|
Applying the determination algorithm to the above NFA, the following determination table is obtained:
| 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 | 1, 2, 3, 4, 6, 7 | 1 |
| 0 | b | 5 | 1, 2, 4, 5, 6, 7 | 2 |
| 1 | a | 3 | 1, 2, 3, 4, 6, 7 | 1 |
| 1 | b | 5 | 1, 2, 4, 5, 6, 7 | 2 |
| 2 | a | 3 | 1, 2, 3, 4, 6, 7 | 1 |
| 2 | b | 5 | 1, 2, 4, 5, 6, 7 | 2 |
Graphically, the DFA is represented as follows:
HTTP-Response:
error code: 1015
Diagram-Code:
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];
start -> 0
0 -> 1 [label="a"]
0 -> 2 [label="b"]
1 -> 1 [label="a"]
1 -> 2 [label="b"]
2 -> 1 [label="a"]
2 -> 2 [label="b"]
fontsize=10
}
Given the minimization tree to the right, the final minimal DFA is: HTTP-Response:
error code: 1015
Diagram-Code:
digraph dfamin {
{ node [shape=circle style=invis] start }
rankdir=LR; ratio=0.5
node [shape=doublecircle,fixedsize=true,width=0.4,fontsize=10]; 012
node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
start -> 012
012 -> 012 [label="a"]
012 -> 012 [label="b"]
fontsize=10
/*label="DFA for (a|b)*"*/
}
|
The minimization tree is as follows. As can be seen, the states are indistinguishable.
HTTP-Response:
error code: 1015
Diagram-Code:
digraph mintree {
node [shape=none,fixedsize=true,width=0.2,fontsize=10]
" {0, 1, 2}" -> "{}" [label="NF",fontsize=10]
" {0, 1, 2}" -> "{0, 1, 2}" [label="F",fontsize=10]
"{0, 1, 2}" -> "{0, 1, 2} " [label="a,b",fontsize=10]
fontsize=10
/*label="Minimization tree"*/
}
|
|---|