| Line 10: | Line 10: | ||
<graph> | <graph> | ||
digraph nfa { | digraph nfa { | ||
| − | { node [shape=circle style=invis] | + | { node [shape=circle style=invis] s } |
rankdir=LR; ratio=0.5 | rankdir=LR; ratio=0.5 | ||
node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 17 | node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 17 | ||
node [shape=circle,fixedsize=true,width=0.2,fontsize=10]; | node [shape=circle,fixedsize=true,width=0.2,fontsize=10]; | ||
| − | + | s -> 0 | |
0 -> 1 | 0 -> 1 | ||
1 -> 2 | 1 -> 2 | ||
| Line 156: | Line 156: | ||
<graph> | <graph> | ||
digraph dfa { | digraph dfa { | ||
| − | { node [shape=circle style=invis] | + | { node [shape=circle style=invis] s } |
rankdir=LR; ratio=0.5 | rankdir=LR; ratio=0.5 | ||
node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 4 5 6 | node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 4 5 6 | ||
node [shape=circle,fixedsize=true,width=0.2,fontsize=10]; | node [shape=circle,fixedsize=true,width=0.2,fontsize=10]; | ||
| − | + | s -> 0 | |
| − | 0 -> 1 [label="a"] | + | 0 -> 1 [label="a",fontsize=10] |
| − | 0 -> 2 [label="b"] | + | 0 -> 2 [label="b",fontsize=10] |
| − | 1 -> 1 [label="a"] | + | 1 -> 1 [label="a",fontsize=10] |
| − | 1 -> 3 [label="b"] | + | 1 -> 3 [label="b",fontsize=10] |
| − | 2 -> 1 [label="a"] | + | 2 -> 1 [label="a",fontsize=10] |
| − | 2 -> 2 [label="b"] | + | 2 -> 2 [label="b",fontsize=10] |
| − | 3 -> 1 [label="a"] | + | 3 -> 1 [label="a",fontsize=10] |
| − | 3 -> 4 [label="b"] | + | 3 -> 4 [label="b",fontsize=10] |
| − | 4 -> 5 [label="a"] | + | 4 -> 5 [label="a",fontsize=10] |
| − | 4 -> 6 [label="b"] | + | 4 -> 6 [label="b",fontsize=10] |
| − | 5 -> 5 [label="a"] | + | 5 -> 5 [label="a",fontsize=10] |
| − | 5 -> 6 [label="b"] | + | 5 -> 6 [label="b",fontsize=10] |
| − | 6 -> 5 [label="a"] | + | 6 -> 5 [label="a",fontsize=10] |
| − | 6 -> 6 [label="b"] | + | 6 -> 6 [label="b",fontsize=10] |
fontsize=10 | fontsize=10 | ||
//label="DFA for (a|b)*abb(a|b)*" | //label="DFA for (a|b)*abb(a|b)*" | ||
| Line 183: | Line 183: | ||
<graph> | <graph> | ||
digraph dfamin { | digraph dfamin { | ||
| − | { node [shape=circle style=invis] | + | { node [shape=circle style=invis] s } |
rankdir=LR; ratio=0.5 | rankdir=LR; ratio=0.5 | ||
node [shape=doublecircle,fixedsize=true,width=0.4,fontsize=10]; 456 | node [shape=doublecircle,fixedsize=true,width=0.4,fontsize=10]; 456 | ||
| − | node [shape=circle,fixedsize=true,width=0. | + | node [shape=circle,fixedsize=true,width=0.3,fontsize=10]; |
| − | + | s -> 02 | |
| − | 02 -> 1 [label="a"] | + | 02 -> 1 [label="a",fontsize=10] |
| − | 02 -> 02 [label="b"] | + | 02 -> 02 [label="b",fontsize=10] |
| − | 1 -> 1 [label="a"] | + | 1 -> 1 [label="a",fontsize=10] |
| − | 1 -> 3 [label="b"] | + | 1 -> 3 [label="b",fontsize=10] |
| − | 3 -> 1 [label="a"] | + | 3 -> 1 [label="a",fontsize=10] |
| − | 3 -> 456 [label="b"] | + | 3 -> 456 [label="b",fontsize=10] |
| − | 456 -> 456 [label="a"] | + | 456 -> 456 [label="a",fontsize=10] |
| − | 456 -> 456 [label="b"] | + | 456 -> 456 [label="b",fontsize=10] |
fontsize=10 | fontsize=10 | ||
//label="DFA for (a|b)*abb(a|b)*" | //label="DFA for (a|b)*abb(a|b)*" | ||
Use Thompson's algorithm to build the NFA for the following regular expression. Build the corresponding DFA and minimize it.
The following is the result of applying Thompson's algorithm.
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, 15 | 1, 2, 4, 5, 6, 7, 11, 12, 14, 15, 16, 17 | 6 |
| 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 |
| 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.
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. |
|---|