(→NFA) |
(→NFA) |
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Line 45: | Line 45: | ||
<runphp> | <runphp> | ||
− | echo '<div id="mynetwork"></div><script type="text/javascript"> | + | echo '<div id="mynetwork"></div><script type="text/javascript">var container = document.getElementById("mynetwork"); var data = { dot: \'digraph nfa { |
− | var container = document.getElementById("mynetwork"); | + | { node [shape=circle style=invis] s } |
− | var data = { dot: \' | + | rankdir=LR; ratio=0.5 |
− | var network = new vis.Network(container, data);</script>'; | + | node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 3 8 14 |
+ | node [shape=circle,fixedsize=true,width=0.2,fontsize=10]; | ||
+ | |||
+ | s -> 0 | ||
+ | |||
+ | 0 -> 1 | ||
+ | 1 -> 2 [label="a",fontsize=10] | ||
+ | 2 -> 3 [label="b",fontsize=10] | ||
+ | |||
+ | 0 -> 4 | ||
+ | 4 -> 5 [label="a",fontsize=10] | ||
+ | 5 -> 6 | ||
+ | 5 -> 8 | ||
+ | 6 -> 7 [label="b",fontsize=10] | ||
+ | 7 -> 6 | ||
+ | 7 -> 8 | ||
+ | |||
+ | 0 -> 9 | ||
+ | 9 -> 10 | ||
+ | 9 -> 12 | ||
+ | 10 -> 11 [label="a",fontsize=10] | ||
+ | 12 -> 13 [label="b",fontsize=10] | ||
+ | 11 -> 14 | ||
+ | 13 -> 14 | ||
+ | fontsize=10 | ||
+ | /*label="NFA for (a|b)*abb(a|b)*"*/ | ||
+ | }\' }; var network = new vis.Network(container, data);</script>'; | ||
</runphp> | </runphp> | ||
Compute the non-deterministic finite automaton (NFA) by using Thompson's algorithm. Compute the minimal deterministic finite automaton (DFA).
The alphabet is Σ = { a, b }. Indicate the number of processing steps for the given input string.
The following is the result of applying Thompson's algorithm. State 3 recognizes the first expression (token T1); state 8 recognizes token T2; and state 14 recognizes token T3.
The following is the result of applying Thompson's algorithm. State 3 recognizes the first expression (token T1); state 8 recognizes token T2; and state 14 recognizes token T3.
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, 4, 9, 10, 12 | 0 |
0 | a | 2, 5, 11 | 2, 5, 6, 8, 11, 14 | 1 (T2) |
0 | b | 13 | 13, 14 | 2 (T3) |
1 | a | - | - | - |
1 | b | 3, 7 | 3, 6, 7, 8 | 3 (T1) |
2 | a | - | - | - |
2 | b | - | - | - |
3 | a | - | - | - |
3 | b | 7 | 6, 7, 8 | 4 (T2) |
4 | a | - | - | - |
4 | b | 7 | 6, 7, 8 | 4 (T2) |
Graphically, the DFA is represented as follows:
The minimization tree is as follows. Note that before considering transition behavior, states are split according to the token they recognize.
The tree expansion for non-splitting sets has been omitted for simplicity ("a" transition for final states {1, 4}).
Given the minimization tree, the final minimal DFA is exactly the same as the original DFA (all leaf sets are singular).
In | Input | In+1 / Token |
---|---|---|
0 | abaabb$ | 1 |
1 | baabb$ | 3 |
3 | aabb$ | T1 |
0 | aabb$ | 1 |
1 | abb$ | T2 |
0 | abb$ | 1 |
1 | bb$ | 3 |
3 | b$ | 4 |
4 | $ | T2 |
The input string abaabb is, after 9 steps, split into three tokens: T1 (corresponding to lexeme ab), T2 (a), and T2 (abb).