(→NFA) |
(→NFA) |
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<runphp> | <runphp> | ||
− | echo '<nowiki><div id="mynetwork2" style="height: 600px;"></div><script type="text/javascript">var container = document.getElementById("mynetwork2"); var nodes = [ {id: "s", label: "s"}, {id: 0, label: "0"}, {id: 1, label: "1"}, {id: 2, label: "2"}, {id: 3, label: "3final"}, {id: 4, label: "4"}, {id: 5, label: "5"}, {id: 6, label: "6"}, {id: 7, label: "7"}, {id: 8, label: "8final"}, {id: 9, label: "9"}, {id: 10, label: "10"}, {id: 11, label: "11"}, {id: 12, label: "12"}, {id: 13, label: "13"}, {id: 14, label: "14final"} ]; var edges = [ {from: "s", to: 0}, {from: 0, to: 1}, {from: 1, to: 2, label: "a" }, {from: 2, to: 3, label: "b" }, {from: 0, to: 4}, {from: 4, to: 5, label: "a" }, {from: 5, to: 6}, {from: 5, to: 8}, {from: 6, to: 7, label: "b" }, {from: 7, to: 6}, {from: 7, to: 8}, {from: 0, to: 9}, {from: 9, to: 10}, {from: 9, to: 12}, {from: 10, to: 11, label: "a" }, {from: 12, to: 13, label: "b" }, {from: 11, to: 14}, {from: 13, to: 14}, ]; var options = { nodes: { color: { background: "white", border: "#2B7CE9" }, shape: "circle", radius: 30 }, | + | echo '<nowiki><div id="mynetwork2" style="height: 600px;"></div><script type="text/javascript">var container = document.getElementById("mynetwork2"); var nodes = [ {id: "s", label: "s"}, {id: 0, label: "0"}, {id: 1, label: "1"}, {id: 2, label: "2"}, {id: 3, label: "3final"}, {id: 4, label: "4"}, {id: 5, label: "5"}, {id: 6, label: "6"}, {id: 7, label: "7"}, {id: 8, label: "8final"}, {id: 9, label: "9"}, {id: 10, label: "10"}, {id: 11, label: "11"}, {id: 12, label: "12"}, {id: 13, label: "13"}, {id: 14, label: "14final"} ]; var edges = [ {from: "s", to: 0}, {from: 0, to: 1}, {from: 1, to: 2, label: "a" }, {from: 2, to: 3, label: "b" }, {from: 0, to: 4}, {from: 4, to: 5, label: "a" }, {from: 5, to: 6}, {from: 5, to: 8}, {from: 6, to: 7, label: "b" }, {from: 7, to: 6}, {from: 7, to: 8}, {from: 0, to: 9}, {from: 9, to: 10}, {from: 9, to: 12}, {from: 10, to: 11, label: "a" }, {from: 12, to: 13, label: "b" }, {from: 11, to: 14}, {from: 13, to: 14}, ]; var options = { nodes: { color: { background: "white", border: "#2B7CE9" }, shape: "circle", radius: 30 }, }; var data = { nodes: nodes, edges: edges }; var network = new vis.Network(container, data, options);</script></nowiki>'; |
</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.
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).