(→NFA) |
(→NFA) |
||
Line 56: | Line 56: | ||
var nodes = [ | var nodes = [ | ||
− | {id: | + | {id: "s", label: "s"}, |
{id: 0, label: "0"}, | {id: 0, label: "0"}, | ||
{id: 1, label: "1"}, | {id: 1, label: "1"}, | ||
Line 74: | Line 74: | ||
]; | ]; | ||
var edges = [ | var edges = [ | ||
− | {from: 1, to: 2}, | + | {from: "s", to: 0}, |
− | {from: | + | {from: 0, to: 1}, |
− | {from: | + | {from: 1, to: 2, label: "a" }, |
− | {from: | + | {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 }, hierarchicalLayout: { layout: "direction" } }; var data = { | + | var options = { nodes: { color: { background: "white", border: "#2B7CE9" }, shape: "circle", radius: 30 }, hierarchicalLayout: { layout: "direction" } }; var data = { nodes: nodes, edges: edges }; var network = new vis.Network(container, data, options);</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.
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).