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.
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).