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The input string ''aababb'' is, after 10 steps, split into three tokens: '''T2''' (corresponding to lexeme ''aa''), '''T1''' (''b''), '''T1''' (''a''), and '''T1''' (''bb''). | The input string ''aababb'' is, after 10 steps, split into three tokens: '''T2''' (corresponding to lexeme ''aa''), '''T1''' (''b''), '''T1''' (''a''), and '''T1''' (''bb''). | ||
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[[en:Theoretical Aspects of Lexical Analysis]] | [[en:Theoretical Aspects of Lexical Analysis]] | ||
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 8 recognizes the first expression (token T1); state 12 recognizes token T2; and state 20 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, 2, 4, 5, 7, 8, 9, 10, 12, 13, 14, 16, 17, 19, 20 | 0 (T1) |
| 0 | a | 3, 11, 18 | 3, 8, 10, 11, 12, 17, 18, 19, 20 | 1 (T1) |
| 0 | b | 6, 15 | 5, 6, 7, 8, 15, 20 | 2 (T1) |
| 1 | a | 11, 18 | 10, 11, 12, 17, 18, 19, 20 | 3 (T2) |
| 1 | b | - | - | - |
| 2 | a | - | - | - |
| 2 | b | 6 | 5, 6, 7, 8 | 4 (T1) |
| 3 | a | 11, 18 | 10, 11, 12, 17, 18, 19, 20 | 3 (T2) |
| 3 | b | - | - | - |
| 4 | a | - | - | - |
| 4 | b | 6 | 5, 6, 7, 8 | 4 (T1) |
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 (transitions for super-state {2,4}).
Given the minimization tree, the final minimal DFA is as follows. Note that states 1 and 3 cannot be the same since they recognize different tokens.
| In | Input | In+1 / Token |
|---|---|---|
| 0 | aababb$ | 1 |
| 1 | ababb$ | 3 |
| 3 | babb$ | T2 (aa) |
| 0 | babb$ | 24 |
| 24 | abb$ | T1 (b) |
| 0 | abb$ | 1 |
| 1 | bb$ | T1 (a) |
| 0 | bb$ | 24 |
| 24 | b$ | 24 |
| 24 | $ | T1 (bb) |
The input string aababb is, after 10 steps, split into three tokens: T2 (corresponding to lexeme aa), T1 (b), T1 (a), and T1 (bb).