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The input string ''aaabaaaaa'' is, after 16 steps, split into three tokens: '''T1''' (corresponding to lexeme ''aa''), '''T3''' (''a''), '''T3''' (''b''), '''T2''' (''aaaa''), and '''T3''' (''a''). | The input string ''aaabaaaaa'' is, after 16 steps, split into three tokens: '''T1''' (corresponding to lexeme ''aa''), '''T3''' (''a''), '''T3''' (''b''), '''T2''' (''aaaa''), and '''T3''' (''a''). | ||
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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 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, 11, 14 | 1 (T3) |
| 0 | b | 13 | 13, 14 | 2 (T3) |
| 1 | a | 3, 6 | 3, 6 | 3 (T1) |
| 1 | b | - | - | - |
| 2 | a | - | - | - |
| 2 | b | - | - | - |
| 3 | a | 7 | 7 | 4 |
| 3 | b | - | - | - |
| 4 | a | 8 | 8 | 5 (T2) |
| 4 | b | - | - | - |
| 5 | a | - | - | - |
| 5 | b | - | - | - |
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 sets {0, 4} and {1, 2} was only tested for "a" (sufficient).
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 | aaabaaaaa$ | 1 |
| 1 | aabaaaaa$ | 3 |
| 3 | abaaaaa$ | 4 |
| 4 | baaaaa$ | error (backtracking) |
| 3 | abaaaaa$ | T1 (aa) |
| 0 | abaaaaa$ | 1 |
| 1 | baaaaa$ | T3 (a) |
| 0 | baaaaa$ | 2 |
| 2 | aaaaa$ | T3 (b) |
| 0 | aaaaa$ | 1 |
| 1 | aaaa$ | 3 |
| 3 | aaa$ | 4 |
| 4 | aa$ | 5 |
| 5 | a$ | T2 (aaaa) |
| 0 | a$ | 1 |
| 1 | $ | T3 (a) |
The input string aaabaaaaa is, after 16 steps, split into three tokens: T1 (corresponding to lexeme aa), T3 (a), T3 (b), T2 (aaaa), and T3 (a).