Theoretical Aspects of Lexical Analysis/Exercise 2
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Problem
Use Thompson's algorithm to build the NFA for the following regular expression. Build the corresponding DFA and minimize it.
- (a*|b*)*
Solution
The non-deterministic finite automaton (NFA), built by applying Thompson's algorithm to the regular expression (a*|b*)* is the following:
Applying the determination algorithm to the above NFA, the following determination table is obtained:
| In | α∈Σ | move(In, α) | ε-closure(move(In, α)) | In+1 = ε-closure(move(In, α)) |
|---|---|---|---|---|
| - | - | 0 | 0, 1, 2, 3, 5, 6, 7, 9, 10, 11 | 0 |
| 0 | a | 4 | 1, 2, 3, 4, 5, 6, 7, 9, 10, 11 | 1 |
| 0 | b | 8 | 1, 2, 3, 5, 6, 7, 8, 9, 10, 11 | 2 |
| 1 | a | 4 | 1, 2, 3, 4, 5, 6, 7, 9, 10, 11 | 1 |
| 1 | b | 8 | 1, 2, 3, 5, 6, 7, 8, 9, 10, 11 | 2 |
| 2 | a | 4 | 1, 2, 3, 4, 5, 6, 7, 9, 10, 11 | 1 |
| 2 | b | 8 | 1, 2, 3, 5, 6, 7, 8, 9, 10, 11 | 2 |
Graphically, the DFA is represented as follows:
 Given the minimization tree to the right, the final minimal DFA is:  |
The minimization tree is as follows. As can be seen, the states are indistinguishable.
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