## Theoretical Aspects of Lexical Analysis/Exercise 12

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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.

• G = { b*|a, a*, b|a* }, input string = aababb

## NFA

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.

## DFA

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.

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.

## Input Analysis

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).