(→NFA) |
(→DFA) |
||
Line 43: | Line 43: | ||
== DFA == | == DFA == | ||
− | Determination table for the above NFA | + | {{CollapsedCode|Determination table for the above NFA| |
− | |||
{| cellspacing="2" | {| cellspacing="2" | ||
! style="padding-left: 20px; padding-right: 20px; background: wheat;" | I<sub>n</sub> | ! style="padding-left: 20px; padding-right: 20px; background: wheat;" | I<sub>n</sub> | ||
Line 131: | Line 130: | ||
|- | |- | ||
|} | |} | ||
+ | }} | ||
− | + | {{CollapsedCode|Graphical representation of the DFA| | |
− | |||
<graph> | <graph> | ||
digraph dfa { | digraph dfa { | ||
Line 150: | Line 149: | ||
} | } | ||
</graph> | </graph> | ||
+ | }} | ||
− | + | {{CollapsedCode|DFA minimization tree| | |
+ | Note that before considering transition behavior, states are split according to the token they recognize. | ||
<graph> | <graph> | ||
Line 173: | Line 174: | ||
Given the minimization tree, the final minimal DFA is exactly the same as the original DFA (all leaf sets are singular). | Given the minimization tree, the final minimal DFA is exactly the same as the original DFA (all leaf sets are singular). | ||
+ | }} | ||
== Input Analysis == | == Input 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.
NFA built by Thompson's algorithm |
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State 3 recognizes the first expression (token T1); state 8 recognizes token T2; and state 14 recognizes token T3.
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Determination table for the above NFA |
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{ |
Graphical representation of the DFA |
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|
DFA minimization tree |
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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).