(→Input Analysis) |
|||
Line 180: | Line 180: | ||
! style="font-weight: normal; align: center; background: #ffffcc;" | 0 | ! style="font-weight: normal; align: center; background: #ffffcc;" | 0 | ||
! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>aaabaaaaa$</tt> | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>aaabaaaaa$</tt> | ||
− | ! style="font-weight: normal; align: center; background: #ffffcc;" | | + | ! style="font-weight: normal; align: center; background: #ffffcc;" | 1 |
|- | |- | ||
− | ! style="font-weight: normal; align: center; background: #ffffcc;" | | + | ! style="font-weight: normal; align: center; background: #ffffcc;" | 1 |
− | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>$</tt> | + | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>aabaaaaa$</tt> |
− | ! style="font-weight: normal; align: center; background: #ffffcc;" | | + | ! style="font-weight: normal; align: center; background: #ffffcc;" | 3 |
|- | |- | ||
− | ! style="font-weight: normal; align: center; background: #ffffcc;" | | + | ! style="font-weight: normal; align: center; background: #ffffcc;" | 3 |
− | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>$</tt> | + | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>abaaaaa$</tt> |
− | ! style="font-weight: normal; align: center; background: #ffffcc;" | | + | ! style="font-weight: normal; align: center; background: #ffffcc;" | 4 |
|- | |- | ||
− | ! style="font-weight: normal; align: center; background: # | + | ! style="font-weight: normal; align: center; background: #e6bbaf;" | 4 |
− | ! style="font-weight: normal; text-align: right; background: # | + | ! style="font-weight: normal; text-align: right; background: #e6bbaf;" | <tt>baaaaa$</tt> |
− | ! style="font-weight: normal; align: center; background: # | + | ! style="font-weight: normal; align: center; background: #e6bbaf;" | error (backtracking) |
|- | |- | ||
− | ! style="font-weight: normal; align: center; background: # | + | ! style="font-weight: normal; align: center; background: #ffffcc;" | 3 |
− | ! style="font-weight: normal; text-align: right; background: # | + | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>abaaaaa$</tt> |
− | ! style="font-weight: normal; align: center; background: # | + | ! style="font-weight: normal; align: center; background: #ffffcc;" | '''T1''' (aa) |
|- | |- | ||
− | ! style="font-weight: normal; align: center; background: #ffffcc;" | | + | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 0 |
− | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>$</tt> | + | ! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>abaaaaa$</tt> |
− | ! style="font-weight: normal; align: center; background: #ffffcc;" | | + | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 1 |
+ | |- | ||
+ | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 1 | ||
+ | ! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>baaaaa$</tt> | ||
+ | ! style="font-weight: normal; align: center; background: #e6e6e6;" | '''T3''' (a) | ||
+ | |- | ||
+ | ! style="font-weight: normal; align: center; background: #ffffcc;" | 0 | ||
+ | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>baaaaa$</tt> | ||
+ | ! style="font-weight: normal; align: center; background: #ffffcc;" | 2 | ||
+ | |- | ||
+ | ! style="font-weight: normal; align: center; background: #ffffcc;" | 2 | ||
+ | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>aaaaa$</tt> | ||
+ | ! style="font-weight: normal; align: center; background: #ffffcc;" | '''T3''' (b) | ||
+ | |- | ||
+ | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 0 | ||
+ | ! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>aaaaa$</tt> | ||
+ | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 1 | ||
+ | |- | ||
+ | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 1 | ||
+ | ! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>aaaa$</tt> | ||
+ | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 3 | ||
+ | |- | ||
+ | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 3 | ||
+ | ! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>aaa$</tt> | ||
+ | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 4 | ||
+ | |- | ||
+ | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 4 | ||
+ | ! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>aa$</tt> | ||
+ | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 5 | ||
|- | |- | ||
− | ! style="font-weight: normal; align: center; background: # | + | ! style="font-weight: normal; align: center; background: #e6e6e6;" | 5 |
− | ! style="font-weight: normal; text-align: right; background: # | + | ! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>a$</tt> |
− | ! style="font-weight: normal; align: center; background: # | + | ! style="font-weight: normal; align: center; background: #e6e6e6;" | '''T2''' (aaaa) |
|- | |- | ||
− | ! style="font-weight: normal; align: center; background: #ffffcc;" | | + | ! style="font-weight: normal; align: center; background: #ffffcc;" | 0 |
− | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>$</tt> | + | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>a$</tt> |
− | ! style="font-weight: normal; align: center; background: #ffffcc;" | | + | ! style="font-weight: normal; align: center; background: #ffffcc;" | 1 |
|- | |- | ||
− | ! style="font-weight: normal; align: center; background: #ffffcc;" | | + | ! style="font-weight: normal; align: center; background: #ffffcc;" | 1 |
! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>$</tt> | ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>$</tt> | ||
− | ! style="font-weight: normal; align: center; background: #ffffcc;" | | + | ! style="font-weight: normal; align: center; background: #ffffcc;" | '''T3''' (a) |
|} | |} | ||
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) |