| Line 7: | Line 7: | ||
The following is the result of applying Thompson's algorithm. State '''8''' recognizes the first expression (token '''T1'''); state '''13''' recognizes token '''T2'''; and state '''17''' recognizes token '''T3'''. | The following is the result of applying Thompson's algorithm. State '''8''' recognizes the first expression (token '''T1'''); state '''13''' recognizes token '''T2'''; and state '''17''' recognizes token '''T3'''. | ||
| − | < | + | <dot-hack> |
digraph nfa { | digraph nfa { | ||
{ node [shape=circle style=invis] s } | { node [shape=circle style=invis] s } | ||
| Line 44: | Line 44: | ||
fontsize=10 | fontsize=10 | ||
} | } | ||
| − | </ | + | </dot-hack> |
== DFA == | == DFA == | ||
| Line 126: | Line 126: | ||
Graphically, the DFA is represented as follows: | Graphically, the DFA is represented as follows: | ||
| − | < | + | <dot-hack> |
digraph dfa { | digraph dfa { | ||
{ node [shape=circle style=invis] s } | { node [shape=circle style=invis] s } | ||
| Line 142: | Line 142: | ||
fontsize=10 | fontsize=10 | ||
} | } | ||
| − | </ | + | </dot-hack> |
The minimization tree is as follows. Note that before considering transition behavior, states are split according to the token they recognize. | The minimization tree is as follows. Note that before considering transition behavior, states are split according to the token they recognize. | ||
| − | < | + | <dot-hack> |
digraph mintree { | digraph mintree { | ||
node [shape=none,fixedsize=true,width=0.3,fontsize=10] | node [shape=none,fixedsize=true,width=0.3,fontsize=10] | ||
| Line 154: | Line 154: | ||
"{0, 1, 2, 3, 4} " -> "{3}" [label=" T2",fontsize=10] | "{0, 1, 2, 3, 4} " -> "{3}" [label=" T2",fontsize=10] | ||
"{0, 1, 2, 3, 4} " -> "{4}" [label=" T3",fontsize=10] | "{0, 1, 2, 3, 4} " -> "{4}" [label=" T3",fontsize=10] | ||
| − | "{0, 1, 2}" -> "{0, 1}" | + | "{0, 1, 2}" -> "{0, 1}" |
"{0, 1, 2}" -> "{2}" [label=" a",fontsize=10] | "{0, 1, 2}" -> "{2}" [label=" a",fontsize=10] | ||
"{0, 1}" -> "{0}" | "{0, 1}" -> "{0}" | ||
"{0, 1}" -> "{1}" [label=" b",fontsize=10] | "{0, 1}" -> "{1}" [label=" b",fontsize=10] | ||
fontsize=10 | fontsize=10 | ||
| − | |||
} | } | ||
| − | </ | + | </dot-hack> |
The tree expansion for non-splitting sets has been omitted for simplicity ("a" transitions for super-state {0, 1}). | The tree expansion for non-splitting sets has been omitted for simplicity ("a" transitions for super-state {0, 1}). | ||
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 8 recognizes the first expression (token T1); state 13 recognizes token T2; and state 17 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, 2, 3, 5, 6, 8, 9, 14, 15, 17 | 0 (T1) |
| 0 | a | 4 | 3, 4, 5, 8 | 1 (T1) |
| 0 | b | 7, 10, 16 | 7, 8, 10, 11, 13, 15, 16, 17 | 2 (T1) |
| 1 | a | 4 | 3, 4, 5, 8 | 1 (T1) |
| 1 | b | - | - | - |
| 2 | a | 12 | 11, 12, 13 | 3 (T2) |
| 2 | b | 16 | 15, 16, 17 | 4 (T3) |
| 3 | a | 12 | 11, 12, 13 | 3 (T2) |
| 3 | b | - | - | - |
| 4 | a | - | - | - |
| 4 | b | 16 | 15, 16, 17 | 4 (T3) |
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 non-splitting sets has been omitted for simplicity ("a" transitions for super-state {0, 1}).
Given the minimization tree, the DFA is already minimal.
| In | Input | In+1 / Token |
|---|---|---|
| 0 | aababb$ | 1 |
| 1 | ababb$ | 1 |
| 1 | babb$ | T1 (aa) |
| 0 | babb$ | 2 |
| 2 | abb$ | 3 |
| 3 | bb$ | T2 (ba) |
| 0 | bb$ | 2 |
| 2 | b$ | 4 |
| 4 | $ | T3 (bb) |
The input string aababb is, after 9 steps, split into three tokens: T1 (corresponding to lexeme aa), T2 (ba), and T3 (bb).