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In computer science, Thompson's construction algorithm, also called the McNaughton–Yamada–Thompson algorithm, [1] is a method of transforming a regular expression into an equivalent nondeterministic finite automaton (NFA). [2]
Glushkov's algorithm can be used to transform it into an NFA, which furthermore is small by nature, as the number of its states equals the number of symbols of the regular expression, plus one. Subsequently, the NFA can be made deterministic by the powerset construction and then be minimized to get an optimal automaton corresponding to the ...
Union (cf. picture); that is, if the language L 1 is accepted by some NFA A 1 and L 2 by some A 2, then an NFA A u can be constructed that accepts the language L 1 ∪L 2. Intersection; similarly, from A 1 and A 2 an NFA A i can be constructed that accepts L 1 ∩L 2. Concatenation; Negation; similarly, from A 1 an NFA A n can be constructed ...
The NFA below has four states; state 1 is initial, and states 3 and 4 are accepting. Its alphabet consists of the two symbols 0 and 1, and it has ε-moves. The initial state of the DFA constructed from this NFA is the set of all NFA states that are reachable from state 1 by ε-moves; that is, it is the set {1,2,3}.
A GNFA must have only one transition between any two states, whereas a NFA or DFA both allow for numerous transitions between states. In a GNFA, a state has a single transition to every state in the machine, although often it is a convention to ignore the transitions that are labelled with the empty set when drawing generalized nondeterministic ...
An example of a deterministic finite automaton that accepts only binary numbers that are multiples of 3. The state S 0 is both the start state and an accept state. For example, the string "1001" leads to the state sequence S 0, S 1, S 2, S 1, S 0, and is hence accepted.
Now if the machine is in the state S 1 and receives an input of 0 (first column), the machine will transition to the state S 2. In the state diagram, the former is denoted by the arrow looping from S 1 to S 1 labeled with a 1, and the latter is denoted by the arrow from S 1 to S 2 labeled with a 0.
Reversing the transitions of a non-deterministic finite automaton (NFA) and switching initial and final states [note 1] produces an NFA for the reversal of the original language. Converting this NFA to a DFA using the standard powerset construction (keeping only the reachable states of the converted DFA) leads to a DFA M D R {\displaystyle M_{D ...