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The algorithm works recursively by splitting an expression into its constituent subexpressions, from which the NFA will be constructed using a set of rules. [3] More precisely, from a regular expression E, the obtained automaton A with the transition function Δ [clarification needed] respects the following properties:
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}.
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 ...
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 ...
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 ...
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.
A two-way nondeterministic finite automaton (2NFA) may have multiple transitions defined in the same configuration. Its transition function is : ({,}) {,}. Like a standard one-way NFA, a 2NFA accepts a string if at least one of the possible computations is accepting.
While an exhaustive search may minimize an NFA, there is no polynomial-time algorithm to minimize general NFAs unless P = PSPACE, an unsolved conjecture in computational complexity theory that is widely believed to be false. However, there are methods of NFA minimization that may be more efficient than brute force search. [11]