Intuition
To simulate the operation of a DFA on a given input string, one needs to keep track of a single state at any time: the state that the automaton will reach after seeing a prefix of the input. In contrast, to simulate an NFA, one needs to keep track of a set of states: all of the states that the automaton could reach after seeing the same prefix of the input, according to the nondeterministic choices made by the automaton. If, after a certain prefix of the input, a set of states can be reached, then after the next input symbol the set of reachable states is a deterministic function of and . Therefore, the sets of reachable NFA states play the same role in the NFA simulation as single DFA states play in the DFA simulation, and in fact the sets of NFA states appearing in this simulation may be re-interpreted as being states of a DFA.Construction
The powerset construction applies most directly to an NFA that does not allow state transformations without consuming input symbols (aka: "ε-moves"). Such an automaton may be defined as a 5-tuple , in which is the set of states, is the set of input symbols, is the transition function (mapping a state and an input symbol to a set of states), is the initial state, and is the set of accepting states. The corresponding DFA has states corresponding to subsets of . The initial state of the DFA is , the (one-element) set of initial states. The transition function of the DFA maps a state (representing a subset of ) and an input symbol to the set , the set of all states that can be reached by an -transition from a state in . A state of the DFA is an accepting state if and only if at least one member of is an accepting state of the NFA. In the simplest version of the powerset construction, the set of all states of the DFA is the powerset of , the set of all possible subsets of . However, many states of the resulting DFA may be useless as they may be unreachable from the initial state. An alternative version of the construction creates only the states that are actually reachable.NFA with ε-moves
For an NFA with ε-moves (also called an ε-NFA), the construction must be modified to deal with these by computing the ''ε- closure'' of states: the set of all states reachable from some given state using only ε-moves. Van Noord recognizes three possible ways of incorporating this closure computation in the powerset construction: # Compute the ε-closure of the entire automaton as a preprocessing step, producing an equivalent NFA without ε-moves, then apply the regular powerset construction. This version, also discussed by Hopcroft and Ullman, is straightforward to implement, but impractical for automata with large numbers of ε-moves, as commonly arise in natural language processing application. # During the powerset computation, compute the ε-closure of each state that is considered by the algorithm (and cache the result). # During the powerset computation, compute the ε-closure of each subset of states that is considered by the algorithm, and add its elements to .Multiple initial states
If NFAs are defined to allow for multiple initial states, the initial state of the corresponding DFA is the set of all initial states of the NFA, or (if the NFA also has ε-moves) the set of all states reachable from initial states by ε-moves.Example
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 . A transition from by input symbol 0 must follow either the arrow from state 1 to state 2, or the arrow from state 3 to state 4. Additionally, neither state 2 nor state 4 have outgoing ε-moves. Therefore, (,0) = , and by the same reasoning the full DFA constructed from the NFA is as shown below. As can be seen in this example, there are five states reachable from the start state of the DFA; the remaining 11 sets in the powerset of the set of NFA states are not reachable.Complexity
Because the DFA states consist of sets of NFA states, an -state NFA may be converted to a DFA with at most states. For every , there exist -state NFAs such that every subset of states is reachable from the initial subset, so that the converted DFA has exactly states, giving Θ() worst-case time complexity.. A simple example requiring nearly this many states is the language of strings over the alphabet in which there are at least characters, the th from last of which is 1. It can be represented by an -state NFA, but it requires DFA states, one for each -character suffix of the input; cf. picture for .Applications
Brzozowski's algorithm for DFA minimization uses the powerset construction, twice. It converts the input DFA into an NFA for the reverse language, by reversing all its arrows and exchanging the roles of initial and accepting states, converts the NFA back into a DFA using the powerset construction, and then repeats its process. Its worst-case complexity is exponential, unlike some other known DFA minimization algorithms, but in many examples it performs more quickly than its worst-case complexity would suggest. Safra's construction, which converts a non-deterministic Büchi automaton with states into a deterministic Muller automaton or into a deterministicReferences
Further reading
* {{cite book , last=Anderson , first=James Andrew , date=2006 , title=Automata theory with modern applications , publisher=Cambridge University Press , isbn=978-0-521-84887-9 , pages=43–49 , url=https://books.google.com/books?id=ikS8BLdLDxIC&pg=PA43 Finite automata