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The instances of the DFA minimization problem that cause the worst-case behavior are the same as for Hopcroft's algorithm. The number of steps that the algorithm performs can be much smaller than n , so on average (for constant s ) its performance is O ( n log n ) or even O ( n log log n ) depending on the random distribution on automata chosen ...
On the other hand, finite-state automata are of strictly limited power in the languages they can recognize; many simple languages, including any problem that requires more than constant space to solve, cannot be recognized by a DFA. The classic example of a simply described language that no DFA can recognize is bracket or Dyck language, i.e ...
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.
At each step of the simulation, the active set of NFA states forms a new DFA state. If the new state is identical to an existing DFA state, it is discarded and replaced with the existing one, and the current branch of simulation terminates. Otherwise the new state is added to the growing set of DFA states and simulation from this state continues.
In automata theory (a branch of theoretical computer science), NFA minimization is the task of transforming a given nondeterministic finite automaton (NFA) into an equivalent NFA that has a minimum number of states, transitions, or both. While efficient algorithms exist for DFA minimization, NFA minimization is PSPACE-complete. [1]
An early application of partition refinement was in an algorithm by Hopcroft (1971) for DFA minimization. In this problem, one is given as input a deterministic finite automaton, and must find an equivalent automaton with as few states as possible. Hopcroft's algorithm maintains a partition of the states of the input automaton into subsets ...
On the other hand, some problems are easily solved on DFAs and not on UFAs. For example, given an automaton A, an automaton A ′ which accepts the complement of A can be computed in linear time when A is a DFA, whereas it is known that this cannot be done in polynomial time for UFAs. Hence UFAs are a mix of the worlds of DFA and of NFA; in ...
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.