Some fact: For each automaton an equivalent

deterministic automaton can be created.

For an assertion graph G, we construct a corresponding finite-state automaton [M.sub.G] such that [L.sup.*](G) = [[summation].sup.*] - [L.sup.*]([M.sub.G]) and [L.sup.[omega]](G) = [[summation].sup.[omega]] - [L.sup.[omega]]([M.sub.G]), the number of states in [M.sub.G] is only twice as many as that in G, and a deterministic assertion graph is transformed to a

deterministic automaton. On the other hand, for an arbitrary finite-state automaton M, we construct a corresponding assertion graph [G.sub.M] such that [L.sup.*]([G.sub.M]) = [[summation].sup.*] - [L.sup.*](M) and [L.sup.[omega]]([G.sub.M]) = [[summation].sup.[omega]] - [L.sup.[omega]](M), the number of states in [G.sub.M] is the same as that in M, and a

deterministic automaton is transformed to a deterministic assertion graph.

Hopcroft (1971) has given an algorithm that computes the minimal automaton of a given deterministic automaton. The running time of the algorithm is O([absolute value of A] x n log n) where [absolute value of A] is the cardinality of the alphabet and n is the number of states of the given automaton.

Given a deterministic automaton A, Hopcroft's algorithm computes the coarsest congruence which saturates the set F of final states.

Cal designed an essentially

deterministic automaton (Figure 9).

It is shown that an elementary soliton graph defines a

deterministic automaton iff it reduces to a graph not containing even-length cycles.

1985] built on a string S is a

deterministic automaton able to recognize all the substrings of S.

In contrast, a

deterministic automaton has a single run on w.

Proof: For each n [greater than or equal to] 0 we construct a

deterministic automaton that accepts all words of length at most n in the complement of [Q.sub.k].

Let [T.sub.U] be the ordinary trie representing the set U, seen as a finite

deterministic automaton (Q, [delta], [epsilon], T) where the set of states is Q = Pref (u) (prefixes of words in u), the initial state is [epsilon], the set of final states is T = [A.sup.*] [intersection] Pref (u) and the transition function [delta] is defined on Pref (U) x A by

Let D = (Q, [q.sub.0], [Delta], F) be a

deterministic automaton. Let [f.sub.w] be the function induced by a word w on Q.

Because they are

deterministic automatons, computers struggle to generate numbers that are truly random.