Prove property (12.3.3).
To prove:
That there mutually disjoint subsets in the set of states in a finite-state automaton partitioned by the k− equivalence classes.
Given information:
For a finite-state automaton, there are k− equivalence classes for each integer k≥0.
Proof:
Let A be finite state automaton that has a set of states S. Suppose s and t are two states of S.
If s is k− equivalent to t, sRkt then the eventual function Nk for an input string w that the length is less than or equal k.
Nk(s,w)⇔Nk(t,w)
If Nk(s,w) is a nonaccepting state Nk(t,w) is also a nonaccepting state
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