Using the concept in the previous page, we now define an indistinguishability relation as follows:

Definition : States p and q are indistinguishable if iff , and is denoted as . It is easy to see that indistingushability is an equivalence relation.

In other words we say that sattes p and q are "distinguishable" if s.t and and is denoted as .

we say that, states p and q of a DFA M accepting a language L can be merged safely (i.e. without changing the accepted language L) if i.e. if p and q are indistinguishable. we can prove this by showing that when p and q are merged. Then they correspond to the same state in .

Formally, iff , and x y.

Proof : (only if) Let , and for some . Now, for any we have