if, for all w1,w2 in Σ*, w1uw2 ∈ L iff w1u′w2 ∈ L
Similarly (and more generally), if f is a function from Σ* to any set, its Myhill equivalence is defined by: u =M u′
if, for all w1,w2 in Σ*, f(w1uw2) = f(w1u′w2)
See also Nerode equivalence.
An important fact is that L is regular iff the equivalence relation =M is of finite index (i.e. there are finitely many equivalence classes). Indeed, L is regular iff it is a union of classes of any equivalence relation of finite index. In addition =M is a congruence on Σ*, i.e. u =M u′ and v =M v′ implies uv =M u′v′
The equivalence classes therefore can be concatenated consistently and form a semigroup. This is in fact the semigroup of the minimal machine for L (or f).
"Myhill equivalence." A Dictionary of Computing. . Encyclopedia.com. (March 20, 2019). https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/myhill-equivalence
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