Residual Tree Languages

By the Myhill-Nerode Theorem, a tree language L is recognisable if and only if ≡L has finite index. The minimal bottom-up DFTA of L is then defined using the congruence classes of ≡L as states. Exercise 1 (Bottom-up Residuals). Let L ⊆ T (F) and t ∈ T (F). The bottom-up residual of L by t is the set of all contexts C such that C[t] ∈ L: t−1L def = {C ∈ C(F) | C[t] ∈ L} . 1. Show that L is recognisable if and only if {t−1L | t ∈ T (F)} is finite. Show that [1] the minimal bottom-up DFTA for L has |{t−1L | t ∈ T (F)}| many states.