Fuzzy finite automata and fuzzy monadic second-order logic

We introduce fuzzy monadic second-order (LMSO-) logic and prove that the behaviours of fuzzy finite automata with membership values in an MV-algebra are precisely the fuzzy languages definable with sentences of our LMSO logic. This generalizes Buchipsilas and Elgotpsilas fundamental theorems to fuzzy logic setting. We also consider fuzzy first-order logic and show that star-free fuzzy languages and aperiodic fuzzy languages introduced here coincide with the fuzzy first-order definable ones.