Logics for Reversible Regular Languages and Semigroups with Involution

We present MSO and FO logics with predicates between' and neighbour’ that characterise various fragments of the class of regular languages that are closed under the reverse operation. The standard connections that exist between MSO and FO logics and varieties of finite semigroups extend to this setting with semigroups extended with an involution. The case is different for FO with neighbour relation where we show that one needs additional equations to characterise the class.