Hamming metric and products of modal logics

With a set S of words in an alphabet A we associate the frame (S,H), where sHt iff s and t are words of the same length and h(s,t) = 1 for the Hamming distance h, and investigate unimodal logics of such frames. These logics are closely related to many-dimensional modal logics: if we consider the n-th power of the inequality frame over a given alphabet A, then the Hamming box-operator on words of length n acts as the conjunction of all box-operators of the product; on the other hand, we show that all modalities of the logic of the n-th power of A with the universal relation can be expressed in the unimodal language with the Hamming box-operator. We present results on (un)decidability, complexity, (non-)finite axiomatizability, and completeness for these logics. Joint work with Andrey Kudinov and Valentin Shehtman.