On modal logics of Hamming spaces
Kudinov Andrey, Valentin Borisovich Shehtman, Ilya Borisovich Shapirovsky · Advances in Modal Logic · 2012
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. We investigate some unimodal logics of these frames. We show that if the length of words n is fixed and finite, the logics are closely related to many-dimensional products S5, so in many cases they are undecidable and not finitely axiomatizable. The relation H can be extended to infinite sequences. In this case we prove some completeness theorems characterizing the well-known modal logics DB and TB in terms of the Hamming distance.