Products, or How to Create Modal Logics of High Complexity
Maarten Marx, Szabolcs Mikulas

The aim of this paper is to exemplify the complexity of the
satisfiability problem of products of modal logics.  Our main goal is
to arouse interest for the main open problem in this area: a tight
complexity bound for the satisfiability problem of the product
\textbf{K}$\times$\textbf{K}.  At present, only non-elementary
decision procedures for this problem are known.  Our modest
contribution is two-fold.  We show that the problem of deciding
\textbf{K}$\times$\textbf{K}-satisfiability of formulas of modal depth
two is already hard for nondeterministic exponential time, and provide
a matching upper bound.  For the full language, a new proof for
decidability is given which combines filtration and selective
generation techniques from modal logic.  We put products of modal
logics into an historic perspective and review the most important
results.  

Keyword(s): modal logic, computational complexity