A Simplest Undecidable Modal Logic
  Modal logics are widely used in computer science. The complexity of their satisfiability problems has been an active field of research since the 1970s. We prove that even very "simple" modal logics can be undecidable: We show that there is an undecidable modal logic that can be obtained by restricting the allowed models with a first-order formula in which only universal quantifiers appear. 
Edith Hemaspaandra and Henning Schnoor
cs.LO cs.CC