The Undecidability of Iterated Modal Relativization

Abstract

In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that three fragments of logic of iterated relativization and transitive closure are $\Sigma^1_1$-complete. Of these, two fragments do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is $\Sigma^0_1$-complete. These results go via reduction to problems concerning domino systems.