Article

Studia Logica

, Volume 79, Issue 3, pp 373-407

The Undecidability of Iterated Modal Relativization

  • Joseph S. MillerAffiliated withDepartment of Mathematics, Indiana University Email author 
  • , Lawrence S. MossAffiliated withDepartment of Mathematics, Indiana University

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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 for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are fi1 Σ1 1–complete. Two of these 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 fi0 Σ0 1–complete. These results go via reduction to problems concerning domino systems.

Keywords

Dynamic epistemic logic iterated relativization modal logic undecidability