Tractable Transformations from Modal Provability Logics into First-Order Logic

We deene a class of modal logics LF by uniformly extending a class of modal logics L. Each logic L is characterised by a class of rst-order deenable frames, but the corresponding logic LF is sometimes characterised by classes of modal frames that are not rst-order deenable. The class LF includes provability logics with deep arithmeti-cal interpretations. Using Belnap's proof-theoretical framework Display Logic we characterise the \pseudo-displayable" subclass of LF and show how to deene polynomial-time transformations from each such LF into the corresponding L, and hence into rst-order classical logic. Theorem provers for classical rst-order logic can then be used to mechanise deduction in these \psuedo-displayable second order" modal logics.