Normal modal logics can be defined axiomatically as
Hilbert systems, or semantically in terms of Kripke's possible
worlds and accessibility relations. Unfortunately there are Hilbert
axioms which do not have corresponding first-order properties for the
accessibility relation. For these logics the standard semantics-based
theorem proving techniques, in particular, the relational translation
into first-order predicate logic, do not work.
There is an alternative translation, the so-called functional
translation, in which the accessibility relations are replaced by certain
terms which intuitively can be seen as functions mapping worlds to
accessible worlds. In this paper we show that from a certain point of
view this functional language
is more expressive than the relational language, and that certain
second-order frame properties
can be mapped to first-order formulae expressed in the
functional language.
Moreover, we show how these formulae can be computed automatically
from the Hilbert axioms.
This extends the applicability of the functional translation method.