Characterizing Interpolation Pairs in Infinitary Graded Logics

In this paper the problem of interpolation for the family of countable infinitary graded modal logics is considered. It is well known that interpolation fails in general for these logics and it is then natural to ask for a semantical characterization (stronger than entailment) of pairs of graded formulae having an interpolant. This is obtained using the notion of entailment along elementary equivalence. More precisely, we prove that if L is a graded modal logic then a pair (ø, ψ) of graded formulae in L have an interpolant in L if, and only if, ø entails ψ along elementary equivalence with respect to L. This characterization is obtained by adapting to graded modal logics the method of consistency property modulo bisimulation, which was previously used in Infinitary Logic and Infinitary Modal Logic. In the case of full Countable Infinitary Graded Modal Logic we improve this result and show that this logic enjoys Craig interpolation. This is done using a characterization of graded bisimulation between models via isomorphism of their unravellings.