A Note on Constructive Interpolation for the Multi-Modal Logic Km

The Craig Interpolation Theorem is a well-known property in the mathematical logic curricula, with many domain applications, such as in the modularization of formal specifications and ontologies. This property states the following: given an implication, say formula ϕ implies another formula ψ, then there is a formula β, called the interpolant, in the common language of ϕ and ψ, such that ϕ also implies β, as well as β implies ψ. Although it is already known that the propositional multi-modal logic K_m enjoys Craig interpolation, we are not aware of method providing an explicit construction of interpolants. We describe in this paper a constructive proof of the Craig interpolation property on the multi-modal logic K_m. Interpolants can be explicitly computed from the proof. Furthermore, we also describe an upper bound for the computation of interpolants. The proof is based on the application of Maehara technique on a tree-hypersequent calculus. As a corollary of interpolation, we also show Beth definability and Robinson joint consistency.