Unifying Functional Interpretations: Past and Future

This article surveys work done in the last six years on the unification of various functional interpretations including G\"odel's dialectica interpretation, its Diller-Nahm variant, Kreisel modified realizability, Stein's family of functional interpretations, functional interpretations "with truth", and bounded functional interpretations. Our goal in the present paper is twofold: (1) to look back and single out the main lessons learnt so far, and (2) to look forward and list several open questions and possible directions for further research.Comment: 18 page

Modal Functional (Dialectica) Interpretation

We adapt our light Dialectica interpretation to usual and light modal formulas (with universal quantification on boolean and natural variables) and prove it sound for a non-standard modal arithmetic based on Goedel's T and classical S4. The range of this light modal Dialectica is the usual (non-modal) classical Arithmetic in all finite types (with booleans); the propositional kernel of its domain is Boolean and not S4. The `heavy' modal Dialectica interpretation is a new technique, as it cannot be simulated within our previous light Dialectica. The synthesized functionals are at least as good as before, while the translation process is improved. Through our modal Dialectica, the existence of a realizer for the defining axiom of classical S5 reduces to the Drinking Principle (cf. Smullyan)