On bounded functional interpretations

A unified view over well-known interpretations of intuitionistic logic, such as Gödel’s dialectica interpretation [6], Diller-Nahm interpretation [1] and Kreisel’s modified realizability [7] was achieved through a parametrised interpretation in the intuitionistic logic context [8] but also, very elegantly, in the linear logic setting (see [10], [9] and [3]). In this talk we report on work in progress concerning a general framework to the unification of the bounded interpretations of intuitionistic logic. This unification should include the known bounded functional interpretations whose bounds occur at the level of the interpretation of formulas, namely: bounded functional interpretation [5], bounded modified realizability [4] and confined modified realizability [2]. Similarly to the study of the interpretations that focuses in precise witnesses, in the bounded environment we also outline two different approaches towards the unification. One in the context of intuitionistic logic and the other via intuitionistic linear logic. This is joint work with Paulo Oliva.