Principles of Constructive Provability Logic

We present a novel formulation of the modal logic CPL, a constructive logic of provability that is closely connected to the Gödel-Löb logic of provability. Our logical formulation allows modal operators to talk about both provability and non-provability of propositions at reachable worlds. We are interested in the applications of CPL to logic programming; however, this report focuses on the presentation of a minimal fragment (in the sense of minimal logic) of CPL and on the formalization of minimal CPL and its metatheory in the Agda programming language. We present both a natural deduction system and a sequent calculus for minimal CPL and show that the presentations are equivalent. Support for this research was provided by an X10 Innovation Award from IBM, a National Science Foundation Graduate Research Fellowship for the first author, and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the Carnegie Mellon Portugal Program under Grants NGN-44 and SFRH / BD / 33763 / 2009. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of these supporting groups.