Rewrites as Terms through Justification Logic

Abstract

Justification Logic is a refinement of modal logic where the modality is annotated with a reason s for “knowing” A and written . The expression s is a proof of A that may be encoded as a lambda calculus term of type A, according to the propositions-as-types interpretation. Our starting point is the observation that terms of type are reductions between lambda calculus terms. Reductions are usually encoded as rewrites essential tools in analyzing the reduction behavior of lambda calculus and term rewriting systems, such as when studying standardization, needed strategies, Lévy permutation equivalence, etc. We explore a new propositions-as-types interpretation for Justification Logic, based on the principle that terms of type are proof terms encoding reductions (with source s). Note that this provides a logical language to reason about rewrites.