Liron Cohen and Robert Constable: Intuitionistic Ancestral Logic as a Dependently Typed Abstract Programming Language

In this paper we define a constructive version of AL, pure intuitionistic ancestral logic (iAL), extending pure intuitionistic first-order logic (iFOL). This logic is a dependently typed abstract programming language with computational functionality beyond iFOL, given by its realizer for the transitive closure operator TC, which corresponds to recursive programs. We derive this operator from the natural type theoretic definition of TC using intersection type. We show that provable formulas in iAL are uniformly realizable, thus iAL is sound with respect to constructive type theory. We further demonstrate how iAL can serve as a natural framework for reasoning about programs.