Global semantic typing for inductive and coinductive computing

Abstract

Inductive and coinductive types are commonly construed as ontological(Church-style) types, denoting canonical data-sets such as natural numbers,lists, and streams. For various purposes, notably the study of programs in thecontext of global semantics, it is preferable to think of types as semanticalproperties (Curry-style). Intrinsic theories were introduced in the late 1990sto provide a purely logical framework for reasoning about programs and theirsemantic types. We extend them here to data given by any combination ofinductive and coinductive definitions. This approach is of interest because itfits tightly with syntactic, semantic, and proof theoretic fundamentals offormal logic, with potential applications in implicit computational complexityas well as extraction of programs from proofs. We prove a Canonicity Theorem,showing that the global definition of program typing, via the usual (Tarskian)semantics of first-order logic, agrees with their operational semantics in theintended model. Finally, we show that every intrinsic theory is interpretablein a conservative extension of first-order arithmetic. This means thatquantification over infinite data objects does not lead, on its own, toproof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theoriesare perfectly amenable to formulas-as-types Curry-Howard morphisms, and wereused to characterize major computational complexity classes Their extensionsdescribed here have similar potential which has already been applied