Bootstrapping Inductive and Coinductive Types in HasCASL

We discuss the treatment of initial datatypes and final process types in the wide-spectrum language HasCASL. In particular, we present specifications that illustrate how datatypes and process types arise as bootstrapped concepts using HasCASL's type class mechanism, and we describe constructions of types of finite and infinite trees that establish the conservativity of datatype and process type declarations adhering to certain reasonable formats. The latter amounts to modifying known constructions from HOL to avoid unique choice; in categorical terminology, this means that we establish that quasitoposes with an internal natural numbers object support initial algebras and final coalgebras for a range of polynomial functors, thereby partially generalising corresponding results from topos theory. Moreover, we present similar constructions in categories of internal complete partial orders in quasitoposes

Implicit complexity for coinductive data: a characterization of corecurrence

We propose a framework for reasoning about programs that manipulate coinductive data as well as inductive data. Our approach is based on using equational programs, which support a seamless combination of computation and reasoning, and using productivity (fairness) as the fundamental assertion, rather than bi-simulation. The latter is expressible in terms of the former. As an application to this framework, we give an implicit characterization of corecurrence: a function is definable using corecurrence iff its productivity is provable using coinduction for formulas in which data-predicates do not occur negatively. This is an analog, albeit in weaker form, of a characterization of recurrence (i.e. primitive recursion) in [Leivant, Unipolar induction, TCS 318, 2004].Comment: In Proceedings DICE 2011, arXiv:1201.034

Global semantic typing for inductive and coinductive computing

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 the context of global semantics, it is preferable to think of types as semantical properties (Curry-style). Intrinsic theories were introduced in the late 1990s to provide a purely logical framework for reasoning about programs and their semantic types. We extend them here to data given by any combination of inductive and coinductive definitions. This approach is of interest because it fits tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as 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 the intended model. Finally, we show that every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were used to characterize major computational complexity classes Their extensions described here have similar potential which has already been applied