Basic properties of data types with inequational refinements

Abstract

In this work we propose a typed functional wide-spectrum (i.e. both programming and specification) language Final, an extension of Cardelli-Wegner's Fun enriched with the fixed-point construction on expressions and inequational refinements (assertions) for types. The inequational assertion has the form ∀ x ₁∶σ ₁⋯∀ x _n∶σ _n. e ₁≤e ₂∶τ and its intuitive meaning is that if the expression e ₁ terminates, then it gives a value that e ₂ does but the computation of e ₁ may diverge. Hence it may be seen, in some sense, a weaker form of equations in algebraic specification of abstract data types. The type system has a subtype relation as Fun does, and the subtype relation reflects the strength of assertions. We demonstrate that inequationally refined record types nicely model (the representation type of) abstract data types à la algebraic specification by giving some examples. Then we show that the type system is a conservative extension of the original one, and give a domain-theoretic semantics to its sublanguage without existential types. This semantics assures that we can freely use general recursive functions over inequationally refined types, that is, for any function f∶τ →τ, its least fixed-point belongs to τ even if τ is a refined type. This shows that our language, Final, is an appropriate base language in designing modular wide-spectrum languages with both general recursions and equationally specified data types which allow us to manipulate implementation modules of abstract data types with general recursion as we do for ordinal data like integers.