Which data types have ω-complete initial algebra specifications?

Abstract

An algebraic specification is called ω-complete or inductively complete if all (open as well as closed) equations valid in its initial model are equationally derivable from it, i.e., if the equational theory of the initial model is identical to the equational theory of the specification. As the latter is recursively enumerable, the initial model of an ω-complete algebraic specification is a data type with a recursively enumerable equational theory. We show that if hidden sorts and functions are allowed in the specification, the converse is also true: every data type with a recursively enumerable equational theory has an ω-complete initial algebra specification with hidden sorts and functions. We also show that in the case of finite data types the hidden sorts can be dispensed with.