Abstract Computability, Algebraic Specification and Initiality

computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable functions on any many-sorted algebra; and (ii) all functions effectively approximable by abstract computable functions on any metric algebra. We show that there exist universal algebraic specifications for all the classically computable functions on the set R of real numbers. The algebraic specifications used are mainly bounded universal equations and conditional equations. We investigate the initial algebra semantics of these specifications, and derive situations where algebraic specifications define precisely the computable functions.