The Independence of Control Structures in Abstract Programming Systems

Abstract An instance of a control structure is a mapping which takes one or more programs into a new program whose behavior is based on that of the original programs. An instance of a control structure is effective iff it is effectively computable. In order to study the interrelationships of control structures, . we consider abstract programming systems (numberings of the partial recursive functions) in which some control structures, effective or otherwise, are present, but others are not. This paper uses the techniques of recursive function theory, including recursion theorems and priority arguments to prove the independence of certain control structures in abstract programming systems. For example, we have obtained the following results. In effective numberings of the partial recursive functions, the one-one effective Kleene recursion theorem and the one-one effective (partial) if-then-else control structure are independent, but together, they yield all effective control structures. In any effective numbering, the effective Kleene form of the double recursion theorem yields all effective control structures.