Programming Systems

Abstract

Most of the constructions in the later chapters involve programming within various programming systems for subrecursive classes. To carry out this programming one needs to know something of the “control structure” properties of these systems. This chapter is primarily concerned with establishing such control structure properties for a wide variety of programming systems. First, we show how certain useful control structures can be built from certain other basic ones (Theorems 4.4 and 4.6). We also establish our hybrid recursion theorem (Theorem 4.9). We next identify four broad classes of programming systems, clocked (Section 4.2), provably explicitly bounded (Section 4.3.1), provably implicitly bounded (Section 4.3.2), and reduction induced (Section 4.4), each of which can be thought of as being derived from other programming systems. For the case of clocked systems, Theorem 4.16 provides a means of making sure examples exist for many important complexity classes. We show how clocked and reduction induced systems inherit control structure properties from programming systems from which they are derived. We further show that clocked systems and provably explicit bounded systems are essentially equivalent, and we compare these to the provably implicitly bounded systems.