Restricted classes of programming systems (Gödel numberings) are studied, where a programming system is in a given class if every programming system can be translated into it by functions in a given restricted class. For pairs of systems in various “natural” classes, results are given on the existence of isomorphisms (one-to-one and onto translations) between them from the corresponding classes of functions. The results with the most computational significance concern polynomial time programming systems. It is shown that if $\mathcal{P} = \mathcal{N}\mathcal{P}$ then every two polynomial time programming systems are isomorphic via a polynomial time computable function. If $\mathcal{P} \ne \mathcal{N}\mathcal{P}$ this result points the way to the possible existence of “natural” but intractable computational problems concerning programming systems. Results are also given concerning the relationship between the complexity of certain important and commonly used properties of programming systems (such as effective composition of programs) and the complexity of translations into the systems.

We study restricted classes of programming systems (Godel numberings), where a programming system is in a given class if every programming system can be translated into it by functions in a given restricted class. For pairs of systems in various “natural” classes we give results on the existence of isomorphisms (one-to-one and onto translations) between them from the appropriate class of functions. Our results with the most computational significance concern polynomial time programming systems. We show that if P=NP then every two polynomial time programming systems are isomorphic via a polynomial time computable function. IfP

A classification of all the computable functions is given in terms of subrecursive programming languages. These classes are those which also arise from the relation “primitive recursive in.” By distinguishing between honest and dishonest classes the classification is related to the computational complexity of the functions classified, and the classification has a wide degree of measure invariance.

This paper deals with a fundamental problem in the theory of recursion on initial segments of the ordinal numbers which was originated by Kripke [4] and Platek [8]. Following Kreisel-Sacks [3], we define a notion of relative recursiveness for this abstract recursion theory which allows us to speak of a function ƒ being weakly recursive in a set A via a partial recursive function g; this makes it reasonable to speak of partial recursive functions as ‘reduction procedures’. We say that a reduction procedure g is intrinsically consistent if for any set A there is a partial function ƒ which is weakly recursive in A via g. It is an elementary result of ordinary recursion theory that an arbitrary reduction procedure can be replaced in a uniform manner by an intrinsically consistent reduction procedure without the loss of any function f being reducible to any set A. In §3 we show that for recursion theory on the ordinals less than any infinite cardinal the analogous result is consistent with the axioms of set theory. In §5 we show that for all other cases of abstract recursion theory the analogous result fails in a very strong way to be true.