Simple Gödel Numberings, Isomorphisms, and Programming Properties

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.