Characteristics of Minimal Effective Programming Systems

The Rogers semilattice of effective programming systems (epses) is the collection of all effective numberings of the partial computable functions ordered such that θ ≤ ψ whenever θ-programs can be algorithmically translated into ψ-programs. Herein, it is shown that an eps ψ is minimal in this ordering if and only if, for each translation function t into ψ, there exists a computably enumerable equivalence relation (ceer) R such that (i) R is a subrelation of ψ’s program equivalence relation, and (ii) R equates each ψ-program to some program in the range of t. It is also shown that there exists a minimal eps for which no single such R does the work for all such t. In fact, there exists a minimal eps ψ such that, for each ceer R, either R contradicts ψ’s program equivalence relation, or there exists a translation function t into ψ such that the range of t fails to intersect infinitely many of R’s equivalence classes.