Characteristics of minimal effective programming systems

• Authors: Samuel E. Moelius
• Venue: CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
• Year: 2012

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 epsψ 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 epsψ such that, for each ceerR, either R contradicts ψ's program equivalence relation, or there exists a translation function t into ψ such that the range of tfails to intersect infinitely many of R's equivalence classes.