Characteristics of Minimal Effective Programming Systems

TL;DR

This paper characterizes minimal effective programming systems within the Rogers semilattice, linking their minimality to properties of computably enumerable equivalence relations and translation functions.

Contribution

It provides a precise characterization of minimal epses using ceers and shows the existence of minimal epses with complex translation properties.

Findings

A minimal eps  characterized by ceers and translation functions.

Existence of a minimal eps with no single ceer working for all translation functions.

Some minimal epses have translation functions that fail to intersect infinitely many ceer classes.

Abstract

The Rogers semilattice of effective programming systems (epses) is the collection of all effective numberings of the partial computable functions ordered such that \theta\ is less than or equal to \psi\ whenever \theta-programs can be algorithmically translated into \psi-programs. Herein, it is shown that an eps \psi\ is minimal in this ordering if and only if, for each translation function t into \psi, there exists a computably enumerable equivalence relation (ceer) R such that (i) R is a subrelation of \psi's program equivalence relation, and (ii) R equates each \psi-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 \psi\ such that, for each ceer R, either R contradicts \psi's program equivalence relation, or there exists a translation function t into \psi\ such that the range of t fails to intersect infinitely many of R's equivalence classes.