Helping and the Meet of Pairs of Honest Subrecursive Classes

A notion of one computable function helping the computation of another by an amount equal to a third is defined in terms of the lattices of honest subrecursive classes. It is said that two honest computable functions help each other's computation by an amount equal to a third honest function if the intersection (meet) of the subrecursive classes which the first two generate is equal to the class generated by the third; that is, two functions help each other's computation by an amount equal to a third if the information content they have in common is that given by the third. A technical characterization is given of those pairs of honest subrecursive classes for which there exist nontrivial third honest subrecursive classes whose information content in common with the first class is equal to the second class. Further, it is shown that for every pair of honest subrecursive classes with the first properly containing the second there is an effective, increasing sequence of honest subrecursive classes which have information content in common with the first class equal to the second class, and this sequence is cofinal upward with the set of all honest subrecursive classes which have information content in common with the first class equal to the second class. Although there is always an upper bound to such a sequence, the sequence may or may not be eventually constant (and therefore there may or may not be a maximal class whose information content in common with the first class is equal to the second).