Eeectivizing Inseparability

Smullyan's notion of eeectively inseparable pairs of sets is not the best eeec-tive/constructive analog of Kleene's notion of pairs of sets inseparable by a recursive set. We present a corrected notion of eeectively inseparable pairs of sets, prove a characterization of our notion, and show that the pairs of index sets eeectively inseparable in Smullyan's sense are the same as those eeectively inseparable in ours. In fact we characterize the pairs of index sets eeectively inseparable in either sense thereby generalizing Rice's Theorem. For subrecursive index sets we have suucient conditions for various inseparabilities to hold. For inseparability by sets in the same subrecursive class we have a characterization. The latter essentially generalizes Kozen's (and Royer's later) Subrecursive Rice Theorem, and the proof of each result about subrecursive index sets is presented \Rogers style" with care to observe subrecursive restrictions. There are pairs of sets eeectively inseparable in Smullyan's sense, but not eeec-tively inseparable in ours. The proof of this involves a non-eeective construction by nite extensions with the unusual and interesting feature that alternate stages in the construction apply an instance of Smullyan's Double Recursion Theorem eeective in the previous stage. Our construction yields as a corollary that the pairs of sets eeec-tively inseparable in Smullyan's sense, but not in ours, are plentiful in the sense of Baire Category. By way of contrast with the previous result we show that, for pairs of r.e. sets, our notion and Smullyan's are coextensive. We call our notion eeective 0 1-inseparability and generalize it, Smullyan's notion, and all our results (except those about subrecursive index sets) to the 0 n level, for each n > 1. (Royer and Case apply eeective 0 2-inseparability to obtain results in structural complexity theory.) For sub-recursive index sets we have a suucient condition for eeective 0 2-inseparability. The proof of this latter result is made compact by an application of Royer and Case's Hybrid Recursion Theorem which facilitates interaction between a subrecursive programming system and a programming system for the partial limiting-recursive functions. This paper is dedicated to the memory of John Myhill. It grew out of a joint monograph with Royer, and owes an especial intellectual debt to that work and the methods in Myhill's paper on creative sets and Smullyan's monograph. A suggestion of Royer's provided a cleaner version of Theorem 3. He also made some helpful suggestions regarding presentation.