On the commutativity of jumps

Journal of Symbolic Logic 65 (4):1725-1748 (2000)

Abstract

We study the following classes: Q* (r 1 A 1 ,..., r kA k ) which is defined to be the collection of all sets that can be computed by a Turing machine that on any input makes a total of r i queries to A i for all i ∈ {1,..., k}. Q(r 1A 1 ,...,r kA k ) which is defined like Q* (r 1A 1 ,..., r kA k ) except that queries to A i must be made before queries to A i+1 for all i ∈ {1,..., k - 1}. QC(r 1A 1 ,..., r kA k ) which is defined like Q(r 1A 1 ,..., r kA k ) except that the Turing machine must halt even if given incorrect answers to some of its queries. We show that if A 1 ,..., A k are jumps that are not too close together, then all three of these classes are identical and are not changed if we permute (r 1A 1 ,..., r kA k ). This extends a result of Beigel's [1]. Since the second class is not affected by permutations, we say that these sets commute with each other. We also show that jumps that are too close together may not commute. We also characterize the commutative sequences of sets obtained by iterating the jump operation through an ordinal notation

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