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On the commutativity of jumps

Published online by Cambridge University Press:  12 March 2014

Timothy H. McNicholl
Timothy H. McNicholl*
Affiliation:
Department of Mathematics, University of Dallas, Irving, Texas 75062, USA, E-mail:tmcnicho@acad.udallas.edu
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Abstract

We study the following classes:

● Q* (r1A1…..rkAk) 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 ri, queries to Ai, for all i ∈ {1..… k}.

● Q(r1A1…..rkAk) which is defined like Q* (r1A1….. rkAk) except that queries to Ai, must be made before queries to Ai+1 for all i ∈ {1….. k – 1}.

● QC(r1A1….. rkAk) which is defined like Q{r1A1….. rkAk) except that the Turing machine must halt even if given incorrect answers to some of its queries.

We show that if A1 ….. Ak are jumps that are not too close together, then all three of these classes are identical and are not changed if we permute (r1…..rkAk). 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 4 , December 2000 , pp. 1725 - 1748
Copyright
Copyright © Association for Symbolic Logic 2000

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References

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