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The complexity of ODDnA

Published online by Cambridge University Press:  12 March 2014

Richard Beigel
Affiliation:
Department of Eecs (M/C 154), University of Illinois at Chicago, 851 S. Morgan St. - 1120 Seo, Chicago. IL, 60607-7053U.S.A., E-mail: beigel@uic.edu
William Gasarch
Affiliation:
Department of Computer Science, and Institute For Advanced Computing Studies, University of Maryland, College Park. MD 20742, U.S.A., E-mail: gasarch@cs.umd.edu
Martin Kummer
Affiliation:
Technische Universität Chemnitz-Zwickau, Fakultät Für Informatik, StraΒe Der Nationen 62, 09107 Chemnitz, Germany, Eu, E-mail: martin.kummer@informatik.tu-chemnitz.de
Georgia Martin
Affiliation:
12602 Goodhill Road, Wheaton, MD 20906, U.S.A.
Timothy Mcnicholl
Affiliation:
Department of Mathematics, University of Dallas, Irving, TX 75062, U.S.A., E-mail: tmcnicho@acad.udallas.edu
Frank Stephan
Affiliation:
Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, GermanyEU, E-mail: fstephan@math.uni-heidelberg.de

Abstract

For a fixed set A. the number of queries to A needed in order to decide a set S is a measure of S's complexity. We consider the complexity of certain sets defined in terms of A:

and, for m > 2,

where #nA. (x1….. xn) = A(x1) + A(xn)(We identify with , where χA is the characteristic function of A.)

If A is a nonrecursive semirecursive set or if A is a jump, we give tight bounds on the number of queries needed in order to decide ODDnA and MODmnA:

• ODDnA can be decided with n parallel queries to A, but not with n − 1.

• ODDnA can be decided with ⌈log(n + 1)⌉ sequential queries to A but not with ⌈log(n + 1)⌉ − 1.

• MODmnA can be decided with ⌈n/m⌉ + ⌊n/m⌋ parallel queries to A but not with ⌈n/m⌉ + ⌊n/m⌋ − 1.

• MODmnA can be decided with ⌈log(⌈n/m⌉ + ⌊n/m⌋ + 1)⌉ sequential queries to A but not with ⌈log(⌈n/m⌉ + ⌊n/m⌋ + 1)⌉ − 1.

The lower bounds above hold for nonrecursive recursively enumerable sets A as well. (Interestingly, the lower bounds for recursively enumerable sets follow by a general result from the lower bounds for semirecursive sets.)

In particular, every nonzero truth-table degree contains a set A such that ODDnA cannot be decided with n − 1 parallel queries to A. Since every truth-table degree also contains a set B such that ODDnB can be decided with one query to B, a set's query complexity depends more on its structure than on its degree.

For a fixed set A,

Q(n, A) = {S: S can be decided with n sequential queries to A}.

Q (n, A) = {S : S can be decided with n parallel queries to A}.

We show that if A is semirecursive or recursively enumerable, but is not recursive, then these classes form non-collapsing hierarchies:

• Q(0,A) ⊂ Q (1, A) ⊂ Q(2, A) ⊂ …

Q (0, A) ⊂ Q (1, A) ⊂ Q (2, A) ⊂ …

The same is true if A is a jump.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2000

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