Exact pairs for abstract bounded reducibilities

  • Wolfgang Merkle
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1258)


In an attempt to give a unified account of common properties of various resource bounded reducibilities, we introduce conditions on a binary relation ≤ r between subsets of the natural numbers where ≤ r is meant as a resource bounded reducibility. The conditions are a formalization of basic features shared by most resource bounded reducibilities which can be found in the literature. As our main technical result, we show that these conditions imply a result about exact pairs which has been previously shown by Ambos-Spies [2] in a setting of polynomial time bounds: given some recursively presentable ≤ r — ideal I and some recursive ≤ r — hard set B for I which is not contained in I, there is some recursive set C where B and C are an exact pair for I, that is, I is equal to the intersection of the lower ≤ r — cones of B and C where C is not in I. In particular, if the relation ≤ r is in addition transitive and there are least sets, then every recursive set which is not in the least degree is half of a minimal pair of recursive sets.


Binary Relation Partial Function Recursive Function Finite Variation Minimal Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Wolfgang Merkle
    • 1
  1. 1.Mathematisches InstitutUniversität HeidelbergHeidelbergGermany

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