A Hierarchy for \( BPP //\log \!\star \) Based on Counting Calls to an Oracle

  • Edwin Beggs
  • Pedro Cortez
  • José Félix CostaEmail author
  • John V Tucker
Part of the Emergence, Complexity and Computation book series (ECC, volume 24)


Algorithms whose computations involve making physical measurements can be modelled by Turing machines with oracles that are physical systems and oracle queries that obtain data from observation and measurement. The computational power of many of these physical oracles has been established using non-uniform complexity classes; in particular, for large classes of deterministic physical oracles, with fixed error margins constraining the exchange of data between algorithm and oracle, the computational power has been shown to be the non-uniform class \( BPP //\log \!\star \). In this paper, we consider non-deterministic oracles that can be modelled by random walks on the line. We show how to classify computations within \( BPP //\log \!\star \) by making an infinite non-collapsing hierarchy between \( BPP //\log \!\star \) and \( BPP \). The hierarchy rests on the theorem that the number of calls to the physical oracle correlates with the size of the responses to queries.


Random Walk Polynomial Time Turing Machine Advice Function Oracle Query 
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.



The research of José Félix Costa is supported by Fundação para a Ciência e Tecnologia, projeto FCT I.P.:UID/FIL/00678/2013.


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Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Edwin Beggs
    • 1
  • Pedro Cortez
    • 2
  • José Félix Costa
    • 2
    Email author
  • John V Tucker
    • 1
  1. 1.College of Science, Swansea UniversityWalesUK
  2. 2.Department of Mathematics, Instituto Superior Técnico and Centro de Filosofia das Ciências da Universidade de LisboaLisboaPortugal

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