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Ergodic Theorems and Converses for PSPACE Functions

Authors Satyadev Nandakumar , Subin Pulari

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ACM Subject Classification
  • Theory of computation → Constructive mathematics
  • Theory of computation → Complexity theory and logic
  • Mathematics of computing → Probability and statistics
Keywords
  • Ergodic Theorem
  • Resource-bounded randomness
  • Computable analysis
  • Complexity theory

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Abstract

We initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow [Ulrich Krengel, 1978]. In contrast, we show that for a class of PSPACE L¹ functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x.

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Satyadev Nandakumar and Subin Pulari. Ergodic Theorems and Converses for PSPACE Functions. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 80:1-80:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.MFCS.2021.80

Author Details

Satyadev Nandakumar
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, India
Subin Pulari
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, India

Acknowledgements

The authors wish to thank anonymous reviewers for helpful suggestions.

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