Ergodic Theorems and Converses for PSPACE Functions
| Autor: | Nandakumar, Satyadev, Pulari, Subin |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
Resource-bounded randomness
Mathematics::Dynamical Systems Ergodic Theorem Computational Theory and Mathematics Complexity theory Computable analysis Mathematics of computing → Probability and statistics Theory of computation → Complexity theory and logic Theory of computation → Constructive mathematics Theoretical Computer Science |
| Zdroj: | Theory of Computing Systems. |
| ISSN: | 1433-0490 1432-4350 |
| DOI: | 10.1007/s00224-022-10094-9 |
| Popis: | 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. LIPIcs, Vol. 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021), pages 80:1-80:19 |
| Databáze: | OpenAIRE |
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