Lax Lean Archive

Randomized Complexity Classes

lax-666725·formalized by Édouard Bonnet · Codex·registered·created ·GitHub @31864d7·Lean v4.33.0 epoch · mathlib db584cd6d46c

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    Abstract

    We define RP\mathrm{RP}, coRP\mathrm{coRP}, BPP\mathrm{BPP}, ZPP\mathrm{ZPP}, and PP\mathrm{PP} as classes of the binary languages of Classical Complexity Classes. The definitions use finite probabilistic Turing machines and exact probabilities over independent fair bits. Zero-error polynomial time uses the bounded-time formulation allowing a "don't know" answer.

    We prove ZPP⊆RP∩coRP\mathrm{ZPP}\subseteq\mathrm{RP}\cap\mathrm{coRP}, RP∪coRP⊆BPP\mathrm{RP}\cup\mathrm{coRP}\subseteq\mathrm{BPP}, ZPP⊆BPP\mathrm{ZPP}\subseteq\mathrm{BPP}, and BPP⊆PP\mathrm{BPP}\subseteq\mathrm{PP}.

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    This is only the formalizers. The authors of the formalized results may be different (see References).

    @misc{lax-666725,
  author = {Édouard Bonnet and Codex},
  title = {Randomized Complexity Classes},
  year = {2026},
  howpublished = {Lax Archive, lax-666725},
  url = {https://laxarchive.org/lax-666725/},
}

    References

    1. Jonathan Katz. Notes on Complexity Theory: Lecture 12. 2011. cs.umd.edu/~jkatz/complexity/f11/lecture12.pdf

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