Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories

Authors Melissa Antonelli , Ugo Dal Lago , Davide Davoli, Isabel Oitavem , Paolo Pistone

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Author Details

Melissa Antonelli
  • Helsinki Institute for Information Technology, Finland
Ugo Dal Lago
  • Bologna University, Italy
  • Inria, Université Côte d'Azur, Sophia Antipolis, France
Davide Davoli
  • Inria, Université Côte d'Azur, Sophia Antipolis, France
Isabel Oitavem
  • Center for Mathematics and Applications (NOVA Math), NOVA FCT, Caparica, Portugal
  • Department of Mathematics, NOVA FCT, Caparica, Portugal
Paolo Pistone
  • Bologna University, Italy

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Melissa Antonelli, Ugo Dal Lago, Davide Davoli, Isabel Oitavem, and Paolo Pistone. Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We consider a minimal extension of the language of arithmetic, such that the bounded formulas provably total in a suitably-defined theory à la Buss (expressed in this new language) precisely capture polytime random functions. Then, we provide two new characterizations of the semantic class BPP obtained by internalizing the error-bound check within a logical system: the first relies on measure-sensitive quantifiers, while the second is based on standard first-order quantification. This leads us to introduce a family of effectively enumerable subclasses of BPP, called BPP_T and consisting of languages captured by those probabilistic Turing machines whose underlying error can be proved bounded in T. As a paradigmatic example of this approach, we establish that polynomial identity testing is in BPP_T, where T = IΔ₀+Exp is a well-studied theory based on bounded induction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Proof theory
  • Bounded Arithmetic
  • Randomized Computation
  • Implicit Computational Complexity


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