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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

Funding

The first, second, third and fifth authors' work is supported by the European Research Council through the project DIAPASoN ERC COoG 818616, and by the French "Agence Nationale de la Recherche" through the project PPS ANR-19-C48-0014. The first author’s work is supported by the Helsinki Institute for Information Technology. The third author’s work is supported by the French "Agence Nationale de la Recherche" through the project UCA DS4H ANR-17-EURE-0004. The fourth author’s work is supported by national funds through the "FCT-Fundação para a Ciência e a Tecnologia, I.P.", through the projects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics and Applications).

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.CSL.2024.10

Abstract

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
Keywords
  • Bounded Arithmetic
  • Randomized Computation
  • Implicit Computational Complexity

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