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DOI: 10.4230/LIPIcs.MFCS.2024.10

A New Characterization of FAC⁰ via Discrete Ordinary Differential Equations

Authors: Melissa Antonelli, Arnaud Durand, and Juha Kontinen

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract

Implicit computational complexity is an active area of theoretical computer science, which aims at providing machine-independent characterizations of relevant complexity classes. One of the seminal works in this field appeared in 1965, when Cobham introduced a function algebra closed under bounded recursion on notation to capture FP. Later on, several complexity classes have been characterized using limited recursion schemas. In this context, a new approach was recently introduced, showing that ordinary differential equations (ODEs) offer a natural tool for algorithmic design and providing a characterization of FP by an ODE-schema. The overall goal of the present work is precisely that of generalizing this approach to parallel computation, obtaining an original ODE-characterization for the small circuit classes FAC⁰ and FTC⁰.

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Melissa Antonelli, Arnaud Durand, and Juha Kontinen. A New Characterization of FAC⁰ via Discrete Ordinary Differential Equations. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{antonelli_et_al:LIPIcs.MFCS.2024.10,
  author =	{Antonelli, Melissa and Durand, Arnaud and Kontinen, Juha},
  title =	{{A New Characterization of FAC⁰ via Discrete Ordinary Differential Equations}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{10:1--10:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.10},
  URN =		{urn:nbn:de:0030-drops-205664},
  doi =		{10.4230/LIPIcs.MFCS.2024.10},
  annote =	{Keywords: Implicit computational complexity, parallel computation, ordinary differential equations, circuit complexity}
}

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DOI: 10.4230/LIPIcs.CSL.2024.10

Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories

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

Published in: LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

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.

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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)

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@InProceedings{antonelli_et_al:LIPIcs.CSL.2024.10,
  author =	{Antonelli, Melissa and Dal Lago, Ugo and Davoli, Davide and Oitavem, Isabel and Pistone, Paolo},
  title =	{{Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories}},
  booktitle =	{32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)},
  pages =	{10:1--10:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-310-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{288},
  editor =	{Murano, Aniello and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.10},
  URN =		{urn:nbn:de:0030-drops-196538},
  doi =		{10.4230/LIPIcs.CSL.2024.10},
  annote =	{Keywords: Bounded Arithmetic, Randomized Computation, Implicit Computational Complexity}
}

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A New Characterization of FAC⁰ via Discrete Ordinary Differential Equations

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Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories

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