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arXiv · 2311.15003

Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories

Melissa Antonelli·Ugo Dal Lago·Davide Davoli·Isabel Oitavem·Paolo Pistone
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Abstract

We consider a minimal extension of the language of arithmetic, such that the bounded formulas provably total in a suitably-defined theory \`a 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 the theory T. As a paradigmatic example of this approach, we establish that polynomial identity testing is in BPP_T where T=$\mathrm{I}\Delta_0+\mathrm{Exp}$ is a well-studied theory based on bounded induction.

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Melissa Antonelli, Ugo Dal Lago, Davide Davoli, Isabel Oitavem, Paolo Pistone. 2023-11-25. Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories. https://arxiv.org/abs/2311.15003

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