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Learning Read-Constant Polynomials of Constant Degree Modulo Composites

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Abstract

Boolean functions that have constant degree polynomial representation over a fixed finite ring form a natural and strict subclass of the complexity class ACC0. They are also precisely the functions computable efficiently by programs over fixed and finite nilpotent groups. This class is not known to be learnable in any reasonable learning model.

In this paper, we provide a deterministic polynomial time algorithm for learning Boolean functions represented by polynomials of constant degree over arbitrary finite rings from membership queries, with the additional constraint that each variable in the target polynomial appears in a constant number of monomials. Our algorithm extends to superconstant but low degree polynomials and still runs in quasipolynomial time.

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Notes

  1. In an equivalence query, the learning algorithm emits some representation of a Boolean function and receives as answer either a Boolean assignment where it differs from the target function, or “yes” if no such assignment exists.

  2. Alternatively one could fix the same ordering, say 1,…,n for all monomials. However we find it natural to identify monomials that are identical up to a permutation of the variables.

  3. A DNF formula is read-k if every variable appears at most k times; a DNF formula is satisfy-j if no assignment satisfies more than j terms simultaneously.

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Acknowledgements

A. Chattopadhyay was at the University of Toronto when this work was written up, partially supported by a Natural Sciences and Engineering Research Council (NSERC) postdoctoral fellowship, an Ontario Ministry of Research and Innovation fellowship and research grants of Prof. T. Pitassi. R. Gavaldà is partially funded by the Spanish MICINN projects TIN-2008-06582-C03-01 (SESAAME) and TIN2011-27479-C04-03 (BASMATI), by the Generalitat de Catalunya SGR2009-1428 (LARCA), and by the EU PASCAL2 Network of Excellence (FP7-ICT-216886). We thank the anonymous reviewers for their useful comments that improved the presentation. In particular, Remark 5 was pointed out by a reviewer.

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Authors and Affiliations

  1. Tata Institute of Fundamental Research, Mumbai, India

    Arkadev Chattopadhyay

  2. Universitat Politècnica de Catalunya, Barcelona, Catalonia, Spain

    Ricard Gavaldà

  3. Aarhus University, Aarhus, Denmark

    Kristoffer Arnsfelt Hansen

  4. McGill University, Montreal, Quebec, Canada

    Denis Thérien

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  1. Arkadev Chattopadhyay
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  2. Ricard Gavaldà
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  3. Kristoffer Arnsfelt Hansen
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  4. Denis Thérien
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Correspondence to Ricard Gavaldà.

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Chattopadhyay, A., Gavaldà, R., Hansen, K.A. et al. Learning Read-Constant Polynomials of Constant Degree Modulo Composites. Theory Comput Syst 55, 404–420 (2014). https://doi.org/10.1007/s00224-013-9488-6

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