Learning read-constant polynomials of constant degree modulo composites

Authors:
Arkadev Chattopadhyay;Ricard Gavaldà;Kristoffer Arnsfelt Hansen;Denis Thérien
Affiliations:
University of Toronto;Universitat Politècnica de Catalunya;Aarhus University;McGill University
Venue:
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
Year:
2011

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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.