Learning Read-Constant Polynomials of Constant Degree Modulo Composites

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.

[1]  Klim Efremenko,et al.  3-Query Locally Decodable Codes of Subexponential Length , 2008 .

[2]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[3]  Denis Thérien,et al.  On the Languages Recognized by Nilpotent Groups (a translation of "Sur les Langages Reconnus par des Groupes Nilpotents") , 2001, Electron. Colloquium Comput. Complex..

[4]  Parikshit Gopalan.  Constructing Ramsey Graphs from Boolean Function Representations , 2006, Computational Complexity Conference.

[5]  Eyal Kushilevitz,et al.  On Learning Read-k-Satisfy-j DNF , 1998, SIAM J. Comput..

[6]  Marek Karpinski,et al.  Learning read-once formulas with queries , 1993, JACM.

[7]  David A. Mix Barrington,et al.  Bounded-width polynomial-size branching programs recognize exactly those languages in NC1 , 1986, STOC '86.

[8]  Ran Raz,et al.  Pseudorandom Generators for Regular Branching Programs , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[9]  Howard Straubing,et al.  Superlinear Lower Bounds for Bounded-Width Branching Programs , 1995, J. Comput. Syst. Sci..

[10]  Pavel Pudlák,et al.  Pseudorandom generators for group products: extended abstract , 2011, STOC '11.

[11]  Arkadev Chattopadhyay.  Multilinear Polynomials Modulo Composites , 2010, Bull. EATCS.

[12]  Joshua Brody,et al.  The Coin Problem and Pseudorandomness for Branching Programs , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[13]  Vince Grolmusz.  On the Weak mod m Representation of Boolean Functions , 1995, Chic. J. Theor. Comput. Sci..

[14]  Howard Straubing,et al.  Non-Uniform Automata Over Groups , 1990, Inf. Comput..

[15]  David A. Mix Barrington,et al.  Representing Boolean functions as polynomials modulo composite numbers , 1994, computational complexity.

[16]  Nader H. Bshouty,et al.  Learning Matrix Functions over Rings , 1998, Algorithmica.

[17]  Eyal Kushilevitz,et al.  Learning functions represented as multiplicity automata , 2000, JACM.

[18]  Gábor Tardos,et al.  A lower bound on the mod 6 degree of the OR function , 1998, computational complexity.

[19]  A. Razborov.  Lower bounds on the size of bounded depth circuits over a complete basis with logical addition , 1987 .

[20]  Denis Thérien,et al.  An Algebraic Perspective on Boolean Function Learning , 2009, ALT.

[21]  Linda Sellie,et al.  Learning sparse multivariate polynomials over a field with queries and counterexamples , 1993, COLT '93.

[22]  J. Bourgain.  Estimation of certain exponential sums arising in complexity theory , 2005 .

[23]  Manfred K. Warmuth,et al.  Learning Nested Differences of Intersection-Closed Concept Classes , 1989, COLT '89.