Learning Circuits with few Negations

Authors Eric Blais, Clément L. Canonne, Igor C. Oliveira, Rocco A. Servedio, Li-Yang Tan

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Eric Blais
Clément L. Canonne
Igor C. Oliveira
Rocco A. Servedio
Li-Yang Tan

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Eric Blais, Clément L. Canonne, Igor C. Oliveira, Rocco A. Servedio, and Li-Yang Tan. Learning Circuits with few Negations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 512-527, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, establishing nearly matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A.A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions).
  • Boolean functions
  • monotonicity
  • negations
  • PAC learning


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