Learning circuits with few negations

TL;DR

This paper investigates the complexity of Boolean functions based on the minimum number of negations in their circuits, providing new structural insights and learnability bounds that bridge monotone and general functions.

Contribution

It introduces a structural characterization of negation-limited circuits and establishes near-matching upper and lower bounds on their learnability, extending classical results.

Findings

Structural characterization of negation-limited circuits

Near-matching upper and lower bounds on learnability

New Fourier-analytic lower bounds for monotone functions

Abstract

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, giving near-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).