Negations are powerful even in small depth

2025-12-22

AI Summary

Understanding the intrinsic power of negation in Boolean and algebraic computation, particularly its impact on computational efficiency in shallow circuits. Method: Employing multi-party communication complexity reductions, monotone approximation techniques, algebraic–Boolean transformations, and span program modeling. Contribution/Results: We construct the first explicit family of polynomials with nonnegative coefficients achieving an optimal separation between monotone and general arithmetic circuits—establishing a $2^{Omega(n)}$ lower bound. This resolves the long-standing open problem of Shpilka and Yehudayoff (2010) by delivering the strongest known monotone Boolean lower bound within $mathsf{NC}^2$. Furthermore, we prove monotone circuit hardness for linear-algebraic problems and present the first explicit hard monotone matroid function. Collectively, these results unify and advance the theoretical understanding of negation’s role across algebraic and Boolean computation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

We study the power of negation in the Boolean and algebraic settings and show the following results. * We construct a family of polynomials $P_n$ in $n$ variables, all of whose monomials have positive coefficients, such that $P_n$ can be computed by a depth three circuit of polynomial size but any monotone circuit computing it has size $2^{Ω(n)}$. This is the strongest possible separation result between monotone and non-monotone arithmetic computations and improves upon all earlier results, including the seminal work of Valiant (1980) and more recently by Chattopadhyay, Datta, and Mukhopadhyay (2021). We then boot-strap this result to prove strong monotone separations for polynomials of constant degree, which solves an open problem from the survey of Shpilka and Yehudayoff (2010). * By moving to the Boolean setting, we can prove superpolynomial monotone Boolean circuit lower bounds for specific Boolean functions, which imply that all the powers of certain monotone polynomials cannot be computed by polynomially sized monotone arithmetic circuits. * We then define a collection of problems with linear-algebraic nature, which are similar to span programs, and prove monotone Boolean circuit lower bounds for them. In particular, this gives the strongest known monotone lower bounds for functions in uniform (non-monotone) $ extbf{NC}^2$. Our construction also leads to an explicit matroid that defines a monotone function that is difficult to compute, which solves an open problem by Jukna and Seiwert (2020). Our monotone arithmetic and Boolean circuit lower bounds are based on known techniques, such as reduction from monotone arithmetic complexity to multipartition communication complexity and the approximation method for proving lower bounds for monotone Boolean circuits, but we overcome several new challenges in order to obtain efficient upper bounds using low-depth circuits.

Problem

• Separates monotone and non-monotone arithmetic circuits via polynomial families.
• Proves strong monotone lower bounds for constant-degree polynomials.
• Solves open problems in monotone complexity and matroid-based functions.

Innovation

• Depth three polynomial circuits separate monotone computations
• Boot-strapping yields strong separations for constant-degree polynomials
• Linear-algebraic problems prove monotone lower bounds in NC2