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Constant-Depth Circuits vs. Monotone Circuits

Authors Bruno P. Cavalar , Igor C. Oliveira

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Author Details

Bruno P. Cavalar
  • University of Warwick, Coventry, UK
Igor C. Oliveira
  • University of Warwick, Coventry, UK

Funding

This work received support from the Royal Society University Research Fellowship URF⧵R1⧵191059, the EPSRC New Horizons Grant EP/V048201/1, and the Centre for Discrete Mathematics and its Applications (DIMAP) at the University of Warwick.

Acknowledgements

We thank Arkadev Chattopadhyay for several conversations about the AC⁰ versus mSIZE[poly] problem and related questions. We are also grateful to Denis Kuperberg for explaining to us the results from [Denis Kuperberg, 2021; Denis Kuperberg, 2022]. The first author thanks Ninad Rajgopal for helpful discussions about depth reduction. Finally, we thank Gernot Salzer for the code used to generate Figures 1, 2, and 3.

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Bruno P. Cavalar and Igor C. Oliveira. Constant-Depth Circuits vs. Monotone Circuits. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 29:1-29:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CCC.2023.29

Abstract

We establish new separations between the power of monotone and general (non-monotone) Boolean circuits:  
- For every k ≥ 1, there is a monotone function in AC⁰ (constant-depth poly-size circuits) that requires monotone circuits of depth Ω(log^k n). This significantly extends a classical result of Okol'nishnikova [Okol'nishnikova, 1982] and Ajtai and Gurevich [Ajtai and Gurevich, 1987]. In addition, our separation holds for a monotone graph property, which was unknown even in the context of AC⁰ versus mAC⁰. 
- For every k ≥ 1, there is a monotone function in AC⁰[⊕] (constant-depth poly-size circuits extended with parity gates) that requires monotone circuits of size exp(Ω(log^k n)). This makes progress towards a question posed by Grigni and Sipser [Grigni and Sipser, 1992].  These results show that constant-depth circuits can be more efficient than monotone formulas and monotone circuits when computing monotone functions.
In the opposite direction, we observe that non-trivial simulations are possible in the absence of parity gates: every monotone function computed by an AC⁰ circuit of size s and depth d can be computed by a monotone circuit of size 2^{n - n/O(log s)^{d-1}}. We show that the existence of significantly faster monotone simulations would lead to breakthrough circuit lower bounds. In particular, if every monotone function in AC⁰ admits a polynomial size monotone circuit, then NC² is not contained in NC¹.
Finally, we revisit our separation result against monotone circuit size and investigate the limits of our approach, which is based on a monotone lower bound for constraint satisfaction problems (CSPs) established by Göös, Kamath, Robere and Sokolov [Göös et al., 2019] via lifting techniques. Adapting results of Schaefer [Thomas J. Schaefer, 1978] and Allender, Bauland, Immerman, Schnoor and Vollmer [Eric Allender et al., 2009], we obtain an unconditional classification of the monotone circuit complexity of Boolean-valued CSPs via their polymorphisms. This result and the consequences we derive from it might be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • circuit complexity
  • monotone circuit complexity
  • bounded-depth circuis
  • constraint-satisfaction problems

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