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Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0 (Extended Abstract)

Authors Yuval Filmus , Or Meir , Avishay Tal

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Author Details

Yuval Filmus
  • Technion - Israel Institute of Technology, Haifa, Israel
Or Meir
  • Department of Computer Science, University of Haifa, Israel
Avishay Tal
  • Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, CA, USA


A.T. would like to thank Igor Carboni Oliveira for bringing the question of proving formula size lower bounds for AC⁰ to his attention. We are also grateful to Robin Kothari for posing this open question on "Theoretical Computer Science Stack Exchange" [Kothari, 2011], and to Kaveh Ghasemloo and Stasys Jukna for their feedback on this question. We would like to thank Anna Gál for very helpful discussions.

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Yuval Filmus, Or Meir, and Avishay Tal. Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0 (Extended Abstract). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 89:1-89:7, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


Håstad showed that any De Morgan formula (composed of AND, OR and NOT gates) shrinks by a factor of O(p²) under a random restriction that leaves each variable alive independently with probability p [SICOMP, 1998]. Using this result, he gave an Ω̃(n³) formula size lower bound for the Andreev function, which, up to lower order improvements, remains the state-of-the-art lower bound for any explicit function. In this work, we extend the shrinkage result of Håstad to hold under a far wider family of random restrictions and their generalization - random projections. Based on our shrinkage results, we obtain an Ω̃(n³) formula size lower bound for an explicit function computed in AC⁰. This improves upon the best known formula size lower bounds for AC⁰, that were only quadratic prior to our work. In addition, we prove that the KRW conjecture [Karchmer et al., Computational Complexity 5(3/4), 1995] holds for inner functions for which the unweighted quantum adversary bound is tight. In particular, this holds for inner functions with a tight Khrapchenko bound. Our random projections are tailor-made to the function’s structure so that the function maintains structure even under projection - using such projections is necessary, as standard random restrictions simplify AC⁰ circuits. In contrast, we show that any De Morgan formula shrinks by a quadratic factor under our random projections, allowing us to prove the cubic lower bound. Our proof techniques build on the proof of Håstad for the simpler case of balanced formulas. This allows for a significantly simpler proof at the cost of slightly worse parameters. As such, when specialized to the case of p-random restrictions, our proof can be used as an exposition of Håstad’s result.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • De Morgan formulas
  • KRW Conjecture
  • shrinkage
  • random restrictions
  • random projections
  • bounded depth circuits
  • constant depth circuits
  • formula complexity


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