LIPIcs.CCC.2023.19.pdf
- Filesize: 2 MB
- 24 pages
Document
Criticality of AC⁰-Formulae
Authors
Prahladh Harsha 
,
-
Part of:
Volume:
38th Computational Complexity Conference (CCC 2023)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Computational Complexity Conference (CCC) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2023-07-10
File
Document Identifiers
Author Details
Acknowledgements
The first and the second authors spent several years thinking about this problem and we are indebted to several people along the way. First and foremost, we thank Jaikumar Radhakrishnan and Ramprasad Saptharishi for spending innumerable hours in the various stages of this project going over several failed attempts and potential proofs and giving us very helpful feedback along the way. We are grateful to Ben Rossman for discussions in the early stages of this project as well as pointing out an error in the previous version of this proof. We would also like to thank Srikanth Srinivasan, Siddharth Bhandari, Yuval Filmus and Mrinal Kumar for their comments and feedback.
Cite As Get BibTex
Prahladh Harsha, Tulasimohan Molli, and Ashutosh Shankar. Criticality of AC⁰-Formulae. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CCC.2023.19
Abstract
Rossman [In Proc. 34th Comput. Complexity Conf., 2019] introduced the notion of criticality. The criticality of a Boolean function f : {0,1}ⁿ → {0,1} is the minimum λ ≥ 1 such that for all positive integers t and all p ∈ [0,1], Pr_{ρ∼ℛ_p}[DT_{depth}(f|_ρ) ≥ t] ≤ (pλ)^t, where ℛ_p refers to the distribution of p-random restrictions.
Håstad’s celebrated switching lemma shows that the criticality of any k-DNF is at most O(k). Subsequent improvements to correlation bounds of AC⁰-circuits against parity showed that the criticality of any AC⁰-circuit of size S and depth d+1 is at most O(log S)^d and any regular AC⁰-formula of size S and depth d+1 is at most O((1/d)⋅log S)^d. We strengthen these results by showing that the criticality of any AC⁰-formula (not necessarily regular) of size S and depth d+1 is at most O((log S)/d)^d, resolving a conjecture due to Rossman.
This result also implies Rossman’s optimal lower bound on the size of any depth-d AC⁰-formula computing parity [Comput. Complexity, 27(2):209-223, 2018.]. Our result implies tight correlation bounds against parity, tight Fourier concentration results and improved #SAT algorithm for AC⁰-formulae.
Subject Classification
ACM Subject Classification
- Theory of computation → Circuit complexity
Keywords
- AC⁰ circuits
- AC⁰ formulae
- criticality
- switching lemma
- correlation bounds
Metrics
- Access Statistics
-
Total Accesses (updated on a weekly basis)
0PDF Downloads
References
- Miklós Ajtai. Σ₁¹-formulae on finite structures. Ann. Pure Appl. Logic, 24(1):1-48, July 1983. URL: https://doi.org/10.1016/0168-0072(83)90038-6.
- Paul Beame. A switching lemma primer. Technical Report UW-CSE-95-07-01, University of Washington, 1994. URL: http://www.cs.washington.edu/homes/beame/papers/primer.ps.
- Paul Beame, Russell Impagliazzo, and Srikanth Srinivasan. Approximating AC⁰ by small height decision trees and a deterministic algorithm for #AC⁰-SAT. In Venkatesan Guruswami, editor, Proc. 27th IEEE Conf. on Comput. Complexity, pages 117-125, 2012. URL: https://doi.org/10.1109/CCC.2012.40.
- Merrick L. Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17(1):13-27, 1984. (Preliminary version in 22nd FOCS, 1981). URL: https://doi.org/10.1007/BF01744431.
- Johan Håstad. Almost optimal lower bounds for small depth circuits. In Silvio Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 143-170. JAI Press, Greenwich, Connecticut, 1989. (Preliminary version in 18th STOC 1986). URL: http://www.csc.kth.se/~johanh/largesmalldepth.pdf.
- Johan Håstad. On the correlation of parity and small-depth circuits. SIAM J. Comput., 43(5):1699-1708, 2014. URL: https://doi.org/10.1137/120897432.
- Russell Impagliazzo, William Matthews, and Ramamohan Paturi. A satisfiability algorithm for AC⁰. In Yuval Rabani, editor, Proc. 23rd Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 961-972, 2012. URL: https://doi.org/10.1137/1.9781611973099.77.
- Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, Fourier transform, and learnability. J. ACM, 40(3):607-620, 1993. (Preliminary version in 30th FOCS, 1989). URL: https://doi.org/10.1145/174130.174138.
- Alexander A. Razborov. Нжние оценки размера схем ограниченной глубины в полном базисе, содержащем функцию логического сложения (Russian) [Lower bounds on the size of bounded depth circuits over a complete basis with logical addition]. Mathematicheskie Zametki, 41(4):598-607, 1987. (English translation in Mathematical Notes of the Academy of Sciences of the USSR, 41(4):333-338, 1987). URL: https://doi.org/10.1007/BF01137685.
- Alexander A. Razborov. Bounded arithmetic and lower bounds in Boolean complexity. In Peter Clote and Jeffrey B. Remmel, editors, Feasible Mathematics II, volume 13 of Progress in Computer Science and Applied Logic, pages 344-387. Birkhäuser, 1995. URL: https://doi.org/10.1007/978-1-4612-2566-9_12.
- Alexander A. Razborov and Steven Rudich. Natural proofs. J. Comput. Syst. Sci., 55(1):24-35, 1997. (Preliminary version in 26th STOC, 1991). URL: https://doi.org/10.1006/jcss.1997.1494.
- Benjamin Rossman. An entropy proof of the switching lemma and tight bounds on the decision-tree size of AC⁰. (manuscript), 2017. URL: https://users.cs.duke.edu/~br148/logsize.pdf.
- Benjamin Rossman. The average sensitivity of bounded-depth formulas. Comput. Complexity, 27(2):209-223, 2018. (Preliminary version in 56th FOCS, 2015). URL: https://doi.org/10.1007/s00037-017-0156-0.
- Benjamin Rossman. Criticality of regular formulas. In Amir Shpilka, editor, Proc. 34th Comput. Complexity Conf., volume 137 of LIPIcs, pages 1:1-1:28. Schloss Dagstuhl, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.1.
- Michael Sipser. Borel sets and circuit complexity. In David S. Johnson, Ronald Fagin, Michael L. Fredman, David Harel, Richard M. Karp, Nancy A. Lynch, Christos H. Papadimitriou, Ronald L. Rivest, Walter L. Ruzzo, and Joel I. Seiferas, editors, Proc. 15th ACM Symp. on Theory of Computing (STOC), pages 61-69, 1983. URL: https://doi.org/10.1145/800061.808733.
- Roman Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Alfred V. Aho, editor, Proc. 19th ACM Symp. on Theory of Computing (STOC), pages 77-82, 1987. URL: https://doi.org/10.1145/28395.28404.
- Avishay Tal. Tight bounds on the Fourier Spectrum of AC⁰. In Ryan O'Donnell, editor, Proc. 32nd Comput. Complexity Conf., volume 79 of LIPIcs, pages 15:1-15:31. Schloss Dagstuhl, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.15.
- Andrew Chi-Chih Yao. Separating the polynomial-time hierarchy by oracles (preliminary version). In Manuel Blum, John Hopcroft, Jeff Lagarias, Tom Leighton, Charles Rackoff, Larry Ruzzo, Larry Stockmeyer, Robert Tarjan, and Frances Yao, editors, Proc. 26th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 1-10, 1985. URL: https://doi.org/10.1109/SFCS.1985.49.