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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

What is the power of constant-depth circuits with MOD_m gates, that can count modulo m? Can they efficiently compute MAJORITY and other symmetric functions? When m is a constant prime power, the answer is well understood. In this regime, Razborov and Smolensky proved in the 1980s that MAJORITY and MOD_m require super-polynomial-size MOD_q circuits, where q is any prime power not dividing m. However, relatively little is known about the power of MOD_m gates when m is not a prime power. For example, it is still open whether every problem decidable in exponential time can be computed by depth-3 circuits of polynomial-size and only MOD_6 gates.
In this paper, we shed some light on the difficulty of proving lower bounds for MOD_m circuits, by giving new upper bounds. We show how to construct MOD_m circuits computing symmetric functions with non-prime power m, with size-depth tradeoffs that beat the longstanding lower bounds for AC^0[m] circuits when m is a prime power. Furthermore, we observe that our size-depth tradeoff circuits have essentially optimal dependence on m and d in the exponent, under a natural circuit complexity hypothesis.
For example, we show that for every ε > 0, every symmetric function can be computed using MOD_m circuits of depth 3 and 2^{n^ε} size, for a constant m depending only on ε > 0. In other words, depth-3 CC^0 circuits can compute any symmetric function in subexponential size. This demonstrates a significant difference in the power of depth-3 CC^0 circuits, compared to other models: for certain symmetric functions, depth-3 AC^0 circuits require 2^{Ω(√n)} size [Håstad 1986], and depth-3 AC^0[p^k] circuits (for fixed prime power p^k) require 2^{Ω(n^{1/6})} size [Smolensky 1987]. Even for depth-2 MOD_p ∘ MOD_m circuits, 2^{Ω(n)} lower bounds were known [Barrington Straubing Thérien 1990].

Brynmor Chapman and R. Ryan Williams. Smaller ACC0 Circuits for Symmetric Functions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 38:1-38:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chapman_et_al:LIPIcs.ITCS.2022.38, author = {Chapman, Brynmor and Williams, R. Ryan}, title = {{Smaller ACC0 Circuits for Symmetric Functions}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {38:1--38:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.38}, URN = {urn:nbn:de:0030-drops-156342}, doi = {10.4230/LIPIcs.ITCS.2022.38}, annote = {Keywords: ACC, CC, circuit complexity, symmetric functions, Chinese Remainder Theorem} }

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**Published in:** LIPIcs, Volume 118, 29th International Conference on Concurrency Theory (CONCUR 2018)

We study the growth behaviour of rational linear recurrence sequences. We show that for low-order sequences, divergence is decidable in polynomial time. We also exhibit a polynomial-time algorithm which takes as input a divergent rational linear recurrence sequence and computes effective fine-grained lower bounds on the growth rate of the sequence.

Shaull Almagor, Brynmor Chapman, Mehran Hosseini, Joël Ouaknine, and James Worrell. Effective Divergence Analysis for Linear Recurrence Sequences. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 42:1-42:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{almagor_et_al:LIPIcs.CONCUR.2018.42, author = {Almagor, Shaull and Chapman, Brynmor and Hosseini, Mehran and Ouaknine, Jo\"{e}l and Worrell, James}, title = {{Effective Divergence Analysis for Linear Recurrence Sequences}}, booktitle = {29th International Conference on Concurrency Theory (CONCUR 2018)}, pages = {42:1--42:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-087-3}, ISSN = {1868-8969}, year = {2018}, volume = {118}, editor = {Schewe, Sven and Zhang, Lijun}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2018.42}, URN = {urn:nbn:de:0030-drops-95802}, doi = {10.4230/LIPIcs.CONCUR.2018.42}, annote = {Keywords: Linear recurrence sequences, Divergence, Algebraic numbers, Positivity} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

What sort of code is so difficult to analyze that every potential analyst can discern essentially no information from the code, other than its input-output behavior? In their seminal work on program obfuscation, Barak, Goldreich, Impagliazzo, Rudich, Sahai, Vadhan, and Yang (CRYPTO 2001) proposed the Black-Box Hypothesis, which roughly states that every property of Boolean functions which has an efficient "analyst" and is "code independent" can also be computed by an analyst that only has black-box access to the code. In their formulation of the Black-Box Hypothesis, the "analysts" are arbitrary randomized polynomial-time algorithms, and the "codes" are general (polynomial-size) circuits. If true, the Black-Box Hypothesis would immediately imply NP ̸ ⊂ BPP.
We consider generalized forms of the Black-Box Hypothesis, where the set of "codes" 𝒞 and the set of "analysts" 𝒜 may correspond to other efficient models of computation, from more restricted models such as AC⁰ to more general models such as nondeterministic circuits. We show how lower bounds of the form 𝒞 ̸ ⊂ 𝒜 often imply a corresponding Black-Box Hypothesis for those respective codes and analysts. We investigate the possibility of "complete" problems for the Black-Box Hypothesis: problems in 𝒞 such that they are not in 𝒜 if and only if their corresponding Black-Box Hypothesis is true. Along the way, we prove an equivalence: for nondeterministic circuit classes 𝒞, the "𝒞-circuit satisfiability problem" is not in 𝒜 if and only if the Black-Box Hypothesis is true for analysts in 𝒜.

Brynmor K. Chapman and R. Ryan Williams. Black-Box Hypotheses and Lower Bounds. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 29:1-29:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chapman_et_al:LIPIcs.MFCS.2021.29, author = {Chapman, Brynmor K. and Williams, R. Ryan}, title = {{Black-Box Hypotheses and Lower Bounds}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {29:1--29:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.29}, URN = {urn:nbn:de:0030-drops-144698}, doi = {10.4230/LIPIcs.MFCS.2021.29}, annote = {Keywords: Black-Box hypothesis, circuit complexity, lower bounds} }

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