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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We study the formula complexity of the word problem Word_{S_n,k} : {0,1}^{kn²} → {0,1}: given n-by-n permutation matrices M₁,… ,M_k, compute the (1,1)-entry of the matrix product M₁⋯ M_k. An important feature of this function is that it is invariant under action of S_n^{k-1} given by (π₁,… ,π_{k-1})(M₁,… ,M_k) = (M₁π₁^{-1},π₁M₂π₂^{-1},… ,π_{k-2}M_{k-1}π_{k-1}^{-1},π_{k-1}M_k).
This symmetry is also exhibited in the smallest known unbounded fan-in {and,or,not}-formulas for Word_{S_n,k}, which have size n^O(log k).
In this paper we prove a matching n^{Ω(log k)} lower bound for S_n^{k-1}-invariant formulas computing Word_{S_n,k}. This result is motivated by the fact that a similar lower bound for unrestricted (non-invariant) formulas would separate complexity classes NC¹ and Logspace.
Our more general main theorem gives a nearly tight n^d(k^{1/d}-1) lower bound on the G^{k-1}-invariant depth-d {maj,and,or,not}-formula size of Word_{G,k} for any finite simple group G whose minimum permutation representation has degree n. We also give nearly tight lower bounds on the G^{k-1}-invariant depth-d {and,or,not}-formula size in the case where G is an abelian group.

William He and Benjamin Rossman. Symmetric Formulas for Products of Permutations. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 68:1-68:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{he_et_al:LIPIcs.ITCS.2023.68, author = {He, William and Rossman, Benjamin}, title = {{Symmetric Formulas for Products of Permutations}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {68:1--68:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.68}, URN = {urn:nbn:de:0030-drops-175717}, doi = {10.4230/LIPIcs.ITCS.2023.68}, annote = {Keywords: circuit complexity, group-invariant formulas} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We establish nearly tight bounds on the expected shrinkage of decision lists and DNF formulas under the p-random restriction R_p for all values of p ∈ [0,1]. For a function f with domain {0,1}ⁿ, let DL(f) denote the minimum size of a decision list that computes f. We show that E[DL(f ↾ R_p)] ≤ DL(f)^log_{2/(1-p)}((1+p)/(1-p)). For example, this bound is √{DL(f)} when p = √5-2 ≈ 0.24. For Boolean functions f, we obtain the same shrinkage bound with respect to DNF formula size plus 1 (i.e., replacing DL(⋅) with DNF(⋅)+1 on both sides of the inequality).

Benjamin Rossman. Shrinkage of Decision Lists and DNF Formulas. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{rossman:LIPIcs.ITCS.2021.70, author = {Rossman, Benjamin}, title = {{Shrinkage of Decision Lists and DNF Formulas}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {70:1--70:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.70}, URN = {urn:nbn:de:0030-drops-136098}, doi = {10.4230/LIPIcs.ITCS.2021.70}, annote = {Keywords: shrinkage, decision lists, DNF formulas} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

We define the criticality of a boolean function f : {0,1}^n -> {0,1} as the minimum real number lambda >= 1 such that Pr [DT_{depth}(f|R_p) >= t] <= (p lambda)^t for all p in [0,1] and t in N, where R_p is the p-random restriction and DT_{depth} is decision-tree depth. Criticality is a useful parameter: it implies an O(2^((1- 1/(2 lambda))n)) bound on the decision-tree size of f, as well as a 2^{-Omega(k/lambda)} bound on Fourier weight of f on coefficients of size >= k.
In an unpublished manuscript [Rossmann, 2018], the author showed that a combination of Håstad’s switching and multi-switching lemmas [Håstad, 1986; Håstad, 2014] implies that AC^0 circuits of depth d+1 and size s have criticality at most O(log s)^d. In the present paper, we establish a stronger O(1/d log s)^d bound for regular formulas: the class of AC^0 formulas in which all gates at any given depth have the same fan-in. This result is based on
(i) a novel switching lemma for bounded size (unbounded width) DNF formulas, and
(ii) an extension of (i) which analyzes a canonical decision tree associated with an entire depth-d formula.
As corollaries of our criticality bound, we obtain an improved #SAT algorithm and tight Linial-Mansour-Nisan Theorem for regular formulas, strengthening previous results for AC^0 circuits due to Impagliazzo, Matthews, Paturi [Impagliazzo et al., 2012] and Tal [Tal, 2017]. As a further corollary, we increase from o(log n /(log log n)) to o(log n) the number of quantifier alternations for which the QBF-SAT (quantified boolean formula satisfiability) algorithm of Santhanam and Williams [Santhanam and Williams, 2014] beats exhaustive search.

Benjamin Rossman. Criticality of Regular Formulas. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 1:1-1:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{rossman:LIPIcs.CCC.2019.1, author = {Rossman, Benjamin}, title = {{Criticality of Regular Formulas}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {1:1--1:28}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.1}, URN = {urn:nbn:de:0030-drops-108230}, doi = {10.4230/LIPIcs.CCC.2019.1}, annote = {Keywords: AC^0 circuits, formulas, criticality} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Previous work of the author [Rossmann'08] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC0 formula size of the colored subgraph isomorphism problem. Formally, we show the following: if a first-order sentence of quantifier-rank k is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence of quantifier-rank poly(k). Quantitatively, this improves the result of [Rossmann'08], where the upper bound on quantifier-rank is a non-elementary function of k.

Benjamin Rossman. An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{rossman:LIPIcs.ITCS.2017.27, author = {Rossman, Benjamin}, title = {{An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {27:1--27:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.27}, URN = {urn:nbn:de:0030-drops-81435}, doi = {10.4230/LIPIcs.ITCS.2017.27}, annote = {Keywords: circuit complexity, finite model theory} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

This paper gives the first separation between the power of formulas and circuits of equal depth in the AC^0[\oplus] basis (unbounded fan-in AND, OR, NOT and MOD_2 gates). We show, for all d(n) <= O(log n/log log n), that there exist polynomial-size depth-d circuits that are not equivalent to depth-d formulas of size n^{o(d)} (moreover, this is optimal in that n^{o(d)} cannot be improved to n^{O(d)}). This result is obtained by a combination of new lower and upper bounds for Approximate Majorities, the class of Boolean functions {0,1}^n to {0,1} that agree with the Majority function on 3/4 fraction of inputs.
AC^0[\oplus] formula lower bound.
We show that every depth-d AC^0[\oplus] formula of size s has a (1/8)-error polynomial approximation over F_2 of degree O((log s)/d)^{d-1}. This strengthens a classic $O(log s)^{d-1}$ degree approximation for circuits due to Razborov. Since the Majority function has approximate degree Theta(\sqrt n), this result implies an \exp(\Omega(dn^{1/2(d-1)})) lower bound on the depth-d AC^0[\oplus] formula size of all Approximate Majority functions for all d(n) <= O(log n).
Monotone AC^0 circuit upper bound.
For all d(n) <= O(log n/log log n), we give a randomized construction of depth-d monotone AC^0 circuits (without NOT or MOD_2 gates) of size \exp(O(n^{1/2(d-1)}))} that compute an Approximate Majority function. This strengthens a construction of formulas of size \exp(O(dn^{1/2(d-1)})) due to Amano.

Benjamin Rossman and Srikanth Srinivasan. Separation of AC^0[oplus] Formulas and Circuits. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{rossman_et_al:LIPIcs.ICALP.2017.50, author = {Rossman, Benjamin and Srinivasan, Srikanth}, title = {{Separation of AC^0\lbrackoplus\rbrack Formulas and Circuits}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {50:1--50:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.50}, URN = {urn:nbn:de:0030-drops-73904}, doi = {10.4230/LIPIcs.ICALP.2017.50}, annote = {Keywords: circuit complexity, lower bounds, approximate majority, polynomial method} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

The n-variable PARITY function is computable (by a well-known recursive construction) by AC^0 formulas of depth d+1 and leaf size n2^{dn^{1/d}}. These formulas are seen to possess a certain symmetry: they are syntactically invariant under the subspace P of even-weight elements in {0,1}^n, which acts (as a group) on formulas by toggling negations on input literals. In this paper, we prove a 2^{d(n^{1/d}-1)} lower bound on the size of syntactically P-invariant depth d+1 formulas for PARITY. Quantitatively, this beats the best 2^{Omega(d(n^{1/d}-1))} lower bound in the non-invariant setting.

Benjamin Rossman. Subspace-Invariant AC^0 Formulas. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 93:1-93:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{rossman:LIPIcs.ICALP.2017.93, author = {Rossman, Benjamin}, title = {{Subspace-Invariant AC^0 Formulas}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {93:1--93:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.93}, URN = {urn:nbn:de:0030-drops-74235}, doi = {10.4230/LIPIcs.ICALP.2017.93}, annote = {Keywords: lower bounds, size-depth tradeoff, parity, symmetry in computation} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

This paper gives the first correlation bounds under product distributions, including the uniform distribution, against the class mNC^ of polynomial-size O(log(n))-depth monotone circuits. Our main theorem, proved using the pathset complexity framework introduced in [Rossmann,arXiv:1312.0355], shows that the average-case k-CYCLE problem (on Erdös-Renyi random graphs with an appropriate edge density) is 1/2 + 1/poly(n) hard for mNC^1. Combining this result with O'Donnell's hardness amplification theorem [O'Donnell,2002], we obtain an explicit monotone function of n variables (in the class mSAC^1) which is 1/2 + n^(-1/2+epsilon) hard for mNC^1 under the uniform distribution for any desired constant epsilon > 0. This bound is nearly best possible, since every monotone function has agreement 1/2 + Omega(log(n)/sqrt(n)) with some function in mNC^1 [O'Donnell/Wimmer,FOCS'09].
Our correlation bounds against mNC^1 extend smoothly to non-monotone NC^1 circuits with a bounded number of negation gates. Using Holley's monotone coupling theorem [Holley,Comm. Math. Physics,1974], we prove the following lemma: with respect to any product distribution, if a balanced monotone function f is 1/2 + delta hard for monotone circuits of a given size and depth, then f is 1/2 + (2^(t+1)-1)*delta hard for (non-monotone) circuits of the same size and depth with at most t negation gates. We thus achieve a lower bound against NC^1 circuits with (1/2-epsilon)*log(n) negation gates, improving the previous record of 1/6*log(log(n)) [Amano/Maruoka,SIAML J. Comp.,2005]. Our bound on negations is "half" optimal, since \lceil log(n+1) \rceil negation gates are known to be fully powerful for NC^1 [Ajtai/Komlos/Szemeredi,STOC'83; Fischer,GI'75].

Benjamin Rossman. Correlation Bounds Against Monotone NC^1. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 392-411, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{rossman:LIPIcs.CCC.2015.392, author = {Rossman, Benjamin}, title = {{Correlation Bounds Against Monotone NC^1}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {392--411}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.392}, URN = {urn:nbn:de:0030-drops-50785}, doi = {10.4230/LIPIcs.CCC.2015.392}, annote = {Keywords: circuit complexity, average-case complexity} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10061, Circuits, Logic, and Games (2010)

From 07/02/10 to 12/02/10, the Dagstuhl Seminar 10061 ``Circuits, Logic, and Games '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Benjamin Rossman, Thomas Schwentick, Denis Thérien, and Heribert Vollmer. 10061 Abstracts Collection – Circuits, Logic, and Games. In Circuits, Logic, and Games. Dagstuhl Seminar Proceedings, Volume 10061, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{rossman_et_al:DagSemProc.10061.1, author = {Rossman, Benjamin and Schwentick, Thomas and Th\'{e}rien, Denis and Vollmer, Heribert}, title = {{10061 Abstracts Collection – Circuits, Logic, and Games}}, booktitle = {Circuits, Logic, and Games}, pages = {1--8}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10061}, editor = {Benjamin Rossman and Thomas Schwentick and Denis Th\'{e}rien and Heribert Vollmer}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.10061.1}, URN = {urn:nbn:de:0030-drops-25280}, doi = {10.4230/DagSemProc.10061.1}, annote = {Keywords: Computational complexity theory, Finite model theory, Boolean circuits, Regular languages, Finite monoids, Ehrenfeucht-Fra\{\backslash''i\}ss\'{e}-games} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10061, Circuits, Logic, and Games (2010)

In the same way as during the first seminar on "Circuits, Logic, and Games"(Nov.~2006, 06451), the organizers aimed to bring together researchers from the areas of finite model theory and computational complexity theory, since they felt that perhaps not all developments in circuit theory and in logic had been explored fully in the context of lower bounds. In fact, the interaction between
the areas has flourished a lot in the past 2-3 years, as can be exemplified by the following lines
of research.

Benjamin Rossman, Thomas Schwentick, Denis Thérien, and Heribert Vollmer. 10061 Executive Summary – Circuits, Logic, and Games. In Circuits, Logic, and Games. Dagstuhl Seminar Proceedings, Volume 10061, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{rossman_et_al:DagSemProc.10061.2, author = {Rossman, Benjamin and Schwentick, Thomas and Th\'{e}rien, Denis and Vollmer, Heribert}, title = {{10061 Executive Summary – Circuits, Logic, and Games}}, booktitle = {Circuits, Logic, and Games}, pages = {1--5}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10061}, editor = {Benjamin Rossman and Thomas Schwentick and Denis Th\'{e}rien and Heribert Vollmer}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.10061.2}, URN = {urn:nbn:de:0030-drops-25279}, doi = {10.4230/DagSemProc.10061.2}, annote = {Keywords: Computational complexity theory, finite model theory, Boolean circuits, regular languages, finite monoids, Ehrenfeucht-Fra\backslash"\backslashi ss\backslash'e-games} }

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