AND testing and robust judgement aggregation

Yuval Filmus, Noam Lifshitz, Dor Minzer, Elchanan Mossel

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations


A function fg¶{0,1}n→ {0,1} is called an approximate AND-homomorphism if choosing x,ygn uniformly at random, we have that f(xg§ y) = f(x)g§ f(y) with probability at least 1-ϵ, where xg§ y = (x1g§ y1,...,xng§ yn). We prove that if fg¶ {0,1}n → {0,1} is an approximate AND-homomorphism, then f is -close to either a constant function or an AND function, where (ϵ) → 0 as ϵ→ 0. This improves on a result of Nehama, who proved a similar statement in which δdepends on n. Our theorem implies a strong result on judgement aggregation in computational social choice. In the language of social choice, our result shows that if f is ϵ-close to satisfying judgement aggregation, then it is (ϵ)-close to an oligarchy (the name for the AND function in social choice theory). This improves on Nehama's result, in which δdecays polynomially with n. Our result follows from a more general one, in which we characterize approximate solutions to the eigenvalue equation f = λ g, where is the downwards noise operator f(x) = y[f(x g§ y)], f is [0,1]-valued, and g is {0,1}-valued. We identify all exact solutions to this equation, and show that any approximate solution in which f and λ g are close is close to an exact solution.

Original languageEnglish
Title of host publicationSTOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
EditorsKonstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy
PublisherAssociation for Computing Machinery
Number of pages12
ISBN (Electronic)9781450369794
StatePublished - 8 Jun 2020
Externally publishedYes
Event52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 - Chicago, United States
Duration: 22 Jun 202026 Jun 2020

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020
Country/TerritoryUnited States

Bibliographical note

Publisher Copyright:
© 2020 ACM.


∗This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 802020-ERC-HARMONIC. †Supported partially by NSF grant CCF-1412958 and Rothschild Fellowship. ‡Supported by NSF Award DMS-1737944, by AROMURI W911NF1910217 and by a Simons Investigator Award in Mathematics (622132).

FundersFunder number
AROMURIW911NF1910217, 622132
National Science FoundationDMS-1737944, CCF-1412958, 1737944, 1412958, 802020
Horizon 2020 Framework Programme


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