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In computational complexity, the Sensitivity Theorem, proved by Hao Huang in 2019,^[1] states that the sensitivity of a Boolean function $f\colon \{0,1\}^{n}\to \{0,1\}$ is at least the square root of its degree, thus settling a conjecture posed by Nisan and Szegedy in 1992.^[2]

Background

Several papers in the late 1980s and early 1990s^[3]^[4]^[5]^[6] showed that various decision tree complexity measures of Boolean functions are polynomially related, meaning that if $\alpha (f),\beta (f)$ are two such measures then $\alpha (f)\leq C\beta (f)^{C}$ for some constant $C>0$ . Nisan and Szegedy^[7] showed that degree and approximate degree are also polynomially related to all these measures. Their proof went via yet another complexity measure, block sensitivity, which had been introduced by Nisan.^[6] Block sensitivity generalizes a more natural measure, (critical) sensitivity, which had appeared before.^[8]^[9]^[10]

Nisan and Szegedy asked^[11] whether block sensitivity is polynomially bounded by sensitivity (the other direction is immediate since sensitivity is at most block sensitivity). This is equivalent to asking whether sensitivity is polynomially related to the various decision tree complexity measures, as well as to degree, approximate degree, and other complexity measures which have been shown to be polynomially related to these along the years.^[12] This became known as the sensitivity conjecture.^[13]

Along the years, several special cases of the sensitivity conjecture were proven.^[14]^[15] The sensitivity theorem was finally proven in its entirety by Huang,^[1] using a reduction of Gotsman and Linial.^[16]

Statement

Every Boolean function $f\colon \{0,1\}^{n}\to \{0,1\}$ can be expressed in a unique way as a multilinear polynomial. The degree of $f$ is the degree of this unique polynomial, denoted $\deg(f)$ .

The sensitivity of the Boolean function $f$ at the point $x\in \{0,1\}^{n}$ is the number of indices $i\in [n]$ such that $f(x^{\oplus i})\neq f(x)$ , where $x^{\oplus i}$ is obtained from $x$ by flipping the $i$ 'th coordinate. The sensitivity of $f$ is the maximum sensitivity of $f$ at any point $x\in \{0,1\}^{n}$ , denoted $s(f)$ .

The sensitivity theorem states that

s(f)\geq {\sqrt {\deg(f)}}.

In the other direction, Tal,^[17] improving on an earlier bound of Nisan and Szegedy,^[2] showed that

s(f)\leq \deg(f)^{2}.

The sensitivity theorem is tight for the AND-of-ORs function:^[18]

\bigwedge _{i=1}^{m}\bigvee _{j=1}^{m}x_{ij}

This function has degree $m^{2}$ and sensitivity $m$ .

Proof

Let $f\colon \{0,1\}^{n}\to \{0,1\}$ be a Boolean function of degree $d$ . Consider any maxonomial of $f$ , that is, a monomial of degree $d$ in the unique multilinear polynomial representing $f$ . If we substitute an arbitrary value in the coordinates not mentioned in the monomial then we get a function $F$ on $d$ coordinates which has degree $d$ , and moreover, $s(f)\geq s(F)$ . If we prove the sensitivity theorem for $F$ then it follows for $f$ . So from now on, we assume without loss of generality that $f$ has degree $n$ .

Define a new function $g\colon \{0,1\}^{n}\to \{0,1\}$ by

g(x_{1},\dots ,x_{n})=f\oplus x_{1}\oplus \cdots \oplus x_{n}.

It can be shown that since $f$ has degree $n$ then $g$ is unbalanced (meaning that $|g^{-1}(0)|\neq |g^{-1}(1)|$ ), say $|g^{-1}(1)|>2^{n-1}$ . Consider the subgraph $G$ of the hypercube (the graph on $\{0,1\}^{n}$ in which two vertices are connected if they differ by a single coordinate) induced by $S=g^{-1}(1)$ . In order to prove the sensitivity theorem, it suffices to show that $G$ has a vertex whose degree is at least ${\sqrt {n}}$ . This reduction is due to Gotsman and Linial.^[16]

Huang^[1] constructs a signing of the hypercube in which the product of the signs along any square is $-1$ . This means that there is a way to assign a sign to every edge of the hypercube so that this property is satisfied. The same signing had been found earlier by Ahmadi et al.^[19] in a different context.

Let $A$ be the signed adjacency matrix corresponding to the signing. The property that the product of the signs in every square is $-1$ implies that $A^{2}=nI$ , and so half of the eigenvalues of $A$ are ${\sqrt {n}}$ and half are $-{\sqrt {n}}$ . In particular, the eigenspace of ${\sqrt {n}}$ (which has dimension $2^{n-1}$ ) intersects the space of vectors supported by $S$ (which has dimension $>2^{n-1}$ ), implying that there is an eigenvector $v$ of $A$ with eigenvalue ${\sqrt {n}}$ which is supported on $S$ . (This is a simplification of Huang's original argument due to Shalev Ben-David.^[20])

Consider a point $x\in S$ maximizing $|v_{x}|$ . On the one hand, $Av={\sqrt {n}}v$ . On the other hand, $Av$ is at most the sum of absolute values of all neighbors of $x$ in $S$ , which is at most $\deg _{G}(x)\cdot |v_{x}|$ . Hence $\deg _{G}(x)\geq {\sqrt {n}}$ .

Notes

^ ^a ^b ^c Huang 2019.
^ ^a ^b Nisan & Szegedy 1994.
^ Blum & Impagliazzo 1987.
^ Hartmanis & Hemachandra 1991.
^ Tardos 1989.
^ ^a ^b Nisan 1991.
^ Nisan & Szegedy.
^ Wegener 1987, pp. 373–410.
^ Cook, Dwork & Reischuk 1986.
^ Simon 1983, pp. 439–444.
^ Nisan & Szegedy 1992, p. 311.
^ Buhrman & de Wolf 2002.
^ Hatami, Kulkarni & Pankratov 2011.
^ Bafna et al. 2016.
^ C. S. et al. 2016.
^ ^a ^b Gotsman & Linial 1992.
^ Tal 2013, pp. 441–454.
^ Hatami, Kulkarni & Pankratov 2011, Example 5.2.
^ Ahmadi et al. 2013.
^ Ben-David 2019.

References

Ahmadi, Bahman; Alinaghipour, Fatemeh; Cavers, Michael S.; Fallat, Shaun; Meagher, Karen; Nasserasr, Shahla (September 2013). "Minimum number of distinct eigenvalues of graphs". Electronic Journal of Linear Algebra. 26. International Linear Algebra Society: 673–691. ISSN 1081-3810.
Bafna, Mitali; Lokam, Satyanarayana V.; Tavenas, Sébastien; Velingker, Ameya; Faliszewski, Piotr; Muscholl, Anca; Niedermeier, Rolf (2016). "On the Sensitivity Conjecture for Read- $k$ Formulas". 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). p. 14 pages. doi:10.4230/LIPICS.MFCS.2016.16. ISSN 1868-8969.
Ben-David, Shalev (3 July 2019). "Comment #35 to Sensitivity Conjecture resolved".
Blum, Manuel; Impagliazzo, Russell (1987). "Generic oracles and oracle classes". 28th Annual Symposium on Foundations of Computer Science (sfcs 1987). IEEE. doi:10.1109/sfcs.1987.30.
Buhrman, Harry; de Wolf, Ronald (2002). "Complexity measures and decision tree complexity: a survey". Theoretical Computer Science. 288 (1). Elsevier BV: 21–43. doi:10.1016/s0304-3975(01)00144-x. ISSN 0304-3975.
C. S., Karthik; Tavenas, Sébastien; Lal, Akash; Akshay, S.; Saurabh, Saket; Sen, Sandeep (2016). "On the Sensitivity Conjecture for Disjunctive Normal Forms". 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). p. 15 pages. doi:10.4230/LIPICS.FSTTCS.2016.15. ISSN 1868-8969.
Cook, Stephen; Dwork, Cynthia; Reischuk, Rüdiger (1986). "Upper and Lower Time Bounds for Parallel Random Access Machines without Simultaneous Writes". SIAM Journal on Computing. 15 (1): 87–97. doi:10.1137/0215006. ISSN 0097-5397.
Gotsman, Chaim; Linial, Nati (1992). "The equivalence of two problems on the cube". Journal of Combinatorial Theory, Series A. 61 (1). Elsevier BV: 142–146. doi:10.1016/0097-3165(92)90060-8. ISSN 0097-3165.
Hartmanis, Juris; Hemachandra, Lane A. (1991). "One-way functions and the nonisomorphism of NP-complete sets". Theoretical Computer Science. 81 (1): 155–163. doi:10.1016/0304-3975(91)90323-T.
Hatami, Pooya; Kulkarni, Raghav; Pankratov, Denis (2011). "Variations on the Sensitivity Conjecture". Theory of Computing. 1 (1): 1–27. doi:10.4086/toc.gs.2011.004. ISSN 1557-2862.
Huang, Hao (2019-11-01). "Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture". Annals of Mathematics. 190 (3). doi:10.4007/annals.2019.190.3.6. ISSN 0003-486X.
Nisan, Noam (1991). "CREW PRAMs and Decision Trees". SIAM Journal on Computing. 20 (6): 999–1007. doi:10.1137/0220062. ISSN 0097-5397.
Nisan, Noam; Szegedy, Mario (1994). "On the degree of boolean functions as real polynomials". Computational Complexity. 4 (4): 301–313. doi:10.1007/BF01263419. ISSN 1016-3328.
Simon, Hans-Ulrich (1983). "A tight Ω(loglog n)-bound on the time for parallel Ram's to compute nondegenerated boolean functions". Foundations of Computation Theory. Vol. 158. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/3-540-12689-9_124. ISBN 978-3-540-12689-8.
Tal, Avishay (2013-01-09). "Properties and applications of boolean function composition". ITCS '13: Proceedings of the 4th conference on Innovations in Theoretical Computer Science. ACM. doi:10.1145/2422436.2422485. ISBN 978-1-4503-1859-4.
Tardos, G. (1989). "Query complexity, or why is it difficult to separate $NP^{A}\cap coNP^{A}$ from $P^{A}$ by random oracles $A$ ?". Combinatorica. 9 (4): 385–392. doi:10.1007/BF02125350. ISSN 0209-9683.
Wegener, Ingo (1987). The Complexity of Boolean Functions. Stuttgart Chichester New York Brisbane [etc.]: John Wiley & Sons. ISBN 0-471-91555-6.

[FOOTNOTEHuang2019-1] Huang 2019.

[FOOTNOTENisanSzegedy1994-2] Nisan & Szegedy 1994.

[FOOTNOTEBlumImpagliazzo1987-3] Blum & Impagliazzo 1987.

[FOOTNOTEHartmanisHemachandra1991-4] Hartmanis & Hemachandra 1991.

[FOOTNOTETardos1989-5] Tardos 1989.

[FOOTNOTENisan1991-6] Nisan 1991.

[FOOTNOTENisanSzegedy-7] Nisan & Szegedy.

[FOOTNOTEWegener1987373–410-8] Wegener 1987, pp. 373–410.

[FOOTNOTECookDworkReischuk1986-9] Cook, Dwork & Reischuk 1986.

[FOOTNOTESimon1983439–444-10] Simon 1983, pp. 439–444.

[FOOTNOTENisanSzegedy1992311-11] Nisan & Szegedy 1992, p. 311.

[FOOTNOTEBuhrmande_Wolf2002-12] Buhrman & de Wolf 2002.

[FOOTNOTEHatamiKulkarniPankratov2011-13] Hatami, Kulkarni & Pankratov 2011.

[FOOTNOTEBafnaLokamTavenasVelingker2016-14] Bafna et al. 2016.

[FOOTNOTEC._S.TavenasLalAkshay2016-15] C. S. et al. 2016.

[FOOTNOTEGotsmanLinial1992-16] Gotsman & Linial 1992.

[FOOTNOTETal2013441–454-17] Tal 2013, pp. 441–454.

[FOOTNOTEHatamiKulkarniPankratov2011Example_5.2-18] Hatami, Kulkarni & Pankratov 2011, Example 5.2.

[FOOTNOTEAhmadiAlinaghipourCaversFallat2013-19] Ahmadi et al. 2013.

[FOOTNOTEBen-David2019-20] Ben-David 2019.

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@@ Line 24: / Line 24: @@
 :<math>\bigwedge_{i=1}^m \bigvee_{j=1}^m x_{ij} </math>
 This function has degree <math>m^2</math> and sensitivity <math>m</math>.
+== Proof ==
+Let <math>f\colon \{0,1\}^n \to \{0,1\}</math> be a Boolean function of degree <math>d</math>. Consider any ''maxonomial'' of <math>f</math>, that is, a monomial of degree <math>d</math> in the unique multilinear polynomial representing <math>f</math>. If we substitute an arbitrary value in the coordinates not mentioned in the monomial then we get a function <math>F</math> on <math>d</math> coordinates which has degree <math>d</math>, and moreover, <math>s(f) \geq s(F)</math>. If we prove the sensitivity theorem for <math>F</math> then it follows for <math>f</math>. So from now on, we assume without loss of generality that <math>f</math> has degree <math>n</math>.
+Define a new function <math>g\colon \{0,1\}^n \to \{0,1\}</math> by
+:<math>g(x_1,\dots,x_n) = f \oplus x_1 \oplus \cdots \oplus x_n.</math>
+It can be shown that since <math>f</math> has degree <math>n</math> then <math>g</math> is unbalanced (meaning that <math>|g^{-1}(0)| \neq |g^{-1}(1)|</math>), say <math>|g^{-1}(1)| > 2^{n-1}</math>. Consider the subgraph <math>G</math> of the hypercube (the graph on <math>\{0,1\}^n</math> in which two vertices are connected if they differ by a single coordinate) induced by <math>S = g^{-1}(1)</math>. In order to prove the sensitivity theorem, it suffices to show that <math>G</math> has a vertex whose degree is at least <math>\sqrt{n}</math>. This reduction is due to Gotsman and Linial.{{sfn|Gotsman|Linial|1992}}
+Huang{{sfn|Huang|2019}} constructs a ''signing of the hypercube'' in which the product of the signs along any square is <math>-1</math>. This means that there is a way to assign a sign to every edge of the hypercube so that this property is satisfied. The same signing had been found earlier by Ahmadi et al.{{sfn|Ahmadi|Alinaghipour|Cavers|Fallat|2013}} in a different context.
+Let <math>A</math> be the signed adjacency matrix corresponding to the signing. The property that the product of the signs in every square is <math>-1</math> implies that <math>A^2=nI</math>, and so half of the eigenvalues of <math>A</math> are <math>\sqrt{n}</math> and half are <math>-\sqrt{n}</math>. In particular, the eigenspace of <math>\sqrt{n}</math> (which has dimension <math>2^{n-1}</math>) intersects the space of vectors supported by <math>S</math> (which has dimension <math>>2^{n-1}</math>), implying that there is an eigenvector <math>v</math> of <math>A</math> with eigenvalue <math>\sqrt{n}</math> which is supported on <math>S</math>. (This is a simplification of Huang's original argument due to Shalev Ben-David.{{sfn|Ben-David|2019}})
+Consider a point <math>x \in S</math> maximizing <math>|v_x|</math>. On the one hand, <math>Av = \sqrt{n}v</math>.
+On the other hand, <math>Av</math> is at most the sum of absolute values of all neighbors of <math>x</math> in <math>S</math>, which is at most <math>\deg_G(x) \cdot |v_x|</math>. Hence <math>\deg_G(x) \geq \sqrt{n}</math>.
 ==See also==
@@ Line 32: / Line 47: @@
 ==References==
+* {{cite journal | last1=Ahmadi | first1=Bahman | last2=Alinaghipour | first2=Fatemeh | last3=Cavers | first3 = Michael S. | last4=Fallat | first4=Shaun | last5=Meagher | first5=Karen | last6=Nasserasr | first6=Shahla | title=Minimum number of distinct eigenvalues of graphs | journal=Electronic Journal of Linear Algebra | issn=1081-3810 | publisher=International Linear Algebra Society | volume=26 | pages=673–691 | date=September 2013}}
 * {{cite conference | last=Bafna | first=Mitali | last2=Lokam | first2=Satyanarayana V. | last3=Tavenas | first3=Sébastien | last4=Velingker | first4=Ameya | last5=Faliszewski | first5=Piotr | last6=Muscholl | first6=Anca | last7=Niedermeier | first7=Rolf | title=On the Sensitivity Conjecture for Read-{{mvar|k}} Formulas | book-title=41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016) | date=2016 | issn=1868-8969 | doi=10.4230/LIPICS.MFCS.2016.16 | page=14 pages}}
+* {{cite web | last=Ben-David | first=Shalev | url=https://scottaaronson.blog/?p=4229#comment-1813084 | title=Comment #35 to Sensitivity Conjecture resolved | date=3 July 2019}}
 * {{cite conference | last=Blum | first=Manuel | last2=Impagliazzo | first2=Russell | title=Generic oracles and oracle classes | book-title=28th Annual Symposium on Foundations of Computer Science (sfcs 1987) | publisher=IEEE | date=1987 | doi=10.1109/sfcs.1987.30}}
 * {{cite journal | last=Buhrman | first=Harry | last2=de Wolf | first2=Ronald | title=Complexity measures and decision tree complexity: a survey | journal=Theoretical Computer Science | publisher=Elsevier BV | volume=288 | issue=1 | year=2002 | issn=0304-3975 | doi=10.1016/s0304-3975(01)00144-x | pages=21–43}}