On the relationship between energy complexity and other boolean function measures
We focus on energy complexity , a Boolean function measure related closely to Boolean circuit design. Given a circuit C over the standard basis { ∨ 2 , ∧ 2 , ¬ } , the energy complexity of C , denoted by EC ( C ) , is the maximum number of its activated inner gates over all inputs. The energy comple...
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| Published in: | Journal of combinatorial optimization Vol. 43; no. 5; pp. 1470 - 1492 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
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Springer US
01.07.2022
Springer Nature B.V |
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| ISSN: | 1382-6905, 1573-2886 |
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| Abstract | We focus on
energy complexity
, a Boolean function measure related closely to Boolean circuit design. Given a circuit
C
over the standard basis
{
∨
2
,
∧
2
,
¬
}
, the energy complexity of
C
, denoted by
EC
(
C
)
, is the maximum number of its activated inner gates over all inputs. The energy complexity of a Boolean function
f
, denoted by
EC
(
f
)
, is the minimum of
EC
(
C
)
over all circuits
C
computing
f
. Recently, K. Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) gave
EC
(
f
)
an upper bound by the decision tree complexity,
EC
(
f
)
=
O
(
D
(
f
)
3
)
. On the input size
n
, They also showed that
EC
(
f
)
is at most
3
n
-
1
. For the lower bound side, they showed that
EC
(
f
)
≥
1
3
psens
(
f
)
, where
psens
(
f
)
is called
positive sensitivity
. A remained open problem is whether the energy complexity of a Boolean function has a polynomial relationship with its decision tree complexity.
Our results for energy complexity can be listed below.
For the lower bound, we prove the equation that
EC
(
f
)
=
Ω
(
D
(
f
)
)
, which answers the above open problem.
For upper bounds,
EC
(
f
)
≤
min
{
1
2
D
(
f
)
2
+
O
(
D
(
f
)
)
,
n
+
2
D
(
f
)
-
2
}
holds.
For non-degenerated functions, we also provide another lower bound
EC
(
f
)
=
Ω
(
log
n
)
where
n
is the input size.
We also discuss the energy complexity of two specific function classes,
OR
functions and
ADDRESS
functions, which implies the tightness of our two lower bounds respectively. In addition, the former one answers another open question in Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) asking for non-trivial lower bound for energy complexity of
OR
functions. |
|---|---|
| AbstractList | We focus on energy complexity, a Boolean function measure related closely to Boolean circuit design. Given a circuit C over the standard basis {∨2,∧2,¬}, the energy complexity of C, denoted by EC(C), is the maximum number of its activated inner gates over all inputs. The energy complexity of a Boolean function f, denoted by EC(f), is the minimum of EC(C) over all circuits C computing f. Recently, K. Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) gave EC(f) an upper bound by the decision tree complexity, EC(f)=O(D(f)3). On the input size n, They also showed that EC(f) is at most 3n-1. For the lower bound side, they showed that EC(f)≥13psens(f), where psens(f) is called positive sensitivity. A remained open problem is whether the energy complexity of a Boolean function has a polynomial relationship with its decision tree complexity.Our results for energy complexity can be listed below.For the lower bound, we prove the equation that EC(f)=Ω(D(f)), which answers the above open problem.For upper bounds, EC(f)≤min{12D(f)2+O(D(f)),n+2D(f)-2} holds.For non-degenerated functions, we also provide another lower bound EC(f)=Ω(logn) where n is the input size.We also discuss the energy complexity of two specific function classes, OR functions and ADDRESS functions, which implies the tightness of our two lower bounds respectively. In addition, the former one answers another open question in Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) asking for non-trivial lower bound for energy complexity of OR functions. We focus on energy complexity , a Boolean function measure related closely to Boolean circuit design. Given a circuit C over the standard basis { ∨ 2 , ∧ 2 , ¬ } , the energy complexity of C , denoted by EC ( C ) , is the maximum number of its activated inner gates over all inputs. The energy complexity of a Boolean function f , denoted by EC ( f ) , is the minimum of EC ( C ) over all circuits C computing f . Recently, K. Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) gave EC ( f ) an upper bound by the decision tree complexity, EC ( f ) = O ( D ( f ) 3 ) . On the input size n , They also showed that EC ( f ) is at most 3 n - 1 . For the lower bound side, they showed that EC ( f ) ≥ 1 3 psens ( f ) , where psens ( f ) is called positive sensitivity . A remained open problem is whether the energy complexity of a Boolean function has a polynomial relationship with its decision tree complexity. Our results for energy complexity can be listed below. For the lower bound, we prove the equation that EC ( f ) = Ω ( D ( f ) ) , which answers the above open problem. For upper bounds, EC ( f ) ≤ min { 1 2 D ( f ) 2 + O ( D ( f ) ) , n + 2 D ( f ) - 2 } holds. For non-degenerated functions, we also provide another lower bound EC ( f ) = Ω ( log n ) where n is the input size. We also discuss the energy complexity of two specific function classes, OR functions and ADDRESS functions, which implies the tightness of our two lower bounds respectively. In addition, the former one answers another open question in Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) asking for non-trivial lower bound for energy complexity of OR functions. |
| Author | Sun, Yuan Sun, Xiaoming Wu, Kewen Xia, Zhiyu |
| Author_xml | – sequence: 1 givenname: Xiaoming surname: Sun fullname: Sun, Xiaoming organization: CAS Key Lab of Network Data Science and Technology, Institute of Computing Technology, Chinese Academy of Sciences, University of Chinese Academy of Sciences – sequence: 2 givenname: Yuan orcidid: 0000-0002-2012-2019 surname: Sun fullname: Sun, Yuan email: sunyuan2016@ict.ac.cn organization: CAS Key Lab of Network Data Science and Technology, Institute of Computing Technology, Chinese Academy of Sciences, University of Chinese Academy of Sciences – sequence: 3 givenname: Kewen surname: Wu fullname: Wu, Kewen organization: School of Electronics Engineering and Computer Science, Peking University – sequence: 4 givenname: Zhiyu surname: Xia fullname: Xia, Zhiyu organization: CAS Key Lab of Network Data Science and Technology, Institute of Computing Technology, Chinese Academy of Sciences, University of Chinese Academy of Sciences |
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| Cites_doi | 10.4007/annals.2019.190.3.6 10.1016/j.tcs.2011.02.042 10.1007/11549345_11 10.1007/BF02579140 10.1007/978-3-662-48054-0_8 10.1016/j.tcs.2020.09.003 10.1016/j.tcs.2010.08.006 10.1016/S0304-3975(01)00144-X 10.1134/S1995080215040277 10.1016/0022-0000(93)90001-D 10.1016/j.tcs.2012.11.006 10.1016/0020-0190(93)90041-7 10.1016/j.tcs.2012.11.039 10.1016/j.tcs.2008.07.028 10.1007/BF01263419 10.1145/2554797.2554826 10.1137/0217031 10.1007/978-3-319-94776-1_61 10.1016/0020-0190(84)90019-X 10.1162/neco.2006.18.12.2994 10.1016/j.tcs.2010.11.022 10.1016/0304-3975(76)90053-0 10.1145/3406325.3451047 10.1007/BF01272517 |
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| Keywords | Energy complexity Boolean function Decision tree Circuit complexity |
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| References | Dinesh K, Otiv S, Sarma J (2018) New bounds for energy complexity of boolean functions. In: International computing and combinatorics conference, Springer, Berlin, 738–750 Lovasz L, Young N (2002) Lecture notes on evasiveness of graph properties. arXiv preprint arXiv:cs/0205031 HajnalAMaassWPudlákPSzegedyMTuránGThreshold circuits of bounded depthJ Comput Syst Sci1993462129154121715310.1016/0022-0000(93)90001-D Nikolaevich VM (1961) On the power of networks of functional elements. In: Proceedings of the USSR academy of sciences. Volume 139., Russian Academy of Sciences 320–323 SuzukiAUchizawaKZhouXEnergy and fan-in of logic circuits computing symmetric boolean functionsTheoret Comput Sci20135057480310756210.1016/j.tcs.2012.11.039 GaoYMaoJSunXZuoSOn the sensitivity complexity of bipartite graph propertiesTheoret Comput Sci20134688391300377010.1016/j.tcs.2012.11.006 Karpas I (2016) Lower bounds for sensitivity of graph properties. arXiv preprint arXiv:1609.05320 BuhrmanHDe WolfRComplexity measures and decision tree complexity: a surveyTheoret Comput Sci200228812143193488810.1016/S0304-3975(01)00144-X DuDZKoKITheory of computational complexity2001New YorkWiley1433.68001 DineshKOtivSSarmaJNew bounds for energy complexity of boolean functionsTheoret Comput Sci20208455975416565610.1016/j.tcs.2020.09.003 YaoACCMonotone bipartite graph properties are evasiveSIAM J Comput198817351752094194210.1137/0217031 RivestRLVuilleminJOn recognizing graph properties from adjacency matricesTheoret Comput ence19763337138452107510.1016/0304-3975(76)90053-0 UchizawaKTakimotoEExponential lower bounds on the size of constant-depth threshold circuits with small energy complexityTheoret Comput Sci20084071–3474487246302910.1016/j.tcs.2008.07.028 Amano K, Maruoka A (2005) On the complexity of depth-2 circuits with threshold gates. In: international symposium on mathematical foundations of computer science, Springer, Berlin, 107–118 LozhkinSShupletsovMSwitching activity of boolean circuits and synthesis of boolean circuits with asymptotically optimal complexity and linear switching activityLobachevskii J Math2015364450460343120810.1134/S1995080215040277 NisanNSzegedyMOn the degree of boolean functions as real polynomialsComput Complex199444301313131353110.1007/BF01263419 SunXAn improved lower bound on the sensitivity complexity of graph propertiesTheoret Comput Sci20114122935243529283969510.1016/j.tcs.2011.02.042 UchizawaKDouglasRMaassWOn the computational power of threshold circuits with sparse activityNeural Comput2006181229943008226520910.1162/neco.2006.18.12.2994 Aaronson S, Ben-David S, Kothari R, Rao S, Tal A (2020) Degree vs. approximate degree and quantum implications of huang’s sensitivity theorem Antoniadis A, Barcelo N, Nugent M, Pruhs K, Scquizzato M (2014) Energy-efficient circuit design. In: Proceedings of the 5th conference on Innovations in theoretical computer science, ACM 303–312 Huang H (2019) Induced subgraphs of hypercubes and a proof of the sensitivity conjecture TuránGThe critical complexity of graph propertiesInf Process Lett198418315115376036710.1016/0020-0190(84)90019-X UchizawaKTakimotoENishizekiTSize-energy tradeoffs for unate circuits computing symmetric boolean functionsTheoret Comput Sci20114128–10773782279692410.1016/j.tcs.2010.11.022 RazborovAWigdersonAnΩ(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{\Omega (\log n)}$$\end{document} lower bounds on the size of depth-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3$$\end{document} threshold cicuits with AND gates at the bottomInf Process Lett199345630330710.1016/0020-0190(93)90041-7 HåstadJGoldmannMOn the power of small-depth threshold circuitsComput Complex199112113129113494510.1007/BF01272517 Kasim-ZadeOMOn a measure of active circuits of functional elementsMath Probl Cybernet1992421822812175020809.94030 UchizawaKNishizekiTTakimotoEEnergy and depth of threshold circuitsTheoret Comput Sci201041144–4639383946276878010.1016/j.tcs.2010.08.006 KahnJSaksMSturtevantDA topological approach to evasivenessCombinatorica19844429730677989010.1007/BF02579140 Barcelo N, Nugent M, Pruhs K, Scquizzato M (2015) Almost all functions require exponential energy. In: international symposium on mathematical foundations of computer science, Springer 90–101 Hatami P, Kulkarni R, Pankratov D (2010) Variations on the sensitivity conjecture. arXiv preprint arXiv:1011.0354 Y Gao (689_CR9) 2013; 468 S Lozhkin (689_CR18) 2015; 36 J Kahn (689_CR14) 1984; 4 A Razborov (689_CR21) 1993; 45 K Dinesh (689_CR6) 2020; 845 OM Kasim-Zade (689_CR16) 1992; 4 689_CR7 N Nisan (689_CR20) 1994; 4 X Sun (689_CR23) 2011; 412 RL Rivest (689_CR22) 1976; 3 689_CR15 ACC Yao (689_CR30) 1988; 17 K Uchizawa (689_CR29) 2011; 412 689_CR13 689_CR12 689_CR19 689_CR17 J Håstad (689_CR11) 1991; 1 G Turán (689_CR25) 1984; 18 K Uchizawa (689_CR27) 2006; 18 689_CR4 689_CR3 689_CR2 689_CR1 DZ Du (689_CR8) 2001 H Buhrman (689_CR5) 2002; 288 A Suzuki (689_CR24) 2013; 505 K Uchizawa (689_CR26) 2008; 407 K Uchizawa (689_CR28) 2010; 411 A Hajnal (689_CR10) 1993; 46 |
| References_xml | – reference: YaoACCMonotone bipartite graph properties are evasiveSIAM J Comput198817351752094194210.1137/0217031 – reference: Hatami P, Kulkarni R, Pankratov D (2010) Variations on the sensitivity conjecture. arXiv preprint arXiv:1011.0354 – reference: DineshKOtivSSarmaJNew bounds for energy complexity of boolean functionsTheoret Comput Sci20208455975416565610.1016/j.tcs.2020.09.003 – reference: Dinesh K, Otiv S, Sarma J (2018) New bounds for energy complexity of boolean functions. In: International computing and combinatorics conference, Springer, Berlin, 738–750 – reference: BuhrmanHDe WolfRComplexity measures and decision tree complexity: a surveyTheoret Comput Sci200228812143193488810.1016/S0304-3975(01)00144-X – reference: HajnalAMaassWPudlákPSzegedyMTuránGThreshold circuits of bounded depthJ Comput Syst Sci1993462129154121715310.1016/0022-0000(93)90001-D – reference: KahnJSaksMSturtevantDA topological approach to evasivenessCombinatorica19844429730677989010.1007/BF02579140 – reference: DuDZKoKITheory of computational complexity2001New YorkWiley1433.68001 – reference: HåstadJGoldmannMOn the power of small-depth threshold circuitsComput Complex199112113129113494510.1007/BF01272517 – reference: RazborovAWigdersonAnΩ(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{\Omega (\log n)}$$\end{document} lower bounds on the size of depth-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3$$\end{document} threshold cicuits with AND gates at the bottomInf Process Lett199345630330710.1016/0020-0190(93)90041-7 – reference: SunXAn improved lower bound on the sensitivity complexity of graph propertiesTheoret Comput Sci20114122935243529283969510.1016/j.tcs.2011.02.042 – reference: UchizawaKTakimotoEExponential lower bounds on the size of constant-depth threshold circuits with small energy complexityTheoret Comput Sci20084071–3474487246302910.1016/j.tcs.2008.07.028 – reference: Antoniadis A, Barcelo N, Nugent M, Pruhs K, Scquizzato M (2014) Energy-efficient circuit design. 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| Snippet | We focus on
energy complexity
, a Boolean function measure related closely to Boolean circuit design. Given a circuit
C
over the standard basis
{
∨
2
,
∧
2
,
¬... We focus on energy complexity, a Boolean function measure related closely to Boolean circuit design. Given a circuit C over the standard basis {∨2,∧2,¬}, the... |
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| SubjectTerms | Boolean Boolean algebra Boolean functions Circuit design Combinatorial analysis Combinatorics Complexity Computation Convex and Discrete Geometry Decision trees Energy Gates (circuits) Lower bounds Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Polynomials Theory of Computation Tightness Upper bounds |
| Title | On the relationship between energy complexity and other boolean function measures |
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