On the enumeration of Boolean functions with distinguished variables
Boolean functions have a fundamental role in neural networks and machine learning. Enumerating these functions and significant subclasses is a highly complex problem. Therefore, it is of interest to study subclasses that escape this limitation and can be enumerated by means of sequences depending on...
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| Veröffentlicht in: | Soft computing (Berlin, Germany) Jg. 25; H. 19; S. 12627 - 12640 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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01.10.2021
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| ISSN: | 1432-7643, 1433-7479 |
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| Abstract | Boolean functions have a fundamental role in neural networks and machine learning. Enumerating these functions and significant subclasses is a highly complex problem. Therefore, it is of interest to study subclasses that escape this limitation and can be enumerated by means of sequences depending on the number of variables. In this article, we obtain seven new formulas corresponding to enumerations of some subclasses of Boolean functions. The versatility of these functions does the problem interesting to several different fields as game theory, hypergraphs, reliability, cryptography or logic gates. |
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| AbstractList | Boolean functions have a fundamental role in neural networks and machine learning. Enumerating these functions and significant subclasses is a highly complex problem. Therefore, it is of interest to study subclasses that escape this limitation and can be enumerated by means of sequences depending on the number of variables. In this article, we obtain seven new formulas corresponding to enumerations of some subclasses of Boolean functions. The versatility of these functions does the problem interesting to several different fields as game theory, hypergraphs, reliability, cryptography or logic gates. |
| Author | Freixas, Josep |
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| Keywords | Complete games Boolean functions Enumeration of Boolean functions Dedekind’s problem 2-Monotonic Boolean functions Simple games |
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| References | FreixasJMolineroXRouraSComplete voting systems with two types of voters: weightedness and countingAnn Oper Res2012193273287287476610.1007/s10479-011-0863-x TaylorADPacelliAMathematics and politics20082New YorkSpringer10.1007/978-0-387-77645-3 CheungWSNgTWA three-dimensional voting system in Hong KongEur J Oper Res2014236292297317312210.1016/j.ejor.2014.01.020 KilgourMA formal analysis of the amending formula of Canada’s Constitution ActCan J Polit Sci19831677177710.1017/S0008423900028560 HardyGHRamanujan: twelve lectures on subjects suggested by his life and work19993New YorkChelsea0027.19608 HuSTThreshold logic1965BerkeleyUniversity of California Press10.1525/9780520329560 IsbellJRA class of simple gamesDuke Mat J1958254234399702810.1215/S0012-7094-58-02537-7 SiuKRoychowdhuryVKailathTDiscrete neural computation: a theoretical foundation1995Englewood CliffsPrentice-Hall0838.68034 Freixas J, Samaniego D (2020) On the enumeration of bipartite simple games. Discrete Appl Math (submitted) MurogaSThreshold logic and its applications1971NewYorkWiley0243.94014 von NeumannJMorgensternOTheory of games and economic behavior1944PrincetonPrinceton University Press0063.05930 PictonPDNeural networks19912IndianapolisMacmillan KurzSNapelSA dimension of the Lisbon voting rules in the EU Council: a challenge and new world recordOptim Lett201610612451256352984410.1007/s11590-015-0917-0 TaylorADZwickerWSWeighted voting, multicameral representation, and powerGames Econ Behav19935170181120147910.1006/game.1993.1009 TaylorADZwickerWSSimple games: desirability relations, trading, and pseudoweightings1999PrincetonPrinceton University Press0943.91005 CarrerasFFreixasJComplete simple gamesMath Soc Sci199632139155140885810.1016/0165-4896(96)00815-3 TaylorADZwickerWSA characterization of weighted votingProc Am Math Soc199211510891094109292710.1090/S0002-9939-1992-1092927-0 FreixasJThe dimension for the European Union Council under the Nice rulesEur J Oper Res200415641541910.1016/S0377-2217(02)00903-7 KurzSTautenhahnNOn Dedekind’s problem for complete simple gamesInt J Game Theory201342411437305390210.1007/s00182-012-0327-9 Boros E, Hammer PL, Ibaraki T, Kawakawi K (1991) Identifying 2-monotonic positive Boolean functions in polynomial time. In: Hsu WL, Lee RCT (eds) ISA’91 algorithms, Springer Lecture Notes in Computer Sciences (LNCS), vol 557, pp 104–115 FreixasJPuenteMAReliability importance measures of the components in a system based on semivalues and probabilistic valuesAnn Oper Res2002109331342192973810.1023/A:1016368606348 KuoWZhuXImportance measures is reliability, risk, and optimization: principles and applications2012West SussexWiley10.1002/9781118314593 PetersHGame theory: a multi-leveled approach20152AmsterdamSpringer10.1007/978-3-662-46950-7 StinsonDRAn explication of secret sharing schemesDes Codes Crypt19922357390119477610.1007/BF00125203 FreixasJKurzSThe golden number and Fibonacci sequences in the design of voting systemsEur J Oper Res201322624625710.1016/j.ejor.2012.10.017 GvozdevaTHameedASlinkoAWeightedness and structural characterization of hierarchical simple gamesMath Soc Sci201365181189302919510.1016/j.mathsocsci.2012.11.007 Le BretonMMonteroMZaporozhetsVVoting power in the EU council of ministers and fair decision making in distributive politicsMath Soc Sci201263159173288897310.1016/j.mathsocsci.2011.11.004 EinyEThe desirability relation of simple gamesMath Soc Sci19851015516880604010.1016/0165-4896(85)90032-0 Sloane NJA (1964) www.oeis.org. Encyclopedia On-Line of Integer Sequences KurzSOn minimum sum representations for weighted voting gamesAnn Oper Res20121961361369294658410.1007/s10479-012-1108-3 BorosEHammerPLIbarakiTKawakawiKPolynomial time recognition of 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2$$\end{document}-monotonic positive functions given by an oracleSIAM J Comput19972693109143124710.1137/S0097539793269089 Simmons GJ (1990) How to (really) share a secret. In: Proceedings of the 8th annual international cryptology conference on advances in cryptology, pp 390–448. Springer, London, UK ChvátalVHammerPLAggregation of inequalities in integer programmingAnn Discrete Math1977114516247938410.1016/S0167-5060(08)70731-3 MarziFRossiFSmriglioSComputational study of separation algorithms for clique inequalitiesSoft Comput2019233013302710.1007/s00500-019-03769-y FelsenthalDSMachoverMThe measurement of voting power1998CheltenhamEdward Elgar Publishing Limited10.4337/9781840647761 RoychowdhuryVSiuKOrlitskyATheoretical advances in neural computation and learning1994NorwellKluwer10.1007/978-1-4615-2696-4 LeechDVoting power in the governance of the International Monetary FundAnn Oper Res2002109375397192974110.1023/A:1016324824094 CramaYDualization of regular Boolean functionsDiscrete Appl Math198716798586974010.1016/0166-218X(87)90056-4 FreixasJFreixasMKurzSOn the characterization of weighted simple gamesTheor Decis2017834469498373632710.1007/s11238-017-9606-z TassaTHierarchical threshold secret sharingJ Cryptol200720237264234412110.1007/s00145-006-0334-8 FreixasJMolineroXSimple games and weighted games: a theoretical and computational viewpointDiscrete Appl Math200915714961508251023010.1016/j.dam.2008.09.009 RamamurthyKGCoherent structures and simple games, theory and decision series C1990AmsterdamSpringer10.1007/978-94-009-2099-6 ReitermanJRödlVŠiňajováETumaMThreshold hypergraphsDiscrete Math19855419320079166010.1016/0012-365X(85)90080-9 WS Cheung (5422_CR4) 2014; 236 J Freixas (5422_CR10) 2013; 226 GH Hardy (5422_CR17) 1999 V Chvátal (5422_CR5) 1977; 1 E Boros (5422_CR2) 1997; 26 T Tassa (5422_CR38) 2007; 20 S Kurz (5422_CR22) 2012; 196 ST Hu (5422_CR18) 1965 J Reiterman (5422_CR32) 1985; 54 E Einy (5422_CR7) 1985; 10 Y Crama (5422_CR6) 1987; 16 S Kurz (5422_CR24) 2013; 42 AD Taylor (5422_CR41) 1993; 5 M Kilgour (5422_CR20) 1983; 16 J Freixas (5422_CR12) 2002; 109 D Leech (5422_CR26) 2002; 109 5422_CR1 W Kuo (5422_CR21) 2012 M Le Breton (5422_CR25) 2012; 63 J Freixas (5422_CR14) 2012; 193 KG Ramamurthy (5422_CR31) 1990 AD Taylor (5422_CR42) 1999 AD Taylor (5422_CR39) 2008 DS Felsenthal (5422_CR8) 1998 J Freixas (5422_CR11) 2009; 157 T Gvozdeva (5422_CR16) 2013; 65 F Marzi (5422_CR27) 2019; 23 J Freixas (5422_CR15) 2017; 83 JR Isbell (5422_CR19) 1958; 25 S Muroga (5422_CR28) 1971 K Siu (5422_CR35) 1995 AD Taylor (5422_CR40) 1992; 115 S Kurz (5422_CR23) 2016; 10 V Roychowdhury (5422_CR33) 1994 F Carreras (5422_CR3) 1996; 32 5422_CR34 5422_CR13 DR Stinson (5422_CR37) 1992; 2 H Peters (5422_CR29) 2015 5422_CR36 J von Neumann (5422_CR43) 1944 J Freixas (5422_CR9) 2004; 156 PD Picton (5422_CR30) 1991 |
| References_xml | – reference: RoychowdhuryVSiuKOrlitskyATheoretical advances in neural computation and learning1994NorwellKluwer10.1007/978-1-4615-2696-4 – reference: TassaTHierarchical threshold secret sharingJ Cryptol200720237264234412110.1007/s00145-006-0334-8 – reference: FreixasJPuenteMAReliability importance measures of the components in a system based on semivalues and probabilistic valuesAnn Oper Res2002109331342192973810.1023/A:1016368606348 – reference: TaylorADPacelliAMathematics and politics20082New YorkSpringer10.1007/978-0-387-77645-3 – reference: FelsenthalDSMachoverMThe measurement of voting power1998CheltenhamEdward Elgar Publishing Limited10.4337/9781840647761 – reference: Boros E, Hammer PL, Ibaraki T, Kawakawi K (1991) Identifying 2-monotonic positive Boolean functions in polynomial time. In: Hsu WL, Lee RCT (eds) ISA’91 algorithms, Springer Lecture Notes in Computer Sciences (LNCS), vol 557, pp 104–115 – reference: IsbellJRA class of simple gamesDuke Mat J1958254234399702810.1215/S0012-7094-58-02537-7 – reference: StinsonDRAn explication of secret sharing schemesDes Codes Crypt19922357390119477610.1007/BF00125203 – reference: KuoWZhuXImportance measures is reliability, risk, and optimization: principles and applications2012West SussexWiley10.1002/9781118314593 – reference: ChvátalVHammerPLAggregation of inequalities in integer programmingAnn Discrete Math1977114516247938410.1016/S0167-5060(08)70731-3 – reference: EinyEThe desirability relation of simple gamesMath Soc Sci19851015516880604010.1016/0165-4896(85)90032-0 – reference: ReitermanJRödlVŠiňajováETumaMThreshold hypergraphsDiscrete Math19855419320079166010.1016/0012-365X(85)90080-9 – reference: KilgourMA formal analysis of the amending formula of Canada’s Constitution ActCan J Polit Sci19831677177710.1017/S0008423900028560 – reference: KurzSTautenhahnNOn Dedekind’s problem for complete simple gamesInt J Game Theory201342411437305390210.1007/s00182-012-0327-9 – reference: CarrerasFFreixasJComplete simple gamesMath Soc Sci199632139155140885810.1016/0165-4896(96)00815-3 – reference: RamamurthyKGCoherent structures and simple games, theory and decision series C1990AmsterdamSpringer10.1007/978-94-009-2099-6 – reference: CramaYDualization of regular Boolean functionsDiscrete Appl Math198716798586974010.1016/0166-218X(87)90056-4 – reference: HuSTThreshold logic1965BerkeleyUniversity of California Press10.1525/9780520329560 – reference: HardyGHRamanujan: twelve lectures on subjects suggested by his life and work19993New YorkChelsea0027.19608 – reference: LeechDVoting power in the governance of the International Monetary FundAnn Oper Res2002109375397192974110.1023/A:1016324824094 – reference: CheungWSNgTWA three-dimensional voting system in Hong KongEur J Oper Res2014236292297317312210.1016/j.ejor.2014.01.020 – reference: Freixas J, Samaniego D (2020) On the enumeration of bipartite simple games. Discrete Appl Math (submitted) – reference: von NeumannJMorgensternOTheory of games and economic behavior1944PrincetonPrinceton University Press0063.05930 – reference: Le BretonMMonteroMZaporozhetsVVoting power in the EU council of ministers and fair decision making in distributive politicsMath Soc Sci201263159173288897310.1016/j.mathsocsci.2011.11.004 – reference: TaylorADZwickerWSWeighted voting, multicameral representation, and powerGames Econ Behav19935170181120147910.1006/game.1993.1009 – reference: MurogaSThreshold logic and its applications1971NewYorkWiley0243.94014 – reference: FreixasJMolineroXSimple games and weighted games: a theoretical and computational viewpointDiscrete Appl Math200915714961508251023010.1016/j.dam.2008.09.009 – reference: MarziFRossiFSmriglioSComputational study of separation algorithms for clique inequalitiesSoft Comput2019233013302710.1007/s00500-019-03769-y – reference: TaylorADZwickerWSSimple games: desirability relations, trading, and pseudoweightings1999PrincetonPrinceton University Press0943.91005 – reference: PetersHGame theory: a multi-leveled approach20152AmsterdamSpringer10.1007/978-3-662-46950-7 – reference: SiuKRoychowdhuryVKailathTDiscrete neural computation: a theoretical foundation1995Englewood CliffsPrentice-Hall0838.68034 – reference: FreixasJThe dimension for the European Union Council under the Nice rulesEur J Oper Res200415641541910.1016/S0377-2217(02)00903-7 – reference: Simmons GJ (1990) How to (really) share a secret. In: Proceedings of the 8th annual international cryptology conference on advances in cryptology, pp 390–448. Springer, London, UK – reference: TaylorADZwickerWSA characterization of weighted votingProc Am Math Soc199211510891094109292710.1090/S0002-9939-1992-1092927-0 – reference: GvozdevaTHameedASlinkoAWeightedness and structural characterization of hierarchical simple gamesMath Soc Sci201365181189302919510.1016/j.mathsocsci.2012.11.007 – reference: PictonPDNeural networks19912IndianapolisMacmillan – reference: BorosEHammerPLIbarakiTKawakawiKPolynomial time recognition of 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2$$\end{document}-monotonic positive functions given by an oracleSIAM J Comput19972693109143124710.1137/S0097539793269089 – reference: FreixasJKurzSThe golden number and Fibonacci sequences in the design of voting systemsEur J Oper Res201322624625710.1016/j.ejor.2012.10.017 – reference: FreixasJMolineroXRouraSComplete voting systems with two types of voters: weightedness and countingAnn Oper Res2012193273287287476610.1007/s10479-011-0863-x – reference: FreixasJFreixasMKurzSOn the characterization of weighted simple gamesTheor Decis2017834469498373632710.1007/s11238-017-9606-z – reference: KurzSNapelSA dimension of the Lisbon voting rules in the EU Council: a challenge and new world recordOptim Lett201610612451256352984410.1007/s11590-015-0917-0 – reference: KurzSOn minimum sum representations for weighted voting gamesAnn Oper Res20121961361369294658410.1007/s10479-012-1108-3 – reference: Sloane NJA (1964) www.oeis.org. 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| Title | On the enumeration of Boolean functions with distinguished variables |
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