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Decentralized Controllability

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Workshop presentation given by Niels Lohmann on March 23, 2009 in Eindhoven, The Netherlands at the Berlin-Eindhoven Service Technology Colloquium 2009 (B.E.S.T. 2009).

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Transcript of "Decentralized Controllability"

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  2. 2. 1 !"#"$%&'()*"+ ,-$%&-(('.)()%/ !"#$%"&'(")*"&+,-%"./"$)0/"*+"'#1/")* !"#$%&'()*+,, -./0&1($$(23"3*&4556&▪&.",7)(8#,&▪&49&:+;<)&4556 !""#$%%&'()*+',"'+!-./.012.(0 UNIVERSITÄT ROSTOCK
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  6. 6. !"#$%"&&'()&)$*+,-$$)#./ = <#,>;+$"?#7 R P = 7"%>;"@3>#7 P1 R P2 ! <((;7",+>#7&A;3,B>"*#&<((;7",+>"(,C ! 7#<#,>;+$"?#7&A@3"$7B>"*#&<((;7",+>"(,C ! +3>(,(*(3%&A,(&<((;7",+>"(,C 4
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  9. 9. !"#$%"&&'()&)$*+,-$$)#./ = <#,>;+$"?#7 ! R P = 7"%>;"@3>#7 P1 R P2 ! <((;7",+>#7&A;3,B>"*#&<((;7",+>"(,C ! ! 7#<#,>;+$"?#7&A@3"$7B>"*#&<((;7",+>"(,C ! +3>(,(*(3%&A,(&<((;7",+>"(,C ! 4
  10. 10. 0-1-#$%'&)2-3+!"#$%"&&'()&)$* = D"8#,&+&0"&1)#"2E">)&3-&%0 P1 R P2 = E#&%#+;<)&F(;&%43("02(F&0"&1)#"02A(,#&F(;&#+<)&G(;>C&%3<)& >)+>&>)#&<(*G(%">"(,&"%&#-53'%).(" "&%#;8"<#%&+;#&<((;7",+>#7&+>&@3"$7B>"*# 9
  11. 11. 0-1-#$%'&)2-3+!"#$%"&&'()&)$* = D"8#,&+&0"&1)#"2E">)&3-&%0 P1 R P2 = E#&%#+;<)&F(;&%43("02(F&0"&1)#"02A(,#&F(;&#+<)&G(;>C&%3<)& >)+>&>)#&<(*G(%">"(,&"%&#-53'%).(" [ P1 , P2 ] [ P3 , P4 ] [ P1 , P4 ] "&%#;8"<#%&+;#&<((;7",+>#7&+>&@3"$7B>"*# 9
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  13. 13. 455&)1'$)"#+67+43'5$-%/ = D"8#,I&",<(*G+>"@$#&%#;8"<#%J&>;+,%F(;*+>"(,&;3$#% C R T P 6 %&'$07-&5'%)-$2&4("0&K&#-$%&-(("&2L&'+'3%"& "&+7+G>#;&%M,>)#%"%&L&<#,>;+$"?#7&<(,>;($$+@"$">M H
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  15. 15. 455&)1'$)"#+67+43'5$-%/ = D"8#,I&",<(*G+>"@$#&%#;8"<#%J&>;+,%F(;*+>"(,&;3$#% C R T P 6 %&'$07-&5'%)-$2&4("0&K&#-$%&-(("&2L&'+'3%"& "&+7+G>#;&%M,>)#%"%&L&<#,>;+$"?#7&<(,>;($$+@"$">M H
  16. 16. 455&)1'$)"#+67+43'5$-%/ = D"8#,I&",<(*G+>"@$#&%#;8"<#%J&>;+,%F(;*+>"(,&;3$#% C R T P 6 %&'$07-&5'%)-$2&4("0&K&#-$%&-(("&2L&'+'3%"& "&+7+G>#;&%M,>)#%"%&L&<#,>;+$"?#7&<(,>;($$+@"$">M H
  17. 17. 455&)1'$)"#+67+43'5$-%/ = D(+$I&O,7&+7+G>+@$#&%#;8"<#%&F(;&+&D"8#,&0"&1)#" = >)#;#&+;#&>E(&P(G#,P&G(;>%I = E">)&7#<#,>;+$"?#7&<(,>;($$+@"$">MJ&E#&O,7&>3G$#%&(F& <(,>;($$#;%&+,7&G;(8"7#;&%#;8"<#% N "&P<$(%3;#P&(G#;+>(;&F(;&%#;8"<#&;#G(%">(;"#%
  18. 18. 455&)1'$)"#+67+43'5$-%/ = D(+$I&O,7&+7+G>+@$#&%#;8"<#%&F(;&+&D"8#,&0"&1)#" = >)#;#&+;#&>E(&P(G#,P&G(;>%I R T = E">)&7#<#,>;+$"?#7&<(,>;($$+@"$">MJ&E#&O,7&>3G$#%&(F& <(,>;($$#;%&+,7&G;(8"7#;&%#;8"<#% N "&P<$(%3;#P&(G#;+>(;&F(;&%#;8"<#&;#G(%">(;"#%
  19. 19. 455&)1'$)"#+67+43'5$-%/ = D(+$I&O,7&+7+G>+@$#&%#;8"<#%&F(;&+&D"8#,&0"&1)#" = >)#;#&+;#&>E(&P(G#,P&G(;>%I R T N
  20. 20. 455&)1'$)"#+67+43'5$-%/ = D(+$I&O,7&+7+G>+@$#&%#;8"<#%&F(;&+&D"8#,&0"&1)#" = >)#;#&+;#&>E(&P(G#,P&G(;>%I R T = E">)&7#<#,>;+$"?#7&<(,>;($$+@"$">MJ&E#&O,7&>3G$#%&(F& <(,>;($$#;%&+,7&G;(8"7#;&%#;8"<#% [ C1 , P1 ] [ C2 , P2 ] N
  21. 21. 455&)1'$)"#+67+43'5$-%/ = D(+$I&O,7&+7+G>+@$#&%#;8"<#%&F(;&+&D"8#,&0"&1)#" = >)#;#&+;#&>E(&P(G#,P&G(;>%I R T = E">)&7#<#,>;+$"?#7&<(,>;($$+@"$">MJ&E#&O,7&>3G$#%&(F& <(,>;($$#;%&+,7&G;(8"7#;&%#;8"<#% [ C1 , P1 ] [ C2 , P2 ] N "&P<$(%3;#P&(G#;+>(;&F(;&%#;8"<#&;#G(%">(;"#%
  22. 22. 455&)1'$)"#+87+!9"%-".%'59*+:-'&)2'()&)$* = D"8#,I&#,7G(",>%&+,7&+,&)$%"&'#%)-$23&-%-#-( E1 E2 E3 = R#+$"?+@"$">MI&1+,&>)"%&",>#;+<>"(,&G;(>(<($&@#& "*G$#*#,>#7&@M&0"&1)#"0S = +$$(E%&F(;&<(,>;+<>B@MB7#%"D,&%<#,+;"(% Q
  23. 23. 455&)1'$)"#+87+!9"%-".%'59*+:-'&)2'()&)$* = D"8#,I&#,7G(",>%&+,7&+,&)$%"&'#%)-$23&-%-#-( E1 E2 E3 = R#+$"?+@"$">MI&1+,&>)"%&",>#;+<>"(,&G;(>(<($&@#& "*G$#*#,>#7&@M&0"&1)#"0S S1 S2 S3 = +$$(E%&F(;&<(,>;+<>B@MB7#%"D,&%<#,+;"(% Q
  24. 24. 455&)1'$)"#+87+!9"%-".%'59*+:-'&)2'()&)$* = +%%3*#&+&*(,">(;&>)+>&$(D%&>)#&",>#;+<>"(,% E1 E2 E3 = ;#*#*@#;&G(;>%S "&;#+$"?+@"$">M&L&7#<#,>;+$"?#7&<(,>;($$+@"$">M T
  25. 25. 455&)1'$)"#+87+!9"%-".%'59*+:-'&)2'()&)$* = +%%3*#&+&*(,">(;&>)+>&$(D%&>)#&",>#;+<>"(,% E1 E2 E3 M = ;#*#*@#;&G(;>%S "&;#+$"?+@"$">M&L&7#<#,>;+$"?#7&<(,>;($$+@"$">M T
  26. 26. 455&)1'$)"#+87+!9"%-".%'59*+:-'&)2'()&)$* = +%%3*#&+&*(,">(;&>)+>&$(D%&>)#&",>#;+<>"(,% E1 E2 E3 M = ;#*#*@#;&G(;>%S M "&;#+$"?+@"$">M&L&7#<#,>;+$"?#7&<(,>;($$+@"$">M T
  27. 27. 4&."%)$9; = (@%#;8+>"(,I&<(*G(%">"(,&(F&7#<#,>;+$"?#7&<(,>;($$#;%&"%&+& <#,>;+$"?#7&<(,>;($$#; U
  28. 28. 4&."%)$9; = (@%#;8+>"(,I&<(*G(%">"(,&(F&7#<#,>;+$"?#7&<(,>;($$#;%&"%&+& <#,>;+$"?#7&<(,>;($$#; = +$D(;">)*I VW %>+;>&E">)&*(%>BG#;*"%%"8#&<#,>;+$"?#7&<(,>;($$#; 4W ;#*(8#&P3,B7#<#,>;+$"?+@$#P&G+;>% 9W G;(X#<>&;#%3$>&>(&G(;>% U
  29. 29. 4&."%)$9; = (@%#;8+>"(,I&<(*G(%">"(,&(F&7#<#,>;+$"?#7&<(,>;($$#;%&"%&+& <#,>;+$"?#7&<(,>;($$#; = +$D(;">)*I VW %>+;>&E">)&*(%>BG#;*"%%"8#&<#,>;+$"?#7&<(,>;($$#; 4W ;#*(8#&P3,B7#<#,>;+$"?+@$#P&G+;>% 9W G;(X#<>&;#%3$>&>(&G(;>% = E(;Y%&F(;&+<M<$"<&%#;8"<#%&Z&>#;*",+>"(,&,(>&D3+;+,>##7&",& F(;&<M<$"<&%#;8"<#% U
  30. 30. <=';5&-+6 = ",&+&G#;F#<>&E(;$7J&E#&X3%>&,##7&>(&G;(X#<>&>)# <#,>;+$"?#7&<(,>;($$#; A B ! 6
  31. 31. <=';5&-+6 = ",&+&G#;F#<>&E(;$7J&E#&X3%>&,##7&>(&G;(X#<>&>)# <#,>;+$"?#7&<(,>;($$#; A !A !B B !B !A ! centralized controller 6
  32. 32. <=';5&-+6 = ",&+&G#;F#<>&E(;$7J&E#&X3%>&,##7&>(&G;(X#<>&>)# <#,>;+$"?#7&<(,>;($$#; !A !B projections A !A !B B !B !A ! centralized controller 6
  33. 33. <=';5&-+6 = ",&+&G#;F#<>&E(;$7J&E#&X3%>&,##7&>(&G;(X#<>&>)# <#,>;+$"?#7&<(,>;($$#; !A !B projections A !A !B B !B !A !A !B ! centralized controller !B !A composition of the projections 6
  34. 34. <=';5&-+6 = ",&+&G#;F#<>&E(;$7J&E#&X3%>&,##7&>(&G;(X#<>&>)# <#,>;+$"?#7&<(,>;($$#; !A !B projections A !A !B B !B !A !A !B ! centralized controller !B !A composition of ! the projections 6
  35. 35. <=';5&- = [&@3>&>)#&E(;$7&"%&,(>&+$E+M%&G#;F#<> A B ! V5
  36. 36. <=';5&- = [&@3>&>)#&E(;$7&"%&,(>&+$E+M%&G#;F#<> A B !A !B ! centralized controller V5
  37. 37. <=';5&- = [&@3>&>)#&E(;$7&"%&,(>&+$E+M%&G#;F#<> A B !A !B !A !B ! centralized controller projections V5
  38. 38. <=';5&- = [&@3>&>)#&E(;$7&"%&,(>&+$E+M%&G#;F#<> A B !A !B !A !B ! centralized controller projections !A !B !B !A composition of projections V5
  39. 39. <=';5&- = [&@3>&>)#&E(;$7&"%&,(>&+$E+M%&G#;F#<> A B !A !B !A !B ! centralized controller projections !A !B !B !A # composition of projections V5
  40. 40. 0-5-#3-#1*+67+<#'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&"$'.("2+,&+<>"(,&(F& +,(>)#;&G(;> A B ! VV
  41. 41. 0-5-#3-#1*+67+<#'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&"$'.("2+,&+<>"(,&(F& +,(>)#;&G(;> A B ! = ;"D)>&G+;>,#;&<+,,(>&Y,(E&E)#,&>(&%#,7&8 VV
  42. 42. 0-5-#3-#1*+67+<#'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&"$'.("2+,&+<>"(,&(F& +,(>)#;&G(;> A ?A B !B ! centralized controller = ;"D)>&G+;>,#;&<+,,(>&Y,(E&E)#,&>(&%#,7&8 VV
  43. 43. 0-5-#3-#1*+67+<#'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&"$'.("2+,&+<>"(,&(F& +,(>)#;&G(;> A ?A B !B ! # centralized controller = ;"D)>&G+;>,#;&<+,,(>&Y,(E&E)#,&>(&%#,7&8 VV
  44. 44. 0-5-#3-#1*+67+<#'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&"$'.("2+,&+<>"(,&(F& +,(>)#;&G(;> A ?A # B !B ! # centralized controller = ;"D)>&G+;>,#;&<+,,(>&Y,(E&E)#,&>(&%#,7&8 VV
  45. 45. 0-5-#3-#1*+67+<#'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&"$'.("2+,&+<>"(,&(F& +,(>)#;&G(;> A # ?A # B !B ! # centralized controller = ;"D)>&G+;>,#;&<+,,(>&Y,(E&E)#,&>(&%#,7&8 VV
  46. 46. 0-5-#3-#1*+87+0)/'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&+)0'.("2+,&+<>"(,&(F& +,(>)#;&G(;> A B ! V4
  47. 47. 0-5-#3-#1*+87+0)/'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&+)0'.("2+,&+<>"(,&(F& +,(>)#;&G(;> A B ! = G+;>,#;%&,##7&>(&@#&<((;7",+>#7 V4
  48. 48. 0-5-#3-#1*+87+0)/'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&+)0'.("2+,&+<>"(,&(F& +,(>)#;&G(;> A B !A !B ! centralized controller = G+;>,#;%&,##7&>(&@#&<((;7",+>#7 V4
  49. 49. 0-5-#3-#1*+87+0)/'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&+)0'.("2+,&+<>"(,&(F& +,(>)#;&G(;> A B !A !B ! # centralized controller = G+;>,#;%&,##7&>(&@#&<((;7",+>#7 V4
  50. 50. 0-5-#3-#1*+87+0)/'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&+)0'.("2+,&+<>"(,&(F& +,(>)#;&G(;> !A A B !A !B ! # centralized controller = G+;>,#;%&,##7&>(&@#&<((;7",+>#7 V4
  51. 51. 0-5-#3-#1*+87+0)/'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&+)0'.("2+,&+<>"(,&(F& +,(>)#;&G(;> !A A B !A !B ! centralized controller = G+;>,#;%&,##7&>(&@#&<((;7",+>#7 V4
  52. 52. 0-5-#3-#1*+87+0)/'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&+)0'.("2+,&+<>"(,&(F& +,(>)#;&G(;> !A A B !A !B ! # centralized controller = G+;>,#;%&,##7&>(&@#&<((;7",+>#7 V4
  53. 53. 0-5-#3-#1*+87+0)/'(&)#. = +,&+<>"(,&(F&(,#&G(;>&*3%>&,(>&+)0'.("2+,&+<>"(,&(F& +,(>)#;&G(;> !A A B !A !B ! # centralized controller !B = G+;>,#;%&,##7&>(&@#&<((;7",+>#7 V4
  54. 54. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> A B C ! V9
  55. 55. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> A B C ! = >)#&(;7#;&(F&9&+,7&82<+,,(>&@#&(@%#;8#7&@M&>)#&G+;>,#;% V9
  56. 56. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> A B A B B A C C A C centralized controller ! = >)#&(;7#;&(F&9&+,7&82<+,,(>&@#&(@%#;8#7&@M&>)#&G+;>,#;% V9
  57. 57. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> A B A B B A C $ $C not equivalent A C centralized controller ! = >)#&(;7#;&(F&9&+,7&82<+,,(>&@#&(@%#;8#7&@M&>)#&G+;>,#;% V9
  58. 58. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> A B A B B A C $ $C not equivalent A C # centralized controller ! = >)#&(;7#;&(F&9&+,7&82<+,,(>&@#&(@%#;8#7&@M&>)#&G+;>,#;% V9
  59. 59. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> A B A B B A C # $ $C not equivalent A C # centralized controller ! = >)#&(;7#;&(F&9&+,7&82<+,,(>&@#&(@%#;8#7&@M&>)#&G+;>,#;% V9
  60. 60. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> A B A B B A C # $ $C not equivalent A C # # centralized controller ! = >)#&(;7#;&(F&9&+,7&82<+,,(>&@#&(@%#;8#7&@M&>)#&G+;>,#;% V9
  61. 61. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> A B A B B A C # $ $C # not equivalent A C # # centralized controller ! = >)#&(;7#;&(F&9&+,7&82<+,,(>&@#&(@%#;8#7&@M&>)#&G+;>,#;% V9
  62. 62. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> A B A B A # C B # # $ $C not equivalent A C # # centralized controller ! = >)#&(;7#;&(F&9&+,7&82<+,,(>&@#&(@%#;8#7&@M&>)#&G+;>,#;% V9
  63. 63. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> A B A B A # C B ## # $ $C not equivalent A C # # centralized controller ! = >)#&(;7#;&(F&9&+,7&82<+,,(>&@#&(@%#;8#7&@M&>)#&G+;>,#;% V9
  64. 64. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> A B A B # A# C B ## # $ $C A not equivalent C # # centralized controller ! = >)#&(;7#;&(F&9&+,7&82<+,,(>&@#&(@%#;8#7&@M&>)#&G+;>,#;% V9
  65. 65. 0-5-#3-#1*+>7+?"#@-AB)C'&-#1- = %>+>#%&;#+<)+@$#&@M&982+,7&892*3%>&@#&#23"8+$#,> # A B A B # A# C B ## # $ $C A not equivalent C # # centralized controller ! = >)#&(;7#;&(F&9&+,7&82<+,,(>&@#&(@%#;8#7&@M&>)#&G+;>,#;% V9
  66. 66. D%"(&-;/+E)$9+!*1&)1+,-%C)1-/ = ",&<M<$#%J&7#$#>"(,&*"D)>& ,(>&@#&+,&(G>"(, A disables B B disables A A B VH
  67. 67. D%"(&-;/+E)$9+!*1&)1+,-%C)1-/ = ",&<M<$#%J&7#$#>"(,&*"D)>& ,(>&@#&+,&(G>"(, = 7#<"%"(,%&)+8#&>(&@#& <((;7",+>#7&+D+",&+,7& +D+", A disables B B disables A A B VH
  68. 68. D%"(&-;/+E)$9+!*1&)1+,-%C)1-/ = ",&<M<$#%J&7#$#>"(,&*"D)>& ,(>&@#&+,&(G>"(, = 7#<"%"(,%&)+8#&>(&@#& <((;7",+>#7&+D+",&+,7& +D+", "&B+;<% A disables B B disables A A B χ χ A B VH
  69. 69. D%"(&-;/+E)$9+!*1&)1+,-%C)1-/ = ",&<M<$#%J&7#$#>"(,&*"D)>& = %>+>#%&*+M&)+8#&*(;#& ,(>&@#&+,&(G>"(, >)+,&(,#&G;#7#<#%%(; = 7#<"%"(,%&)+8#&>(&@#& <((;7",+>#7&+D+",&+,7& +D+", A enables C B disables A "&B+;<% A disables B A B B disables A A B χ χ B C A B VH
  70. 70. D%"(&-;/+E)$9+!*1&)1+,-%C)1-/ = ",&<M<$#%J&7#$#>"(,&*"D)>& = %>+>#%&*+M&)+8#&*(;#& ,(>&@#&+,&(G>"(, >)+,&(,#&G;#7#<#%%(; = 7#<"%"(,%&)+8#&>(&@#& "&%G$">]<(GM&%>+>#%&@#F(;#& <((;7",+>#7&+D+",&+,7& &>+Y",D&+,&+<>"(, +D+", A enables C B disables A "&B+;<% A disables B A B B disables A A B χ χ B C A B VH
  71. 71. D%"(&-;/+E)$9+!*1&)1+,-%C)1-/ = ",&<M<$#%J&7#$#>"(,&*"D)>& = %>+>#%&*+M&)+8#&*(;#& ,(>&@#&+,&(G>"(, >)+,&(,#&G;#7#<#%%(; = 7#<"%"(,%&)+8#&>(&@#& "&%G$">]<(GM&%>+>#%&@#F(;#& <((;7",+>#7&+D+",&+,7& &>+Y",D&+,&+<>"(, +D+", A enables C B disables A "&B+;<% A B χ χ A disables B B disables A A B χ χ B C A B B A B B C VH
  72. 72. F%'5+B5 = 7"%>;"@3>#7&<(,>;($$+@"$">M&)+%&,"<#&+GG$"<+>"(,%I ! +7+G>+@"$">M ! ;#+$"?+@"$">M&(F&<)(;#(D;+G)"#% = @3"$7B>"*#&<((;7",+>"(,&*+Y#%&E(;$7&<(*G$"<+>#7 = <3;;#,>&+$D(;">)*&*+M&,(>&>#;*",+># VN
  73. 73. F%'5+B5 = 7"%>;"@3>#7&<(,>;($$+@"$">M&)+%&,"<#&+GG$"<+>"(,%I ! +7+G>+@"$">M ! ;#+$"?+@"$">M&(F&<)(;#(D;+G)"#% = @3"$7B>"*#&<((;7",+>"(,&*+Y#%&E(;$7&<(*G$"<+>#7 = <3;;#,>&+$D(;">)*&*+M&,(>&>#;*",+># Thank you! Any questions? VN

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