Expression 28: "G" equals left bracket, "g" Subscript, "b" "l" "o" "c" "k" , Baseline left parenthesis, "b" Subscript, "B" "l" "o" "c" "k" , Baseline , right parenthesis , "g" Subscript, "b" "i" "g" "S" "t" "e" "p" , Baseline left parenthesis, "b" Subscript, "B" "i" "g" "S" "t" "e" "p" , Baseline , right parenthesis , "g" Subscript, "s" "m" "a" "l" "l" "S" "t" "e" "p" , Baseline left parenthesis, "b" Subscript, "S" "m" "a" "l" "l" "S" "t" "e" "p" , Baseline , right parenthesis , right bracketG=gblockbBlock,gbigStepbBigStep,gsmallStepbSmallStep
equals=
1248132612481326
1231980912319809
1521054415210544
28
Expression 29: "W" equals left bracket, "w" Subscript, "b" "l" "o" "c" "k" , Baseline left parenthesis, "b" Subscript, "B" "l" "o" "c" "k" , Baseline , right parenthesis , "w" Subscript, "b" "i" "g" "S" "t" "e" "p" , Baseline left parenthesis, "b" Subscript, "B" "i" "g" "S" "t" "e" "p" , Baseline , right parenthesis , "w" Subscript, "s" "m" "a" "l" "l" "S" "t" "e" "p" , Baseline left parenthesis, "b" Subscript, "S" "m" "a" "l" "l" "S" "t" "e" "p" , Baseline , right parenthesis , right bracketW=wblockbBlock,wbigStepbBigStep,wsmallStepbSmallStep
equals=
35891353589135
52212315221231
1188455211884552
29
Expression 30:
30
challenge bond equation
31
Expression 32: "s" left parenthesis, "l" , "p" , "a" , right parenthesis equals left brace, "l" greater than "L" : 0 , "p" "g" left parenthesis, Start sum from "m" equals "l" to "L" , end sum, left parenthesis, "a" Superscript, left parenthesis, "m" minus "l" , right parenthesis , Baseline left parenthesis, left parenthesis, "a" minus 1 , right parenthesis "G" left bracket, "m" , right bracket plus "a" "W" left bracket, "m" , right bracket , right parenthesis , right parenthesis minus "c" Subscript, "l" "a" "y" "e" "r" "Z" "e" "r" "o" , Baseline , right parenthesis , right bracesl,p,a=l>L:0,pgL∑m=lam−la−1Gm+aWm−clayerZero
32
Expression 33:
33
resource ratio given level, current gas price, gas price used for stake calculation, ratio used for stake calculation
34
the adversary has best ratio in the smallstep level when p > P_s, best in block level otherwise
35
the min(1,L-l) term is just "if not small step level, honest must create a refinement node"
36
Expression 37: "R" left parenthesis, "l" , "p" , "p" Subscript, "s" , Baseline , "a" , right parenthesis equals StartFraction, "p" "g" left parenthesis, "G" left bracket, "l" , right bracket plus "c" Subscript, "l" "a" "y" "e" "r" "Z" "e" "r" "o" , Baseline , right parenthesis plus "s" left parenthesis, "l" , "p" Subscript, "s" , Baseline , "a" , right parenthesis Over "p" "g" left parenthesis, "W" left bracket, "l" , right bracket plus "G" left bracket, "l" , right bracket plus min left parenthesis, 1 , "L" minus "l" , right parenthesis "c" Subscript, "l" "a" "y" "e" "r" "Z" "e" "r" "o" , Baseline , right parenthesis plus "s" left parenthesis, "l" plus 1 , "p" Subscript, "s" , Baseline , "a" , right parenthesis , EndFractionRl,p,ps,a=pgGl+clayerZero+sl,ps,apgWl+Gl+min1,L−lclayerZero+sl+1,ps,a
37
Expression 38:
38
stake sizes given some P and A
39
"P"P
"A"A
"s" left parenthesis, 1 , "P" , "A" , right parenthesiss1,P,A
"s" left parenthesis, 2 , "P" , "A" , right parenthesiss2,P,A
"s" left parenthesis, 3 , "P" , "A" , right parenthesiss3,P,A
100100
10.9 6 210.962
computed cell: 3599.9 2 63599.926
computed cell: 326.8 5 9 0 6326.85906
computed cell: 28.1 2 7 5 228.12752
200200
8.6 9 48.694
computed cell: 3600.2 1 2 13600.2121
computed cell: 411.0 8 2 1 8411.08218
computed cell: 43.9 6 4 7 0 443.964704
500500
6.3 7 96.379
computed cell: 3569.4 3 6 43569.4364
computed cell: 552.2 7 9 8 2552.27982
computed cell: 78.5 4 9 1 8 578.549185
800800
5.4 6 55.465
computed cell: 3599.6 1 6 63599.6166
computed cell: 647.2 9 1 1 1647.29111
computed cell: 105.8 6 6 7 6105.86676
computed cell:
computed cell:
computed cell:
40
Expression 41:
41
visualize resource ratio as function of current gas price
42
Expression 43: left brace, "x" greater than or equal to 0 : min left parenthesis, "R" left parenthesis, 1 , "x" , "P" left bracket, 1 , right bracket , "A" left bracket, 1 , right bracket , right parenthesis , "R" left parenthesis, "L" , "x" , "P" left bracket, 1 , right bracket , "A" left bracket, 1 , right bracket , right parenthesis , right parenthesis , right bracex≥0:minR1,x,P1,A1,RL,x,P1,A1
43
Expression 44: left brace, "x" greater than or equal to 0 : min left parenthesis, "R" left parenthesis, 1 , "x" , "P" left bracket, 2 , right bracket , "A" left bracket, 2 , right bracket , right parenthesis , "R" left parenthesis, "L" , "x" , "P" left bracket, 2 , right bracket , "A" left bracket, 2 , right bracket , right parenthesis , right parenthesis , right bracex≥0:minR1,x,P2,A2,RL,x,P2,A2
44
Expression 45: left brace, "x" greater than or equal to 0 : min left parenthesis, "R" left parenthesis, 1 , "x" , "P" left bracket, 3 , right bracket , "A" left bracket, 3 , right bracket , right parenthesis , "R" left parenthesis, "L" , "x" , "P" left bracket, 3 , right bracket , "A" left bracket, 3 , right bracket , right parenthesis , right parenthesis , right bracex≥0:minR1,x,P3,A3,RL,x,P3,A3
45
Expression 46: left brace, "x" greater than or equal to 0 : min left parenthesis, "R" left parenthesis, 1 , "x" , "P" left bracket, 4 , right bracket , "A" left bracket, 4 , right bracket , right parenthesis , "R" left parenthesis, "L" , "x" , "P" left bracket, 4 , right bracket , "A" left bracket, 4 , right bracket , right parenthesis , right parenthesis , right bracex≥0:minR1,x,P4,A4,RL,x,P4,A4
Expression 50: "C" left parenthesis, "l" , right parenthesis equals "P" Subscript, "c" , Baseline "g" left parenthesis, "G" left bracket, "l" , right bracket plus "W" left bracket, "l" , right bracket plus "c" Subscript, "l" "a" "y" "e" "r" "Z" "e" "r" "o" , Baseline plus "c" Subscript, "c" "o" "n" "f" "i" "r" "m" "B" "y" "T" "i" "m" "e" , Baseline , right parenthesisCl=PcgGl+Wl+clayerZero+cconfirmByTime
50
"x" Subscript, 1 , Baselinex1
"C" left parenthesis, "x" Subscript, 1 , Baseline , right parenthesisCx1
"P" Subscript, "c" , Baseline "g" left parenthesis, "G" left bracket, "x" Subscript, 1 , Baseline , right bracket plus "W" left bracket, "x" Subscript, 1 , Baseline , right bracket , right parenthesisPcgGx1+Wx1
11
computed cell: 8.3 5 7 1 3 0 58.3571305
computed cell: 8.0 3 5 2 3 0 58.0352305
22
computed cell: 9.0 9 2 4 29.09242
computed cell: 8.7 7 0 5 28.77052
33
computed cell: 13.8 6 9 4 4 813.869448
computed cell: 13.5 4 7 5 4 813.547548
computed cell:
computed cell:
51
Expression 52:
52
Expression 53: "g" equals 10 to the negative 9th powerg=10−9