Albedo

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Line 1: {{mylangbegin}} {{mylang\|Piet\|~~Intermediate~~Pro}} {{mylang\|Julia\|~~Beginner~~Intermediate}} {{mylang\|Cardinal\|Pro}} {{mylang\|DUP\|Intermediate}} {{mylang\|FALSE\|Intermediate}} {{mylang\|Beeswax\|Creator}} {{mylang\|Glee\|Beginner}} {{mylang\|Q'Nial\|Beginner}} {{mylangend}} There aren’t any ways to upload images at the moment, so ~~I’ll~~all ~~post~~examples ~~links~~are torendered ~~the~~as ~~cloud~~wikitables. ~~storage place of~~See my ~~Piet~~PNG ~~examples~~to onwikitable myconversion ~~user~~code ~~page,~~written ~~with~~in ~~explanations~~Julia ~~how~~at ~~they~~the ~~work—hopefully~~bottom ~~for later integration in~~of the ~~appropriate places on Rosettacode~~page. ==Explanation of shorthand code for Piet examples== To shrink down the size of larger problems, I invented a shorthand text version for explaining the general program flow in a more compact form: NOP ADD DIV GRT DUP INC END 0 . + / > = c ~ PSH SUB MOD PTR ROL OUN X - % # @ N POP MUL NOT SWI INN OUC ? * ! $ n C ==Ackermann Function== [[Ackermann_function#Piet]] {\| style="border-collapse: collapse; border-spacing: 0; font-family: courier-new,courier,monospace; font-size: 10px; line-height: 1.2em; padding: 0px" \| style="background-color:#ffc0c0; color:#ffc0c0;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#c00000; color:#c00000;" \| ww \| style="background-color:#ffffc0; color:#ffffc0;" \| ww \| style="background-color:#c000c0; color:#c000c0;" \| ww \| style="background-color:#00ffff; color:#00ffff;" \| ww \| style="background-color:#00ffff; color:#00ffff;" \| ww \| style="background-color:#00c0c0; color:#00c0c0;" \| ww \| style="background-color:#c0ffff; color:#c0ffff;" \| ww \| style="background-color:#ffff00; color:#ffff00;" \| ww \| style="background-color:#ff00ff; color:#ff00ff;" \| ww \| style="background-color:#c000c0; color:#c000c0;" \| ww \| style="background-color:#00c0c0; color:#00c0c0;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \| style="background-color:#ffffc0; color:#ffffc0;" \| ww \| style="background-color:#00c0c0; color:#00c0c0;" \| ww \| style="background-color:#ffc0c0; color:#ffc0c0;" \| ww \| style="background-color:#ffc0c0; color:#ffc0c0;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#c00000; color:#c00000;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \| style="background-color:#c0ffc0; color:#c0ffc0;" \| ww \| style="background-color:#00ff00; color:#00ff00;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#c0c000; color:#c0c000;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c00000; color:#c00000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#0000c0; color:#0000c0;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c00000; color:#c00000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffc0c0; color:#ffc0c0;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \| style="background-color:#00c0c0; color:#00c0c0;" \| ww \| style="background-color:#00ffff; color:#00ffff;" \| ww \| style="background-color:#00c0c0; color:#00c0c0;" \| ww \| style="background-color:#00c0c0; color:#00c0c0;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c0ffff; color:#c0ffff;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#0000c0; color:#0000c0;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#00ffff; color:#00ffff;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffc0ff; color:#ffc0ff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#00c000; color:#00c000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c0c000; color:#c0c000;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c00000; color:#c00000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c0ffc0; color:#c0ffc0;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c000c0; color:#c000c0;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c000c0; color:#c000c0;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffc0c0; color:#ffc0c0;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c000c0; color:#c000c0;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ff00ff; color:#ff00ff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffff00; color:#ffff00;" \| ww \| style="background-color:#ffff00; color:#ffff00;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c000c0; color:#c000c0;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \| style="background-color:#c0ffff; color:#c0ffff;" \| ww \| style="background-color:#00c0c0; color:#00c0c0;" \| ww \| style="background-color:#ff00ff; color:#ff00ff;" \| ww \| style="background-color:#ffc0ff; color:#ffc0ff;" \| ww \| style="background-color:#c000c0; color:#c000c0;" \| ww \| style="background-color:#c000c0; color:#c000c0;" \| ww \| style="background-color:#00ff00; color:#00ff00;" \| ww \| style="background-color:#ff00ff; color:#ff00ff;" \| ww \| style="background-color:#ffffc0; color:#ffffc0;" \| ww \| style="background-color:#c0c000; color:#c0c000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffc0ff; color:#ffc0ff;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ff00ff; color:#ff00ff;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#c0ffff; color:#c0ffff;" \| ww \| style="background-color:#00c000; color:#00c000;" \| ww \| style="background-color:#00ff00; color:#00ff00;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \| style="background-color:#0000c0; color:#0000c0;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#c0ffff; color:#c0ffff;" \| ww \| style="background-color:#00c0c0; color:#00c0c0;" \| ww \|- \|} Program flow (see explanation of the shorthand codes at the top of the page), with labels: l1 l3 ↓ ↓ 1X2X-nn2X1X@=1X=->!#2X1X@=1X=- . l2→? > . 2 ! ................X1-X1?# . X l4↑2←l5 . 1 X . X 1 . @ X . = @ . 1 = . ~ X 3 . N = X . + - 1 . X 2 X . 1 X @ . l6→? 1 1 .+X1@X1X2#>@X X . l7↑ - ...............-X1@X1X3 Explanation with stack for the Ackermann Function(2,0): (for the meaning of the square brackets, please read below) [A] [B] 1 1 1 1 1 0 1 1 1 0 2 2 2 2 2 2 1 0 2 2 0 0 0 0 0 1 0 0 0 [2] 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 1 [1] 2 2 2 2 2 [0] 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 1 [1] 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ——————————————————————————————————————————————————————————————————————————————————————— 1X2X - n n 2X 1X @ = 1X = - > ! # 2X 1X @ = 1X = - > ! # ? 1X - 1X ↑ ↑ ↑ l1 l3 l4 --------------------------------------------------------------------------------------- explanation: correct order m>0? yes n>0? no A(m-1,1) of m,n [B] [C] 1 1 1 1 1 1 1 0 1 1 1 0 1 3 3 1 3 3 1 2 2 1 1 1 1 1 0 2 2 1 1 1 1 1 0 2 2 1 1 1 1 1 0 0 0 1 1 [0] 1 1 [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 [1] 1 1 [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 [0] -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 —————————————————————————————————————————————————————————————————————————————————————————————————————— 2X 1X @ = 1X = - > ! # 2X 1X @ = 1X = - > ! # 2X 1X @ = 3X 1X @ 1X - 3X 1X @ 1X - ↑ ↑ ↑ l1 l3 l5 ------------------------------------------------------------------------------------------------------- correct order m>0? yes n>0? yes A(m-1,A(m,n-1)) of m,n [D] 1 1 1 1 1 1 0 1 1 1 0 1 1 0 2 2 1 1 1 1 1 0 2 2 0 0 0 0 0 1 2 2 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 [1] 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 [0] 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [0] 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ————————————————————————————————————————————————————————————————————————————————————————————————————— 2X 1X @ = 1X = - > ! # 2X 1X @ = 1X = - > ! # ? 1X - 1X 2X 1X @ = 1X = - > ! # ↑ ↑ ↑ ↑ l1 l3 l4 l1 ----------------------------------------------------------------------------------------------------- correct order m>0? yes n>0? no A(m-1,1) correct order m>0? of m,n for inner inner for inner f. function function 1 1 2 2 1 1 1 1 0 0 0 0 1 1 1 1 0 2 2 0 0 0 0 0 0 0 0 2 2 1 2 2 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 ——————————————————————————————————————————————————————————————————————————————————— ? 2X 1X @ = 1X = - 2X 1X @ > # 2X 1X @ 1X + 2X 1X @ = 1X = - > ! # ↑ ↑ ↑ l2 l7 l1 output 3 ————————————————————————————————————————————————————— 1 1 2 2 1 1 1 0 0 0 -1 2 2 -1 -1 -1 -1 -1 -1 0 1 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 2 2 2 2 2 2 2 2 2 2 2 2 3 ————————————————————————————————————————————————————— ? 2X 1X @ = 1X = - 2X 1X @ > # ? 1X + N ~ ↑ ↑ l2 l6 Which leads to the result Ackermann(2,0) = 3 What the program actually does is working through the whole steps of calculating the A(m,n) functions, according to: A(2,0) = A(1,1) ,A(m,n)= A(m-1,1) if m>0 and n=0 A(1,1) = A(0,A(1,0)) ,A(m,n)= A(m-1,A(m,n-1)) if m>0 and n>0 A(0,A(1,0)) = A(0,A(0,1)) ,A(m,n)= A(m,n-1) if m>1 and n=0, for the inner function A(0,A(0,1)) = A(0,2) ,A(m,n)= n+1 if m=0, for the inner function A(0,2) = 3 ,A(m,n)= n+1 if m=0 The coefficients and the results can be found on the stack. See the places marked with square brackets [] and Labels in the program/stack flow above. At each turn, the upper two values on the stack are flipped, so they are in the proper order for computing inner functions. For highly nested A(...,(A(A(A(...))))) functions the stack can grow to enormous proportions very fast, then computing its way outwards again, from innermost functions outwards. The innermost coefficients are always on top of the stack. Checks for the appropriate substitute function and branching structures take the largest part of the program. ==Binary Digits== [[Binary digits#Piet]] png image download: [Image:https://copy.com/Ixp3nc7QJQTCrcDb] Rendered as wikitable {\| style="border-collapse: collapse; border-spacing: 0; font-family: courier-new,courier,monospace; font-size: 18px; line-height: 1.2em; padding: 0px" \| style="background-color:#ffc0c0; color:#ffc0c0;" \| ww \| style="background-color:#ffc0c0; color:#ffc0c0;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \| style="background-color:#c0ffc0; color:#c0ffc0;" \| ww \| style="background-color:#c0ffc0; color:#c0ffc0;" \| ww \| style="background-color:#00ff00; color:#00ff00;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#00ff00; color:#00ff00;" \| ww \| style="background-color:#00c000; color:#00c000;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#ffff00; color:#ffff00;" \| ww \| style="background-color:#0000c0; color:#0000c0;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#00c0c0; color:#00c0c0;" \| ww \| style="background-color:#c0c000; color:#c0c000;" \| ww \| style="background-color:#ffffc0; color:#ffffc0;" \| ww \|- \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#0000c0; color:#0000c0;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \|- \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#00ff00; color:#00ff00;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \| style="background-color:#0000c0; color:#0000c0;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#c0ffc0; color:#c0ffc0;" \| ww \| style="background-color:#00c000; color:#00c000;" \| ww \| style="background-color:#00c000; color:#00c000;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \|- \| style="background-color:#c000c0; color:#c000c0;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#c0c0ff; color:#c0c0ff;" \| ww \|- \| style="background-color:#ffc0ff; color:#ffc0ff;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#ff00ff; color:#ff00ff;" \| ww \| style="background-color:#ffff00; color:#ffff00;" \| ww \| style="background-color:#ffffc0; color:#ffffc0;" \| ww \| style="background-color:#ffffc0; color:#ffffc0;" \| ww \| style="background-color:#ffffc0; color:#ffffc0;" \| ww \| style="background-color:#ffffc0; color:#ffffc0;" \| ww \| style="background-color:#ffffc0; color:#ffffc0;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffff00; color:#ffff00;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#ffffc0; color:#ffffc0;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffff; color:#ffffff;" \| ww \| style="background-color:#ffffc0; color:#ffffc0;" \| ww \|- \|} Example output: ? 3 11 ? 13 1101 Program flow (explanation of codes at the top of the page): 2Xn=2X/=1X!>#?N=1X . . 2 . > . @ X . . . X1X2%X2@X1 . . ? . . C+=X5............# . . .... Example, with stack and output, for base-10 number 3: 1 1 \n \| Output ———————————————————————————————————————————————————————————+——————— 10 \| Stack 10 1 1 2 0 000 \| 2 1 111 2 2 2 2 2 1 100 0000 1 \| 3 311 1111 1 13 31 1 111 111 11111 1 10 1 5 \| 33 333 3333 3 31 11 1 111 111 1111111 111 2 21 55 10 \| 222 222 2222 2 22 22 2 222 222 2222222 22222 222 22 2 2 \| ———————————————————————————————————————————————————————————+ 2Xn=2X/=1X!>#2X1X@2X%2X1X@=2X/=1X!>#?N=1X>#N=1X>#5X= + C ? ... 2Xn=2X ... (return to beginning) The program executes short division with remainder. l0: 2X : acts as marker for the end of the binary output n : input base-10 number l1: = : duplicate number 2X/= : divide n by 2 and duplicate the result 1X!>#: is the result >0? (is n>=2?) It is, thus rotate the pointer by 1 (downwards, continue at l2) and go through the loop l2: 2X1X@: move the result down (needed for further processing), and move n to the top of the stack 2X% : n mod 2 (remainder, last binary digit) 2X1X@: move binary digit down, move result of division up loop is finished, back to the beginning (l1) l1: = : duplicate result of division 2X/= : divide n by 2 and duplicate the result 1X!>#: is the result >0? (is n>=2?) It is not, thus don’t rotate the pointer and move on (towards l3) ? : pop division result from the stack l3: N : output first bit: 1 (12^1) = : duplicate next bit 1X># : is top of stack >1? It isn’t (end marker check). Thus, back to beggining of loop (l3) N : output next bit: 1 (12^0) = : duplicate next bit (which, in this case, is the marker) 1X># : is top of stack >1? It is (end marker reached). Thus, move on to (l4) l4: 5X=+ : 5+5=10, 10 is the ASCII value for the newline character (\n) C? : output top of stack as character (newline) and pop the marker. End reached, leading back to (l0) l0: 2Xn=2X/=........... Flow diagram, with labels l0 to l3: l0 l1 l3 ↓ ↓ ↓ 2Xn=2X/=1X!>#?N=1X . . . . . . . l2→2 . > . @ X . . . X1X2%X2@X1 . . ? . . C+=X5............# ↑ . . l4 .... ==Integer Sequence== [[Integer_sequence#Piet]] PNG image download: [Image:https://copy.com/TQuwy3dwBRl7nEOL] Rendered as wikitable: {\| style="border-collapse: collapse; border-spacing: 0; font-family: courier-new,courier,monospace; font-size: 20px; line-height: 1.2em; padding: 0px" \| style="background-color:#ffc0c0; color:#ffc0c0;" \| ww \| style="background-color:#ff0000; color:#ff0000;" \| ww \| style="background-color:#c0ffc0; color:#c0ffc0;" \| ww \| style="background-color:#ffc0c0; color:#ffc0c0;" \| ww \| style="background-color:#ff00ff; color:#ff00ff;" \| ww \| style="background-color:#ff00ff; color:#ff00ff;" \| ww \| style="background-color:#c000c0; color:#c000c0;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#ff00ff; color:#ff00ff;" \| ww \| style="background-color:#ff00ff; color:#ff00ff;" \| ww \| style="background-color:#ff00ff; color:#ff00ff;" \| ww \| style="background-color:#00c0c0; color:#00c0c0;" \| ww \|- \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#000000; color:#000000;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \| style="background-color:#00c0c0; color:#00c0c0;" \| ww \| style="background-color:#00ffff; color:#00ffff;" \| ww \| style="background-color:#0000ff; color:#0000ff;" \| ww \|} (7x3 codels) Line 33 ⟶ 782: 1 X ! = N 5 X @ 0. + = X 1 C + Line 55 ⟶ 804: \| +————————————————————————————————— !1=0 (Sequence begins at 0) ==Create Wikitable from PNG files== This is the Julia code I use to generate wikitables from Piet program png files (codel size 1). I just copy/paste the console output of the program. Usage: convert("filename.png", blocksize) using Images, ImageView function convert(name::String,blocksize::Int) img=Images.imread("$name") view(img) println("{\| style=\"border-collapse: collapse; border-spacing: 0; font-family: courier-new,courier,monospace; font-size: $(blocksize)px; line-height: 1.2em; padding: 0px\"") for y=1:height(img) for x=1:width(img) r=hex(int(img[x,y].r255)) g=hex(int(img[x,y].g255)) b=hex(int(img[x,y].b*255)) r=="0" ? r="00" : nothing g=="0" ? g="00" : nothing b=="0" ? b="00" : nothing print("\| style=\"background-color:#$r$g$b; color:#$r$g$b;\" \| ww\n") x==width(img) ? print("\|-\n\n"):nothing end end print("\|}") end