The Knackeredman, or gag for short, is a function defined as \(\mathrm{gag}(n) = A(n,n)\), where \(A(x,y)\) stands for the Ackermann function. The name came from Alistair Cockburn (of fuga- glory) in one of his forum exchanges.[1] Due to Ackermann function's relation with Arrow notation , \(\text{Gag-x} = 2\uparrow^{x-2}(x+3)-3\). Since this function diagnalizes the Ackermann function, Knackeredman is not primitive recursive.
Unlike the gar-, fz-, and (mega)fuga-, Alistair Cockburn never mentioned gag (but did mention hyperoperations) in A fuga really big numbers[2]. He also never used gag as a prefix despite rumors of him doing so; the prefix form was coined by Sbiis Saibian, also known for coining booga-[3].
Approximations
Notation | Approximation |
---|---|
Arrow notation | \(2\uparrow^{x-2}(x+3)-3\) |
Bowers' Exploding Array Function | \(\{2,(n-2),(n+3)\}\) |
Bird's array notation | \(\{2,(n-2),(n+3)\}\) |
Fast-growing hierarchy | \(f_\omega(n)\) |
Hardy hierarchy | \(H_{\omega^\omega}(n)\) |
Slow-growing hierarchy | \(g_{\varphi(\omega, 0)}(n)\) |
Sources[]
See also[]
Gar-: garone · gartwo · garthree · garfour · garfive · garsix · garseven · gareight · garnine · garten · garhundred · garmillion · gargoogol · gareceton · gartrialogue · gargoogolplex · gargiggol
Fz-: fzone · fztwo · fzthree · fzfour · fzfive · fzsix · fzseven · fzeight · fznine · fzten · fztwenty · fzthirty · fzhundred · fzthousand · fzmillion · fzgoogol · fzmilliplexion · fzgoogolplex · fzgargoogolplex · fzgiggol
Fuga-: fugaone · fugatwo · fugathree · fugafour · fugafive · fugasix · fugaseven · fugaeight · fuganine · fugaten · fugahundred · fugagoogol · fugagoogolplex · fugagargoogolplex · fugagiggol
Megafuga-: megafugaone · megafugatwo · megafugathree · megafugafour · megafugafive · megafugasix · megafugaseven · megafugaeight · megafuganine · megafugaten · megafugahundred · megafugagoogol · megafugagoogolplex · megafugagargoogolplex · megafugagargantugoogolplex · megafugagrangol
Extensions:
Booga-:: boogaone · boogatwo · boogathree · boogafour · boogafive · boogasix · boogaten · boogahundred · boogagoogol · boogagoogolplex
Gag-:: gagone · gagtwo · gagthree · gagfour · gagfive · gagsix · gagseven · gageight · gagnine · gagten · gaggoogol · gaggoogolplex