Revised Pehan Notation is a notation created by User:Licorneuhh.[1]
Basic Level
The rules for basic level are as follows:
- 10(m) = um , where u0 = 1 and un+1 = m↑unm , using the Knuth up-arrow notation
- ak(n) = ak-1n(n) for all a, k > 0
- a0(n) = a-1n(n) for all a > 1
Geometric level
- ab(n) = 1ab(n)
- |a|b(n) = 2ab(n)
- gab(n) = a(in a g-gon)b(n)
- g10(n) = g-110(n) for all g>1
Simple array level
- 1[1]0(n) = nnn(n)
- 1[a]0(n) = nn[a-1]n(n) for all a > 1
Multi-entries array level
Multi-entry rules have to be followed in the described order :
At each changement (so when you reach 1[l]0(n) point, where l is a list of entries) :
1. All non-array arguments are replaced by n
2. In the array, the leftest non-0 entry decrease by 1
3. In the array, if there is one or more 0s chain at the left, they're all replaced by the n argument of the function
4. In the array, if the rightest entry decrease to 0, then the entry is removed.
Multi-square-brackets array level
- ga[ [[[[[...k...[l]...]]]]]]b(n) = ga[(k)[l]]b(n)
- 1[(k)[l]]0(n) = nn[(k-1)[n,n,n...n]]n(n) with n n's in the array
Multi-arrays level
When one array disappear, the array at its left take over, and has n terms of n value with n square brackets. The fact that there is one ore more array is denoted as empty arrays "[ ]" (different from array that have 0 !). The priority goes to the rightest array. The numbers of empty array can be descibed by a number in supersupscript after one empty array.
- 1[ ] [1]0(n) = nn[(n)[n,n,n,n,n,...n,n,n]]n(n), with n terms in the bracket.
Superior array levels level
The L level of array is put in the supersupscript at the left of the array, noted as {L}, with the level 1 as basic level :
ga[{1}l]b(n) = ga[l]b(n)
The basic relation between level of arrays is as follows :
- 1[{L}1]0(n) = nn[{L-1} ]n [(n)[{L-1}n,n,n...,n]]n(n)
Stars level
- 1*0(n) = nn[{n} ]n-1 [(n)[{n}n,n,...,n]]n(n) (with n n's in the array)
- 1*k 0(n) = nn*k-1[{n} ]n-1 [(n)[{n}n,n,...,n]] n(n) (with n n's in the array, and with k = number of stars)
Yogh function
Ȝ(n) = 1*n 0(n)
Examples
- 10(0) = 0
- 10(1) = 1↑1 = 1
- 10(2) = 2↑2↑22 = 2↑42 = 2↑↑↑↑2 = 4
- 10(3) = 3↑3↑3↑333 = 3↑3↑2733
- 11(4) = 10(10(10(10(4)))) = 10(10(10(4↑4↑4↑256444)))
- 21(3) = 202(122(112(102(3↑3↑2733))))
- |1|2(2) = |1|1(|1|1(2)) = |1|1(|1|0(22(2))) = |1|1(|1|0(21(20(11(10(10(2))))))) = |1|1(|1|0(21(20(11(4↑4↑4↑256444)))))
- 510(4) =444(4) = 4434(4) = 443(443(443(443(4))))
- 1[1]0(5) = 555(5)
- 1[6]0(3) = 33[5]3(3)
- 1[0,1]0(2) = 22[2,0]2(2) = 22[2]2(2) = |2[2]|1(|2[2]|0(|1[2]|1(|1[2]|0(22[1]2(2)))))
- 1[0,43,0,145,1]0(2) = 1[2,42,0,145,1]0(2)
- 1[0,0,0,145,1]0(10100) = 1[10100,10100,10100,144,1]0(2)
- 1[0,0,0,0,0,1]0(150) = 1[150,150,150,150,150]0(150)
- 1[ [1] ]0(4) = 44[4,4,4,4]4(4)
- 1[(3)[1]]0(4) = 44[ [4,4,4,4]]4(4)
- 1[(6)[1]]0(8) = 88[(5)[8,8,8,8,8,8,8,8]]8(8)
- 1[ ] [1]0(6) = 66[(6)[6,6,6,6,6,6]]6(6)
- 1[ ] [ ] [1]0(5) = 55[ ] [(5)[5,5,5,5,5]]5(5)
- 1[ ]87 [1]0(5) = 55[ ]86 [(5)[5,5,5,5,5]]5(5)
- 1[{2}1]0(6) = 66[{1} ]6 [(6)[{1}6,6,6,6,6,6]]6(6) = 66[ ]6 [(6)[6,6,6,6,6,6]]6(6)