Analysis (and possible extension) to Fish's s(n) map

See s(n) map.

Definition

s(1)[f(n)] = fⁿ(n)

s(n)[f(m)] = s(n-1)^m[f(m)] if n > 1

With only two arguments, this reaches ω^ω-level recursion.

It's standard to set f(n) = n+1 in this case--using a pre-defined function would be rather naive.

Comparison to FGH if f(n) = n+1

f(n) = n+1 = f₀(n)

f²(n) = n+2

f³(n) = n+3

f^a(n) = n+a

s(1)[f(n)] = fⁿ(n) = 2n = f₁(n)

s(1)²[f(n)] = s(1)[fⁿ(n)] = s(1)[2n] = 2ⁿ×n = f₂(n)

s(1)³[f(n)] = s(1)[2ⁿ×n] = f₃(n) > (2↑)ⁿ↑n

s(1)⁴[f(n)] = f₄(n) > (2↑↑)ⁿ↑↑n

s(1)^a[f(n)] = f_a(n) > (2↑^a)ⁿ↑^an

s(2)[f(n)] = s(1)ⁿ[f(n)] = f_ω(n)

s(1)[s(2)[f(n)]] = s(2)[f(s(2)[f(s(2)[f(...s(2)[f(s(2)[f(n)])]...)])])] with n "s(2)"s = f_ω+1(n)

s(1)²[s(2)[f(n)]] = f_ω+2(n)

s(1)^a[s(2)[f(n)]] = f_ω+a(n)

s(2)²[f(n)] = s(1)ⁿ[s(2)[f(n)]] = f_ω2(n)

s(1)^a[s(2)²[f(n)]] = f_ω2+a(n)

s(2)³[f(n)] = s(1)ⁿ[s(2)²[f(n)]] = f_ω3(n)

s(2)^a[f(n)] = s(1)ⁿ[s(2)^a-1[f(n)]] = f_ωa(n)

s(1)^a[s(2)^b[f(n)]] = f_ωb+a(n)

s(3)[f(n)] = s(2)ⁿ[f(n)] = f_ω²(n)

s(1)^a[s(3)[f(n)]] = f_ω²+a(n)

s(2)^a[s(3)[f(n)]] = f_ω²+ωa(n)

s(3)^a[f(n)] = f_ω²a(n)

s(4)[f(n)] = s(3)ⁿ[f(n)] = f_ω³(n)

s(5)[f(n)] = s(4)ⁿ[f(n)] = f_ω⁴(n)

s(a)[f(n)] = s(a-1)ⁿ[f(n)] = f_ω^a-1(n)

s(1)^a[s(2)^b[s(3)^c[s(4)^d[s(5)^e[...]]]]] = f_{...+ω⁴e+ω³d+ω²c+ωb+a}(n)

Limit: s(n+1)[f(n)] = f_ω^ω(n)

Extension

In the definition of Fish number 3, an extension is made called ss(n) map. It is defined as:

ss(1)[f(n)] = s(n)[f(n)]

ss(n)[f(m)] = ss(n-1)^m[f(m)] if m > 1

Unfortunately, at this point, this function can no longer be compared directly to FGH, due to the fact that:

s(n+1)[f(n)] = f_ωⁿ(n) = f_ω^ω(n)

s(n)[f(n)] = f_ω^n-1(n) ≠ f_ω^ω(n)

But that's not necessary.

In this extension, ss(n) can just be expressed as s(n,2). Then, we can define the following extension based off of this:

s(a,1)[f(n)] = s(a)[f(n)]

s(1,b)[f(n)] = s(n,b-1)[f(n)]

s(a,b)[f(n)] = s(a-1,b)ⁿ[f(n)]

Or, actually, generalize this to any number of arguments:

(@ represents the rest of the array, or none)

s(@,1)[f(n)] = s(@)[f(n)]

s(1,1,1,...,1,1,1,a,@)[f(n)] = s(n,n,n,...,n,n,n,a-1,@)[f(n)]

s(a,@)[f(n)] = s(a-1,@)ⁿ[f(n)]

Comparison to FGH if f(n) = n+1

Note: All expressions are actually equal to significantly less than their approximations. s(1,2)[f(n)] = s(n)[f(n)] = f_ω^n-1(n) ≈ f_ω^ω(n)

s(1)[s(1,2)[f(n)]] ≈ f_ω^ω+1(n)

s(2)[s(1,2)[f(n)]] ≈ f_ω^ω+ω(n)

s(3)[s(1,2)[f(n)]] ≈ f_ω^ω+ω²(n)

s(a)[s(1,2)[f(n)]] ≈ f_ω^ω+ω^a-1(n)

s(1,2)^a[f(n)] ≈ f_ω^ωa(n)

s(2,2)[f(n)] = s(1,2)ⁿ[f(n)] ≈ f_ω^ω+1(n)

s(a,2)[f(n)] = s(a-1,2)ⁿ[f(n)] ≈ f_ω^ω+a-1(n)

s(1,3)[f(n)] = s(n,2)[f(n)] ≈ f_ω^ω+n-1(n) ≈ f_ω^ω2(n)

s(a)[s(1,3)[f(n)]] ≈ f_{ω^ω2+ω^a-1}(n)

s(a,2)[s(1,3)[f(n)]] ≈ f_{ω^ω2+ω^ω+a-1}(n)

s(2,3)[f(n)] = s(1,3)ⁿ[f(n)] ≈ f_ω^ω2+1(n)

s(a,3)[f(n)] = s(a-1,3)ⁿ[f(n)] ≈ f_ω^ω2+a-1(n)

s(1,4)[f(n)] = s(n,3)[f(n)] ≈ f_ω^ω3(n)

s(1,a)[f(n)] = s(n,a-1)[f(n)] ≈ f_ω^ω(a-1)(n)

s(1,1,2)[f(n)] = s(n,n)[f(n)] ≈ f_ω^ω(n-1)+n(n) = f_ω^ωn(n) = f_ω^ω2(n)

s(a)[s(1,1,2)[f(n)]] ≈ f_{ω^ω2+ω^a-1}(n)

s(1,a)[s(1,1,2)[f(n)]] ≈ f_{ω^ω2+ω^ω(a-1)}(n)

s(2,1,2)[f(n)] = s(1,1,2)ⁿ[f(n)] ≈ _^fωω2+1(n)

s(a,1,2)[f(n)] = s(a-1,1,2)ⁿ[f(n)] ≈ _^fωω2+a-1(n)

s(1,2,2)[f(n)] = s(n,1,2)[f(n)] ≈ _^fωω2+n-1(n) ≈ _^fωω2+ω(n)

s(2,2,2)[f(n)] = s(1,2,2)ⁿ[f(n)] ≈ _^fωω2+ω+1(n)

s(1,3,2)[f(n)] = s(n,2,2)[f(n)] ≈ _^fωω2+ω2(n)

s(1,4,2)[f(n)] = s(n,3,2)[f(n)] ≈ _^fωω2+ω3(n)

s(1,a,2)[f(n)] = s(n,a-1,2)[f(n)] ≈ _{^{fωω2+ω(a-1)}}(n)

s(a,b,2)[f(n)] = s(a-1,b,2)ⁿ[f(n)] ≈ _{^{fωω2+ω(b-1)+a}}(n)

s(1,1,3)[f(n)] = s(n,n,2)[f(n)] ≈ _{^{fωω2+ω(n-1)+n}}(n) = _^fωω2+ωn(n) = _^fωω22(n)

s(1,1,4)[f(n)] = s(n,n,3)[f(n)] ≈ _^fωω23(n)

s(1,1,a)[f(n)] = s(n,n,a-1)[f(n)] ≈ _^fωω2(a-1)(n)

s(1,1,1,2)[f(n)] = s(n,n,n)[f(n)] ≈ _^fωω2(n-1)(n) ≈ f_ω^ω3(n)

s(1,1,1,1,2)[f(n)]  = s(n,n,n,n)[f(n)] ≈ f_ω^ω4(n)

Limit: s(n,n,n,...(n "n"s)...,n,n,n)[f(n)] ≈ f_ω^ωn(n) = f_ω^ωω(n)

Well, that was disappointing. I expected f_ε0(n). Oh well. Still a proper and valid extension.