Measuring the computational complexity of non-primitive recursive algorithms.
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If the ordinal is a successor (e.g., $1, 2, 3...$), we use functional iteration. $$f_\alpha+1(n) = f_\alpha^n(n)$$ Translation for the calculator: Apply the previous function $f_\alpha$ to $n$ repeatedly, $n$ times. fast growing hierarchy calculator
fα(n)=fα[n](n)f sub alpha of n equals f sub alpha open bracket n close bracket end-sub of n
. These functions are defined by how they build upon one another: fα(n)=fα[n](n)f sub alpha of n equals f sub
An FGH calculator is, in a sense, a partial time machine. It lets you skip past the puny exponentials, past the Knuth arrows, past Conway chains, past the busy beaver of low-level recursion, and stare directly at the boundary where computation itself begins to falter.
def fgh(alpha, n): """Basic Fast Growing Hierarchy Calculator (Wainer)""" if n == 0: return 0 # Convention for f_a(0) if isinstance(alpha, int): # Finite ordinal if alpha == 0: return n + 1 else: result = n for _ in range(n): result = fgh(alpha - 1, result) return result scientific notation fails
In the realm of googology—the study of mind-bogglingly large numbers—standard mathematical notation quickly breaks down. Writing out zeros becomes impossible, scientific notation fails, and even advanced systems like Knuth’s up-arrows eventually lose their grip.
Outstrips the standard FGH entirely, requiring extensions past the Church-Kleene ordinal ( ω1CKomega sub 1 raised to the cap C cap K power Practical Applications of the Hierarchy