User:Ikosarakt1/Optimized notation

This page is just a training place where I try to create compact and optimized set of rules in order to reach faster and faster growing functions. "Limit growth rate" means growth rate of the function which diagonalizes through all defined structures.

Limit growth rate = $$f_1(n) = 2n$$

 * (0,n) = n
 * 1) (N+1,n) = (N,n+1)

Looks like first 2 rules in Hardy hierarchy.

Limit growth rate = $$f_\omega(n) \approx n \uparrow^n n$$

 * (0,n) = n
 * 1) (N+1,n) = (N,n+1)
 * (N,n) = (N[n],n)
 * 1) (N+M)[n] = N+(M[n])
 * 2) A(0) = 1
 * 3) A(N+1)[0] = 0
 * 4) A(N+1)[n+1] = A(N)+A(N+1)[n]

Note that the rule 4 looks like Cantor's normal form rule for ordinals, but if we should allow things like $$\omega+1+\omega$$ (Why not, if $$\omega$$ is just a notation symbol? We don't need to waste rules in order to remove such expressions from legal list.)

Limit growth rate = $$f_{\varepsilon_0}(n) \approx \text{Hydra}(n)$$

 * (0,n) = n
 * 1) (N+1,n) = (N,n+1)
 * (N,n) = (N[n],n)
 * 1) (N+M)[n] = N+(M[n])
 * 2) A(0) = 1
 * 3) A(N+1)[0] = 0
 * 4) A(N+1)[n+1] = A(N)+A(N+1)[n]
 * 5) A(N)[n] = A(N[n])

Adding only one simple rule can get us from $$f_\omega(n)$$ to $$f_{\varepsilon_0}(n)$$. By that we surpassed Conway's chained arrows, Bowers' linear arrays, multidimensional, superdimensional, trimensional... all the way up to tetrational arrays. Note that no multiplication and exponentiation is defined: it wastes rules and we can express it using A-function and addition. No direct definition of iteration is necessary because it is covered by rule 7. But we can't remove addition as it is the fundamental binary function.

Limit growth rate = $$f_{\psi(\Omega_\omega)}$$

 * (0,n) = n
 * 1) (N+1,n) = (N,n+1)
 * (N,n) = (N[n],n)
 * 1) (N+M)[n] = N+(M[n])
 * 2) A(B(0),0) = 1
 * 3) A(B(N+1),0) = B(N)
 * 4) A(B(N),M+1)[0] = 0
 * 5) A(B(N),M+1)[n+1] = A(B(N),M)+A(B(N),M+1)[n]
 * 6) A(B(N),M)[n] = A(B(N),M[n])
 * 7) B(N+1)[0] = 0
 * 8) B(N+1)[n+1] = A(B(N+1),M+B(N+1)[n])

B(N) works like $$\Omega_\alpha$$ and A(B(N),M) is like $$\psi_{\Omega_\alpha}(\beta)$$.

Limit growth rate = $$f_{\psi(\psi_I(0))}$$

 * (0,n) = n
 * 1) (N+1,n) = (N,n+1)
 * (N,n) = (N[n],n)
 * 1) (N+M)[n] = N+(M[n])
 * A(0,0) = 1
 * 1) A(B(N+1),0) = B(N)
 * 2) A(B(N),M+1)[0] = 0
 * 3) A(B(N),M+1)[n+1] = A(B(N),M)+A(B(N),M+1)[n]
 * 4) A(B(N),M)[n] = A(B(N),M[n])
 * 5) B(N)[0] = 0
 * 6) B(N)[n+1] = A(B(N),M+B(N)[n]) (N = 0 | N = K+1)
 * 7) B(N)[n] = B(N[n])

The last rule is similar to $$(\Omega_\alpha)[n] = \Omega_{\alpha[n]}$$ for limit $$\alpha$$.