A Short Program in MuPAD that Computes in the Limit a Function <em>f</em> : N → N Which Eventually Dominates Every Computable Function<em> g</em> : N → N

For n ∈ N, let En = {1 = xk, xi + xj = xk, xi · xj = xk : i, j, k ∈ {0, . . . , n}}. For n ∈ N, f (n) denotes the smallest b ∈ N such that if a system of equations S ⊆ En has a solution in Nn+1, then S has a solution in {0, . . . , b}n+1. The author proved earlier that the function f : N → N is computable in the limit and eventually dominates every computable function g : N → N. We present a short program in MuPAD which for n ∈ N prints the sequence { fi(n)}∞ i=0 of non-negative integers converging to f (n). For n ∈ N, β(n) denotes the smallest b ∈ N such that if a system of equations S ⊆ En has a unique solution in Nn+1, then this solution belongs to {0, . . . , b}n+1. The author proved earlier that the function β : N → N is computable in the limit and eventually dominates every function δ : N → N with a single-fold Diophantine representation. We present a short program in MuPAD which for n ∈ N prints the sequence {βi(n)}∞ i=0 of non-negative integers converging to β(n).