N primitive

N primitive (N原始 in Japanese, N PRIMITIV in German) is the generic name of difference sequence systems created by the Googology Wiki user Nayuta Ito. They are intended to surpass Bashicu matrix system version 2.3 with respect to koteitan's classification and admit simple isomorphisms between the set of standard expressions below \((0,1,3)\) and the set of standard Bashicu matrices whose restriction to the subset of standard expressions below \((0,1,2,4)\) is the identity map onto the subset of standard primitive sequences.

Version Name
There are several versions of N primitive. Here is a table of names of versions of N primitive.

Definition
The original definitions versions called N1.1, N1.1½, and N3.0 are open.

The Python version of N1.1 is called N1.1.π, and its source code is also open.

Here, we explain the bad root searching rule for N1.1 in the sense of the terminilogy in the article of difference sequence system. Let \(\textrm{FinSeq}\) denote the set of finite sequence of natural numbers. The bad root searching rule for N1.1 is the partial computable function \begin{eqnarray*} \textrm{Parent} \colon \textrm{FinSeq} \times \mathbb{N} & \to & \mathbb{N} \\ (a,i) & \mapsto & \textrm{Parent}(a,i) \end{eqnarray*} defined in the following recursive way: By the definition, the restriction of \(\textrm{Parent}\) to the direct product of the set of standard primitive sequences and \(\mathbb{N}\) coincides with [the bad root searching rule for primitive sequence system]. The main difference from the bad root searching rule for primitive sequence system is that \(((1,1,2),1)\) belongs to the domain of \(\textrm{Parent}\). Actually, we have \(\textrm{Parent}((1,1,2),1) = 0\).
 * 1) Denote by \(L\) the length of \(a\).
 * 2) For each \(j \in \mathbb{N}\) smaller than \(L\), denote by \(a_j\) the \((1+j)\)-th entry of \(a\).
 * 3) If \(i \geq L\), then \(\textrm{Parent}(a,i)\) is not defined.
 * 4) Suppose \(i < L\).
 * 5) Suppose that there exists a \(j \in \mathbb{N}\) larger than \(i\) such that \(\textrm{Parent}(a,j)\) is defined and coincides with \(i\).
 * 6) Denote by \(k\) the maximum of such a \(j\).
 * 7) If \(a_0 \geq 1\) and \(a_i = a_k-1 < a_{L-1}\), then put \(b := 1\).
 * 8) Otherwise, put \(b:= 0\).
 * 9) Otherwise, put \(b := 0\).
 * 10) Suppose \(b = 0\).
 * 11) If there exists an \(m \in \mathbb{N}\) smaller than \(i\) such that \(a_m < a_i\), then \(\textrm{Parent}(a,i)\) is the maximum of such an \(m\).
 * 12) Otherwise, \(\textrm{Parent}(a,i)\) is not defined.
 * 13) Suppose \(b = 1\).
 * 14) If there exists an \(m \in \mathbb{N}\) smaller than \(i\) such that \(a_m \leq a_i\), then \(\textrm{Parent}(a,i)\) is the maximum of such an \(m\).
 * 15) Otherwise, \(\textrm{Parent}(a,i)\) is not defined.

Dead N primitive
A version of N primitive is said to be dead if it does not work as intended. For example, N1.1 is dead because it does not seem to admit a simple isomorphism between the set of standard expressions below \((0,1,3)\) and the set of standard Bashicu matrices. It does not mean that N1.1 is known to be weaker than Bashicu matrix system version 2.3 or to admit an infinite loop. Here is a table of the status of versions of N primitive.

All functions defined by N primitive are computable and hence are weaker than busy beaver function.