Hyper primitive sequence system

Hyper primitive sequence system is a notation defined by a Japanese googologist Yukito. It is an extension of primitive sequence system, which is a subsystem of Bashicu matrix system, using difference sequences. The limit of this notation is \(\psi_0(\Omega_{\omega})\) with respect to FGH and Buchholz's function.

Definition
A term in the notation is a finite sequence \(S = (S_i)_{i=1}^{m}\) of natural numbers followed by a natural number \(n\) in a bracket, i.e. an expression of the form \((S_1,\ldots,S_m)[n]\). It is evaluated in the following recursive way: The function \((0,\omega)[n]\) defined as \((0,n)[n]\) gives the limit of this notation.
 * 1) If \(m = 0\), then set \(S[n] = n\).
 * 2) Suppose \(m > 0\).
 * 3) For an \(i \in \mathbb{N}\) satisfying \(i \leq m\), put \(P_i = \{j \in \mathbb{N} \mid 1 \leq j < i \land S_j < S_i\}\).
 * 4) Put \(k = 0\).
 * 5) If \(k = 0\), then put \(m_k = m\).
 * 6) If \(k > 0\), then put \(m_k = \max P_{m_{k-1}}\).
 * 7) If \(P_{m_k} \neq \emptyset\), then increment \(k\), and go back to the line 2-3.
 * 8) Define a finite sequence \(\textrm{Expand}(S,n)\) in the following way:
 * 9) If \(k = 0\), then put \(\textrm{Expand}(S,n) = (S_1,\ldots,S_{m-1})\).
 * 10) Suppose \(k > 0\).
 * 11) Denote by \(N = (N_i)_{i=0}^{k-1}\) the difference sequence \((m_{i+1}-m_i)_{i=0}^{k-1}\) of \((m_i)_{i=0}^{k}\).
 * 12) Define the bad root \(r\) and the ascention level \(\delta\) in the following way:
 * 13) Suppose \(N_0 = 1\).
 * 14) Put \(r = m_1\).
 * 15) Put \(\delta = 0\).
 * 16) Suppose \(N_0 \neq 1\).
 * 17) Suppose that there does not exist an \(i \in \mathbb{N}\) such that \(0 < i \leq k-1\) and \(N_i < N_0\).
 * 18) Put \(r = 1\).
 * 19) Put \(\delta = S_m - 1\).
 * 20) Suppose that such an \(i\) exists.
 * 21) Denote by \(p\) the minimum of such an \(i\).
 * 22) Put \(r = m_p\).
 * 23) Put \(\delta = S_m - S_r - 1\).
 * 24) Put \(G = (S_i)_{i=1}^{r-1}\). (If \(r = 1\), then it is the empty sequence.)
 * 25) For an \(h \in \mathbb{N}\), put \(B(h) = (S_i + h \delta)_{i=r}^{m-1}\).
 * 26) Then \(\textrm{Expand}(S,n)\) is the concaternation of \(G,B(0),\ldots,B(n)\).
 * 27) Set \(S[n] = \textrm{Expand}(S,n)[n+1]\).

Standard form
The set \(OT\) of finite sequences of standard forms in hyper primitive sequence system is defined in the following recursive way: Then \(OT\) is a recursive subset of the set \(T\) of finite sequences of natural numbers, and \(\textrm{Expand}\) gives a primitive recursive map \begin{eqnarray*} (T \setminus \{\}) \times \mathbb{N} \to T, \end{eqnarray*} which preserves standard forms.
 * 1) For any \(n \in \mathbb{N}\), \((0,n) \in OT\).
 * 2) For any \((S,n) \in (OT \setminus \{\}) \times \mathbb{N}\), \(\textrm{Expand}(S,n) \in OT\).

Explanation
Follow the convention and the terminology in the article on difference sequences. Then hyper primitive sequence system employs the bad root searching rule \(\textrm{Parent} \colon \textrm{FinSeq} \times \mathbb{N} \to \textrm{FinSeq}\) for the primitive sequence system explained here. Indeed \((m_i)_{i=0}^{k}\) in and \(N\) in the definition of \(S[n]\) precisely coincide with \(\textrm{Ancestors}(S)\) and \(\textrm{Kaiser}(S)\) respectively.

If \(\textrm{Ancestors}(S)\) is of length \(1\), then \(S\) is a successor term, and \(\textrm{Expand}(S,n)\) is obtained by deleting the rightmost entry of \(S\). If \(\textrm{Ancestors}(S)\) is of length \(> 1\), then apply an expansion rule to \(\textrm{Kaiser}(S)\) quite similar to the one in primitive sequence system; if \(\textrm{Ancestors}(\textrm{Kaiser}(S))\) is of length \(1\), then decrement the rightmost entry of \(\textrm{Kaiser}(S)\), and otherwise, copy the bad part of \(\textrm{Kaiser}(S)\) with respect to \(\textrm{Parent}\). Denote by \(\textrm{Kaiser}(S)(n)\) the resulting finite sequence. As \(\textrm{Kaiser}(S)\) is the differece sequence of \(\textrm{RightNodes}(S)\), \(\textrm{Kaiser}(S)(n)\) is the difference sequence of a unique finite sequence \(\textrm{RightNodes}(S)(n)\) whose first entry is \(\textrm{RightNodes}(S)_0\). Finally, \(\textrm{Expand}(S,n)\) is given by replacing the subsequence \(\textrm{RightNodes}(S)\) of \(S\) by \(\textrm{RightNodes}(S)(n)\) and interpolating the intermidiate entries by the copies of the corresponding entries of \(S\).

Through the interpretation above, hyper primitive sequence system can be regarded as an analogue of the system of two-lined hydra diagrams, which corresponds to pair sequence system.

Termination
For the convention and the terminology, see the following:
 * [Buc1] W. Buchholz, A new system of proof-theoretic ordinal functions, Annals of Pure and Applied Logic, Volume 32, 1986, pp. 195--207.
 * [Buc2] W. Buchholz, Relating ordinals to proofs in a prespicious way, unpublished article.

Let \(T_B\) denote the set of terms in Buchholz's notation, in which \(D_{\omega}\) does not appear. Define a total recursive map \begin{eqnarray*} \textrm{Trans} \colon T_B & \to & T \\ t & \mapsto & \textrm{Trans}(t) \end{eqnarray*} in the following recursive way:
 * 1) If \(t = 0\), then set \(\textrm{Trans}(t) = \).
 * 2) If \(t = t_0 + 0\) for some \(t_0 \in T_B \setminus \{0\}\), then set \(\textrm{Trans}(t) = \textrm{Trans}(t_0)\).
 * 3) Suppose \(t = t_0 + D_u t_1\) for some \((u,t_0,t_1) \in \mathbb{N} \times T_B \times T_B\).
 * 4) If \(t_1 = D_0 0 \cdot m\) for some \(m \in \mathbb{N}\), then \(\textrm{Trans}(t)\) is the concatenation of \(\textrm{Trans}(t_0)\) and \((u,\underbrace{u+1,\ldots,u+1}_{m})\).
 * 5) Otherwise, \(\textrm{Trans}(t)\) is the concatenation of \(\textrm{Trans}(t_0)\), \((u)\), and the finite sequence obtained by adding \(u+1\) to each entri of \(\textrm{Trans}(t_1)\).

Let \(OT_B \subset T_B\) denote the subset of ordinal terms below \(D_1 0\).


 * proof:
 * For an \(m \in \mathbb{N}\), define \(\Sigma_m \subset OT_B\) in the following recursive way:


 * 1) If \(m = 0\), then set \(\Sigma_m = \{D_0 D_u 0) \mid u \in \mathbb{N} \land u > 0\}\).
 * 2) If \(m > 0\), then set \(\Sigma_m = \Sigma_{m-1} \cup \{t[n] \mid (t,n) \in \Sigma_{m-1} \times \mathbb{N}\}\).
 * By [Buc1] 2.2 Lemma (c) and 2.3 Lemma (b), we have \(\bigcup_{m \in \mathbb{N}} \Sigma_m = OT_B\).
 * Let \(t \in OT_B\). By the argument above, there exists an \(m \in \mathbb{N}\) such that \(t \in \Sigma_m\). Denote by \(\mu\) the minimum of such an \(m\). We show \(\textrm{Trans}(t) \in OT\) by induction of \(\mu\).
 * If \(\mu = 0\), then we have \(t = D_0 D_u 0\) for some \(u \in \mathbb{N}\) satisfying \(u > 0\), and hence \(\textrm{Trans}(t) = (0,u) \in OT\). Suppose \(\mu > 0\) in the following. Take a \((t',n) \in \Sigma_{\mu-1} \times \mathbb{N}\) satisfying \(t = t'[n]\). By the induction hypothesis, we have \(\textrm{Trans}(t') \in OT_B\). By \(t \notin \Sigma_{\mu-1}\), we have \(t \neq t'\), and hence \(t' \neq 0\). It implies \(t = t'[n] < t'\) by [Buc1] 3.2 Lemma (a).
 * Take a unique \((u,t_0,t_1) \in \mathbb{N} \times OT_B \times OT_B\) satisfying \(t' = t_0 + D_u t_1\). By the definition of \(\textrm{Expand}\) and \(\textrm{Trans}\), we have \(\textrm{Trans}(t) = \textrm{Expand}(\textrm{Trans}(t'),n) \in OT_B\) unless \(\textrm{dom}(t_1) = T_v\) for some \(v \in \mathbb{N}\) satifying \(u \leq v\). Therefore we may assume \(\textrm{dom}(t_1) = T_v\) for some \(v \in \mathbb{N}\) satifying \(u \leq v\).
 * For each \(m \in \mathbb{N}\), denote by \(t_2(m)\) a unique ordinal term satisfying \(o(t_2(m)) = o(t'(m))\), where \(t'(m)\) is the term obtained by replacing the rightmost appearrence of \(D_v 0\) in \(t'[m]\) by \(0\).ref>The map \(o\) is defined in [Buc1] p. 201. Again by the definition of \(\textrm{Expand}\) and \(\textrm{Trans}\), \(\textrm{Trans}(t_2(m))\) and \(\textrm{Trans}(t'(m))\) coincide with \(\textrm{Expand}(\textrm{Trans}(t'),m)\).
 * We have \(t_2(n) \leq t'(n) < t \leq t_2(n+1) \leq t'(n+1)\), and \(t\) coincides with the term obtained by replacing the righmost appearrence of \(0\) in \(t'(n)\) by \(D_v 0\) because of \(t = t'[n]\). Therefore \(\textrm{Trans}(t)\) coincides with the concatenation of \(\textrm{Trans}(t'(n))\) and \((a+v+1)\), where \(a\) is the next rightmost entry of \(\textrm{RightNodes}(\textrm{Trans}(t))\) with respect to the bad root searching rule \(\textrm{Parent}\), and \(\textrm{Trans}(t'(n+1))\) coincides with the concatenation of \(\textrm{Trans}(t)\) and a finite sequence. Since the deletion of the rightmost entry is given by the application of \(\textrm{Expand}(-,0)\), it implies \(t \in OT\).
 * Thus the restriction of \(\textrm{Trans}\) to \(OT_B\) gives a map to \(OT\). The assertions (1) and (2) immediately follow from the argument above, because of \(\textrm{Trans}(D_0 0 + D_0 0) = (0,0)\) and \(\textrm{Trans}(D_u 0) = (0,u)\) for any \(u \in \mathbb{N}\) satisfying \(u > 0\). □

As is shown in the proof above, hyper primitive sequence system is quite compatible with Buchholz's ordinal notation unlike pair sequence system. By Lemma and [Buc1] 2.2 Lemma (c), we immediately obtain the termination of hyper primitive sequence system restricted to standard forms.

Analysis
By Lemma (2), the limit of hyper primitive sequence is \(\psi_0(\Omega_{\omega})\) in Hardy hierarchy with respect to a minour replacedment of the recursive system of fundamental sequences assciated to Buchholz's ordinal notation. In particular, it is approximated to \(\psi_0(\Omega_{\omega})\) in fast growing hierarchy. The composite \begin{eqnarray*} o \circ (\textrm{Trans} |_{OT_B})^{-1} \colon OT \to \psi_0(\Omega_{\omega}) \end{eqnarray*} gives an interpretation of terms in hyper primitive sequence system of standard form into ordinals below \(\psi_0(\Omega_{\omega})\). The following table exhibits the examples of the correspondence: