User:TGReddy/TGR's Playground

The Slow-Growing Hierarchy

 * \( g_{0}(n) = 0 \)
 * \( g_{\alpha+1}(n) = g_{\alpha}(n) + 1 \)
 * \( g_{\alpha}(n) = g_{\alpha[n]}(n) \) if \( \alpha \) is a limit ordinal.

The Hardy Hierarchy

 * \( H_{0}(n) = n \)
 * \( H_{\alpha+1}(n) = H_{\alpha}(n + 1) \)
 * \( H_{\alpha}(n) = H_{\alpha[n]}(n) \) if \( \alpha \) is a limit ordinal.

The Middle-Growing Hierarchy

 * \( m_{0}(n) = n + 1 \)
 * \( m_{\alpha+1}(n) = m_{\alpha}(m_{\alpha}(n)) \)
 * \( m_{\alpha}(n) = m_{\alpha[n]}(n) \) if \( \alpha \) is a limit ordinal.

The Fast-Growing Hierarchy

 * \( f_{0}(n) = n + 1 \)
 * \( f_{\alpha+1}(n) = f^{n}_{\alpha}(n) \), where \( f^{n} \) denotes recursion.
 * \( f_{\alpha}(n) = f_{\alpha[n]}(n) \) if \( \alpha \) is a limit ordinal.

Terms

 * \( A(n) \) eventually dominates (\( \gg \)) \( B(n) \) if there exists \( k \in \mathbb{N} \) such that \( A(n) > B(n) \) for all \( n > k \)
 * \( A(n) \) has strength \( X \) if \( A(n) \gg f_{\alpha}(n) \) for all \( \alpha < X \), and \( A(n) < f_{X + 1}(n) \)
 * \( A(n) \) is comparable (\( \sim \)) to \( B(n) \) if they have the same strength.

TGRx
Analysis

Convention

 * \( \text{NaturalSeq} \) is the set of finite sequences of natural numbers
 * For \( A \in \text{NaturalSeq} \), \( \text{Len}(A) \in \mathbb{N} \) is the length of sequence \( A \)
 * For \( i \in \mathbb{N} \) and \( i < \text{Len}(A) \), \( A_{i} \in \mathbb{N} \) is \( ( 1 + i ) \)-th member of sequence \( A \)

Find Bad Root
We define a partial computable function \begin{align} \text{Parent} : \text{NaturalSeq} \times \mathbb{N} \rightarrow& \mathbb{N} \\ (A,n) \mapsto& \text{Parent}(A,n) \end{align} in the following recursive way:


 * 1) Denote \( \text{Len}(A) \) by \( L \)
 * 2) If \( i \geq L \), then \( \text{Parent}(A,i) \) is not defined
 * 3) Suppose \( i < L \)
 * 4) Suppose

Lift
We define a partial computable function \begin{align} \text{Lift} : \text{NaturalSeq} \times \mathbb{N}^{2} \rightarrow& \mathbb{N} \\ (A,n,k) \mapsto& \text{Lift}(A,n,k) \end{align} in the following recursive way:

Get Expansion
We define a partial computable function \begin{align} [] : \text{NaturalSeq} \times \mathbb{N} \rightarrow& \text{NaturalSeq} \\ (A,n) \mapsto& A[n] \end{align} in the following recursive way:

Convention

 * \( \text{NaturalSeq} \) is the set of finite sequences of natural numbers
 * For \( A \in \text{NaturalSeq} \), \( \text{Len}(A) \in \mathbb{N} \) is the length of sequence \( A \)
 * For \( i \in \mathbb{N} \) and \( i < \text{Len}(A) \), \( A_{i} \in \mathbb{N} \) is \( ( 1 + i ) \)-th member of sequence \( A \)

Well Ordering
We define a total recursive map \begin{align} \text{Compare} : \text{NaturalSeq}^{2} \rightarrow& \{ -1, 0, 1 \} \\ (A,B) \mapsto& \text{Compare}(A,B) \end{align} in the following recursive way:

\( \text{Compare}(A,B) = \left\{ \begin{array}{ll} -1 & \text{If } A_{0} < B_{0} \\ 1 & \text{If } A_{0} > B_{0} \\ 0 & \text{If } (Len(A) = 1) \land (A = B) \\ \text{Compare}(A_{s},B_{s}) & \text{If } A_{0} = B_{0} \end{array} \right. \)

Get Limiting Sequence
We define a total recursive map \begin{align} \text{Limit} : \text{NaturalSeq} \rightarrow& \text{NaturalSeq} \\ (A) \mapsto& \text{Limit}(A) \end{align} in the following recursive way:

Shift and Expand
We define a total recursive map \begin{align} \text{Expand} : \text{NaturalSeq} \times \mathbb{N} \rightarrow& \text{NaturalSeq} \\ (A,n) \mapsto& \text{Expand}(A,n) \end{align} in the following recursive way:


 * 1) \( \text{Expand}(A,n) = max\{ B : B < \text{Limit}(A) \} \)

Set of Catches

 * Let \( \Upsilon_0 = \{\alpha : f_{\alpha}(n) \sim g_{\alpha}(n) \} \)
 * Let \( \Upsilon_{\alpha + 1} = \{\beta : f_{C_{\alpha}(\beta)}(n) \sim f_{\pi(\beta)}(n) \} \)
 * If \( \alpha \) is a limit ordinal, let \( \Upsilon_{\alpha} = \bigcap_{\beta < \alpha} \Upsilon_{\beta} \)

Catching Function

 * \( C^{0}_{\nu}(\alpha,\beta) = \{ 0,1,\Omega_{\nu} \} \cup \Upsilon_{\alpha} \setminus \{ C_{\nu}(\alpha,\eta) : \eta < \beta \} \)
 * \( C^{n+1}_{\nu}(\alpha,\beta) = C^{n}_{\nu}(\alpha,\beta) \cup \{ \gamma + \delta : \gamma,\delta \in C^{n}_{\nu}(\alpha,\beta) \} \cup \{ \phi_{\eta}(\zeta) : \eta,\zeta \in C^{n}_{\nu}(\alpha,\beta) \land \zeta < \beta \} \)
 * \( C_{\nu}(\alpha,\beta) = \bigcup_{n < \omega} C^{n}_{\nu}(\alpha,\beta) \)
 * \( \phi_{\nu}(\alpha,\beta) = min \{ \gamma : \gamma \notin C_{\nu}(\alpha,\beta) \} \)

Ordinal Table