Higher order randomness
Higher order randomness is already developed. It adapts classical randomness notions, using the analogy between \(\Pi^1_1\) and computably enumerable:
A real \(x\) is \(\Pi^1_1\)-ML-random if it belongs to no set \(A\) of the form $$A=\bigcap_n{[W_n]^\prec}$$ where the sets \(W_n\) are uniformly \(\Pi^1_1\) and \(\mu([W_n])\leq{2^{-n}}\).
It also introduces new notions :
A real \(x\) is \(\Delta^1_1\)-random if it avoids all \(\Delta^1_1\) measure 0 properties.
A real \(x\) is \(\Pi^1_1\)-random if it avoids all \(\Pi^1_1\) measure 0 properties.
A real is \(\Pi^1_1\)-random if and only if it is \(\Delta^1_1\)-random and \(\omega_1^{\mathrm{CK}}=\omega_1^{\mathrm{CK(x)}}\).
We have the following strict inclusion:
$$\Delta^1_1\text{-randomness}\subsetneq\Pi^1_1\text{-ML-randomness}\subsetneq\Pi^1_1\text{-randomness}$$
Randomness for \(\alpha\)-computability and Infinite Time Turing Machine
The extension of the definitions from higher order randomness is straightforward:
A real \(x\) is random over \(L_\alpha\) if it avoids every set \(A\subseteq\mathbb R\), with Borel code in \(L_\alpha\) and of measure 0.
A real \(x\) is ITTM-random if it avoids every set \(A\subseteq\mathbb R\) ITTM-semi-decidable and of measure 0.
A real \(x\) is \(\alpha\)-ML-random if it avoids every set \(A\) of the form $$A=\bigcap_n{[W_n]^\prec}$$ where the sets \(W_n\) are uniformly \(\alpha\)-computably enumerable and \(\mu([W_n])\leq{2^{-n}}\). It is ITTM-ML-random if it is \(\Sigma\)-ML-random.
A real is \(\Pi^1_1\)-random if and only if it is \(\Delta^1_1\)-random and \(\omega_1^{\mathrm{CK}}=\omega_1^{\mathrm{CK(x)}}\)
A real \(x\) is ITTM-random if and only if it is random over \(L_\Sigma\) and \(\Sigma^x=\Sigma\).
We have the following strict inclusion:
$$\Delta^1_1\text{-randomness}\subsetneq\Pi^1_1\text{-ML-randomness}\subsetneq\Pi^1_1\text{-randomness}$$
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For which \(\alpha\) do we have :
$$\text{randomness over }L_\alpha\subsetneq\alpha\text{-ML-randomness}\text{ ?}$$
Do we have
$$\text{randomness over }L_\Sigma\subsetneq\text{ITTM-ML-randomness}\subsetneq\text{ITTM-randomness}\text{ ?}$$
Let \(\alpha\) be such that \(L_\alpha\models\text{"everything is countable"}\). Then the following statements are equivalent:
- \(\alpha\)-ML-randomness is strictly stronger than randomness over \(L_\alpha\).
- \(\alpha\) is projectible in \(\omega\).
- There exists a universal \(\alpha\)-ML-test.
$$\text{Randomness over }L_\lambda\subsetneq\lambda\text{-ML-randomness}$$
$$\text{Randomness over }L_\zeta=\zeta\text{-ML-randomness}$$
$$\text{Randomness over }L_\Sigma\subsetneq\Sigma\text{-ML-randomness}$$
Do we have:
$$\text{Randomness over }L_\Sigma\subsetneq\text{ITTM-ML-randomness}\subsetneq\text{ITTM-randomness}\text{ ?}$$
$$\text{Randomness over }L_\Sigma\subseteq\text{ITTM-randomness}\subsetneq\text{ITTM-ML-randomness}$$
$$\text{Randomness over }L_\Sigma\neq\text{ITTM-randomness ?}$$
Does \(x\) random over \(L_\Sigma\) implies $$L_\zeta[x]\not\prec_2L_\Sigma[x]\text{ ?}$$
If \(x\) is generic over \(L_\Sigma\) then $$L_\zeta[x]\prec_2L_\Sigma[x]$$
We therefore have:
$$\text{Genericity over }L_\Sigma = \text{ITTM-genericity}$$