Infinite Computations
Algorithmic Randomness
in .

Infinite time computation

Usually, a computation halts in a finite number of steps. An algorithm consists of the set of rules to go from one step to the next one.

We now want that the number of steps can be infinite. We shall however keep the notion of next one.

An ordinal is a set that is well ordered by the relation \(\in\).
\(\omega\)
\(\omega+1\)
\(\omega+\omega\)
\(\omega\times\omega\)
\(\omega^{\omega}\)

Three types of infinite time computations

  • We defined several notions of infinite computations:
    • Of length \(\omega_1^{\mathrm{CK}}<\lambda<\zeta<\Sigma\) or any \(\alpha\) admissible,
    • they relativizes, except for \(\alpha\)-computations.
  • These notions yield new complexity notions on sets, much higher than the usual one, but with a similar behaviour!

Infinite Time Turing Machine

An Infinite Time Turing Machine (ITTM) is a Turing Machine with three tapes (input, work and output) with a special state (called limit state). An infinite time computation is a sequence of configurations, indexed by an ordinal, such that:
  • At successor steps, the rules are the same as for a regular Turing machine,
  • At limit steps :
    • The state is the limit state,
    • The head of the tapes is at the leftmost position,
    • Each cell of the memory tape is the \(\limsup\) of its previous values: $$T_\lambda(n)=\limsup_{\alpha<\lambda}T_\alpha(n)$$
A real \(x\) is:
  • Writable if at some step, the machine has \(x\) written on its output tape and halts,
  • Eventually writable if at some step, the machine has \(x\) written on its output tape and never modifies it afterward,
  • Accidentally writable if at some step, the machine has \(x\) written on its output tape.
There is a nice characterization of these classes in terms of definability that also give an implicit bound on the times of computation.
Let \(A\), \(B\) be two \(\mathcal L\)-structure. We say that $$A\prec_n B$$ if for every \(\Phi\) a \(\Sigma_n\) formula with parameters in \(A\), we have: $$ A\models\Phi\quad\iff\quad B\models\Phi$$
Let \(\lambda\), \(\zeta\) and \(\Sigma\) the lexicographically least triplet of ordinals such that: $$ L_{\color{red}\lambda}\prec_1 L_{\color{blue}\zeta} \prec_2 L_{\color{green}\Sigma}$$ Then:
  • \(\lambda\) is the supremum of the halting times and of the writable ordinals by an ITTM
  • \(\zeta\) is the supremum of the stabilisation times and of the eventually writable ordinals by an ITTM
  • \(\Sigma\) is the supremum of the loopless times, and of the accidentally writable ordinals by an ITTM
  • \(\lambda\)-computability and writability coincide,
  • \(\zeta\)-computability and eventual writability coincide,
  • \(\Sigma\)-computability and accidental writability coincide.
  • \(\lambda^x\)-computability and writability relative to \(x\) coincide,
  • \(\zeta^x\)-computability and eventual writability relative to \(x\) coincide,
  • \(\Sigma^x\)-computability and accidental writability relative to \(x\) coincide.
    $${\color{red}\lambda^x}>{\color{red}\lambda}\quad\Longleftrightarrow\quad L_{\color{red}\lambda}[x]\not\prec_1L_{{\color{blue}\zeta}^x}[x]$$
    $${\color{blue}\zeta^x}>{\color{blue}\zeta}\quad\Longleftrightarrow\quad{\color{green}\Sigma^x}>{\color{green}\Sigma}\quad\Longleftrightarrow\quad L_{\color{blue}\zeta}[x]\not\prec_2L_{{\color{green}\Sigma}}[x]$$
We can also use ITTMs to define complexity on subsets of the reals.
For the subsets of reals \(A\subseteq\mathbb R\):
  • ITTM-decidable : \(\forall x\in\mathbb R\), we have \(x\in A\Longleftrightarrow M(x)\downarrow=1\) for some total ITTM \(M\).
  • ITTM-semi-decidable : \(\forall x\in\mathbb R\), we have \(x\in A\Longleftrightarrow M(x)\downarrow\).

Higher Computability

  • A set \(A\subseteq\mathbb N\) is \(\Sigma^1_1\) if it is definable by a formula of the form $$\exists X\in \mathcal P(\mathbb N),\ \psi(X,x)$$ where \(\psi(X,x)\) is arithmetic.
  • A set \(A\subseteq\mathbb N\) is \(\Pi^1_1\) if its complementary is \(\Sigma^1_1\).
  • A set \(A\subseteq\mathbb N\) is \(\Delta^1_1\) if it is both \(\Sigma^1_1\) and \(\Pi^1_1\).
A set \(A\subseteq\mathbb N\) is
  • higher computable relative to \(x\) if it is \(\Delta^1_1\)\(\Delta^1_1(x)\).
  • higher computably enumerable relative to \(x\) if it is \(\Pi^1_1\)\(\Pi^1_1(x)\).
  • higher co-computably enumerable relative to \(x\) if it is \(\Sigma^1_1\)\(\Sigma^1_1(x)\).

Links with infinite time computations

A computable ordinal :
The ordinal \(\omega_1^{\mathrm{CK}}\)\(\omega_1^{x}\) is the smallest ordinal that is not computable relative to \(x\).
A set \(A\subseteq\mathbb N\) is
  • higher computable relative to \(x\) if and only if it is \(\omega_1^{\mathrm{CK}}\)-computable\(\omega_1^{x}\)-calculable,
  • higher c.e. relative to \(x\) if and only if it is \(\omega_1^{\mathrm{CK}}\)-c.e.\(\omega_1^{x}\)-c.e.,
  • higher co-c.e. relative to \(x\) if and only if it is \(\omega_1^{\mathrm{CK}}\)-co-c.e.\(\omega_1^{x}\)-co-c.e.

\(\alpha\)-computability

  • A starting state \(C_0\) defining a machine configuration.
  • Rules A \(\Delta_1\) formula to go from \(C_i\) to \(C_{i+1}\) from \((C_\beta)_{\beta<\lambda}\) to \(C_\lambda\).
From finite time to infinite time:
  • First stage: index machine configurations by an ordinal \(\alpha\).
  • Second stage: replace rules by definability.
Let us define \((L_\alpha)_{\alpha\in\mathrm{Ord}}\) the following way:
  • \(L_0 = \emptyset\)
  • \(L_{\alpha+1} = \{ A\subseteq L_\alpha : \) \(A\) is definable in \(L_\alpha\}\)
  • \(L_\lambda = \bigcup_{\alpha<\lambda}L_\alpha\)
We say that \(A\subseteq\mathbb N\) is
  • \(\alpha\)-computable if \(A\) is \(\Delta_1\)-definable in \(L_\alpha\),
  • \(\alpha\)-c.e. if \(A\) is \(\Sigma_1\)-definable in \(L_\alpha\),
  • \(\alpha\)-co-c.e. if \(A\) is \(\Pi_1\)-definable in \(L_\alpha\).
(\(\omega\)-computability corresponds to classical computability)
Not all ordinals are good to bound a computation time:

We say that \(\alpha\) is admissible if for all \(\alpha\)-computable function \(f:\beta<\alpha\to\alpha\), we have \(\lim_{\gamma<\beta} f(\gamma)<\alpha\).

In other words, given \(\beta<\alpha\) tasks to do, each task taking less than \(\alpha\) steps to do, we can do all tasks without running short in time.

Algorithmic randomness
Study the properties of random reals.
Consider the two following sequences, and their probabilities:
\(\mathbb P(\)
\()=1\)
\()=0.5\)
\()=0.25\)
\()=0.125\)
\() ={2^{-4}}\)
\() ={2^{-5}}\)
\() ={2^{-6}}\)
\() ={2^{-7}}\)
\() ={2^{-8}}\)
\() ={2^{-9}}\)
\() ={2^{-10}}\)
\() ={2^{-11}}\)
\() ={2^{-12}}\)
\() ={2^{-13}}\)
\() ={2^{-14}}\)
\(\mathbb P(\)
\()=1\)
\()=0.5\)
\()=0.25\)
\()=0.125\)
\() ={2^{-4}}\)
\() ={2^{-5}}\)
\() ={2^{-6}}\)
\() ={2^{-7}}\)
\() ={2^{-8}}\)
\() ={2^{-9}}\)
\() ={2^{-10}}\)
\() ={2^{-11}}\)
\() ={2^{-12}}\)
\() ={2^{-13}}\)
\() ={2^{-14}}\)
We remark:
  • Both have the same probabilities.
  • The probability tends toward 0.
  • However, the former seems more random than the latter...
Measure Theory doesn't answer this paradox. Algorithmic randomness does!
An object is random if it doesn't have any discriminatory and simple property.
A real \(x\in {2^{\mathbb N}}\) is weak-\(n\)-random if it has no \(\Pi^0_n\) property of measure 0.
A ML-test is a \(\Pi^0_2\) set \(A\), such that \(A\) is of the form $$\bigcap_{n\in\mathbb N} U_n$$ where \(\forall n,\ \mu(U_n)\leq {2^{-n}}\). A real \(x\in{2^{\mathbb N}}\) is ML-random if it avoids all ML-tests.
  • Each complexity layer induces a randomness notion. We do not define what random real is, but a hierarchy on the reals corresponding to what extent they are random.
  • Each level has its weaknesses :
    • Some reals are strangely ML-random.
    • For all \(n\), there exists an arithmetic weak-\(n\)-random real.
    • This suggests the study of strong randomness notions.

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 \(x\) is ITTM-random if and only if it is random over \(L_\Sigma\) and \(\Sigma^x=\Sigma\).
✓
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 an \(\alpha\)-computable injection \(\alpha\to\omega\))
  • There exists a universal \(\alpha\)-ML-test.
$$\text{Randomness over }L_\lambda\subsetneq\lambda\text{-ML-randomness}$$ $$\text{Randomness over }L_\zeta=\zeta\text{-ML-randomness}$$ $$\text{Randomness over }L_\Sigma\subsetneq\Sigma\text{-ML-randomness}$$
$$\text{Randomness over }L_\Sigma\subseteq\text{ITTM-randomness}\subsetneq\text{ITTM-ML-randomness}$$
  • \(x\) is ITTM-random if and only if \(x\) is random over \(L_\Sigma\) and \(\Sigma^x=\Sigma\)
  • \(\Sigma^x = \Sigma\) if and only if \(L_\zeta[x]\prec_2 L_\Sigma[x]\)
$$\text{Randomness over }L_\Sigma\neq\text{ITTM-randomness ?}$$ Does \(x\) random over \(L_\Sigma\) implies $$L_\zeta[x]\not\prec_2L_\Sigma[x]\text{ ?}$$
  • If \(x\) is random over \(L_{\alpha+1}\) then $$L_\beta\prec_nL_\alpha\quad\Longrightarrow\quad L_\beta[x]\prec_nL_\alpha[x]$$
  • If \(x\) is random over \(L_{\alpha}\) and \(\alpha\) is admissible, then $$L_\beta\prec_2L_\alpha\quad\Longrightarrow\quad L_\beta[x]\prec_2L_\alpha[x]$$
  • If \(x\) is random over \(L_{\alpha}\) and \(\alpha\) is limit, then $$L_\beta\prec_1L_\alpha\quad\Longrightarrow\quad L_\beta[x]\prec_1L_\alpha[x]$$
If \(x\) is generic over \(L_\Sigma\) then $$L_\zeta[x]\prec_2L_\Sigma[x]$$ We therefore have: $$\text{Genericity over }L_\Sigma = \text{ITTM-genericity}$$
\(\infty\)