Infinite time Turing machine

Infinite time Turing machines (ITTMs) are a generalization of Turing machines to infinite computation lengths, first described by Joel David Hamkins and Andy Lewis. They lead to a stronger analog \(\Sigma_\infty\) to the busy beaver function.

Definition
The original model by Hamkins and Lewis has three one-sided, two-color infinite tapes, called input, scratch and output tapes, and a single read-write head which reads one cell from each tape. (Models with one tape have been considered as well — surprisingly, they have been shown to be weaker. To our knowledge, tapes with more than two colors have not been explored.) The transition table takes into account all three colors simultaneously, and writes three at once. The scratch and output tapes are blank initially, and the input tape contains the input to the machine.

ITTMs have a particular state marked as the limit state. At each step, the ITTM proceeds normally with a timer \(t\) starting at zero and incrementing at each step. If it does not halt for any \(t < \omega\), at \(t = \omega\) each cell on the tape is set to the of that cell's symbol for steps \(t = 0, 1, 2, \ldots\). In addition, the heads are set to the initial cell and the state is set to limit. The ITTM continues as before, and if it continues to \(t = \omega2\) without halting, the same thing happens with the limit superior of the tape for \(t = \omega, \omega + 1, \omega + 2, \ldots\). In general, if the ITTM reaches a countable limit ordinal \(\alpha\) without halting, at time \(t = \alpha\), it is set to the limit superior of the tape at \(t = \beta\) for all \(\beta < \alpha\) (i.e. it's set to 0 if, from some point below \(\alpha\) up to \(\alpha\) the cell is set to 0, and it's set to 1 otherwise).

If the ITTM halts, the contents of the output tape at halting time are the output of the machine. If the ITTM does not halt, but after some point the output tape never changes, the output tape after that point is the eventual output of the machine. For both halting and non-halting cases, all of the configurations of the three tapes over the lifetime of the ITTM are the accidental outputs of the machine. It should be noted that outputs and eventual outputs are unique if they exist, but accidental outputs are not.

Formal definition
Formally, a two-color ITTM is a 5-tuple \(M = (Q, \delta, q_0, q_H, q_L)\) with the following components:


 * \(Q\) is a finite, non-empty set of states.
 * \(q_0 \in Q\) is the initial state.
 * \(q_H \in Q\) is the halting state.
 * \(q_L \in Q\) is the limit state.
 * \(\delta : (Q \backslash \{q_H\}) \times \{0, 1\}^3 \rightarrow Q \times \{0, 1\}^3 \times \{L, R\}\) is the transition table.

A configuration of \(M\) is a 3-tuple \((p, \tau, m)\) where \(p \in Q\) is the current state of the machine, \(\tau : (\mathbb{N}\times\{0,1,2\}) \rightarrow \{0, 1\}\) are the contents of tapes, and \(m \in \mathbb{N}\) is the positions of the head. We use \(C(M)\) to denote the set of configurations. We define \(\leadsto\) (a single computation step) as a binary relation over \(C(M)\) as follows. We say that \((p, \tau, m) \leadsto (p', \tau', m')\) iff the following are true:


 * \(\delta(p, (\tau(m,0),\tau(m,1),\tau(m,2))) = (p', (c_0,c_1,c_2), \Delta m)\).
 * For all \(n \in \mathbb{N},i\in\{0,1,2\}\), \(\tau'(n,i) = \left\{ \begin{array}{lr} c_i & : n = m\\ \tau(n,i) & \text{otherwise} \end{array} \right.\).
 * \(m' = \left\{ \begin{array}{lr} m + 1 & : \Delta m = R \\ \max\{m - 1, 0\} & : \Delta m = L \end{array} \right.\)

It can be seen that \(\leadsto\) is a function. That is, \((p', \tau', m')\) is unique.

An input to an ITTM is simply a function \(I:\mathbb{N} \rightarrow \{0, 1\}\) (meant to be the content of the first tape), and an initial contents for that input is \(\tau_0:\mathbb{N}\times\{0,1,2\} \rightarrow \{0,1\}\) such that \(\tau_0(n,0)=I(n),\tau_0(n,1)=\tau_0(n,2)=0\). A computation of \(M\) on an input \(I\) is a function \(U : \mu \rightarrow C(M)\) (where \(\mu\) is a successor ordinal, or the class \(\text{On}\)) that satisfies the following properties:


 * \(U(0) = (q_0, \tau_0, 0)\)
 * For all \(\alpha + 1 \in \mu\), \(U(\alpha) \leadsto U(\alpha + 1)\).
 * For all limit ordinals \(\alpha \in \mu\), \(U(\alpha) = (q_L, \tau_\alpha, 0)\) where for all \(i \in \mathbb{N}\times\{0,1,2\}\), \(\tau_\alpha(i) = \text{lim sup} _{\beta < \alpha} \tau_\beta(i)\) where \(U(\beta) = (\bullet, \tau_\beta, \bullet)\).
 * If \(\mu \neq \text{On}\), \(U(\mu - 1) = (q_H, \bullet, \bullet)\).

If \(\mu\neq\text{On}\), then the machine is halting for \(\tau_0\), and we call \(\mu\) the halting time. Define the output of the computation as the function \(H:\mathbb{N} \rightarrow \{0,1\}\), defined as \(H(n) = \tau_{\mu-1}(n,2)\). It is not hard to show that this output must be unique.

If \(\mu = \text{On}\), then the machine is nonhalting for \(\tau_0\). If there is some \(H : \mathbb{N} \rightarrow \{0,1\}\) such that for all sufficiently large \(\alpha\), \(\forall n : \tau_\alpha(n,2) = H(n)\), then we say that the machine eventually writes \(H\). It is not hard to show that eventual output is also unique if it exists.

For all \(H : \mathbb{N} \rightarrow \{0,1\}\), if there is an \(\alpha \in \mu, i \in \{0, 1, 2\}\) such that \(\forall n: \tau_\alpha(n, i) = H(n)\), we say that the machine accidentally writes \(H\) for the input. Accidental outputs are not unique.

The uniqueness of outputs and eventual outputs allows us to define the partial ITTM-computable functions and partial eventually-computable functions. A partial function \(f : \mathbb{N} \mapsto \mathbb{N}\) is ITTM-computable iff there exists an ITTM \(M\) such that \(f(n) = m\) iff \(M\) produces output \(m\) with input \(n\) (using an encoding scheme for natural numbers such as \(\underbrace{11...1}_{n}00...\)). A partial function \(f : \mathbb{N} \mapsto \mathbb{N}\) is eventually computable iff there exists an ITTM \(M\) such that \(f(n) = m\) iff \(M\) produces eventual output \(m\) with input \(n\).

\(\Sigma_\infty\)
Long and Stanley propose an infinite time Turing machine busy beaver function. \(\Sigma_\infty(n)\) is defined as the maximal finite output \(k\) of an ITTM with at most \(n\) non-halting non-limit states given blank input, where they define output to be a natural number \(k\) if it is of the form \(\underbrace{11...1}_{k}00...\).

They have shown that \(\Sigma_\infty\) eventually dominates every ITTM-computable function, in analogy to domination property of \(\Sigma\) function. Therefore it is an uncomputably fast-growing function.

ITTM ordinals
ITTMs are capable of producing far more outputs than ordinary TMs, including ordinals. This leads to several classes of ordinals:


 * An ordinal \(\alpha\) is writable iff an ITTM with empty input (all zeros) has \(\alpha\) as its output.
 * An ordinal \(\alpha\) is clockable iff an ITTM with empty input halts at time \(\alpha\).
 * An ordinal \(\alpha\) is eventually writable iff an ITTM with empty input has \(\alpha\) as its eventual output.
 * An ordinal \(\alpha\) is accidentally writable iff an ITTM with empty input has \(\alpha\) as an accidental output.

(The first and last two definitions can apply to many mathematical objects other than ordinals) These definitions give rise to several large countable ordinals (the infinite time Turing machine ordinals):


 * \(\lambda\), the supremum of all writable ordinals
 * \(\gamma\), the supremum of all clockable ordinals
 * \(\zeta\), the supremum of all eventually writable ordinals
 * \(\Sigma\), the supremum of all accidentally writable ordinals

It has been shown that \( \ll \lambda = \gamma < \zeta < \Sigma\).

Encoding ordinals in bitstrings
To be able to read and write ITTM ordinals, we must specify an encoding scheme to represent ordinals with countably infinite bitstrings. Here is one such scheme:


 * "1000..." encodes \(0\).
 * If string S encodes \(\alpha\), then the string "0" + S (where + denotes string concatenation) encodes \(\alpha + 1\).
 * If strings \(S_0,S_1,S_2,\ldots\) encode \(\alpha_0,\alpha_1,\alpha_2,\ldots\), then we interleave the strings into a new bitstring \(T\) according to the following scheme:
 * Start with every bit of \(T\) labeled "empty".
 * Write the bits of \(S_0\) in every second bit of \(T\).
 * Write the bits of \(S_1\) in every second remaining empty bit of \(T\).
 * Write the bits of \(S_2\) in every second remaining empty bit of \(T\).
 * Repeat for all of the \(S_i\).
 * Write "1" in the first bit of \(T\).
 * \(T\) encodes \(\sup\{\alpha_i\}\).

This is of course one of many possible encoding schemes, and within this scheme multiple bitstrings represent a single ordinal.

Klev's notations
Ansten Mørch Klev developed two ordinal notations, \(\mathcal{O}^+\) and \(\mathcal{O}^{++}\), which express all writable and eventually writable ordinals, respectively. They are exactly the same as Kleene's \(\mathcal{O}\) but with analogous sets replacing the set of partial recursive functions.