Abstract
In 1962, Hungarian mathematician Tibor Radó introduced in [8] the busy beaver competition for Turing machines: in a class of machines, find one which halts after the greatest number of steps when started on the empty input. In this paper, we generalise the busy beaver competition to the infinite time Turing machines (ITTMs) introduced in [6] by Hamkins and Lewis in 2000. We introduce two busy beaver functions on ITTMs and show both theoretical and experimental results on these functions. We give in particular a comprehensive study, with champions for the busy beaver competition, of the classes of ITTMs with one or two states (in addition to the halt and limit states). The computation power of ITTMs is humongous and thus makes the experimental study of this generalisation of Radó’s competition and functions a daunting challenge. We end this paper with a characterisation of the power of those machines when the use of the tape is restricted in various ways.
The research for this paper has been done thanks to the support of the Agence nationale de la recherche through the RaCAF ANR-15-CE40-0016-01 grant.
The authors would also like to express their thanks to the anonymous referees, who made numerous suggestions and interesting remarks.
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Notes
- 1.
The last transition is not counted to allow for clockable limit ordinals [6].
- 2.
Concerning admissible sets and ordinals, the interested reader is referred to [1].
- 3.
A real x is eventually writable if there is a non-halting infinite time computation, on input 0, which eventually writes x on the output tape and never changes again. A real x is accidentally writable if it appears on one of the tapes during a computation, possibly changing it again later on. An ordinal \(\alpha \) is eventually (resp. accidentally) writable if the real coding a well-order on \(\omega \) of order-type \(\alpha \) is eventually (resp. accidentally) writable.
- 4.
\(^{\omega }00|10^{\omega }\) denotes a tape with its left part filled with 0’s and its right part filled with a 1 (the origin) and then 0’s. In this notation, the origin is the first cell on the right of the symbol |, here a 1.
- 5.
The configuration of a computation at a given stage is a complete description of the state of the machine at this stage. It comprises the state, the position of the head and the complete content of the tapes.
- 6.
Loops can be considered as transient steps followed by an eventual final loop, and it is natural to measure the stage at which it appears and the length of that final loop.
- 7.
By Theorem 1, we know that the sup of looping ordinals, \(\zeta _\infty \), and periods, \(\varSigma _\infty \), are reached by the universal machine. \(\varLambda \) and \(\varPi \) can thus reach these maximum values.
- 8.
A is ITTM computable from B, written \(A \preccurlyeq _\infty B\), if the characteristic function of A is infinite time computable with oracle B. Reals are seen here as subsets of \(\omega \).
- 9.
We denote by \(M_e\) the e th ITTM in a standard recursive enumeration. The two jumps are defined as \(0^\triangledown = \left\{ e \,|\, M_e(0)\downarrow \right\} \) and \(0^\blacktriangledown = \left\{ \langle e,x \rangle \,|\, M_e(x)\downarrow \right\} \).
- 10.
A corollary is that \(\varSigma _\infty \) is not admissible, as is already proven in [9, Corollary 3.4].
- 11.
Finitely many flashes that occur at successor ordinals do not affect the next limit stage but later ones of higher order (cf. cofinally flashing behaviour in the sequel).
- 12.
Indeed, suppose the machine did not halt for \(\omega \) steps and reached the limit state. Either the machine will loop indefinitely on this limit state and is said to be diverging or will eventually halt excluding the middle pattern. Suppose it will eventually halt and go for more than \(\omega \) steps taking the transition that loops on the limit state: either it will halt on the tape, reading a different symbol at a stage \(\omega +c\) for an integer c, or reach the next limit stage at \(\omega \cdot 2\) by reading a uniform tape. At this point, the machine halts if the looping transition modified the origin cell value, or loops indefinitely on the limit state.
- 13.
Note that knowing whether a machine has this property is not decidable. This was also the case with the finite tape restriction.
- 14.
The situation is not the same for stage \(\omega \) than for compound limit ordinal stages, e.g., \(\omega ^2\), since in the first case we take the limit of a recursive sequence of recursive tapes while in the latter, the sequence of tapes is not necessarily recursive.
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Defrain, O., Durand, B., Lafitte, G. (2017). Infinite Time Busy Beavers. In: Kari, J., Manea, F., Petre, I. (eds) Unveiling Dynamics and Complexity. CiE 2017. Lecture Notes in Computer Science(), vol 10307. Springer, Cham. https://doi.org/10.1007/978-3-319-58741-7_22
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