On the simplest and smallest universal Turing machine

Alex Smith has recently been able to prove that a Turing machine conjectured to be capable of universal computation by Wolfram was actually universal (Wolfram 2,3 Turing machine Research Prize).

Part of the challenge was to find an encoding not doing by itself the universal computation that would make the Turing machine universal. Smith succeeded providing an encoding providing a background that while nonperiodic is sufficiently regular to be generated by infinite word written on the tape can be generated by a omega-automaton (not itself universal).

This relaxation of the tape content has been regular in the cellular automaton world, but also in the field of what is called weak-universality with the only difference that the Turing machines are allowed to start with other than blank tapes (blank tapes are periodic tapes with period 1).

An objection might be that such a coupled system could turn a nonuniversal machine into a universal one, but Smith showed that his encoding was capable of restarting the computation itself, something that coupled systems usually are not capable of. In other words, the preparation of the tape in Smith’s case is done in advance and the omega-automaton do not longer interact in any further time, while to have nonuniversal machines to become universal usually requires an automaton intervening at every step (or every certain steps) of a computation.

One may also be surprised by the need of an omega-automaton for infinite strings. But the difference to the traditional blank tape is subtle. Think of how machines operate in the real world. Machines do not run on blank tapes, they usually do so over memory (the cache, RAM or a HD) with all kind of information (that is considered garbage if it is not instantiated by a running program). You may think that this garbage is not infinite, but it is not so a blank tape for a Turing machine, so instead of thinking of providing a Turing machine with blank tape as need it, one can think of providing the Turing machine with a non-blank tape. Now, what is on the tape in the case of Smith is not exactly “garbage” because it plays a role in “helping” the Turing machine to perform its computation, in a kind of support on which the Turing machine that is capable of universal computation actually achieves the computation with the help of a defined background (that doesn’t mean that the machine cannot perform universal with other backgrounds) yet the background is not powerful enough to perform the hardest part of the computation. In the physical world, if processes are seen as computations, computations are performed on a support, on the background of the physical world. So these kinds of relaxation may be closer to actual physical situations than abstract situations in which a blank tape for a computation is assumed or required.

The discussion opened up by Wolfram’s work and motivated by Smith’s answer has generated a fruitful and renovated discussion of universality and complexity of small Turing machines. This is why I think this research s relevant to modern computer science, and not only as an amusing mathematical puzzle:

  • New techniques for proving universality are found (Alex Smith’s novel approach for unbounded computations from arbitrary lengths and non-periodic initial configurations).
  • New universal models of computation have been discovered (cyclic tag- systems, bi-tag systems).
  • Such research provides a better understanding of universality,  its limits, its  underlying principles and its necessary and sufficient conditions.
  • It is a base for actually building universal devices when only a few elements can be used, e.g. in nanotechnology or molecular computation.
  • Simple/small machines may be more easily/effectively embedded in other systems.
  • The old discovery/invention duality question comes to the fore: It sheds light on how simple universality is, how frequently it occurs, whether  it is engineered or not, whether  one builds universal computation or finds it in the universe.
  • It could shed light on the relative feasibility of  universal Turing machines based on different tape configurations (e.g. blank characters, repetitive words, non-repetitive with computationally simple backgrounds) as actual physical systems.  At present it is not at all clear why one ought to  favor blank characters over other possible real-world backgrounds, such as “noise.”
  • Questions of size and complexity  arise: It would be interesting, for instance, to find out whether there is a polynomial (or exponential) trade-off between program size and and the concept of simulating a process.
  • Some questions  on algorithmic complexity arise: Will the encoding always be more complex if the machine is simpler? All theorems in algorithmic information theory depend on additive constants, which depend on the sizes of typical universal Turing machines. What is the impact of different generalizations of universality on algorithmic complexity and what is the role of  encoding in such a measure?
  • Some questions arise on the relation between several variants of universality definitions: Is there an effective and efficient encoding for each non-periodic encoding preserving universality? If so, how does this impact their complexity? Is there a non-periodic encoding with blank characters for each periodic blank word encoding, and what would the impact of such  an encoding be on the size/complexity of the Turing machine in question?

The field is active and still an important area of research. Several computer science conferences include talks on small computational systems such as Computability in Europe (CiE) and Machines, Computations and Universality (MCU) included such talks this year, focusing in particular on reversible cellular automata and universal Turing machines.

Here are some references from the small Turing machine community:

[1] Manfred Kudlek. Small deterministic Turing machines. Theoretical Computer Science, 168(2):241-255, November 1996.
[2] Manfred Kudlek and Yurii Rogozhin. A universal Turing machine with 3 states and 9 symbols. In Werner Kuich, Grzegorz Rozenberg, and Arto Salomaa, editors, Developments in Language Theory (DLT) 2001, vol. 2295 of LNCS, pp. 311-318, Vienna, May 2002. Springer.
[3] Maurice Margenstern and Liudmila Pavlotskaya. On the optimal number of instructions for universality of Turing machines connected with a finite automaton. International Journal of Algebra and Computation, 13(2):133-202, April 2003.
[4] Claudio Baiocchi. Three small universal Turing machines. In Maurice Margenstern and Yurii Rogozhin, editors, Machines, Computations, and Universality (MCU), volume 2055 of LNCS, pp. 1-10, Chisinau Moldavia, May 2001. Springer.
[5] Turlough Neary and Damien Woods. Four small universal Turing machines. Machines, Computations, and Universality (MCU), volume 4664 of LNCS, pp. 242-254, Orleans, France, September 2007. Springer.
[6] Yurii Rogozhin. Small universal Turing machines. Theoretical Computer Science, 168(2):215-240, November 1996.
[7] Shigeru Watanabe. 5-symbol 8-state and 5-symbol 6-state universal Turing machines. Journal of the ACM, 8(4):476-483, October 1961.
[8] Shigeru Watanabe. 4-symbol 5-state universal Turing machines. Journal of Information Processing Society of Japan, 13(9):588-592, 1972.
[9] Stephen Wolfram. A New Kind of Science. Wolfram Media, 2002.

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