We show that Post's system of tag with alphabet $\{0,1\}$, deletion number 3, productions $0\rightarrow 00$ and $1\rightarrow 1101$, and initial string $111\cdots 1$ (330 times) converges to the empty word after a very large number of rewriting steps

Asveld, P.R.J.

English

University of Twente Research Information

1An Inefficient Representation ofthe Empty WordPeter R.J. AsveldDepartment of Computer Science, Twente University of TechnologyP.O. Box 217, 7500 AE Enschede, the Netherlandse-mail: infprja@cs.utwente.nl1 PrefaceFrom July 1978 till the summer of 1980 Paul Vitányi and I shared an office atMathematical Centre (former name of C.W.I.), located at Tweede Boerhaave-straat 49 in the Oosterpark-neighborhood of Amsterdam. A couple of weeksafter I arrived, Paul and I moved together with Arie de Bruin to a freshlypainted room which happened to be the half of a former class room, like manyoffices in that old school building, separated from its companion by a ratherthin wall. Without difficulty or any intention we could hear at least 50% of ourneighbors’ conversations as they could from ours. And the view from our roomwas something special too: a 19th century part of the former Amstel brewerywhich happened to be of too little interest to industrial archeology in order tosurvive in later days.Another well-known feature of the old Mathematical Centre was its library:famous for its outstanding collection of books, journals and reports on math-ematics and computer science, and remarkable for the dishes of rat poison onthe floors, inviting the nightly intruders to commit suicide rather than nibblingat the precious volumes.In 1978 both Paul and I finished our Ph.D. work on parallel rewriting, al-though our theses have very little in common; cf. [1, 7]. But we both had theintention to change our subject into the direction of computational complexity:a subject in which Paul achieved much more than I did; see e.g. [4], which alsogives me the opportunity to turn to a more serious matter that is related to thetitle of this note.2 IntroductionAs we all know from [4], Kolmogorov complexity deals with short and efficientdescriptions of objects, e.g., descriptions of strings over the alphabet {0, 1}. Butin some cases there is a long, involved way to go before we arrive at the objectto be described even in a context that looks anything but complex.To be more precise, we will show in this note that it is possible to describea very simple object (viz. the empty string) by means of a simple method (viz.2a well-known elementary instance of Post’s tag system) in a very complicatedway. To illustrate this point we need a few definitions and some notation.3 Definitions, Examples and a ConjecturePost’s tag system is a rewriting system T = (Σ, d, P, ω0) which consists of analphabet Σ, a natural number d, a function P : Σ → Σ⋆ and an initial stringω0 over Σ. We obtain the string ωi+1 from ωi —denoted by ωi ⇒ ωi+1— fori = 0, 1, . . . as follows. If ωi equals the empty word λ, then ωi+1 = λ. If σ isthe first symbol of a nonempty word ωi, then we append P (σ) to the right endof ωi and we delete the first d symbols from ωiP (σ); this yields ωi+1.As an example —to which we will restrict our attention in the sequel— con-sider T0 = ({0, 1}, 3, P0, ω0) with P0(0) = 00 and P0(1) = 1101. This instanceT0 has originally been introduced by Post in [5, 6], but there are still a lot ofopen questions in connection with T0; cf. e.g. [8, 2, 3].From a dynamical system point of view1, there are three possibilities for theultimate behavior of T0 (as function of the initial string ω0, of course); viz.(i) T0 explodes: for each natural n there is an ωi such that its length exceeds n;(ii) T0 reaches a fixed point: T0’s only fixed point is the empty string λ;(iii) T0 enters a periodic cycle of which the period must be an even number.(And indeed, for each even natural number there exists an initial string thatends in such a cycle; cf. [3]).As examples of (ii) we mention that each string ω0 of the form 0k reduces ink steps to λ: since 0k ⇒ 0k−1, we have 0k ⇒k λ. Less trivially, ω0 = 000100000yields in 13 steps λ because 000100000 ⇒ 10000000 ⇒ 000001101 ⇒ 00110100⇒ 1010000 ⇒ 00001101 ⇒ 0110100 ⇒ 010000 ⇒ 00000 ⇒5 λ.The case ω0 = 100000100 is an example of (iii); the (smallest) period is 6,since ω12 = ω18 = ω24 = . . .; cf. [2].Since no examples of (i) are known, we have the following conjecture.Conjecture 3.1. For each initial string ω0 over {0, 1}, the ultimate behaviorof Post’s tag system T0 is either (ii) or (iii). There are a few similarities between this conjecture and the famous 3n + 1-problem2; see [3]. But apart from the validity of 3.1, it is clear that the onlystring that we can describe in terms of ultimate behavior of T0 is T0’s singlefixed point: the empty string λ.4 The ResultThe advance of modern computers enables us to simulate rewriting systems forquite a number of rewriting steps. A few years ago some people reported inthe appropriate news groups of Internet that, according to their simulations,1It is legitimate to consider T0 as a non-linear discrete deterministic dynamical system; cf.[3] for an explanation.2Also known as Ulam’s problem, Collatz’s problem, or Syracuse problem.3T0 would probably explode for ω0 = (100)110 or, equivalently, for ω0 = 1330,thereby providing a possible counterexample to 3.1. Note that for the invalidityof 3.1 we need a proof and not a simulated initial part of a possibly infinitederivation.But continuing the simulation to the very end shows a different picture: after43 913 328 040 672 rewriting steps T0 reaches the empty string. The maximallength of intermediate strings is 31 299 218; it occurs after 14 392 308 412 264rewriting steps. Of course, the first 110 steps are trivial: for i = 1, . . . , 110 wehave ωi = (100)110−i(1101)i, whereas the end of the rewriting process looks like1328040598 011101000011011101000000000011010000110111011328040599 10100001101110100000000001101000011011101001328040600 000011011101000000000011010000110111010011011328040601 01101110100000000001101000011011101001101001328040602 0111010000000000110100001101110100110100001328040603 101000000000011010000110111010011010000001328040604 0000000000110100001101110100110100000011011328040605 000000011010000110111010011010000001101001328040606 00001101000011011101001101000000110100001328040607 0110100001101110100110100000011010000001328040608 010000110111010011010000001101000000001328040609 00011011101001101000000110100000000001328040610 1101110100110100000011010000000000001328040611 11101001101000000110100000000000011011328040612 010011010000001101000000000000110111011328040613 01101000000110100000000000011011101001328040614 0100000011010000000000001101110100001328040615 000001101000000000000110111010000001328040616 00110100000000000011011101000000001328040617 1010000000000001101110100000000001328040618 00000000000011011101000000000011011328040619 0000000001101110100000000001101001328040620 000000110111010000000000110100001328040621 00011011101000000000011010000001328040622 1101110100000000001101000000001328040623 11101000000000011010000000011011328040624 010000000000110100000000110111011328040625 00000000011010000000011011101001328040626 0000001101000000001101110100001328040627 000110100000000110111010000001328040628 11010000000011011101000000001328040629 100000000110111010000000011011328040630 0000001101110100000000110111011328040631 000110111010000000011011101001328040632 11011101000000001101110100001328040633 1110100000000110111010000110141328040634 0100000000110111010000110111011328040635 000000011011101000011011101001328040636 00001101110100001101110100001328040637 0110111010000110111010000001328040638 011101000011011101000000001328040639 10100001101110100000000001328040640 000011011101000000000011011328040641 01101110100000000001101001328040642 0111010000000000110100001328040643 101000000000011010000001328040644 0000000000110100000011011328040645 000000011010000001101001328040646 00001101000000110100001328040647 0110100000011010000001328040648 010000001101000000001328040649 00000110100000000001328040650 0011010000000000001328040651 101000000000000001328040652 0000000000000011011328040653 000000000001101001328040654 00000000110100001328040655 0000011010000001328040656 001101000000001328040657 10100000000001328040658 000000000011011328040659 00000001101001328040660 0000110100001328040661 011010000001328040662 01000000001328040663 0000000001328040664 000000001328040665 00000001328040666 0000001328040667 000001328040668 00001328040669 0001328040670 001328040671 01328040672 λof which the last 9 steps are again trivial. The first column in this table showsi mod (2 ·109). An overview of the string length as function of the discrete timeparameter i is depicted in Figure 1.The simulation of T0 for ω0 = (100)110 is no simple task; it takes about 80MIPSyears. Clearly, this number depends on the way the simulation is codedbut, as far as I can see, there is not much to be gained. Note that, due to5the sequential character of the rewriting process, parallelism is of no use either.Storing all intermediate strings is out of the question, since this would takeabout 214 · 109 Gb. And still you want to store some useful information formaking snapshots like Figure 1, and for the unlucky moments that your systemcrashes.5 ConclusionWe showed that it is possible to describe the empty string in a very inefficientway using rather simple means only, and that Conjecture 3.1 stands firm asever.Figure 1.References[1] Asveld, P.R.J.: Iterated Context-Independent Rewriting – An Algebraic Ap-proach to Families of Languages (1978), Twente University of Technology,Enschede.[2] Asveld, P.R.J.: On a Post’s system of tag, Bull. Europ. Assoc. Theor. Com-put. Sci. No. 36 (1988) 96–102.[3] Asveld, P.R.J.: Post’s system of tag – A simple discrete nonlinear system,pp. 26–31 in: J.P. van der Weele (ed.): Proceedings of Nonlinear Dynamics– Twente 91 (1992), Center for Theoretical Physics, University of Twente,Enschede, The Netherlands.6[4] Li, M. & Vitányi, P.: An Introduction to Kolmogorov Complexity and ItsApplications (1993), Springer-Verlag, New York, Berlin, Heidelberg, etc.[5] Post, E.M.: Formal reductions of the general combinatorial decision prob-lem, Amer. J. Math. 65 (1943) 197–215.[6] Post, E.M.: Absolutely unsolvable problems and relatively undecidablepropositions — An account of an anticipation, pp. 340–433 in: M. Davis(ed.): The Undecidable — Basic Papers on Undecidable Propositions, Un-solvable Problems and Computable Functions (1965) Raven, New York.[7] Vitányi, P.M.B.: Lindenmayer Systems – Structure, Languages, and GrowthFunctions (1978), Free University, Amsterdam.[8] Wanatabe, S.: Periodicity of Post’s normal process of tag, pp. 83–99 in:Proc. Symp. on Math. Theory of Automata 1962 (1963), Polytechnic Press,Brooklyn, N.Y.

An Inefficient Representation of the Empty Word

Asveld, Peter R.J.

NARCIS 

Absolutely unsolvable problems and relatively undecidable propositions — An account of an anticipation,

An Introduction to Kolmogorov Complexity and Its Applications

Formal reductions of the general combinatorial decision problem,

Iterated Context-Independent Rewriting – An Algebraic Approach to Families of Languages

On a Post’s system of tag,

Periodicity of Post’s normal process of tag,

Post’s system of tag – A simple discrete nonlinear system,

https://ris.utwente.nl/ws/files/198680406/Asveld1996inefficient.pdf

An Inefficient Representation of the Empty Word

Abstract

Similar works

Full text

Available Versions

University of Twente Research Information

University of Twente Research Information

NARCIS