A New Universal Cellular Automaton on the Pentagrid

Abstract

In this paper, we significantly improve a result of the first author, published in an issue of Theoretical Computer Science in 2003. In this paper, the authors showed the existence of a weakly universal cellular automaton on the pentagrid with 22 states. The simulation used a railway circuit which simulates a register machine. In the present paper, using the same simulation tool, we lower the number of states for a weakly universal cellular automaton down to 9. A~t~~b t h h~i s t h a tthiunomctaagle.hmthib,mo m~mthatWIllussofth~bolicphmnon-aecasltIfrtadontyIfthey have a camon p q e n d h h , Parallel Process. Lett. 2009.19:227-246. Downloaded from www.worldscientific.com by AUSTRALIAN NATIONAL UNIVERSITY on 03/15/15. For personal use only.w h M contains the other part of the path. We my that the t r a W is a-j. Wc may mume that in all tho cmbqp, one pith b 1-3 end t L othm is 5-2, lFlguro 12. Tho l o w d m through a c r -:ham from q w r h 6 to qwrhr 2, lhgure 11 repmmta the locomotive going through the path 1-3 of pa a d F w 12 rqmmits tho through tho p t h 8-2.Note that on tine wth 5-2, a new date a p p m in the all 2(l) . w h the front of the locomotive errivee at the centre of the mmhg, b h cell 0. TW ie d e d for the working of the Axsd witch as-will be seen I&. Note that by rotational e m r y , the cell 2(3) ia changed in a sirnfler way when the front d the locomotive Iwm tho ecU l(2) for the dl 4(2), The M e sAll mitehee a e hilt w c d h g to to mmilar pattcm, Throe tr& meat at a cmW, the cemtrd cell of Fig, 13,14 end 16. W o wwm that an active croeelogdm ssriw Parallel Process. Lett. 2009.19:227-246. Downloaded from www.worldscientific.com by AUSTRALIAN NATIONAL UNIVERSITY on 03/15/15. For personal use only.1(1] and 1(3) mdd attract G, d t i n g into t w t o o o~i v e e , whiah is dd out by lha ahuhtlon. Thh b wlp. the cell P(1) h h tbe state X2 whtm the d l(6) is u h r 0. I~B pmmm allaplrs ths cell l(1) L) to that it ~~u n d e a B . N o b % t h e t h p r o b l e m~h h~i f m~u a e t h s~e d 9 i n~d Q I f i o s t L W o f h M i w i n d O . Ffgum 18, TIM hmmotiw c m m m a &wrl rnvitrh from the n o a -s r e w tradr, h m in quytar I . Parallel Process. Lett. 2009.19:227-246. Downloaded from www.worldscientific.com by AUSTRALIAN NATIONAL UNIVERSITY on 03/15/15. For personal use only.

An Algorithmic Approach to Tilings of Hyperbolic Spaces: Universality Results

“…A few years later, the number of states was lowered down to 9 in the pentagrid, see [28] and very recently to 5, again in the pentagrid, see [25]. Then a simulation in the heptagrid was performed with 6 states, see [29]. This latter result was recently lowered down to 4 in the heptagrid, see [20].…”

A Weakly Universal Cellular Automaton in the Pentagrid with Five States

“…The centre of the switch is signalized by the neighbouring of the centre. Figure 4 illustrates the implementation of the crossing and of the switches performed in [6], showing in particular, the feature at which we just pointed. Figure 3 shows the implementation of the verticals, second row in the figure, and of the horizontal, first row.…”

“…Theorem 1 (Margenstern, Song), cf. [6] − There is a planar cellular automaton on the pentagrid with 9-states which is weakly universal and rotation invariant.…”

“…In this paper, we construct a weakly universal cellular automaton on the pentagrid, see Theorem 2 at the end of the paper. Two papers, [1,6] already constructed such a cellular automaton, the first one with 22 states, the second one with 9 states. In this paper, the cellular automaton we construct has five states only.…”

About the Garden of Eden Theorems for Cellular Automata in the Hyperbolic Plane

“…For the other cells, fix side 1 to be the side shared by the cell with its father and, similarly number the other sides by counter-clockwise turning around the cell. Then, a rule of the automaton can be displayed in the following format, see [12,19,18] : η 0 , η 1 , ..., η α → η 1 0 , where η 0 is the current state of the cell, η i , with i ∈ {1..α}, where α = 5 or α = 7, is the current state of neighbour i and η 1 0 is the new state of the cell. We say that a cellular automaton on the pentagrid or on the heptagrid is rotation invariant if and only if for each rule as above, the rule η 0 , η π(1) , ..., η π(α) → η 1 0 , also belongs to the set of rules for any circular permutation π on {1..α}.…”