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A Simple Universal Cellular Automaton and Its One-Way And Complex Syst ems 1 (1987) 1- 16 A Sim ple Universal Cellular Automaton and its One -Way a n d Tota listic Ver sion J iirgen A lbert Lehrstuhl fur Inform atik II, Unjyersitiit Wiirzburg, D-800 Wiirzburg, Germany Karel C ulik II Department of Comput er Science, University of Waterloo, Wate rloo, Ontario N2 L 3G I, Canada Abstract. In this p ap er we first der ive two norm al form construc­ tions for cellular automata to transform any given (one-dime ns ional) cellular automaton into one which is one-way and/ or totalistic. An encod ing of t his rest ricted type of automat on toge ther with any ini­ tial configuration becomes the inpu t of our small universal cellular auto maton, usin g only 14 st ates. This improves well-known results obtained by simulation of sm all universal Turing machines and also some recent results on universal tot alistic cellular automata. 1 . Int r o duction Interest in cellu lar au tomata (CA) has been renewed since their ap plication to the study of complex sys te ms [17,18) 9). In thi s context the universality of C A was d iscussed in [191 an d open problems abo ut universal CA with a small number of states were stated in 1201 . We give some results here. A cellular automaton is said to be un£versal if it ca n simulate every Tur ing Machine (T M) or) even stronger, if it can simulate every CA of the same d imension. T he first un iversa l cellu lar automaton was given in the famous work of J . von Ne uman n on the simulation of self-reproduction. He gave a 2-dimensional un iversa l and self-reproducing CA with 29 states. T his was improved to 20 states in 111 . Also well known is the work on small universal Tu ring Machines (cf. [9,10,13]) . In th is case the goal is to mi nimize simultaneously the number of states and the numbe r of tap e symbols. In 1101 it is shown that there is a universal Tu ring Machine with either 4 symbols and 7 states or with 6 sy mbols and 6 states. Smi th 1141has shown that any Turing Machine with m symb ols an d n states can be simulated by a one-d imensional CA whose © 1987 Complex Systems Publica tions, Inc. 2 J. Albert and K. Culik II uniform cells has m + 2n states. Thus either one of the universal TM from 1101yields CA with 4 + 2 x 7 = 6 + 2 x 6 = 18 states.Here we improve this result and show that there is a universal one-dime nsional CA with 14 states. We believe that some more effort in the detailed "low-level programming" of OUf basic strategy can decrease this by 2 or 3 states. However, the m inimal number of states of a un iversal one-dimensional CA could still be much smaller, since CA with 3 or 4 states already show astonishingly complex behaviour. T he well known "Game of Life" is a "semi-totalistic" CA which means that th e next state of a cell only depends on its own current state and the sum of the states of its neighbors. S. Wolfram also introduced an even more restricted type of CA, ca lled totalistic. The next sta te of a cell of such CA dep ends only on the sum of all the states in its neighborhood, including its own. D. Gordon [7] has shown that totalistic one-dimension al CA can simulate every Tu ring Mach ine and we will strengthen this resu lt a nd show that eve ryone-dime nsiona l CA can b e simu lated by a tot alistic on e. We will use this result in the constr uction of our small universal CA .We will also use the resu lt that everyone-dimensional CA can be simulated by a one-way (unidirectional) one-dimensional CA [4 ,5,16J a nd that one-way CA are equivalent to trellis automata [2 J. We will show our results for CA with the t ransit ion function de pendent on the state of each cell and its immediate left and righ t neighbors (r = 1 in [19]). However, our results ca n easily be extended to larger ne ighborhoods. 2. P reliminaries As we wi ll consider only one-d ime ns iona l a nd hom ogeneous cellula r au­ toma ta in t his paper we will restrict ourselves to t his special case in the following definitions. For mo re gen eral te rm inology see [31or [151. Intuitively, a cellu lar automaton consists of a doubly in finite a rr r ay of cells . All cells are ident ical copies of one single finite automaton. The local transition function of each cell only dep ends on the ac tual states of its left and right neighbor and itself. Thus, a computation of a cellul ar a utomaton can be defined in the straightforward way as t he synchronous application of the local transition function at each level. The set of states al ways contains a so-called quiescent state q with the property: if a cell a nd its left and right ne ighbors a re quiescent at t ime t then this cell is quiescent at time t + 1. As we assume finite ou tput for the cellular automaton this im plies that there is a finite number of nonquiescent cells in the initial configuration and in every subsequent configuration as well. D efinition 1 . A (one-dimensional, homogeneous) cellular automaton is a triple A = (Q,d,q) where Q is a finite set oEstates, d is the local transition [unction, d : Q X Q X Q ---+ Q where the arguments oEd are used in the [01­ Jawing meaning d(sta te ot left neighbor, own state, state of righ t neighbor), and q in Q is th e quiescent state, J.e. it holds d(q ,q,q) = q. Universal Cellular A utomaton 3 D efinition 2. A config uration of A is a mapping C : Z --+ Q, where Z denotes the set of integer numbers such that C(i) = q (qu iescent s ta te) [or all but finitely many i 's. The set ofnonquiescent cells of a configuration C is caIled th e support of C. D efinition 3. A computat ion of A is a sequ ence ofconfig uraUons CO,Cl ,C2 , • •• ,Cn , ••• whe re Co is the init ial config urat ion and each configuration Ci+t is gener­ ated by sim ultaneous in vocation of the transition function d for all cells of A in configuration o.. 3. Simulation by one-way a u tomata For our construction of a universal cellular au tomaton in subsequent chap­ ters we will need an automaton wh ich is totalistic and one-way. A cellu lar automaton A with set of states Q and t ransition funct ion d : Q X Q X Q -> Q is called one-way, if d only depends on the state of t he own cell and the state of its r ight neighbor cell. T hus we can write the transition function of a one-way cellular automato n in the form e : Q x Q --+ Q with the convention that th e arguments of e con sist of the states of t he own cell and those of its right neighbor. In this sect ion we will ou tline a transformation of an arbitra ry cellular automaton t o a one-way cellular automaton in orde r to spe cify bounds for the increase in the nu mb er of states and in t ime-steps nee ded in the simulation. In [16] a similar technique was used for the case of real-time cellular automata, more general cas es were considered in [4] and [5]. The const ruct ion of the one-way automaton can be described br iefly by: 1. mer ging of eac h state with the state of its right neighbor 2. shifting to the left during t he state t ransit ion. Let 8 1 ,82, • .. ,51:. be states in a con figur ation, such that 5 1 f; q an d let to, t l , .. , t J: +l be the states after one t ransit ion as shown below q q q 51 8 2 83 q q q q to t 1 t 2 t3 q q T hen it takes two time-steps in our simula ting au tomaton to produce the transition : q q q 5 1 52 q to t1 t 2 t3 For the given cellu lar automaton A let Q be t he set of states and d : Q x Q x Q --+ Q the transition function. As defined ab ove, in the simulating one-way cellu lar automaton A' the transition function dl will con sider only t he states of its own cell and its right ne ighbor cell. 4 J. Albert an d K. Culik II Let Q' = Q U Q x Q and define d' : Q' X Q' ~ Q' as follows d'(u,v) = uv lor(u,v) i (q,q) d'(q,q) q d'(uv, vw) d(u,v,w) d'(q,qu ) d(q,q,u ) d'(wq,q) = d(w,q,q) for all u , v , w in Q. T hus, the transit ion shown above is completed in two time-steps by A' in the following way 5k- l Sit q q Sk_lSk Skq q q tk tk+ l q q The details of this construction are straightforward, so we can state the following theorem : Theo r em 1.
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