Bilinear Cellular Automata

Bilinear cellular aut omata (CA) are those whose next sta te may be expressed as a bilinear form (inner product) of the neighboring st ates. In this paper it is shown that, unlike linear CA , the bilinear CA over Z~ are 1r-univers al , that is, capable of simula ting any CA of the sam e dimension, and hence also capable of simula t ing any (universal) TUring machin e. Evidence is given that the bilinear CA over Zm, t he integers modulo m , may be universal as well. (Although , like Conway 's Game of Life, this appears to be difficult to establish, even for a prime number of states.) A fairly complete Wolfram classification of the bilinear CA over Zm is also given . 1. Polynomial representations of cellular automaton local rules A linear cellu lar automaton (CA) is one whose next state is given by a linear [multivar iat e] polynomial in the neighboring states. As might be expec ted , the linear (additive) CA have been mor e am enabl e to an alysis [2, 11, 13, 17, 19]. Accordingly, the authors have applied these resu lt s for linear CA to t he study of their multiplicat ive cousins , t he monomial CA [4]. Here, we take the next logical ste p in a progressive algebraic approach to the an alysis of CAs , by invest igating the bilinear CAs , whos e next state is given by a bilinear form, that is, an inner product of the neighboring-st ate vector with itself. T his algebraic approach is motivated by a result in [10], which says that *Electronic mail address: bartlett~sunshine .mat hsci. denison .edu. lE lectronic mail address: gar zonmehermes .ms ci. umem. edu. 456 Ron Bartlett and Max Garzon every CA over a pr ime numb er of states admits a polynomial representation of the local ru le.' In this paper we show that t he bilinear CA over Z~ are zr-universal, that is, capable of simulating any CA of the same dimension. It follows immediate ly that the bilinear CA over Z~ are also T-universal , tha t is, capable of simulat ing any (universal) Turing machine. Hence , there can be no algorithm to predict the dynamics of a bilinear CA over arb itrary commutative rings or modules. This contrasts wit h recent result s in [19], indicating that the linear CA are not T-un iversal, even over arbit ra ry commutat ive rings. However , it appears difficult to establish whether the bilinear CA over Zm are T-universal (witness the proof of the T-universality of the Game of Life [5]), though evidence is given herein that they may be. We also give a phenom enological classification of the bilinear CA over th e state set Zm of integers modulo m along th e lines of the Wolfram Classes [20]. 1.1 Preli minaries A ring is a set tha t is closed under two associative bin ar y operations , where one operation (called multiplication) distributes over the other (called addition , assumed commutative, having a neut ral element and inverses for all its elements). A ring is commutative if mult iplication is commutative. For example, (Z,+, x) , the integers under ordinary addit ion and mult iplication, form a commutat ive ring. Also, (Zm, +, x), the integers under addit ion and multiplication modul o m, form a commutat ive ring. Likewise, (Z[x]' +, x) the polynomials over Z, (i.e. , with int eger coefficients) , in one indetermi nant, under polynomial addition and mult iplication, form a commutat ive ring. As an example of a noncommutative ring, consider the set of square matrices with int eger coefficients under matrix addit ion and matrix multiplication. Next we define an one-dimensional euclidean CA . Let ~ denote a finit e alphabe t . An one-dimensional euclidean configuration space C , is given by C = { s = . . 'S 2S-1S0S1S2"': s, E~} = ~z. When endowed with the met ric p ( s(l ),S(2) ) = I~I -K where K = ~in{i: s?) =1= S}2) } C becomes a topological space equivalent to the product topology, upon which a dynamical system can be defined . Then a CA is a dynamical system T : C f-t C that commutes with the shift , 0: C f-t C , given by o-(Si) = SiH ' That is, T is a CA if it is a cont inuous map and T oo= 00 T . T his is a fundamental resul t from [10] . Further background about CA can be found in [7] and [22]. IThe st udy in [10] was mainly concerne d with symbolic dyn ami cal syste ms in one dimension , th e result is eas ily seen to hold for CA in higher dimensions, since any neighborhood may be ordered in such a way as to prod uce a neighborhood vecto r, and hence a po lynomial representation of th e local CA rule. Bilinear Cellular Automata Definition 1. A CA is defined as T-universal or 7r-universal as follows. 457 1. T-universal if it is cap abl e of simulating an arbit rary (universal) Turing machine. 2. n-univeisel if it is capable of simulat ing an arbitrary CA on the same underlying lat tice tt . Now let 5 : ~n -+ ~ denote the local rule of a CA over a prime number of states (ostensibly, ~ = Zp). Let Xi = ( X i , X i+l , . .. ,Xi+dd be a vector of indeterminat es denoting the corresponding st ates, (herein called the neighborhood state vector). Then there is a unique po lynomial P(X) , in n variables X := X o, . . . ,Xnl , such that P(X) = 5(x) . P may also can be expresse d as a sum of monomials: The ak E Zp are then the coefficients of the monomi al terms. This is anot her result found in [10]. For CA over a compos ite numbe r of states , there may be no polyn omial representation of the local rul e, or there may be more than one polynomial representation of the local ru le. However , we may augment the original state set to obtain a prime number of states, and use a proj ecti on of the local rul e from the larger state set onto the original state set. We shall make use of this technique in Examp le 1. Example 1. The general po lynomial modulo 2 for an elementary CA (with m := 2 states and radius r := 1 (3 neighbors) in dimension one is given by P elem( X -I , X o , X l ) = Co + CIXI + CZXO + C3XI + C4XI X O +CSXIXI + C6X OX I + C7X IXOXI · Since t he number of states is a prime p = 2, each of the 2 = 256 distinct bin ar y assignments for t he coefficients c., correspo nds to a distinct elementary CA rule. We note that the t abl e given in [22] provides boolean expressions for the elementary CA, which is not the same as the polyn omi als representation modulo 2, given here (the difference lies in the XOR operation used here instead of the OR used in standard boolean form s). Definition 2. A bilinear CA is one whose local rul e 5 : ~Zr+l -+ ~ is of the form where ~ is the finite set of states wit h an addit ion and multiplicat ion by a set of scalar coefficients , xT is the t ranspose of X, and B = (bi j ) is the matrix of coefficients wit h entr ies bi j E ~ == Zk. 458 Ron Bartlett and Max Garzon The Elementary Bilinear CA (in Wolfram numbers) 0 6 10 12 18 20 24 30 34 36 40 46 48 54 58 60 66 68 72 78 80 86 90 92 96 102 106 108 114 116 120 126 130 132 136 142 144 150 154 156 160 166 170 172 178 180 184 190 192 198 202 204 210 212 216 222 226 228 232 238 240 246 250 252 Table 1: Wolfram numbers of the elementary bilinear CA. We distinguish bilinear CA from quadrat ic CA , which in addit ion to a sum involving products of pairs of neighb oring states , also have a linear component and a constant term: 5(Xi) XiB xl + L(Xi) + C L L bj ,k X i+j X i +k + L ajxi+j + c. j k j Since x; = Xi (mod 2), we can present the following example. Example 2. A general polyn omial for the elementary bilinear CA is P ( X l , X o , Xl) = C6X OXl + CSX-1Xl + C4X -1 X O + C3X l + C2X o + C1X l (mod 2), which is the polynomial of Example 1, with Co = C7 = o. Hence, there are 2 = 64 elementary bilinear CA. Tab le 1 lists the elementary bilinear CA by their Wolfram numbers. While in [22] it is indicated th at the element ary CA seem to be too simple to be T-universal , [16] indi ca tes that the elementary rule 54 might be T-universal. Hence, the presence of rule 54 in Tabl e 1 suggests that t he bilinear CA over Zm might be T-universal. However , we have been unable to find a bilinear polynomial representation for a known T-universal CA. For example, we have established that John Conway 's Game of Life [5] cannot be expressed as a bilinear polynomial over Zm for any modulus m . (A proof is available from the authors .) 2. n-universality of bilinear cellular automata over Z~ In [19], previously known results ab out linear CA [2, 11, 13, 17] have been extended to linear CA with state sets over arbitrary commutat ive rings. Hence, Bilinear Cellular Automata 459 it is reasonab le to explore the T-universality of bilinear CA over commutative rings ot her than Zm. We show that , in cont ras t, the bilinear CA over Z~ are zr-univeraal , and hence T-universal. Our result relies up on t he original construct ion in [1] of an one-dimensional sr-universal CA (UCA) . In the same pap er , the following was also established . For every CA A with m st ates, there exists an one-way CA A' which simulates A twice slower and A' needs at most m 2 + m states . Hence, there exists an one-way 1r-UCA U, with m = 14 + 14 = 210 states and n = 2 neighbors. If we add one more state , we obtain a prime number of states, p = 211. This also adds 2112 2102 new neighb orhoods on which U are not defined. However , we may obtain a new one-way 1r-UCA U' , over p = 211 states simply by assigning a random next state, (say 0) , to t he new neighborhoods. One is then assured by Theorem 19.1 in [10] that the local rule J(Xi, Xi + l ) = U'( X) i has a polynom ial represent ation P (xo, Xl) over Z 2l 1> such that P( Xi , Xi+l) = J(Xi, Xi+1 )' Now P( xo, Xl ) can be expressed as a bilinear form in the powers of Xo and Xl, with coefficient matr ix B = (bu,v) as follows: P (Xo,xd = :L bu ,vx~x~ (mod p). O:<;U,v < p (1) We therefore expand each cell Xi , to a p-tup le Xi , consist ing of the powers of Xi over Zp, (with the convent ion x? = 1, even when Xi = 0), tha t is, The expansion of Xi to Xi can be illust rat ed for one-dimensional CA , by writ ing Xi vertically under X;: X = ... [ Xi 1 [ Xi+1 ] . . .