Non-Abelian Cellular Automata

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1 Non-Abelian Cellular Automata Cristopher Moore SFI WORKING PAPER: SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. SANTA FE INSTITUTE

2 Non-Abelian Cellular Automata Cristopher Moore Santa Fe Institute September 29, 1995 Abstract We show that a wide variety of non-linear cellular automata can be written as a semidirect product of linear ones, and that these CAs can be predicted in parallel time O(log 2 t). This class includes any CA whose rule, when written as an algebra, is a solvable group. We also show, with an induction on levels of commutators, that CAs based on nilpotent groups can be predicted in parallel time O(log t). 1 Introduction The direct product is a very basic notion in mathematics. Two algebras, dynamical systems, or members of any other category can be paired so that their components act independently of each other. The semidirect product is a notion from group theory, in which the rst component is independent of the second, but not vice-versa. For instance, suppose we taketwo groups A and B. Then if for each a 2 A there is a function f a on B, we can dene the semidirect product A f B with the following multiplication: (a b)(a 0 b 0 )=(aa 0 f a 0(b)b 0 ) This is a group if and only if the f a are automorphisms of B, and if f a1a2 = f a1 f a2 (i.e., f is a homomorphism from A to the automorphism group of B). Specically,ifA and B are subgroups of a group G where their only intersection is the identity, then every element g 2 G can be written uniquely as g = ab where a 2 A and b 2 B. Then if B is a normal subgroup, so that a ;1 ba 2 B, we have (ab)(a 0 b 0 )=aa 0 (a 0;1 ba 0 )b 0 and G is a semidirect product A f B with f a (b) =a ;1 ba. This idea can be extended to dynamical systems, where rather than a direct product where two components evolve independently as in (a b) t+1 = (f(a t ) g(b t )), we have a semidirect product of the form (a b) t+1 =(f(a t ) g at (b t )) 1

3 Figure 1: By blocking together 2r sites, we can transform any CAinto one on a staggered space-time with r = 1=2. We can think of the second component as a non-autonomous dynamical system, which varies with time according to the evolution of the rst component. If a dynamical system can be decomposed in this way, and if we have ecient algorithms to predict both f and the non-autonomous g, then we can predict the system as a whole. Cellular automata (CAs) are dynamical systems which can also be thought of as algebras. A CA is a mapping on sequences, of the form (a) i = (a i;r ::: a i ::: a i+r ) where r is the radius of the rule. By combining blocks of 2r sites together, as shown in gure 1, we can convert any CAinto one with r =1=2 where each site has only two predecessors in a staggered space-time: (a) i = (a i;1=2 a i+1=2 ) We can then think of the CA rule as an algebra, where (a b) =a b. The light-cone below an initial row becomes a 0 a 1 a 2 a 3 a 0 a 1 a 1 a 2 a 2 a 3 (a 0 a 1 )(a 1 a 2 ) (a 1 a 2 )(a 2 a 3 ) ((a 0 a 1 )(a 1 a 2 )(a 1 a 2 )(a 2 a 3 )) and so on. With this approach, we can explore how dierent algebraic properties correspond to properties of the CA, such as ecient prediction [1], partial reversibility [2], and periodicity [3]. In general, predicting a cellular automaton is believed to be no easier than simulating it completely to calculate the nal state we have to ll in the entire light-cone above it, which takes O(t 2 ) serial computation steps (O(t d+1 )ind dimensions) or O(t) in parallel. This prediction problem is easily shown to be P-complete [4], since CAs exist (e.g. [5]) which can simulate universal Turing machines. Many CAs can be predicted in parallel time O(log k t) for some k, putting them in the complexity class NC k [4]. However, if this could be done for all CAs then all problems that could be solved in polynomial serial time could be solved 2

4 in polylogarithmic parallel time, i.e. P = NC. This would be as surprising as, say, if it turned out that P = NP. The CAs that can be predicted in polylogarithmic parallel time, then, seem to occupy a middle position between CAs that are easily predictable, such as elementary rules 90 and 150 [6] that are just addition mod 2, and computationally universal CAs that probably have to be simulated explicitly. In[1]we term these CAs quasi-linear non-linear, but relatively easy to predict. 2 Preliminaries An algebra (A ) is a function from A A to A, written a b or simply ab. The order of an algebra is the number of elements in it. The direct product A B of two algebras is the set of pairs (a b), with (a b)(a 0 b 0 )=(aa 0 bb 0 ). A quasigroup is an algebra whose multiplication table is a Latin square, in which every element of A occurs once in each row and each column. Quasigroups correspond to permutive CAs, in which (a b) is a one-to-one function of each of its inputs (more generally, its leftmost and rightmost inputs) when the other is held xed. An identity is an element 1suchthat1a = a1 =a. An inverse of a is an element a ;1 such that a ;1 a = aa ;1 =1. An algebra is associative if a(bc) =(ab)c for all a b c 2 A. An associative algebra is called a semigroup. An associative quasigroup is a group. Groups have identities and inverses. An algebra is commutative if ab = ba for all a b 2 A. Commutative groups are called Abelian. Thecyclic group Z p = f0 1 2 ::: p; 1g with addition mod p is Abelian. A function f on an algebra is a homomorphism if f(ab) = f(a)f(b). An isomorphism is a one-to-one and onto homomorphism, and an automorphism is an isomorphism from an algebra to itself. Homomorphisms of Abelian groups can be represented as matrices. A subgroup B of a group A is a subset such thatbb 0 2 B for all b b 0 2 B. We say B is normal if a ;1 ba 2 B for all b 2 B and all a 2 A. The subgroup of elements generated by a given subset S is written hsi. For any normal subgroup B of A, there is a factor group A=B and a homomorphism from A to A=B that sends all elements of B to 1. Conversely, the image of A under anyhomomorphism f is A= ker f where ker f = fa 2 Ajf(a) =1g is a normal subgroup of A. In a non-abelian group, the commutator of a and b is [a b] =a ;1 b ;1 ab, so ab = ba[a b]. The commutator subgroup G 0 = h[g G]i of a group G is the set of all elements that can be written as products of commutators. Then G=G 0 is Abelian. 3

5 3 Non-autonomous additive CAs If we have a state space Z p =0 1 ::: p; 1 and a CA rule of the form (a 1 a 2 ::: a 2r+1 )=f 0 + f 1 a 1 + f 2 a f 2r+1 a 2r+1 (1) (mod p) where the f i>0 are scalar coecients and f 0 is a constant, then with a simple Pascal's Triangle method we can predict the CA in O(t) in serial or O(log t) in parallel [7, 1]. Since any Abelian group is the direct product of cyclic groups Z p, this is also true if r =1=2 and (a b) =a + b is an Abelian group. More generally, if the f i are (non-commuting) homomorphisms of an Abelian group (A +), we can represent the CA rule as a polynomial [8] P (x) =f 1 + f 2 x + f 3 x f 2r+1 x 2r+1 after separating out the constant f 0.We can think of P t (x) asthetth row ofa Green's function for the CA, and by using fast algorithms to raise P (x) tothe tth power [9], we can predict the CA in O(t log t) in serial (on a random-access machine) or O(log 2 t) in parallel [1]. We nowshow that a non-autonomous version of (1), where the f i vary in space and time, is still predictable in O(log 2 t) parallel time. Denition. NC k is the class of problems that can be solved in parallel time O(log k n) with a polynomial P (n) number of processors, where n is the length of the input. More precisely, there is a family of circuits C n with P (n) nodes and depth O(log k n), which can be generated by aturing machine using O(log n) space when given n as input [4]. Since the input for predicting a CA is 2rt + 1 initial sites, we say a CA is in NC k if it can be predicted in O(log k t) parallel time. Lemma 1. A non-autonomous CA of the form (1), where the f i>0 are homomorphisms of an Abelian group (A +) and vary in space and time independent of the CA state, is in NC 2. Proof. For simplicity, wewillprove this for r =1=2 a larger radius will simply increase the width of the light-cones, and the computation time, by a constant. Call the states in the light-cone s t x with t = 0 at the initial row, and s t 0 the leftmost state in the tth row asshown in gure 2. Then s t x 's predecessors are s t;1 x and s t;1 x+1, and we write s t x = (s t;1 x s t;1 x+1 )=f t x (s t;1 x )+g t x (s t;1 x+1 )+h t x (where we have replaced f 1, f 2 and f 0 with f, g, and h for clarity) where f t x and g t x are homomorphisms of an Abelian group (A +), varying with t and x but independent ofs t x. 4

6 (0,0) (0,1) (0,2) (0,T) (1,0) (1,1) (2,0) (1,T-1) (t-1,x) f (t,x) (t-1,x+1) g (T,0) Figure 2: The labelling scheme used in the text for the light-cone below the initial row. Nowif(t 0 x 0 ) is in the light-cone below(t x), dene c t 0 x 0 jt x as the coecient of s t x in s t 0 x 0.Thenwehave c t 0 x 0 jt x = (x 0 ; x) ift 0 = t c t 0 x 0 jt x = f t 0 x 0c t 0 ;1 x 0 jt x + g t 0 x 0c t 0 ;1 x 0 +1jt x if t 0 6= t Then it is straightforward to show inductively that the state s T 0 at the bottom of a light-cone T steps high, with initial row s 0 x for 0 x T,is s T 0 = TX x=0 c T 0j0 x (s 0 x )+ TX T X;t t=1 x=0 c T 0jt x (h t x ) (2) Since we are given the h t x,we can add all (T +1)(T +2)=2 of these terms together in parallel time O(log T ) (since n objects can be added in time O(log n) in parallel) but we have to calculate the c T 0jt x rst. In fact, we can calculate any c t 0 x 0 jt x in parallel time proportional to O(log 2 (t 0 ; t)) in the following way. The inuence of s t x on s t 0 x 0 has to go through the intervening sites so for any t 00 where t<t 00 <t 0,we can write min(x x 0 +t;t 00 ) X c t 0 x 0 jt x = c t 0 x 0 jt 00 x 00 c t 00 x 00 jt x (3) x00 =max(x;t 00 +t x 0 ) as shown in gure 3. We can then use induction on increasing time intervals. Assume that the c t 0 x 0 jt 00 x 00 and c t 00 x 00 jt x are known then we can calculate c t 0 x 0 jt x in time 5

7 t (t,x) (t,x) t = (t,x ) t (t,x ) (t,x ) Figure 3: The divide-and-conquer strategy of equation 3. O(log(t 0 ; t)) since the sum in (3) has at most (t 0 ; t)=2 + 1 terms and the multiplications can be done simultaneously in constant time. In particular, if t 00 = b(t 0 ; t)=2c, we can simultaneously calculate all the c t 0 x 0 jt x where t 0 ; t = 1, then for all with t 0 ; t = 2, and so on, doubling each time until we have reached the desired time interval (a divide-and-conquer strategy). This takes time log 1 + log 2 + log 4 + +log(t 0 ; t)=2 =O(log 2 (t 0 ; t)) So in parallel, we can calculate all the c T 0jt x for all t x in the light-cone in time O(log 2 T ) and add the sum in (2) in time O(log T ), nally arriving at the nal state s T 0. We can generalize this further in two ways. First, for these sums to work, (A +) simply needs to be commutative and associative i.e., it can be a commutative semigroup rather than a group. Secondly, this algorithm works in any number of dimensions the number of sites at t 00 between t and t 0 is proportional to (t 0 ; t) d, so each step of the divide-and-conquer strategy takes time O(log(t 0 ; t) d )=O(log(t 0 ; t)). So in full generality we can state the following theorem: Theorem 1. In any number of dimensions, suppose a CA is of the form C = (((C 0 f1 C 1 ) f2 C 2 ) ) fn where C 0 is in NC 2 and the C i>0 are non-autonomous additive CAs of the form i (a 1 a 2 ::: a k )=f i 0 + f i 1 (a 1 )+f i 2 (a 2 )++ f i k (a k ) C n 6

8 where the f i j>0 are homomorphisms of the commutative semigroup (A i +). Then C is in NC 2,i.e.itcan be predicted inparallel time O(log 2 t). Proof. Since C 0 is in NC 2,we can calculate its state, not just at the nal site, but everywhere in the light-cone (isn't parallel time wonderful?) with t 2 times as many processors. This and Lemma 1 tells us how to calculate C 1 everywhere in the light-cone, and so on. Iterate n times. Example 1. If A = f0 1g with addition mod 2, the homomorphisms ofa are zero and the identity. Then the functions expressible as f(a) + g(b) + h are 0 1 a a b b ab and a b. Soany semidirect product C f A where the states of C select among these eight functions on A is in NC 2 if C is. Example 2. If A = Z 3 = f0 1 2g with addition mod 3, the automorphisms of A are f(a) = a. All quasigroups of 3 elements can be expressed as ab+h. Therefore, any permutive CAwhich is a semidirect product C f A, where A has three elements, is in NC 2 if C is. Example 3. Any CA of the form ((a 1 a 2 ::: a k ) (b 1 b 2 ::: b k )) = (P 1 (a 1 b 1 ) P 2 (a 1 a 2 b 1 b 2 ) ::: P k (a 1 ::: a k b 1 ::: b k )) where each P i is a polynomial linear in a i and b i,isinnc 2. 4 Solvable group CAs One interesting class of CAs are those for which (a b) =a b is a non-abelian group. Because of their non-commutativity these CAs do not obey a principle of superposition, so Green's function techniques don't work. In [1] we show an O(log t) algorithm for one such group, the Quaternions Q 8, but other non- Abelian groups such as the permutations of three elements S 3 areleftasan open problem. We cannowshow that a large class of nite groups have CAsinNC 2. First: Lemma 2. The set of algebras whose CAs can be predicted in a given amount of serial or parallel time (up to a multiplicative constant) is closed under nite direct products, subgroups, and homomorphisms. Proof. For nite direct products G 1 G 2, simply predict each of the G i, either sequentially or in parallel. For subgroups, clearly an algorithm that predicts an algebra also predicts any of its subgroups. For a factor group or homomorphic image H = f(g) =G= ker f, take a pre-image f ;1 (h) for each element h 2 H in the initial conditions, use the algorithm for G, and then apply f to return to H this works since f is a homomorphism, e.g. for one time-step h 1 h 2 = f(f ;1 (h 1 )f ;1 (h 2 )). In the language of universal algebra, this makes the set of CAs in a given complexity class an!-variety. Now recall the following denitions from group theory [10]: 7

9 Denition The derived series of a group G is the series of normal subgroups G G 0 G 00 ::: where G 0 = h[g G]i is the derived subgroup of G, G 00 is the derived subgroup of G 0, and so on. A group is solvable if the derived series ends in f1g. For instance, the derived series of S 4, the group of permutations of 4 objects, is S 4 A 4 Z 2 2 f1g where A 4 is the set of even permutations and Z 2 2 is generated by the permutations (12)(34), (13)(24) and (14)(23). Theorem 2. Any CA based on a solvable group is in NC 2. Proof. This follows immediately from a standard theorem. Recall [10] that the wreath product AoB of two groups is a semidirect product B A B where A B is the set of functions from B to A and elements of B permute their components. In other words, (b f)(b 0 f 0 )=(bb 0 b 0f f 0 ) where b 0f(b) =f(b 0 b) Moreover, the wreath product is associative, i.e. A o (B o C) and(a o B) o C are isomorphic for any three groups A B C. Then [11] anysolvable group with derived series G = G 0 G 1 G k = f1g is a subgroup of the wreath product H r o H r;1 ooh 1 where H i = G i;1 =G i are the Abelian factor groups. This is a semidirect product of Abelian groups, so by Lemma 2 a CA with a solvable group as its algebra is in NC 2. For instance, the Quaternions have the derived series Q 8 Z 2 f1g (since [a b] =1 for any a b) soh 1 = Z2 2 and H 2 = Z 2. Then Q 8 is a subgroup of Z 2 o Z2 2, which is a semidirect product Z2 2 Z2 4 of order 64. Now this algorithm can be inconvenient, since a group of order n can result in an exponentially larger wreath product if n =2 k and H i = Z 2, the wreath product has order 2 n;1. The number of processors required to calculate products in a group of this size is O(log 2 n;1 )=O(n), however, so the number of processors is still polynomially bounded in the size of the original group. This means that we actually have annc algorithm that can take any group as input, along with the initial conditions, and predict that group's CA. But in any case, most small groups don't require embedding in such large wreath products. Many small groups are already semidirect products of Abelian groups, including the dihedral groups, groups of order p 3 for p an odd prime, any group of square-free order, all groups of order p 2 q where p and q are primes, and so on [10, 12]. If we call a semidirect product of Abelian groups polyabelian, all groups of order less than 32 are polyabelian except the dicyclic or generalized quaternion groups, which are factor groups of polyabelian groups twice their size, and the binary tetrahedral group of order 24, which isasubgroupofa polyabelian group of order = [14]. The smallest non-solvable group is A 5, the simple group of order 60 (also called the icosahedral group). Since polyabelian groups are easily shown to be solvable (since if G =((A 0 A 1 )A 2 )A 3 ::: then G 0 (A 1 A 2 )A 3 :::), and since subgroups and factors of solvable groups are also solvable, this group's CA 8

10 can not be predicted by these methods. This leaves us with the following open question: is there an algorithm in NC 2,orNC k for some other k, for predicting CAs based on arbitrary nite groups? 5 Nilpotent group CAs We now show that a subset of the solvable groups have CAs in NC 1, i.e. which can be predicted in parallel time O(log t). Recall the following from [10, 11]: Denition. The lower central series of a group G is the series of normal subgroups G = 1 G 2 G where i+1 = h[ i G G]i. In other words, 2 G = G 0, 3 G is the subgroup generated by 3-element commutators [[a b] c], and so on. If the lower central series ends in f1g, wesay that G is nilpotent if k+1 G = f1g, then G is nilpotent of class k and all commutators with more than k elements are 1. The nilpotent groups form a proper subset of the set of solvable groups. For instance, a nilpotent group of class 1 is simply an Abelian group. A nilpotent group of class 2 has commutators which commute with everything, i.e. [[a b] c] = 1 (for instance, the Quaternions, where [a b]=1). And so on. We now show that Theorem 3. CAs based on nilpotent groups are innc 1. Prof. First, consider a nilpotent group of class 2. If its initial conditions are a 0 a 1 ::: a t, the leftmost two columns of its light-cone are a 0 a 1 a 0 a 1 a 1 a 2 a 0 a 2 1 a 2 a 1 a 2 2 a 3 a 0 a 3 1 a3 2 a 3 [a 2 a 1 ] a 1 a 2 2 a2 3 a 4 [a 3 a 2 ] The commutator arises since a 2 a 1 = a 1 a 2 [a 2 a 1 ], so (a 0 a 2 1 a 2)(a 1 a 2 2 a 3)=a 0 a 3 1 a 2[a 2 a 1 ]a 2 2 a 3. Since commutators commute with everything we can move it to the right, leaving the a i in sorted order. Continuing in this way weget a 0 a 4 1 a6 2 a4 3 a 4 [a 3 a 1 ][a 2 a 1 ] 4 [a 3 a 2 ] 4 a 0 a 5 1 a10 2 a10 3 a5 4 a 5 [a 4 a 1 ][a 3 a 1 ] 5 [a 4 a 2 ] 5 [a 2 a 1 ] 10 [a 3 a 2 ] 24 [a 4 a 3 ] 10 a 0 a 6 1 a15 2 a20 3 a15 4 a6 5 a 6 [a 5 a 1 ][a 4 a 1 ] 6 [a 5 a 2 ] 6 [a 3 a 1 ] 15 [a 4 a 2 ] 35 [a 5 a 3 ] 15 [a 2 a 1 ] 20 [a 3 a 2 ] 84 [a 4 a 3 ] 84 [a 5 a 4 ] 20 So each site s t x Q t i=0 a ; t i in a light-cone consists of a product of an \Abelian part" x+i times powers of commutators [a j a i ] where i < j. Since s ; t x is t 1 j ; x the product of s t;1 x, which contains aj,and s t;1 x+1, which contains ; t ; 1 i ; x ; 1 t ; 1 ai,weget t ; 1 j ; x i ; x ; 1 factors of [aj a i ] when we pass these powers of a i and a j through each other. Then these commutators down to the nal 9

11 site s T 0 in s T 0 = T ; t x ways, so Y 0iT a ; T i i Y 0<i<j<T P T P min(t ;t i;1) [a j a i ] t=0 x=max(0 j;t;1) ; T ; t x ; t ; 1 t ; 1 j ; x i ; x ; 1 (4) (If [a b] 2 = 1 for any commutator [a b], this is just the algorithm for Q 8 in [1].) In general, if we have a nilpotent group of class k, then the nal site can always be written s T 0 = Y i a c1 i i Y i j [a i a j ] c2 ij Y i1 i2 ::: i k [[[a i1 a i2 ] ] a ik ] c k i 1 i 2 :::i k (5) for some set of exponents c j i1i2:::i j (1 j k). This is clear by induction: by sorting the a i on the left, we generate 2-element commutators [a i a j ]. By sorting these we generate 3- and 4-element commutators, and so on, until we reach k-element commutators which commute with everything. For each t, then, we have an expression (5) with O(t k ) terms in parallel, these can all be multiplied out in time O(log t k )=O(log t) togive s T 0. It just remains to be shown that this circuit, i.e. all the c j i1i2:::i j, can be generated by aturing machine in O(log t) space. But since j-element commutators are made by crossing commutators with fewer elements, each c j at s t x is a polynomial of c j0 sats t;1 x and s t;1 x+1 where j 0 <j these are polynomialsof smaller commutators, and so on. So to get each c j,wehave to calculate a tree of smaller c j0 but this tree is of constant depth (at most k), and since it ends t in c 1 = which can be calculated in O(log t) space, so can each ofthec j. i i Finally, if each c j can be calculated in logarithmic space, all of them can they can be indexed in logarithmic space since there are only polynomially many. So any such CAisinNC 1. 6 Conclusion Non-linear cellular automata that are easily predictable are akin to exactly solvable models in statistical mechanics, or integrable non-linear partial dierential equations. They help us extend our understanding of dynamical systems into a gray area between the purely linear and those capable of universal computation. CAs based on groups, semigroups, quasigroups and so on are especially interesting, since they allow us to apply a rich and powerful theory of algebras to the problem of predicting the nal state. We have shown here that CAs based on solvable and nilpotent groups can be predicted in parallel time O(log 2 t)and O(log t) respectively. We hope to extend these results in the future, both to arbitrary nite groups and to CAs with looser algebraic structure. 10

12 Acknowledgements. I am grateful to David Rusin for pointing out the theorem that solvable groups are subgroups of wreath products, to Arthur Drisko and Mats Nordahl for their thoughts on the manuscript, and to Elizabeth Hunke and Spootie the Cat for their patience. References [1] C. Moore, \Quasi-Linear Cellular Automata." Submitted to Physica D. [2] K. Eloranta, \Partially Permutive Cellular Automata Analysis via Tilings Subalphabets." Helsinki University of Technology Research report A314 (1992). [3] J. Pedersen, \Cellular Automata as Algebraic Systems." Complex Systems 6 (1992) [4] C.H. Papadimitriou, Computational Complexity. Addison-Wesley, [5] K. Lindgren and M.G. Nordahl, \Universal Computation in Simple One- Dimensional Cellular Automata." Complex Systems 4 (1990) [6] S. Wolfram, \Statistical Mechanics of Cellular Automata." Reviews of Modern Physics 55 (1983) [7] A.D. Robinson, \Fast Computation of Additive Cellular Automata." Complex Systems 1 (1987) [8] O. Martin, A.M. Odlyzko, and S. Wolfram, \Algebraic Properties of Cellular Automata." Communications in Mathematical Physics 93 (1984) [9] D.E. Knuth, Seminumerical Algorithms. Addison-Wesley, [10] M. Hall, The Theory of Groups. Chelsea, [11] M. Suzuki, Group Theory I. Springer-Verlag, [12] W. Burnside, Theory of Groups of Finite Order. Dover, [13] H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups. Springer-Verlag, [14] D. Rusin, personal communication. 11