J. Pedersen, "Cellular Automata as Algebraic Systems." Complex Systems 6 (1992) 237-250.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Here r = 2 and k = 4. This can be a fruitful point of view from which to study CAs; depending on ffl s algebraic properties, we can make statements about how much parallel or serial computation is needed to predict the CA [1, 2] its reversibility or surjectivity [3, 4] or its periodic behavior [5]. Now suppose that we group k sites together into a block; we can look at these blocks as single sites of another CA rule, with a larger alphabet A and a smaller radius r=k (if k divides 2r) In particular, by taking k = 2r we can transform any CA into one with r = 1=2, as shown in figure 2. ....

J. Pedersen, "Cellular Automata as Algebraic Systems." Complex Systems 6 (1992) 237-250.

....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 different algebraic properties correspond to properties of the CA, such as efficient 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 final state we have to fill in the entire light cone above it, which takes O(t ) serial computation steps (O(t d 1 ) in d dimensions) or O(t) in parallel. This prediction ....

J. Pedersen, "Cellular Automata as Algebraic Systems." Complex Systems 6 (1992) 237-250.

....could be fruitful. This approach is particularly attractive when we consider the importance of semigroup theory in the study of regular languages (see [28, 56, 46] Another reason is the very close connection between cellular automata and finite groupoids that has been observed recently. In [55] it is shown how any finite groupoid can be seen as an infinite one dimensional cellular automaton where each cell changes its state according to its current state and that of its left neighbour. Then, cellular automata having periodic behavior are shown to correspond to particular varieties of ....

J. Pedersen, Cellular Automata as Algebraic systems, Complex System 6 (1992) pp.237-250.

....a staggered space time. Then each site has just two predecessors, a 0 i = f(a i 1=2 ; a i 1=2 ) and we can think of the CA rule as a binary algebra, a = f(b; c) b c In fact, any CA can be re written in this form, by lumping blocks of 2r sites together as in gure 1. A number of authors [1, 2, 5, 7 10] enjoy looking at CAs in this way, and have studied properties like reversibility, permutivity, periodicity Fig. 1. By combining blocks of 2r sites, we can transform any CA into one with r = 1=2. Here r = 2. and the computational complexity of predicting the CA s behavior, depending on what ....

J. Pedersen, \Cellular automata as algebraic systems." Complex Systems 6 (1992) 237-250.

....with r = 1=2 by grouping sets of 2r sites together. Each new state is a function of just two predecessors, a 0 i = f(a i Gamma1=2 ; a i 1=2 ) or a = b ffl c b a f(a,b) Figure 1: The staggered space time of an r = 1=2 CA. where ffl is a binary operation on A. This approach was also taken in [2, 3]. Like any dynamical system, we would like to know how hard a particular CA is to predict. In particular, suppose we are given a finite stretch of initial conditions, a 0 : a t . After t time steps, there is a single site at the bottom of a light cone with this initial row on top. Call the ....

....simply identify it with the Pascal s Triangle coefficient and write G x;t = t x . By the Chinese Remainder theorem, any Abelian group can be written as a direct sum A = Z p1 r 1 Phi Z p2 r 1 Phi Delta Delta Delta Phi Z pk r k where the p i are primes so all we need to know [1, 2] is G x;t mod p i r i for each i. But each of these coefficients can be calculated in the following way: since t x 1 = t x (t Gamma x = x 1) we start with G 0;t = 1 and then use this recurrence to go from G x;t to G x 1;t . We calculate G x;t as follows: write G x;t = p ....

J. Pedersen, "Cellular Automata as Algebraic Systems." Complex Systems 6 (1992) 237-250.

....(a 1 a 2 ) a 2 a 3 ) Delta Figure 1: By blocking together 2r sites, we can transform any CA into one on a staggered space time with r 0 = 1=2. Here r = 2. and so on. Several authors have used this approach to explore CA properties such as partial reversibility [11] and periodicity [29]. Predicting a cellular automaton t time steps into the future is believed to be no easier in general than simulating it explicitly. To do this, we have to calculate all the CA states in a light cone of depth t, which takes O(t 2 ) serial computation steps (O(t d 1 ) in d dimensions) or O(t) ....

J. Pedersen, "Cellular automata as algebraic systems." Complex Systems 6 (1992) 237--250.

....Here r = 2 and k = 4. This can be a fruitful point of view from which to study CAs. Depending on ffl s algebraic properties, we can make statements about how much parallel or serial computation is needed to predict the CA [4, 5] its reversibility or surjectivity [1, 3] or its periodic behavior [7]. In fact, any CA is equivalent to one with r = 1=2 through the following block transformation. Treat blocks of k sites as single sites of another CA rule, with a larger alphabet A k and a smaller radius r 0 = r=k (if k divides 2r) Then if k = 2r we get r 0 = 1=2, as shown in figure 2. We ....

....completely non commutative algebras with identity are also cellular. 4 Conclusion By blocking sites together to produce a two site neighborhood, a CA rule can be thought of as a binary algebra. The properties of this block algebra can be related to the CA s dynamical properties in various ways [1, 3, 4, 5, 7]. However, we have shown that if this algebra is associative with identity, an inverse property loop, anti commutative with identity, or commutative, then the original CA rule depends only on its leftmost and rightmost inputs, and the block algebra is simply the direct product of 2r copies of the ....

J. Pedersen, "Cellular automata as algebraic systems." Complex Systems 6 (1992) 237--250.

....body of results in Section 3 come from the author s thesis [3] the results in section 4 are an attempt to use similar tools. The way in which these tools are unusable in the radius 1 paradigm used in Section 4 is then analysed and presented as evidence that the paradigm introduced by Pedersen in [16] but dating back to Hedlund [7] and used by the author and others in [2, 3, 4, 5, 11] has certain strengths and advantages. We proceed first by looking at some elementary cellular automata theory, then looking at the two paradigms more closely. In particular we formulate reversibility conditions ....

....applied combinatorics, using only elementary algebraic concepts. 2 The paradigms In this section we briefly review appropriate cellular automata theory and definitions, before looking at the specific details of reversible cellular automata of radius 1 and 1=2. 2. 1 Cellular automata in brief In [16], J. Pedersen introduced the idea of using radius 1=2 cellular automata as a method of algebraising the theory of cellular automata. In general, he shows that every cellular automata rule can be expressed as a binary operation, leading to the intuitive use of algebraic techniques for the ....

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John Pedersen. Cellular automata as algebraic systems. Complex Systems, 6:237--250, 1992.

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J. Pedersen, "Cellular Automata as Algebraic Systems." Complex Systems 6 (1992) 237-250.

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