Olivier Martin, Andrew M. Odlyzko, and Stephen Wolfram. Algebraic properties of cellular automata. Commun. Math. Phys., 93:219--258, 1984.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 OE(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 Delta Delta Delta f 2r 1 x 2r 1 after separating out the constant f 0 . We can think of P (x) as the tth row of a Green s function for the CA, and by using fast algorithms to raise P (x) to the tth power [9] we can predict the CA in O(t log t) in ....

O. Martin, A.M. Odlyzko, and S. Wolfram, "Algebraic Properties of Cellular Automata." Communications in Mathematical Physics 93 (1984) 219258.

....of one information through the space time diagram. But the inverse question immediately comes: given the evolution of sonhe cellular automaton on sonhe initial configuration, is it possible to understand how it processes information For sonhe of them, in fact the additive ones introduced in [42], an algebraic study is possible (see section 3.3) All these questions have a natural extension to other underlying graphs (especially Z2) even to other cellular automata generalizations. In the Z 2 case, other ways to compute can be considered, for example by simulating no more sonhe Turing ....

....one can prove that it is (in some way) computational universal and thus many problems on its behaviour become undecidable; however, sometimes its dynamics may be understood via special methods. The three main involved methods are algebraic (the most famous example is additive cellular automata of [42]) or concern the order structure of the configurations graph (as for sand piles ) or are physicists like method (with a notion of energy or entropy ) Clearly, such studies on special cellular automata must be carried on and they lead to many open problems. 4.1.1 Algebraic methods in study ....

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O. Martin, A. Odlysko and S. Wolfram, Algebraic Properties of Cellular Automata. Communications in Mathematical Physics, vol. 93, 219-258, 1984.

....detailed analysis of the divisibility properties of these polynomials and apply our results to the study of oe automata. 1 1 Introduction A oe automaton is a simple, non uniform, binary cellular automaton on a directed graph. These automata were first studied by Lindenmayer in [7] and later in [1, 9, 12, 13, 2]. Briefly, a oe automaton consists of a directed graph G = hV; Ei together with a global rule given by oe(X) v) X u2N(v) X(u) mod 2: Here X : V f0; 1g is a pattern or configuration of the automaton and N(v) denotes the open neighborhood f u 2 V j (u; v) 2 E g of vertex v. If the ....

....graphs G, the oe automaton over G is reversible iff the diagram is a disjoint union of cycles. The same comments apply to oe automata. A detailed analysis of the state transition diagrams of rules oe and oe on undirected cycles, as well as higher dimensional analogues, can be found in [9]. We write d(G) for the corank of oe G : C C , and likewise d (G) for rule oe . If the graph G is given explicitly, say, as an adjacency matrix, then it is trivial to determine the corank of the linear maps oe and oe over C in time polynomial in the size of G. However, one is ....

O. Martin, A. M. Odlyzko, and S. Wolfram. Algebraic properties of cellular automata. Commun. Math. Phys., 93:219--258, 1984.

....or of the two neighbors of a cell, and the exclusive or of the neighbors plus the center cell. Another minor variation concerns the type of boundary conditions. We will refer to all such rules generically as rule oe when details are irrelevant. 1 oe automata were studied in great detail in [8] using binary polynomials as the main algebraic tool. The authors represent both the linear operator and the configurations as binary polynomials in some suitable quotient ring F 2 [x] For example, rule 90 with cyclic boundary conditions on a grid of length n can be represented by ....

....the vertex set of the diagram is a pattern space V = 2 n , and there is an edge from vertex v to vertex u if oe(v) u. It is clear from the definition that the components of the transition diagram of oe are unicyclic. In particular, the co orbit of 0 is a tree rooted at the fixed point 0, see [8, 9]. We denote this component K. It follows from the linearity of oe that the branching factor in K is the corank of oe. Moreover, from the results in [1] and [12] it follows that the tree is completely balanced (i.e. all leaves occur at the same level) The height of the tree is plainly the ....

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O. Martin, A. M. Odlyzko, and S. Wolfram. Algebraic properties of cellular automata. Commun. Math. Phys., 93:219--258, 1984.

....values of the lattice at time t, is called the state of the system at time t. The interaction rule is a function which assigns a site value at time t 1 to each site in terms of values at certain of its neighbors at previous time t. Several authors in the past, following the approach initiated in [6], represented states as elements of a Laurent polynomial ring and most frequently studied state transition functions which were linear. For example, see [2, 4, 5, 6] In this paper, we generalize the notion of a linear cellular automaton by viewing the state transition function abstractly as an ....

....each site in terms of values at certain of its neighbors at previous time t. Several authors in the past, following the approach initiated in [6] represented states as elements of a Laurent polynomial ring and most frequently studied state transition functions which were linear. For example, see [2, 4, 5, 6]. In this paper, we generalize the notion of a linear cellular automaton by viewing the state transition function abstractly as an endomorphism acting upon a module of states. This generalizes the approach of several previous contributors and allows a conceptual understanding of the structure of ....

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O. Martin, A. Odlyzko, S. Wolfram, Algebraic Properties of Cellular Automata. Commun. Math. Phys. 93, 219-258 (1984) 18

....two or more subsystems which do not interact under F . Denseness of periodic orbits is an element of regularity for the system. In this paper we consider additive one dimensional CA defined on a finite alphabet of prime cardinality. Additive CA have been studied by several authors (see for example [5,14,19]) Despite their simplicity that allows an algebraic analysis, additive CA exhibit many of the complex features of general CA. They have been used for modeling and approximating many physical phenomena (see for example [21] We prove that they are chaotic according to the definition of chaos ....

.... ; x i ; x i 1 ; x k ) x i 2 Ag = A: Definition 2 f is leftmost [rightmost] permutive if and only if there exists an integer i; Gammak i k; such that ffl i 0 [i 0] ffl f is permutive in the i th variable, and ffl f does not depend on x j ; j i, j i] Definition 3 ([19]) f is additive if and only if it can be written as f(x Gammak ; x k ) 0 k X i= Gammak i x i 1 A mod m; where i 2 A: 5 From now on, we will say that a CA is permutive or additive if the local rule on which it is based is permutive or additive. Let g; g : A A; be any ....

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O. Martin, A. M. Odlyzko, and S. Wolfram, Algebraic Properties of Cellular Automata. Commun. Math. Phys. 93, 219-258, 1984.

....that oe b Delta is continuous. We refer to [Fa, p. 121] for further details on this point. We conclude with the following lemma which is both basic and general; in fact it holds in a context which is more general than where we need it. Such more general contexts occur, e.g. in [St4] Mat] [MOW], and [Od] among other places) But we will still restrict the setting presently to where it is needed below for our proof of Theorem 6.1. Lemma 5.4. Let (R; B) be an affine system in R n (see details in Section 2) with R expansive and B ae R n a finite subset. Let B be given by (5.7) and ....

....dual representation fT g, cf. 6.6) are present for iteration systems which are HARMONIC ANALYSIS OF FRACTAL MEASURES 31 much more general than the affine fractals studied here. As a case in point we mention Matsumoto s [Mat] recent analysis of (von Neumann type) cellular automata (details in [MOW] and [Od] it is based on an S representation which is given by a formula similar to our (6.4) above. There is also an associated endomorphism with an entropy that can be computed; but we stress that for these (and many other) iteration systems, there is typically not a dualitly based on ....

O. Martin, A. M. Odlyzko, and S. Wolfram, Algebraic properties of cellular automata, Commun. Math. Phys. 93 (1984), 219--258.

....by the emergence ratio. For example, it has been demonstrated that in certain highly symmetric classes of one dimensional cellular automata, the single cell at the bottom of a light cone after n time steps can be predicted more quickly than the n(n Gamma 1) 2 steps of a simulation: Linear CA(Martin et al. 1984) have u n ) 2=n; i 1. Quasi linear CA with radius 1 2 (Moore 1993) have u n log 3 log 2 ) 2=n 0:41 ; i 0:41. The proof of the latter result is particularly informative in the direct manner in which it exploits the symmetry of, for example, the quaternion group. 4.1 Research ....

Martin, O., A. Odlyzko, and S. Wolfram (1984). Algebraic properties of cellular automata. Commun. Math. Phys. 93, 219--258.

....defined on N cells rings, the configuration of which are given, at time t, by c (t) x) P N Gamma1 j=0 a (t) j x j with a (t) j = a (t Gamma1) j Gamma1 a (t Gamma1) j 1 and a (t) j 2 Z=2Z. 14 scribed via some polynomials canonically associated to the global functions [48], as it is the case for the examples of Figure 6. Incidentally, let us note here that additive cellular automata have been much studied : their arithmetic nature allows to think them easier to understand although they may have arbitrary complex behaviors (along with [48] see [4] 43] 71] ....

....to the global functions [48] as it is the case for the examples of Figure 6. Incidentally, let us note here that additive cellular automata have been much studied : their arithmetic nature allows to think them easier to understand although they may have arbitrary complex behaviors (along with [48], see [4] 43] 71] 2] 3] and relevant references in these papers) Some attempts to generalize linear cellular automata are set about, with bilinear ones for example [6] 7] The above mentioned difficulty leads to pay attention to the canonical topological structure (evoked in 2:4:3: of ....

Martin O., Odlyzko A. and Wolfram S. Algebraic properties of cellular automata. Comm. Math. Phys., Vol. no. 93: 219--258, 1984.

....automata in finite geometries has been largely limited to numerical work. However, some analytical results concerning the set of temporal cycles have been obtained in a sequence of papers by Jen [23, 24, 25] for some large classes of rules, and for various rules with special characteristics e.g. [29]. In addition, McIntosh [30] has collected a variety of mathematical techniques, which serve, for example, to count one time step preimages of small configurations. Published numerical work on state transition graphs mostly concentrates on lattices small enough that complete enumeration of all ....

O. Martin, A. Odlyzko, and S. Wolfram. Algebraic properties of cellular automata, Commun. Math. Phys., 93:219, (1984)

....jointly periodic points are dense in Sigma A . 2 Proposition 3.2 is a special case of a theorem in [KS] which states that the periodic points are dense in all transitive, d dimensional Markov subgroups. For certain algebraic maps , the periods of points of a given oe period are analyzed in [MOW]. These periods can be very different. 4. Closing maps The following result is a pillar of our proof. The essence of this result is due independently to Kurka [Ku] and Nasu [Na2] We include an exposition in the last section of the paper. Lemma 4.1. BFF] Suppose is a positively expansive ....

O. Martin, A. Odlyzko and S. Wolfram, Algebraic properties of cellular automata, Comm. Math. Phys. 93 (1984), 219-258.

....elementary device, a cell or site, can take only two values and is updated at intervals according to a rule which expresses the actual value from its preceding value and that of its neighbours. Here the values are elements of a field K, finite or infinite. As O. Martin, A. Odlyzko and S. Wolfram [7] emphasized, the behaviour of a cellular automaton with a finite number of cells on a finite field is ultimately periodic. It is natural to consider also automata with cells in a line, which we call one dimensional automata. So a cell is indexed by an integer n 2 Z. At each time all the cells but ....

Olivier Martin, Andrew M. Odlyzko, and Stephen Wolfram. Algebraic properties of cellular automata. Commun. Math. Phys., 93:219--258, 1984.

....t) space. More generally, if a CA is of the form OE(a 0 ; a 1 ; a 2r ) f 0 a 0 f 1 a 1 Delta Delta Delta f 2r a 2r h (2) where the f i are matrix valued homomorphisms of an Abelian group (A; and h is a constant element of A, we can represent the CA rule as a polynomial [20] G(x) f 0 f 1 x f 2 x 2 Delta Delta Delta f 2r x 2r plus the constant h. Then we can think of the coefficients of G t (x) as the t th row of a Green s function for the CA, and for each t the final state is s = t X i=0 G t i a i h t Gamma1 X t 0 =0 t 0 X i=0 G ....

O. Martin, A.M. Odlyzko, and S. Wolfram, "Algebraic properties of cellular automata." Communications in Mathematical Physics 93 (1984) 219--258.

....generalizations, of B trees, and used as data structures in situations where it is desirable to be able to insert and delete records in time that is logarithmic in the total number of records present. Let a n denote the number of 2,3 trees of size (i.e. number of leaves) equal to n. In the paper [13], I showed that a n n 1 q n u(logn) as n , 4.1) where q = 1 5 1 2 ) 2 is the golden ratio, and u(x) is a positive nonconstant continuous function which satisfies u(x) u(x log(u q) for all real x. The proof contains no trace of the use of a symbolic manipulation program. ....

O. Martin, A. M. Odlyzko, and S. Wolfson, Algebraic properties of cellular automata, Commun. Math. Physics 93 (1984), 219-258.

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Olivier Martin, Andrew M. Odlyzko, and Stephen Wolfram. Algebraic properties of cellular automata. Commun. Math. Phys., 93:219--258, 1984.

No context found.

O. Martin, A.M. Odlyzko, and S. Wolfram, "Algebraic Properties of Cellular Automata." Communications in Mathematical Physics 93 (1984) 219258.

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O. Martin, A. Odlyzko, and S. Wolfram. Algebraic properties of cellular automata. Commun. Math. Phys., 93:219, 1984.

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