Wolfram, S.: Computation Theory of Cellular Automata, Comm. Math. Physics, 96(1), 1984, 15--57.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from the fact that cellular automata are computationally universal and can be used to simulate, say, universal Turing machines. Thus it is not surprising that any general classi cation of cellular automata meets with considerable diculties. The heuristic classi cation proposed by Wolfram in [11,14] attempts to distinguish between cellular automata on the basis of the structure of typical orbits of con gurations under long term evolution. Wolfram s classi cation is based on extensive simulations of onedimensional cellular automata and employs clearly discernible visual patterns. Brie y, ....

S. Wolfram. Computation theory of cellular automata. Comm. Math. Physics, 96(1):15-57, 1984.

....accept proper subsets of the acceptance language of the whole machine. This explains an Preprint submitted to Elsevier Preprint 10 December 1996 observation by Wolfram that the minimal automaton associated with a linear cellular automaton may contain additional transient subgraphs, see [24]. It was shown by Beauqier [2] that the minimal Fischer automaton is also minimal in the sense of homomorphisms: there is a homomorphism from any Fischer automaton for a fixed transitive factorial language to the minimal Fischer automaton for that language. This work is motivated in part by a ....

....it appears that for most Wolfram class I and II automata, ae t ) increases polynomially, but for class III and IV , complexities increase exponentially. Here (ae t ) denotes the size of the minimal automaton for L(ae t ) A table of (ae t ) for elementary cellular automata can be found in [24]. It is noted in the reference that (ae t ) usually stays far below the upper bound. To see what this upper bound is, note that there is a transitive semiautomaton B(ae) that accepts L(ae) whose underlying diagram is a de Bruijn graph. This de Bruijn automaton has k w Gamma1 states and k ....

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S. Wolfram. Computation theory of cellular automata. Comm. Math. Physics, 96(1):15--57, 1984.

....applications by Margolus[18, 34] who introduced the Margolus neighborhood [33] and coined the term partitioning cellular automata . Meanwhile, Wolfram had been investigating mainly one dimensional cellular automata in connection with statistical mechanics[45] and computational linguistics[44]. The Information Mechanics group at MIT (chie y Edward Fredkin, Tommaso To oli, Norman Margolus, and, for a while, G erard Vichniac and Charles Bennett) developed both high performance cellular automata machines (cam6[32] and cam8[20] and a vast pool of expertise in programming all sorts of ....

Wolfram, Stephen, \Computation Theory of Cellular Automata," Commun. Math. Phys. 96 (1984), 15-57.

....results to higher dimensions would most likely require a different approach. Since then, for almost twenty years a quest for these generalizations to more than one dimension went on with little success. Invertibility and related properties for the one dimensional case were revisited in [54,87,14,29]. Many equivalent characterizations of ica were given[90,47,48,35] but none that offered a finitary handle on invertibility. Finally, quite recently, Kari proved that Theorem 4.5 (Kari[38,39] There is no effective procedure for deciding whether or not an arbitrary two dimensional cellular ....

Wolfram, Stephen, "Computation Theory of Cellular Automata," Commun. Math. Phys. 96 (1984), 1557.

....is nite, their interactions are synchronous and occur at discrete time steps. Notwithstanding the apparent simplicity of the formal de nition of CA, they display a wide range of interesting dynamical behaviors. And in fact, the problem of their classi cation is a central topic in CA theory. In [23], Wolfram heuristically observes the following behaviors: W 1 : evolution to a homogeneous state; W 2 : evolution to a set of space time patterns which are stable or periodic; W 3 : evolution to an aperiodic or chaotic space time pattern; W 4 : evolution to complex localized structures, ....

....c k ( 1) c k (0) c k (1) c k (2) F k f (c) The study of space time diagrams give some hints on the global qualitative behavior of the CA. Most of the early works on CA classi cation follow this idea [23]. In Remark 1 we saw that surjectivity is necessary for Devaney chaos. Therefore when comparing chaotic CA and random evolutions we are concerned with a subclass of surjective CA. In order to prove our results we need to reformulate surjectivity as a property on the local rule: balance. A CA with ....

S. Wolfram. Computation theory of cellular automata. Comm. in Math. Phys., 96:15-57, 1984. 17

.... a complicated scale invariant structure, and belong to an intermediate class called indexed context free [9] and the iteration of smooth maps in the plane can correspond to universal Turing machines [43] Similarly, the image of a cellular automaton after a nite number of timesteps is regular [60], as is the set of xed points; but limit sets can be contextfree, context sensitive, or the complement of the halting set of a Turing machine [24] The purpose of this paper is to introduce the reader to an analogous hierarchy of two dimensional languages or patterns of symbols. This hierarchy ....

....one can describe sets of con gurations such as periodic sets, nite time sets, and limit sets in terms of their languages of allowed nite blocks. For example, it was shown by Wolfram that any nite time set (set of images allowed at a certain time) of a 1 d CA is described by a regular language [60]. In this Section, we apply some of the results of Section 2 to the dynamics of CA s in two or more dimensions. In particular, we show that the appropriate generalizations of regular languages for xed and periodic points on the one hand, and nite time sets on the other, are LLL s and h(LLL) s ....

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S. Wolfram, \Computation theory of cellular automata." Communications in Mathematical Physics 96 (1984) 15-57. 40

....a high growth dimension (close to the maximum) and short transients, while Class 4 CA tend to have moder11 ate entropy, relatively high mutual information, a moderate growth dimension, and arbitrarily long transients. Researchers have also studied CA from a theoretical perspective. In [34] Wolfram attempted to describe cellular automata behavior in terms of computation theory, viewing cellular automata as computers whose time evolution processes the information specified by their initial configurations. CA have been studied as language recognizers [20] and as continuous functions ....

S. Wolfram. Computation theory of cellular automata. Communications in Mathematical Physics, 96:15--57, 1984.

....interactions are synchronous and occurs at discrete time steps. Notwithstanding the apparent simplicity of the formal definition of CA, a great variety of dynamical behaviors can be observed as in natural systems. This fact makes the problem of classification a central topic in CA theory. In [14], Wolfram heuristically observes the following behaviors: W 1 . evolution to a homogeneous state; 1 W 2 . evolution to a set of space time patterns which are stable or periodic; W 3 . evolution to an aperiodic or chaotic space time pattern; W 4 . evolution to complex localized structures, ....

S. Wolfram. Computation theory of cellular automata. Comm. in Math. Phys., 96:15--57, 1984.

....(Equation 1 means that the transition rule OE is applied to each configuration in Sigma. Let Omega = Phi Sigma be the set of configurations generated after iterations of Phi. For cellular automata transition rules which generate complex non periodic attractors, the following holds [14]: Omega 1 = Phi Omega Omega (2) i.e. the application of the rule Phi to all configurations of the set Omega at iteration results in a set Omega 1 at iteration 1 where the number of configurations is lower or equal to the set Omega . This evolution towards a lower number of ....

....the cellular automaton: Proposition 1: A sub configuration of length l is excluded after iterations of a rule OE if there is no sub configuration from a one dimensional cellular automaton initial configuration of length (l 2) Theta which evolves toward the subconfiguration of length l. see [14] for details) IV. EXPERIMENTAL RESULTS Each of these five parallel procedures has been run on twelve instances of the location allocation combinatorial optimization problem as introduced in [3] Each problem instance is searched using 4, 8 and 16 sequential tabu search programs which are ....

S. Wolfram. Computation Theory of Cellular Automata. Comm. Math. Phys, 96:15--57, 1984.

....solved in terms of differential equations. Cellular automata (CA) are very effective in solving scientific problems because they can capture the essential features of systems in which the global behaviour arises from the collective effect of large numbers of locally interacting simple components [2]. A cellular automaton is a discrete dynamic system composed of a set of cells in a one dimensional or multi dimensional lattice. Each cell in the regular spatial lattice can have any one of a finite number of states. The states of the cells in the lattice are updated according to a local rule ....

S. Wolfram, "Computation Theory of Cellular Automata", Comm. Math. Phys. Vol. 96, 1984, pp. 15-57.

....Computational mechanics is a synthesis of nonlinear dynamics and computation theory, which characterizes patterns and structure occurring in natural processes by means of formal models of computation. Connections between CA and computation theory have been an active area of research for some time. [6 8] One major theme of this research has centered around the problem of designing a CA to behave in some particular way, such as simulating a universal Turing machine, 9] exhibiting particles and computing with them, 10, 11] performing reliable computations in the presence of noise, 12] or a host of ....

S. Wolfram. Computation theory of cellular automata. Comm. Math. Phys., 96:15, 1984.

....of all possible initial conditions up to some time t. It necessarily captures all of the information processing accessible to the CA, which includes correlations and information transmission, as well as computation. Previous investigations of CA patterns have focused on the global machine M t . [18, 19] In terms of dynamical systems theory, though, the global machine describes the entire attractor basin portrait: the collection of all invariant sets and transients, including attractors and separatrices. 1] In many cases, this description is seen to be prohibitively difficult to construct ....

....descriptions are rarely pursued in dynamical systems theory. Even for elementary CA, there are many cases in which the size of M t increases extremely rapidly 17 in time. This is true in particular of rule 18: M( Omega 3 ) has kVk = 143 states; M( Omega 4 ) is estimated to have kVk 20; 000. [18] Furthermore, there are other CA rules for which the growth is even faster. 19] An alternative approach is to identify the structures that dominate and organize the spatial system s behavior. The most basic of these for dissipative systems are attractors, basins, and separatrices. 1] The state ....

S. Wolfram, "Computation theory of cellular automata," Comm. Math. Phys., vol. 96, p. 15, 1984.

....had already been studied in an abstract mathematical context by Hedlund and associates as early as 1963[30, 31] both Richardson s results on invertibility (x4.3) and Patt s search for ica (x5.3) had been anticipated by Hedlund s school. Wolfram s 1983 86 sortie into the cellular automata arena[85, 86, 87, 88, 89], stimulated by that workshop, was in turn a determining factor in introducing a generation of mathematical physicists to the cellular automaton paradigm. Inspired by Fredkin s billiard ball model of computation[19] Margolus arrived in 1983 at a very simple computation universal ica[41] that is ....

....their results to higher dimensions would most likely require a different approach. Since then, for almost twenty years a quest for these generalizations to more than one dimension went on with little success. Invertibility and related properties for the onedimensional case were revisited in [54, 87, 14, 29]. Many equivalent characterizations of ica were given[90, 47, 48, 35] but none that offered a finitary handle on invertibility. Finally, quite recently, Kari proved that Theorem 4.5 (Kari[38, 39] There is no effective procedure for deciding whether or not an arbitrary twodimensional cellular ....

Wolfram, Stephen, "Computation Theory of Cellular Automata, " Commun. Math. Phys. 96 (1984), 15-57.

....As mentioned before, the states of the cells in the lattice are updated according to a local rule called state transition function. That is, the state of a cell at a given time depends only on its own state in the previous time step and the states of its nearby neighbors at the previous time step [7]. All cells of the automaton are updated synchronously in parallel. Thus the state of the entire automaton advances in discrete time steps. Therefore, the transition function plays in cellular automata a role analogous to that of the evolution equation in classical dynamical models. Therefore, a ....

S. Wolfram, Computation Theory of Cellular Automata, Comm. Math. Phys. 96, p. 15 (1984).

....because of the non Markovian behavior of the search process. Even for the case of simpler discrete dynamical systems like the cellular automata , there are indications that their behavior for some classes may in general be determined by no procedure significantly faster than explicit simulation [47]. As a first step in the study, it can be useful to consider a simplified RTS system, in which some free parameters are fixed and the dynamical system driving the search has simple statistical properties. In particular, the dynamics has a task dependent part that changes with the chosen ....

S. Wolfram, "Computation Theory of Cellular Automata," Commun. Math. Phys., vol. 96, pp. 15--57, 1984.

....of the array for a number of consecutive iterations or a limit on the number of iterations has been reached. As is the case in training as well, the second condition is used as a sentinel in order to avoid infinite loops, where the system could possibly exhibit chaotic or periodic behaviour [26]. These kinds of behaviour are discussed with more details in section 3.3 and they are more or less expected since not only the structure is inherited from Cellular Automata. Relaxation The method to decide when a match has been found with the current inputs remains the same as in the learning ....

....[ B D) E] A, E] A, B D) is also checked using the arity two CMM. This procedure is repeated until a match is found or the arity cannot be further decreased. 3.3 Analysis and Testing One of the conclusions which can be derived from S. Walfram s pioneering studies on Cellular Automata [26, 11] is that although their general behaviour can be divided into a distinct number of categories, at the majority of the cases the consequences of their evolution cannot be predicted and can only be revealed by direct observation and simulation. Due to the fact that CANNs are, essentially, dynamic ....

Wolfram S. Computation theory of cellular automata. Communications in Mathematical Physics, 96:15--57, 1984.

....discussion) 4 Future Research . Study the injectivity and surjectivity of GCAs. It is expected that the methods developed for CAs can also be used to give answers for one dimensional GCAs ( 1] 6] Use regular expressions to describe configurations. Apply the methods, found in Wolfram [7] to the class of GCAs. What is the connection between the time and the space evolution in GCAs . Study the connection between CAs and GCAs. Is is possible, given an arbitrary CA, to find a representation as GCA, which is the most natural or most simple one . Is the non locality of GCAs ....

Stephen Wolfram. Computation theory of cellular automata. Communications in Mathematical Physics, 96:15--57, 1984.

....evolution for one time step according to Eq. 5.2) yields a set entropy (5.9) s(t = 1) f log 2 k f 0:88: The irregularity of the transient trees illustrated in Fig. 5 implies a measure entropy s m s. The result (5.9) becomes exact in the limit N . A direct derivation in this limit is given in [17, 18], where it is also shown that the set of infinite configurations generated forms a regular formal language. The set continues to contract with time, so that the set entropy decreases below the value given by Eq. 5.9) 18] Techniques similar to those used in the derivation of Eq. 5.5) may in ....

....exact in the limit N . A direct derivation in this limit is given in [17, 18] where it is also shown that the set of infinite configurations generated forms a regular formal language. The set continues to contract with time, so that the set entropy decreases below the value given by Eq. 5. 9) [18]. Techniques similar to those used in the derivation of Eq. 5.5) may in principle be used to deduce the number of configurations reached after any given number of steps in the evolution of the cellular automaton (5.2) The fraction of configurations L a T E X filename: Algebraic.tex (Paper: 1.2 ....

Wolfram, S.: Computation theory of cellular automata. Institute for Advanced Study preprint (January 1984).

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Wolfram, S.: Computation Theory of Cellular Automata, Comm. Math. Physics, 96(1), 1984, 15--57.

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S. Wolfram. Computation theory of cellular automata. Communications in Mathematical Physics, 96:15--57, 1984.

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Wolfram, S.: Computation theory of cellular automata. Institute for Advanced Study preprint (January 1984).

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S. Wolfram. Computation theory of cellular automata. Comm. Math. Physics, 96(1):15-57, 1984.

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S. Wolfram. Computation Theory of Cellular Automata. Comm. Math. Phys, 96:15--57, 1984. 21

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S. Wolfram, Computation theory of cellular automata. World Scientific, 1987.

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-644 #1984#. #10# Stephen Wolfram, #Computation theory of cellular automata," Communications in Mathematical Physics 96

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