Lindgren, C., Nordahl, M.: Universal Computation in Simple One Dimensional Cellular Automata, Complex Systems, 4, 1990, 299--318.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a simulation may still be possible in more than one dimension. 7 3. What is the minimum number of terms in our two constructions needed to simulate a universal Turing machine The gures given above (24 in two dimensions and p 19 130 = 2470 in one) can probably be reduced by noting as in [7] that not all combinations of state and symbol actually occur in the course of a properly initialized computation. Acknowledgements: C.M. is grateful to Mats Nordahl and Seth Lloyd for helpful discussions, to an anonymous referee for pointing out that the functions in [10] are Lipschitz rather ....

K. Lindgren and M.G. Nordahl, \Universal computation in simple onedimensional cellular automata." Complex Systems 4 (1990) 299-318.

....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 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 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 in ....

K. Lindgren and M.G. Nordahl, "Universal Computation in Simple OneDimensional Cellular Automata." Complex Systems 4 (1990) 299-318.

....in the sense that one of them is able to simulate sonhe universal Turing machine. The question remains 14 to know how to encode the Turing machines in order to get some such one dimensional cellular automaton with a minimal number of states. This num ber is known to lay between 2 and 7 (see [40]) Open Problem 18 : Optimal universality. What is the minimal number of states for a universal 1D CA Open Problem 19 : Background optimal universality. Given a universal 1D CA, can it be always viewed as obtained by background from sonhe simple one Comment This question is connected to ....

K. Lindgren and M. Nodahl, Universal computation in simple one dimensional cellular automata. Complex Systems, vol. 4, 299-318, 1990. 32

....In the fifties, inspired by biological systems, von Neumann decided to study the least complex discrete system capable of self reproduction. The idea of cellular automata was suggested by Ulam as an abstract setting for this problem and many other totally discrete models. A cellular automaton [76, 46] is a discrete space discrete time discrete value dynamic system with only local interactions. It is a collection of many simple finite state machines, the next state of each of which depends only on the states of the machines in some small neighborhood around it. All states are updated ....

Kristian Lindgren and Mats G. Nordahl. Universal Computation in Simple One- Dimensional Cellular Automata. Complex Systems, 4:299-318, 1990.

....systems based on whether the problem of predicting them is in P or NC. We can always predict a cellular automaton in polynomial time by simulating it explicitly, just as we can numerically integrate a di erential equation. Moreover, since cellular automata can easily simulate Turing machines [4], this problem is P complete in general. However, if the cellular automaton has certain algebraic properties, we can predict it much more quickly, in O(log t) or O(log 2 t) time. Thus in special cases the CA prediction problem can be in NC, even for some nonlinear rules [5, 6] On the other ....

K. Lindgren and M.G. Nordahl, \Universal Computation in Simple One-Dimensional Cellular Automata. " Complex Systems 4 (1990) 299-318.

....mind the simple measure of complexity defined by the multiplication of the three parameters that appear in the table, it is clear that, similarly to what we have seen in the case of self reproduction ability, simpler cellular automata have been discovered over time. Noteworthy in this respect is [Lindgren and Nordahl 1990] which, implementing Minsky s [1967] Turing machine with 4 tape symbols and 7 internal states describes the simplest cellular automaton currently known that is capable of universal computation. It is also worth mentioning that due to the complexity of the automaton and the size of the ....

C. Lindgren and M. Nordahl. Universal computation in simple one dimensional cellular automata. Complex Systems, 4:299--318, 1990.

....appropriately designing the rule table, the BCA can be made to perform useful computation. In fact, BCA are computationally universal. A BCA with k 3s symbols can simulate in linear time the operation of a Turing Machine with k tape symbols and s head states the proof is analogous to that in Lindgren and Nordahl (1990). Thus we can conclude that a BCA can be used to answer any 3 The inspiration for this approach comes from the proof of the undecidability of the Tiling Problem (see Grunbaum and Shephard (1986) Chapter 11) 4 BCA (Wolfram 1994) are also known as partitioning CA (Margolus 1984) and as 2 body ....

K. Lindgren and M. Nordahl. Universal computation in simple one-dimensional cellular automata. Complex Systems, 4(3):299--318, 1990.

....McClelland, 1986) p. 119. Dynamical Recognizers 21 Cellular Automata, which we might view as a kind of low density, synchronous, uniform, digital restriction of neural networks, have been studied as dynamical systems (Wolfram, 1984) and proven to be as powerful as universal Turing Machines, e.g. (Lindgren Nordahl, 1990). Furthermore, Moore, 1990) has shown that there are simple mathematical models for dynamical systems which are also universal, and it follows directly that determination of the behavior of such dynamical systems in the limit is undecidable and unpredictable, even with precise initial conditions. ....

Lindgren, K. & Nordahl, M. G. (1990). Universal Computation in simple onedimensional cellular automata. Complex Systems, 4, 299-318.

.... Ising model [21, 27] sandpiles [26] FHP and HPP lattice gases [28] cellular automata with local voting rules [27] and simple deterministic growth models [11] Greenlaw et al. 10] have pointed out that predicting cellular automata is P complete in general, since cellular automata exist (e.g. [18]) which can simulate universal Turing machines. On the other hand, NC algorithms exist for Eden growth [20] the Lorentz lattice gas [31] and cellular automata with certain algebraic properties [29, 30] Even if a speedup to polylogarithmic time isn t possible, we might still hope for a ....

K. Lindgren and M.G. Nordahl, \Universal computation in simple one-dimensional cellular automata." Complex Systems 4 (1990) 299-318.

.... emergent cooperative or collective behavior in complex systems. For discussions of work in all these areas, see, e.g. 4, 26, 30, 37, 46, 59, 44, 55, 71, 82] 3. A Computational Task for Cellular Automata It has been shown that some CAs are capable of universal computation; see, e.g. [3, 50, 67]. The constructions either embed a universal Turing machine s tape states, read write head location, and nite state control in a CA s con gurations and rule or they design a CA rule, supporting propagating and interacting particles, that simulates a universal logic circuit. These constructions ....

K. Lindgren and M. G. Nordahl. Universal computation in a simple one-dimensional cellular automaton. Complex Systems, 4:299-318, 1990.

....(a) represents any neighborhood. Moreover the following result holds, which means that, from a computation point of view, all neighborhoods can be understood as equivalent, which is generally not the case when complexity or architecture are to be taken into account (see for example [16] 69] [46]) 4 Fact 1. Every d cellular automaton can be simulated by a d cellular automaton with the nearest neighbors neighborhood. Another neighborhood on 1 cellular automaton is used which consists of the closest right (left) cell of a given cell. It characterizes cellular automata called one way ....

....minimal number of states still goes on. But, often, less states seems to imply some compensation for example a bigger neighborhood (dimension 2, 2 states but a neighborhood with 85 cells in [16] or some background (dimension 2, 2 states, von Neumann neighborhood and a background [5] See also [46] which recall or prove significant results, and [30] which is recent in the race. A background for a 1 dimensional cellular automaton is a configuration c which is both time and space periodic. Time periodic means that (G t (c) t0 is periodic, and, if oe denotes the shift on configurations ....

Lindgren K. and Nordahl M. Universal computation in simple one dimensional cellular automata. Complex Systems Vol. no. 4: 299--318, 1990.

....automata, one may think that they have a very great number of states. This contradicts our idea that very simple cellular automata with few states (less than fifty) may have very complex behavior. This idea is comforted by the fact that there exists an universal cellular automaton with 8 states [12]. How to reduce the number of states of our constructed automata This optimization is very difficult as shown by the fact that it is not known if the FSSP has a 5 states solution (see 4) We do not know powerful methods. Another interesting point is that parallelism speeds up computations. On the ....

Lindgren K. and Nordahl M. Universal computation in simple one dimensional cellular automata. Complex Systems Vol. no. 4: 299--318, 1990.

.... Gamma2 u Gamma1 E Gamma1 u 0 p 0 u 1 E 1 u 2 E 2 Delta Delta Delta Figure 5: From Turing machine to cellular automaton configuration such that CA simulates the Turing machine in one step (i.e. m = 1) TM (u) CA m (OE(u) Equivalent results can be found in [28, 29, 13]. We present here a homomorphism between these two dynamical systems, allowing cellular automata to simulate any Turing machine. Let us first give the general construction. Take a Turing machine M defined as a dynamical system on Q Theta Gamma Z . We want to translate any state (p; c) 2 Q ....

K. Lindgren and M.G. Nordahl. Universal computation in simple onedimensional cellular automata. Complex Systems, 4:299--318, 1990.

....on the practicality of this approach. 1 TURING MACHINES 1.1 Models of computation A Turing machine [Tu] is a model of computation a way of representing and performing a given computation. Turing machines are mathematically equivalent to many other models of computation cellular automata [Lind], neural networks [Si] and digital computers [Ho] Because none of these models of computation (or any other we have found) is more powerful than the Turing machine model we believe that Turing machines embody what we mean when we say something is computable (Church s Thesis. That is, anything ....

Lindgren, K., Nordahl, M.: Universal Computation in Simple One-Dimensional Cellular Automata. Complex Systems 4 (1990) 299-318.

....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 other tasks. Computational mechanics, in contrast, approaches CA more from the perspective of physical science. Rather than trying to engineer a CA rule table and ....

K. Lindgren and M. Nordahl. Universal computation in simple one-dimensional cellular automata. Complex Systems, 4:299, 1990.

....By appropriately designing the rule table, the BCA can be made to perform useful computation. In fact, BCA are computationally universal. A BCA with k 3s symbols can simulate in linear time the operation of a Turing Machine with k tape symbols and s head states the proof is analogous to that in [Lindgren]. Thus we can conclude that a BCA can be used to answer any question which can be phrased in terms of a computer program. Small BCA have been designed which sort lists of integers, compute primes, and many other tasks. A few more comments are in order concerning the abstract model of blocked ....

Lindgren, K., and M. Nordahl. Universal Computation in Simple One-Dimensional Cellular Automata. Complex Systems 4 (3): 299--318, 1990.

....) then if we allow blocks of the form a b c f(a,b,c) the LLL with simulate the CA s evolution from row to row. In particular, the CA rule can simulate a Turing machine, where special 8 states correspond to the machine s head and internal states, while others correspond to its tape symbols [35]. Then its input appears along the top row, and we can use the LLL to require that it halts or doesn t halt before it reaches the bottom. Thus simple questions about 2 d LLL s can be equivalent to the Halting Problem; this will be our main source of undecidability. 3. Fixed and periodic points ....

....the machine will never halt, so the set of extensible blocks is the complement of the TM s halting set. To give a non recursive example, recall that space time diagrams of onedimensional cellular automata are LLL s in two dimensions. Choose a CA that simulates a universal Turing machine (e.g. [35]) and forbid any neighborhood containing the halt state. Then the set of extensible n 1 rows with the Turing machine properly initialized in the upper left corner are precisely those inputs on which the Turing machine will not halt; this is a non recursive set since the Halting Problem is ....

K. Lindgren and M.G. Nordahl, \Universal computation in simple onedimensional cellular automata." Complex Systems 4 (1990) 299-318.

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Lindgren, C., Nordahl, M.: Universal Computation in Simple One Dimensional Cellular Automata, Complex Systems, 4, 1990, 299--318.

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Lindgren, K., Nordahl, M.G.: Universal computation in simple one-dimensional cellular automata. Complex Systems 4 (1990) 299--318

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K. Lindgren and M.G. Nordahl, "Universal Computation in Simple OneDimensional Cellular Automata." Complex Systems 4 (1990) 299-318.

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K. Lindgren and M. G. Nordahl. Universal computation in a simple one-dimensional cellular automaton. Complex Systems, 4:299-318, 1990.

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K. Lindgren and M. G. Nordahl, "Universal Computation in Simple OneDimensional Cellular Automata," Complex Systems, 4 (1990), 299--318.

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Lindgren, Kristian, and Mats. G. Nordahl, "Universal computation in simple one-dimensional cellular automata ", Complex Systems 4 (1990), 299--318.

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K. Lindgren and M. G. Nordahl, #Universal computation in simple one-dimensional cellular automata," Complex Systems 4, 299 #1990#.

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K. Lindgren and M. G. Nordahl. Universal computation in a simple one-dimensional cellular automaton. Complex Systems, 4:299--318, 1990.

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