Moore, C.: Quasi-linear cellular automata. Physica D 103 (1997) 100--132  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... both of these structures from a general algebra point of view, we have varieties with signatures (3; 3) and (2; 2) One of the most important features of Pedersen s note was that standard algebraic terminology can be used to discuss cellular automata, and this approach has met with some success [4, 5, 14, 15]. For the remainder of this paper I want to look at the two constructions here and to show that although the two are equivalent, that the radius 1=2 is vastly more amenable to analysis. This amenability is by no means restricted to an algebraic viewpoint, for instance in [8] D. Hillman uses ....

Christopher Moore. Quasi--linear cellular automata. Physica D, in press.

....together k = 2r sites, we can transform any CA into one with r = 1=2. 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 = ....

.... transformation create a more linear CA, with simpler scaling behavior or a principle of superposition, presumably by removing irrelevant microscopic details and revealing deeper, more macroscopic behavior In particular, can block transformations move CA rules into classes which were shown in [1, 2] to be easily predictable We will show that for a large class of algebraic systems, they cannot. More precisely, if a blocked form of a CA is a algebra in which a certain identity holds, then the original, unblocked rule must consist of a similar algebra on its leftmost and rightmost inputs, and ....

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C. Moore, "Quasi-linear Cellular Automata." Submitted to Physica D. 9

....below an initial row becomes 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 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 ....

....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; Delta) is a function from A Theta A to A, written a Delta b or simply ab. The order of an algebra is the number of elements in it. The direct product A Theta B of two algebras is the ....

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C. Moore, "Quasi-Linear Cellular Automata." Submitted to Physica D.

....cellular automaton for a polynomial number of steps corresponds to a special case of Circuit Value where the circuit has a periodic structure. Thus these results will also help us tell when there are fast algorithms for predicting cellular automata whose rules correspond to certain algebras, as in [17, 18]. 2 Algebraic preliminaries We will use the following terms. For the theory of quasigroups and loops, we recommend [20] as an introduction. An algebra (also called a groupoid or magma) G; Delta) is a binary operation f : G Theta G G, written f(a; b) a Delta b or simply ab. The order of ....

C. Moore, "Quasi-linear cellular automata." Santa Fe Institute Working Paper 95-09-078, to appear in Physica D, Proceedings of the International Workshop on Lattice Dynamics.

....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 hand, some systems such as lattice gases [7] sandpiles [8] and zero temperature Ising dynamics [9] are P complete to predict, meaning that unless P = NC (in which case all polynomialtime problems are eciently parallelizable) there is no way around simulating them step by step. ....

C. Moore, \Quasi-linear cellular automata." Physica D 103 (1997) 100-132.

....cellular automaton for a polynomial number of steps corresponds to a special case of Circuit Value where the circuit has a periodic structure. Thus these results will also help us tell when there are fast algorithms for predicting cellular automata whose rules correspond to certain algebras, as in [16, 17]. 2 Algebraic preliminaries We will use the following terms. For the theory of quasigroups and loops, we recommend [19] as an introduction. An algebra (also called a groupoid or magma) G; Delta) is a binary operation f : G Theta G G, written f(a; b) a Delta b or simply ab. The order of ....

C. Moore, "Quasi-linear cellular automata." Santa Fe Institute Working Paper 95-09-078, to appear in Physica D, Proceedings of the International Workshop on Lattice Dynamics.

....automaton for a polynomial number of steps corresponds to a special case of Circuit Value where the circuit has a periodic structure. Thus these results will also help us tell when there are fast algorithms for predicting cellular automata whose rules correspond to certain groupoids, as in [22, 23]. The paper is structured as follows. In Section 2 we give an introduction to the algebraic terms and concepts we will use. In Section 3 we de ne Booleancompleteness, the ability to express arbitrary Boolean functions as circuits or expressions. We review existing results on solvability in groups ....

C. Moore, \Quasi-linear Cellular Automata." In Proceedings of the International Workshop on Lattice Dynamics, Physica D 103 (1997) 100-132.

....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 ....

.... Call linear with respect to (some authors prefer additive ) if space time diagrams of s CA can be combined with : a b a b c d c d = a c b d (a c) b d) or in other words (a c) b d) a b) c d) 3) Such principles of superposition are studied in [7]. Equation (3) is a kind of generalized medial identity [4] it is also the interchange rule of horizontal and vertical composition of natural transformations in category theory [6] a fact that may or may not have anything to do with the price of beans. Then we have Theorem 17. If is an ....

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C. Moore, \Quasi-linear cellular automata." Physica D 103 (1997) 100-132, Proceedings of the International Workshop on Lattice Dynamics.

....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 polynomial speedup predicting physical time t in O(t ) parallel time for some 1. For instance, in Ref. 32] it was shown that though ordinary DLA is P complete, on average it can be parallelized to ....

C. Moore, \Quasi-linear cellular automata." Physica D 103 (1997) 100-132, Proceedings of the International Workshop on Lattice Dynamics.

....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) in parallel. However, in [21] we show that CAs whose rules satisfy various algebraic identities can be predicted in parallel time O(log k t) for some k, qualitatively faster than explicit simulation. We term these CAs quasi linear: they are nonlinear, but efficiently predictable nonetheless. It would be highly surprising if ....

....any q, and in particular by ACC[p] gates where p = lcm(p 1 ; p 2 ; p k ) Alternately, a tree of binary gates of depth O(log n) can add the elements in pairs. Now if a CA s algebra is an Abelian group OE(a; b) a b, a simple Green s function method with Pascal s triangle coefficients [21, 31] allows us to predict the CA in O(t) serial time or O(log t) parallel time, i.e. in NC 1 . In fact, this algorithm is in LOGSPACE uniform ACC 0 since we can generate the t th row of Pascal s triangle using O(log t) space. More generally, if a CA is of the form OE(a 0 ; a 1 ; a 2r ) ....

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C. Moore, "Quasi-linear cellular automata." Physica D 103 (1997) 100-132, Proceedings of the International Workshop on Lattice Dynamics.

....in a light cone of depth t ending in that site. This takes time O(t d 1 ) on a serial computer (proportional to the volume of the light cone in d dimensions) or just O(t) if done in parallel. However, for classes of CAs that obey certain algebraic identities, we can do much better than this [5, 6]. These quasi linear systems can be predicted on a parallel computer in time O(log t) or O(log 2 t) much faster than by explicit simulation. This places them in the complexity class NC, the class of problems that can be solved for inputs of size n in time O(log k n) for some k, on a ....

C. Moore, "Quasi-linear cellular automata." Santa Fe Institute Working Paper 95-09-078, to appear in Physica bf D.

....tree is already known could include uniform families of expressions, one of each length. For instance, the problem of predicting a cellular automaton amounts to evaluating a uniform family of circuits, one of each size. This uniformity can significantly simplify the Circuit Value problem, as in [20, 21]. 9 Conclusion and directions for further work We have shown that the very precise relationship between solvability and circuit complexity that holds for associative algebras, has an elegant counterpart in the non associative case, provided that we refine appropriately the notion of algebraic ....

C. Moore, "Quasi-linear cellular automata." Physica D 103 (1997) 100-132, Proceedings of the International Workshop on Lattice Dynamics.

....together k = 2r sites, we can transform any CA into one with r 0 = 1=2. 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 ....

....binary operation, the block algebra of the original CA. We would like to know, then, to what extent block transformations can simplify the analysis of CAs. Can non linear CA rules on several site neighborhoods be equivalent to binary algebras with nice algebraic properties, such as those shown in [4, 5] to allow efficient prediction of the CA We will show that, under many circumstances, they cannot. More precisely, if a CA s block algebra is in one of four large classes of algebras (which include groups, monoids, common types of non associative algebras, and commutative algebras in general) ....

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C. Moore, "Quasi-linear cellular automata." Physica D 103 (1997) 100--132, Proceedings of the International Workshop on Lattice Dynamics.

....order and cannot be efficiently parallelized. Unless NC = P (which would be almost as surprising to computer scientists as if P = NP) then, P complete problems cannot be solved in polylogarithmic parallel time. Some non linear CA s with certain algebraic properties can be predicted in NC [9, 10], as can the Lorentz lattice gas of one particle bouncing off of fixed scatterers [13] But CA Prediction is P complete in general since CA s exist that can perform universal computation [5, 7] Other cellular automata and lattice systems that have been shown to be P complete include ....

C. Moore, "Quasi-linear cellular automata." Physica D 103 (1997) 100-132, Proceedings of the International Workshop on Lattice Dynamics.

.... as NP complete problems are believed to require a super polynomial amount of search, P complete problems are believed to be inherently sequential, so that the work needs to be done in step by step order and cannot be efficiently parallelized [6] Non linear CA s with certain algebraic properties [11, 12] can be predicted in O(log t) or O(log 2 t) placing them in the parallel complexity class NC of efficiently parallelizable problems. But CA Prediction is P complete in general, since CA s exist that can perform universal computation [4, 9] A number of other cellular automata and lattice ....

C. Moore, "Quasi-linear cellular automata." Physica D 103 (1997) 100-132, Proceedings of the International Workshop on Lattice Dynamics.

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Moore, C.: Quasi-linear cellular automata. Physica D 103 (1997) 100--132

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