C. Moore, "Non-Abelian Cellular Automata." Submitted to Physica D.  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. Non--abelian cellular automata. Technical Report 9509 -081, Sante Fe Institute, 1995.

....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 ....

C. Moore, "Non-Abelian Cellular Automata." Submitted to Physica D.

....of its two inputs. Then depending on the algebraic properties of (A; Delta) such as associativity, commutativity, solvability and so on, Expression Evaluation and Circuit Value can have varying complexities. Previous results for the associative case (groups and semigroups) include the following [2, 4, 17]: non solvable NC solvable ACC In this paper, we will extend these results to non associative algebras such as quasigroups and loops, and to some extent to algebras in general. We will show that the idea of solvability generalizes in two important ways in the nonassociative case: ....

....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 ....

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C. Moore, "Non-Abelian cellular automata." Santa Fe Institute Working Paper 9509 -81, or "Predicting non-linear cellular automata quickly by decomposing them into linear ones," submitted to Physica D.

....of its two inputs. Then depending on the algebraic properties of (A; Delta) such as associativity, commutativity, solvability and so on, Expression Evaluation and Circuit Value can have varying complexities. Previous results for the associative case (groups and semigroups) include the following [2, 4, 16]: Expression Evaluation Circuit Value non solvable NC 1 complete P complete solvable ACC 0 ACC 1 DET In this paper, we will extend these results to non associative algebras such as quasigroups and loops, and to some extent to algebras in general. We will show that the idea of ....

....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 ....

[Article contains additional citation context not shown here]

C. Moore, "Non-Abelian cellular automata." Santa Fe Institute Working Paper 9509 -81, or "Predicting non-linear cellular automata quickly by decomposing them into linear ones," submitted to Physica D.

....shown to be P complete [22, 25, 19] Therefore, it seems worthwhile to extend the class of quasi linear CAs as far as possible, to explore what is probably a very rich hierarchy between linear dynamical systems and computationally universal ones. A preliminary version of these results appeared in [23]. 2 Definitions 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 A. We will concern ourselves here with algebras of finite order. The direct product A Theta B of two algebras is the set of pairs (a; ....

C. Moore, "Non-Abelian cellular automata." Santa Fe Institute Working Paper 95-09-081.

....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, "Non-Abelian cellular automata." Santa Fe Institute Working Paper 95-09-081.

....properties of (A; Delta) such as associativity, commutativity, solvability and so on, affect the computational complexity of the problems Expression Evaluation and Circuit Value. Our work is the direct continuation of a series of results dealing with the associative case, including for example [2, 4, 6, 20]: the following table summarizes what is known in that setting. Expression Evaluation Circuit Value non solvable NC 1 complete P complete solvable ACC 0 ACC 1 DET We assume that the reader is familiar with standard complexity classes; in case of emergency, one can consult [15] The ....

....reducible to their algebraic counterparts, since a local transformation can replace and, or and not gates with complexes of algebraic gates. ut In [11] see also [17] it is shown that any non solvable groupoid is Boolean complete. Since the circuit value problem for solvable semigroups is in NC [6, 20], non solvability is both necessary and sufficient for Boolean completeness in the case of semigroups (assuming NC ( P) However, a loop can be solvable and still be Boolean complete. Let (G; Delta) be Delta 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 2 2 3 4 1 6 7 8 5 3 3 4 1 2 7 8 5 6 4 4 1 2 3 8 5 6 7 ....

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C. Moore, "Non-Abelian cellular automata." Santa Fe Institute Working Paper 95-09-81.

....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, "Non-Abelian cellular automata," Santa Fe Institute Working Paper 9509 -081, or "Predicting non-linear cellular automata quickly by decomposing them into linear ones", available as patt-sol/9701008. Submitted to Physica D.

.... 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, "Non-Abelian cellular automata," Santa Fe Institute Working Paper 9509 -081, or "Predicting non-linear cellular automata quickly by decomposing them into linear ones", submitted to Physica D.

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