T. Boykett, "Combinatorial Construction of One-Dimensional Reversible Cellular Automata." Contributions to General Algebra 9 (1995).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 = 2r we can transform any CA into one with r ....

T. Boykett, "Combinatorial Construction of One-Dimensional Reversible Cellular Automata." Contributions to General Algebra 9 (1995).

....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 divides 2r) Then if k = 2r we get r 0 = ....

....completely non commutative algebras with identity are also cellular. 4 Conclusion By blocking sites together to produce a two site neighborhood, a CA rule can be thought of as a binary algebra. The properties of this block algebra can be related to the CA s dynamical properties in various ways [1, 3, 4, 5, 7]. However, we have shown that if this algebra is associative with identity, an inverse property loop, anti commutative with identity, or commutative, then the original CA rule depends only on its leftmost and rightmost inputs, and the block algebra is simply the direct product of 2r copies of the ....

T. Boykett, "Combinatorial construction of one-dimensional reversible cellular automata." Contributions to General Algebra 9 (1995) 81--90.

....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 ....

....if is permutive and ane, i.e. linear up to a constant, then is also ane. We prove a number of lesser results as well. An extensive study of the special case (a b) b c) a b) b c) b where and represent reversible CAs which are each others inverses, is carried out in [1, 2]. 2 Preliminaries A binary algebra is a function f : A A A, written f(a; b) a b. A left (right) identity is an element 1 such that 1 a = a (resp. a 1 = a) for all a. A left (right) zero is an element z such that z a = z (resp. a z = z) for all a. An identity (zero) is both a ....

T. Boykett, \Combinatorial construction of one-dimensional reversible cellular automata. " Contributions to General Algebra 9 (1995) 81-90.

....Let A,B be two sets, and let Q = A Theta B. Define (a 1 ; b 1 ) ffl (a 2 ; b 2 ) a 2 ; b 1 ) 3) a 1 ; b 1 ) ffi (a 2 ; b 2 ) a 1 ; b 2 ) 4) Then (Q; ffl; ffi) is a semicentral bigroupoid. Semicentral bigroupoids are equivalent to reversible one dimensional cellular automata, see e.g. [1]. Definition 2 A Rectangular Structure on a set S, called the base set, is a collection R of ordered pairs of subsets, called rectangles, of S, such that 8(s; t) 2 S 2 9 R 2 R such that (s; t) 2 R (5) 8R; Q 2 R; jR 1 Q 2 j = 1: 6) where we identify R = R 1 ; R 2 ) R 1 Theta R 2 . We ....

....the ffi operation is idempotent, thus both operations are idempotent by the simple note that a ffl a = a ffi a) ffl (a ffi a) a (23) 2 In general a complete equivalence exists between (idempotent) semicentral bigroupoids and rectangular structures, though we do not need that here. See e.g. [1, 2]. From such an idempotent semicentral bigroupoid, we can define a collection of others using a technique called lifting, then determine the exact number of mutually non isomorphic semicentral bigroupoids derivable from a given rectangular structure using knowledge about the automorphism group of ....

Tim Boykett. Combinatorial construction of one dimensional reversible cellular automata. In G. Pilz, editor, Contributions to General Algebra 9, pages 81--90. Holder--Pichler--Tempsky, Teubner, 1995.

....in section 4 are an attempt to use similar tools. The way in which these tools are unusable in the radius 1 paradigm used in Section 4 is then analysed and presented as evidence that the paradigm introduced by Pedersen in [16] but dating back to Hedlund [7] and used by the author and others in [2, 3, 4, 5, 11] has certain strengths and advantages. We proceed first by looking at some elementary cellular automata theory, then looking at the two paradigms more closely. In particular we formulate reversibility conditions for both the radius 1=2 and radius 1 paradigms. Looking at the radius 1=2 paradigm ....

....cellular automata. In general, he shows that every cellular automata rule can be expressed as a binary operation, leading to the intuitive use of algebraic techniques for the analysis and investigation of cellular automata properties. Some papers that have successfully used this technique include [2, 3, 4, 5, 11]. A cellular automaton is defined by its state set S, the local map f of arity m, the global topology T and a neighbourhood E = fj 1 ; j m g of translations of T . In a one dimensional cellular automaton, this topology is the line Z, E is a sequence of integers. The (global) state or ....

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Tim Boykett. Combinatorial construction of one dimensional reversible cellular automata. In G. Pilz, editor, Contributions to General Algebra 9, pages 81--90. Holder--Pichler--Tempsky, Teubner, 1995.

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