Cellular automata and pattern recall: An example

CELLULAR AUTOMATA AND PATTERN RECALL: AN EXAMPLE Burton VooM)ees. Facuity of Arts & Sciences. Athabasca University, Box 10,000 Athabasca, A]berta~ CANADA TOG 2RO. Let (Q,E+) De an additive cellular automaton defined on the set E+ of nail infinite binary sequences. E+ c a n b e c o n s i d e r e d a s t h e s e t o f a l l binary decimals, hence as homeomorphlc to the interval [O, 1]. This is not an isomorphism since, e.g., 1/2 = (10) = (01), just as in decimal notation t/2 = 5 = 4 ~ where underlining is understood to indicate infinite repetition. Since Q:E+-->E + describes the automaton transition rule, this rule is seen to induce a map Q:[O, 1]-->[O, 1]. In this paper the automaton (D,E+) is considered, where in component notation [D(p)] i = ~i + ~i+ 1 defines the automaton transition rule. Let {c i} denote the set of cycles of this automaton. Thegrem: The predecessor states of each c i define a rational dense subset c* i of [0, I] Let {gs] be a set of continuous functions on [0,1] and denote by gs,i the restrict~on of gs tO C* i. Denote the set of all rationals in[O,l]by [0,1]*. A function fD[O,1]*-->[O,1]* is defined by fD(x) = gs(i),i(x)(X) where the notation indicates that in evaulating fD(x) we first determine which c* i contains x and then evauiate a gs on c* i according to some

mapping of indicles choosen so that all gs are evaulated over different c* i, In this way the function fD embeds all of the set {gs } in a single function. Most significantly, each cycle c i of the automaton (D,E+) now becomes an index for a particular function gs This i]lustrates the possibility of "holographically" storing a set of patterns (the {gs}) in a "memory" (the function f) with specific "words" (the c i) as recall cues: ~n order to recall a particular gs one chooses the corresponding c i and computes predecessor states for some sufficiently large number of iterations. The functlon f ls then evaulated on this set of predecessor states and yields an approximation to gs

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