Computing by Temporal Order: Asynchronous Cellular Automata

Abstract

Our concern is the behaviour of the elementary cellular automata with state set 0,1 over the cell set Z/nZ (one-dimensional finite wrap-around case), under all possible update rules (asynchronicity). Over the torus Z/nZ (n<= 11),we will see that the ECA with Wolfram rule 57 maps any v in F_2^n to any w in F_2^n, varying the update rule. We furthermore show that all even (element of the alternating group) bijective functions on the set F_2^n = 0,...,2^n-1, can be computed by ECA57, by iterating it a sufficient number of times with varying update rules, at least for n <= 10. We characterize the non-bijective functions computable by asynchronous rules

Hard, firm, soft … Etherealware:Computing by Temporal Order of Clocking

“…Conjecture 5.10 in [13] about Wolfram rule 57 coincides with our finding that ECA-57 generates the alternating groups From Natural to Artificial Intelligence -Algorithms and Applications on patterns of n bits. They verified this claim for n up to 8, while in [14] this is extended to n ≤ 10 and in this paper up to n ≤ 28.…”

“…Here, continuing the work in [10,14], we again employ CAs as computing devices, whose work comes to an end, when the pattern transformation or function evaluation has been obtained. Also, the clocking, the temporal update rule, is completely deterministic and replaces the usual ways of representing an algorithm, either in software (initial data) or in hardware (choice of ECA and connecting graph).…”

“…The 2 2 3 ¼ 256 ECAs can be arranged into 88 equivalence classes under the symbolic symmetry 0/1 and the chiral symmetry left/right (c iþ1 $ c iÀ1 ); see ( [14], Appendix A). It is sufficient to consider one member per class.…”

“…In [10,14], we considered five universality properties o ðÞto iv ðÞ ,whereeachv ↦ w makes use of a certain update rule AS applied several times. We only give a summary here.…”

“…Eventually, all conversions may happen at all times for some update scheme. This property cannot be satisfied, for no ECA; see ( [14], Thm. 2).…”