One-Dimensional Universal Reversible Cellular Automata
The problem of constructing Turing universal reversible one-dimensional cellular automata (RCAs) is studied. First, methods of designing three-neighbor and two-neighbor reversible partitioned CAs (RPCAs) that directly simulate a given reversible Turing machine (RTM) are shown. Since RPCAs are a subclass of RCAs, we see the class of one-dimensional RCAs is Turing universal even in the twoneighbor case. Next, the problem of finding one-dimensional RPCAs with a small number of states that can simulate any TM is investigated. The first model is a 24-state two-neighbor RPCA with ultimately periodic configurations, and the second one is a 98-state three-neighbor RPCA with finite configurations. Both of them can simulate any cyclic tag system, a kind of a universal string rewriting system. Finally, the computing power of reversible and number-conserving CAs (RNCCAs) are studied. The notion of number-conservation is a property analogous to the conservation law in physics. It is proved that there is a 96-state one-dimensional fourneighbor RNCCA that is Turing universal. Hence, the computing power of RCAs does not decrease even if the constraint of number-conservation is further added.
Keywordsreversible cellular automaton one-dimensional cellular automaton partitioned cellular automaton number-conserving cellular automaton Turing universality
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