Two-Dimensional Universal Reversible Cellular Automata
The problem of finding simple universal two-dimensional reversible cellular automata (RCAs) is studied. Here, three kinds of reversible partitioned CAs (RPCAs) are designed. The first two are 16-state RPCAs in which any circuit composed of Fredkin gates and delay elements is embeddable. Since a finite control and a tape cell of a reversible Turing machine can be composed of Fredkin gates and delay elements, these RPCAs with horizontally ultimately periodic configurations are Turing universal. A cell of any two-dimensional RPCA can also be constructed out of Fredkin gates and delay elements. Hence, these two models are intrinsically universal, as well as Turing universal, in the sense that any two-dimensional RPCA can be simulated in their cellular space. The last model is an 81-state RPCA, which is again both Turing universal, and intrinsically universal. Though the number of states is larger than those of the first two, any reversible two-counter machine is embedded concisely in its finite configurations.
Keywordsreversible cellular automaton two-dimensional partitioned cellular automaton Fredkin gate rotary element reversible counter machine Turing universality intrinsic universality
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