Universal gates with wires in a row

Abstract We give some optimal size generating sets for the group generated by shifts and local permutations on binary full shift. show that a single generator, namely fully asynchronous application of elementary cellular automaton 57 (or, symmetry, ECA 99), suffices in addition to In terminology logical gates, we have reversible gate whose generate all (finitary) gates infinitely many binary-valued wires lie row cannot (a priori) be rearranged. classify pairs words u , v such swapping these two words, together with shift bit flip, generates permutations. As corollary, obtain analogous results case where are arranged cycle, confirming conjecture Macauley-McCammond-Mortveit Vielhaber.