7 research outputs found
Strongly Universal Reversible Gate Sets
It is well-known that the Toffoli gate and the negation gate together yield a
universal gate set, in the sense that every permutation of can be
implemented as a composition of these gates. Since every bit operation that
does not use all of the bits performs an even permutation, we need to use at
least one auxiliary bit to perform every permutation, and it is known that one
bit is indeed enough. Without auxiliary bits, all even permutations can be
implemented. We generalize these results to non-binary logic: If is a
finite set of odd cardinality then a finite gate set can generate all
permutations of for all , without any auxiliary symbols. If the
cardinality of is even then, by the same argument as above, only even
permutations of can be implemented for large , and we show that indeed
all even permutations can be obtained from a finite universal gate set. We also
consider the conservative case, that is, those permutations of that
preserve the weight of the input word. The weight is the vector that records
how many times each symbol occurs in the word. It turns out that no finite
conservative gate set can, for all , implement all conservative even
permutations of without auxiliary bits. But we provide a finite gate set
that can implement all those conservative permutations that are even within
each weight class of .Comment: Submitted to Rev Comp 201
Universal groups of cellular automata
We prove that the group of reversible cellular automata (RCA), on any
alphabet , contains a perfect subgroup generated by six involutions which
contains an isomorphic copy of every finitely-generated group of RCA on any
alphabet . This result follows from a case study of groups of RCA generated
by symbol permutations and partial shifts with respect to a fixed Cartesian
product decomposition of the alphabet. For prime alphabets, we show that this
group is virtually cyclic, and that for composite alphabets it is non-amenable.
For alphabet size four, it is a linear group. For non-prime non-four alphabets,
it contains copies of all finitely-generated groups of RCA. We also obtain that
RCA of biradius one on all large enough alphabets generate copies of all
finitely-generated groups of RCA. We ask a long list of questions.Comment: Major scientific revision: fixed serious problem in the universality
proof and solved the main questions + many smaller improvements. Comments
welcome
Universal gates with wires in a row
We give some optimal size generating sets for the group generated by shifts and local permutations on the binary full shift. We show that a single generator, namely the fully asynchronous application of the elementary cellular automaton 57 (or, by symmetry, ECA 99), suffices in addition to the shift. In the terminology of logical gates, we have a single reversible gate whose shifts generate all (finitary) reversible gates on infinitely many binary-valued wires that lie in a row and cannot (a priori) be rearranged. We classify pairs of words u, v such that the gate swapping these two words, together with the shift and the bit flip, generates all local permutations. As a corollary, we obtain analogous results in the case where the wires are arranged on a cycle, confirming a conjecture of Macauley-McCammond-Mortveit and Vielhaber
Universal groups of cellular automata
We prove that the group of reversible cellular automata (RCA), on any alphabet A, contains a subgroup generated by three involutions which contains an iso-morphic copy of every finitely generated group of RCA on any alphabet B. This result follows from a case study of groups of RCA generated by symbol permutations and par-tial shifts (equivalently, partitioned cellular automata) with respect to a fixed Cartesian product decomposition of the alphabet. For prime alphabets, we show that this group is virtually cyclic, and that for composite alphabets it is non-amenable. For alphabet size four, it is a linear group. For non-prime non-four alphabets, it contains copies of all finitely generated groups of RCA. We also prove this property for the group generated by RCA of biradius one on any full shift with large enough alphabet, and also for some perfect finitely generated groups of RCA