Towards a Noether-like conservation law theorem for one dimensional reversible cellular automata

Evidence and results suggesting that a Noether--like theorem for conservation laws in 1D RCA can be obtained. Unlike Noether's theorem, the connection here is to the maximal congruences rather than the automorphisms of the local dynamics. We take the results of Takesue and Hattori (1992) on the space of additive conservation laws in one dimensional cellular automata. In reversible automata, we show that conservation laws correspond to the null spaces of certain well-structured matrices. It is shown that a class of conservation laws exist that correspond to the maximal congruences of index 2. In all examples investigated, this is all the conservation laws. Thus we conjecture that there is an equality here, corresponding to a Noether--like theorem

Rectangularity

We introduce a condition on arrays in some way maximally distinct from Latin square condition, as well as some other conditions on algebras, graphs and $0,1$-matrices. We show that these are essentially the same structures, generalising a similar collection of models presented by Knuth in 1970. We find ways in which these structures can be made more specific, relating to existing investigations, then show that they are also extremely general; the groupoids satisfy no nontrivial equations. Some construction methods are presented and some conjectures made as to how certain structures are preserved by these constructions. Finally we investigate to what degree partial arrays satisfying our conditions and partial Latin squares overlap

Orderly Algorithm to enumerate central groupoids and their graphs

A graph has the unique path property UPP_n if there is a unique path of length n between any ordered pair of nodes. This paper reiterates Royle and MacKay's technique for constructing orderly algorithms. We wish to use this technique to enumerate all UPP_2 graphs of small orders 9 and 16. We attempt to use the direct graph formalism and find that the algorithm is inefficient. We introduce a generalised problem and derive algebraic and combinatoric structures with appropriate structure. We are able to then design an orderly algorithm to determine all UPP_2 graphs of order 9, which runs fast enough. We hope to be able to determine the UPP_2 graphs of order 16 in the near future.Comment: 21 pages, 6 figure

Efficient Exhaustive Listings of Reversible One Dimensional Cellular Automata

Algebra From a rectangular structure R, using the bijection d from equation (55) above and denoting by R(s; t) the unique rectangle on the pair (s; t) guaranteed by (52), define ffl : S \Theta S ! S (63) (s; t) 7! u where fug = (d \Gamma1 (s)) 2 " (d \Gamma1 (t)) 1 ffi : S \Theta S ! S (64) (s; t) 7! u where fug = d(R(s; t)) as binary operations on S