Some remarks on A_1^{(1)} soliton cellular automata

In this short note, we describe the A_1^{(1)} soliton cellular automata as an evolution of a poset. This allows us to explain the conservation laws for the A_1^{(1)} soliton cellular automata, one given by Torii, Takahashi and Satsuma, and the other given by Fukuda, Okado and Yamada, in terms of the stack permutations of states in a very natural manner. As a biproduct, we can prove a conjectured formula relating these laws.Comment: 10 pages, LaTeX2

A uniform approach to soliton cellular automata using rigged configurations

For soliton cellular automata, we give a uniform description and proofs of the solitons, the scattering rule of two solitons, and the phase shift using rigged configurations in a number of special cases. In particular, we prove these properties for the soliton cellular automata using $B^{r,1}$ when $r$ is adjacent to $0$ in the Dynkin diagram or there is a Dynkin diagram automorphism sending $r$ to $0$.Comment: 37 pages, 3 figures, 4 table

Elementary decomposition of soliton automata

Soliton automata are the mathematical models of certain possible molecular switching devices. In this paper we work out a decomposition of soliton automata through the structure of their underlying graphs. These results lead to the original aim, to give a characterization of soliton automata in general case

Deterministic soliton automata with at most one cycle

AbstractSoliton valves have been proposed as molecular switching elements. Their mathematical model is the soliton graph and the soliton automaton (Dassow and Jürgensen, J. Comput. System Sci.40 (1990), 154–181). In this paper we continue the study of the logic aspects of soliton switching. There are two cases of special importance: those of deterministic and those of strongly deterministic soliton automata. The former have deterministic state transitions in the usual sense of automaton theory. The latter do not only have deterministic state transitions, but also deterministic soliton paths—a much stronger property, as it turns out. In op cit. a characterization of indecomposable, strongly deterministic soliton automata was proved and it was shown that their transition monoids are primitive groups of permutations. Roughly speaking, the main difference between deterministic and strongly deterministic soliton automata is that in the former the underlying soliton graphs may contain cycles of odd lengths while such cycles are not permitted in the soliton graphs belonging to strongly deterministic soliton automata. In the present paper, we focus on a special class of deterministic soliton automata, that of deterministic soliton automata whose underlying graphs contain at most one cycle. For this class we derive structural descriptions. Our main results concern the elimination of certain types of loops, the treatment of soliton paths with repeated edges, the structure of cycles of odd length, and the transition monoid. As an application we show that the memory element proposed in the literature (Carter, in Bioelectronics, edited by Aizawa, Research and Development Report 50, CMC Press, Denver, CO, 1984) can be transformed in into a soliton tree, thus turning a deterministic device into a logically equivalent strongly deterministic device

Lenia and Expanded Universe

We report experimental extensions of Lenia, a continuous cellular automata family capable of producing lifelike self-organizing autonomous patterns. The rule of Lenia was generalized into higher dimensions, multiple kernels, and multiple channels. The final architecture approaches what can be seen as a recurrent convolutional neural network. Using semi-automatic search e.g. genetic algorithm, we discovered new phenomena like polyhedral symmetries, individuality, self-replication, emission, growth by ingestion, and saw the emergence of "virtual eukaryotes" that possess internal division of labor and type differentiation. We discuss the results in the contexts of biology, artificial life, and artificial intelligence.Comment: 8 pages, 5 figures, 1 table; submitted to ALIFE 2020 conferenc

Soliton Cellular Automata Associated With Crystal Bases

We introduce a class of cellular automata associated with crystals of irreducible finite dimensional representations of quantum affine algebras U'_q(\hat{\geh}_n). They have solitons labeled by crystals of the smaller algebra U'_q(\hat{\geh}_{n-1}). We prove stable propagation of one soliton for \hat{\geh}_n = A^{(2)}_{2n-1}, A^{(2)}_{2n}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}. For \gh_n = C^{(1)}_n, we also prove that the scattering matrices of two solitons coincide with the combinatorial R matrices of U'_q(C^{(1)}_{n-1})-crystals.Comment: 29 pages, 1 figure, LaTeX2

Turing Automata and Graph Machines

Indexed monoidal algebras are introduced as an equivalent structure for self-dual compact closed categories, and a coherence theorem is proved for the category of such algebras. Turing automata and Turing graph machines are defined by generalizing the classical Turing machine concept, so that the collection of such machines becomes an indexed monoidal algebra. On the analogy of the von Neumann data-flow computer architecture, Turing graph machines are proposed as potentially reversible low-level universal computational devices, and a truly reversible molecular size hardware model is presented as an example