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Register a new userIssues on drawing the State Transition Diagram for arbitrary Cellular Automata
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This paper proposes several algorithms and their Cellular Automata Machine (CAM) for drawing the State Transition Diagram (STD) of an arbitrary Cellular Automata (CA) Rule (any neighborhood, uniform/ hybrid and null/ periodic boundary) and length of the CA n. It also discusses the novelty, hardware cost and the complexities of these algorithms.
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We present an intuitive formalism for implementing cellular automata on arbitrary topologies. By that means, we identify a symmetry operation in the class of elementary cellular automata. Moreover, we determine the subset of topologically sensitive elementary cellular automata and find that the overall number of complex patterns decreases under increasing neighborhood size in regular graphs. As exemplary applications, we apply the formalism to complex networks and compare the potential of scale-free graphs and metabolic networks to generate complex dynamics.
A transition from asymmetric to symmetric patterns in time-dependent extended systems is described. It is found that one dimensional cellular automata, started from fully random initial conditions, can be forced to evolve into complex symmetrical patterns by stochastically coupling a proportion $p$ of pairs of sites located at equal distance from the center of the lattice. A nontrivial critical value of $p$ must be surpassed in order to obtain symmetrical patterns during the evolution. This strategy is able to classify the cellular automata rules -with complex behavior- between those that support time-dependent symmetric patterns and those which do not support such kind of patterns.
In addition to the $lambda$ parameter, we have found another parameter which characterize the class III, class II and class IV patterns more quantitatively. It explains why the different classes of patterns coexist at the same $lambda$. With this parameter, the phase diagram for an one dimensional cellular automata is obtained. Our result explains why the edge of chaos(class IV) is scattered rather wide range in $lambda$ around 0.5, and presents an effective way to control the pattern classes. oindent PACS: 89.75.-k Complex Systems
We study sources of isomorphisms of additive cellular automata on finite groups (called index-group). It is shown that many isomorphisms (called regular) of automata are reducible to the isomorphisms of underlying algebraic structures (such as the index-group, monoid of automata rules, and its subgroup of reversible elements). However for some groups there exist not regular automata isomorphisms. A complete description of linear automorphisms of the monoid is obtained. These automorphisms cover the most part of all automata isomorphisms for small groups and are represented by reversible matrices M such that for any index-group circulant C the matrix M^{-1}CM is an index-group circulant.
Gauge symmetries play a fundamental role in Physics, as they provide a mathematical justification for the fundamental forces. Usually, one starts from a non-interactive theory which governs `matter, and features a global symmetry. One then extends the theory so as make the global symmetry into a local one (a.k.a gauge-invariance). We formalise a discrete counterpart of this process, known as gauge extension, within the Computer Science framework of Cellular Automata (CA). We prove that the CA which admit a relative gauge extension are exactly the globally symmetric ones (a.k.a the colour-blind). We prove that any CA admits a non-relative gauge extension. Both constructions yield universal gauge-invariant CA, but the latter allows for a first example where the gauge extension mediates interactions within the initial CA.
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