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Register a new userSolutions on 1D and 2D Density Classification Problem Using Programmable Cellular Automata
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This paper presents solutions to Density Classification Task (DCT) using a variant of Cellular Automata (CA) called Programmable Cellular Automata (PCA). The translation property as well as the density preserving property of fundamental CA rules in 1D and 2D, and the advantage of PCA are embedded together to obtain the DCT solution. The advantage of PCA over standard CA is reported. A general 2D DCT of arbitrary shapes and sizes, its applicability and its solution using PCA is newly introduced.
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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.
The mechanism which discriminates the pattern classes at the same $lambda$, is found. It is closely related to the structure of the rule table and expressed by the numbers of the rules which break the strings of the quiescent states. It is shown that for the N-neighbor and K-state cellular automata, the class I, class II, class III and class IV patterns coexist at least in the range, $frac{1}{K} le lambda le 1-frac{1}{K} $. The mechanism is studied quantitatively by introducing a new parameter $F$, which we call quiescent string dominance parameter. It is taken to be orthogonal to $lambda$. Using the parameter F and $lambda$, the rule tables of one dimensional 5-neighbor and 4-state cellular automata are classified. The distribution of the four pattern classes in ($lambda$,F) plane shows that the rule tables of class III pattern class are distributed in larger $F$ region, while those of class II and class I pattern classes are found in the smaller $F$ region and the class IV behaviors are observed in the overlap region between them. These distributions are almost independent of $lambda$ at least in the range $0.25 leq lambda leq 0.75$, namely the overlapping region in $F$, where the class III and class II patterns coexist, has quite gentle $lambda$ dependence in this $lambda$ region. Therefore the relation between the pattern classes and the $lambda$ parameter is not observed. PACS: 89.75.-k Complex Systems
In this paper, linear Cellular Automta (CA) rules are recursively generated using a binary tree rooted at 0. Some mathematical results on linear as well as non-linear CA rules are derived. Integers associated with linear CA rules are defined as linear numbers and the properties of these linear numbers are studied.
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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