On factorisation forests

The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the importance of this approach in the context of automata over infinite words and trees. We extend the theorem of factorisation forest in two directions: we show that it is still valid for any word indexed by a linear ordering; and we show that it admits a deterministic variant for words indexed by well-orderings. A byproduct of this work is also an improvement on the known bounds for the original result. We apply the first variant for giving a simplified proof of the closure under complementation of rational sets of words indexed by countable scattered linear orderings. We apply the second variant in the analysis of monadic second-order logic over trees, yielding new results on monadic interpretations over trees. Consequences of it are new caracterisations of prefix-recognizable structures and of the Caucal hierarchy.Comment: 27 page

A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications

Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation. Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modelling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been introduced in update pattern, lattice structure, neighborhood dependency and local rule. In this survey, we tour to the various types of CAs introduced till date, the different characterization tools, the global behaviors of CAs, like universality, reversibility, dynamics etc. Special attention is given to non-uniformity in CAs and especially to non-uniform elementary CAs, which have been very useful in solving several real-life problems.Comment: 43 pages; Under review in Natural Computin

Complex networks derived from cellular automata

We propose a method for deriving networks from one-dimensional binary cellular automata. The derived networks are usually directed and have structural properties corresponding to the dynamical behaviors of their cellular automata. Network parameters, particularly the efficiency and the degree distribution, show that the dependence of efficiency on the grid size is characteristic and can be used to classify cellular automata and that derived networks exhibit various degree distributions. In particular, a class IV rule of Wolfram's classification produces a network having a scale-free distribution.Comment: 10 pages, 8 figure

The GHZ/W-calculus contains rational arithmetic

Graphical calculi for representing interacting quantum systems serve a number of purposes: compositionally, intuitive graphical reasoning, and a logical underpinning for automation. The power of these calculi stems from the fact that they embody generalized symmetries of the structure of quantum operations, which, for example, stretch well beyond the Choi-Jamiolkowski isomorphism. One such calculus takes the GHZ and W states as its basic generators. Here we show that this language allows one to encode standard rational calculus, with the GHZ state as multiplication, the W state as addition, the Pauli X gate as multiplicative inversion, and the Pauli Z gate as additive inversion.Comment: In Proceedings HPC 2010, arXiv:1103.226

A Combinatorial Parametric Engineering Model for Solid Freeform Fabrication

Fabricated parts are often represented as compact connected smooth 3-manifolds with boundary, where the boundaries consist of compact smooth 2-manifolds. This class of mathematical structures includes topological spaces with enclosed voids and tunnels. Useful information about these structures are coded into level functions (Morse functions) which map points in the 3-manifold onto their height above a fixed plane. By definition, Morse functions are smooth functions, all of whose critical points are nondegenerate. This information is presented by the Reeb graph construction that develops a topologically informative skeleton of the manifold whose nodes are the critical points of the Morse function and whose edges are associated with the connected components between critical slices. This approach accurately captures the SFF process: using a solid geometric model of the part, defining surface boundaries; selecting a part orientation; forming planar slices, decomposing the solid into a sequence of thin cross-sectional polyhedral layers; and then fabricating the part by producing the polyhedra by additive manufacturing. This note will define a qualitative and combinatorial parametric engineering model of the SFF part design process. The objects under study will be abstract simplicial complexes K with boundary ∂K. Systems of labeled 2-surfaces in K, called slices, will be associated with the cross-sectional polyhedral layers. The labeled slices are mapped into a family of digraph automata, which, unlike cellular automata, are defined not on regular lattices with simple connectivities (cells usually have either 4 or 8 cell neighborhoods) but on unrestricted digraphs whose connectivities are irregular and more complicated.Mechanical Engineerin