35,401 research outputs found
Automata, reduced words, and Garside shadows in Coxeter groups
In this article, we introduce and investigate a class of finite deterministic
automata that all recognize the language of reduced words of a finitely
generated Coxeter system (W,S). The definition of these automata is
straightforward as it only requires the notion of weak order on (W,S) and the
related notion of Garside shadows in (W,S), an analog of the notion of a
Garside family. Then we discuss the relations between this class of automata
and the canonical automaton built from Brink and Howlett's small roots. We end
this article by providing partial positive answers to two conjectures: (1) the
automata associated to the smallest Garside shadow is minimal; (2) the
canonical automaton is minimal if and only if the support of all small roots is
spherical, i.e., the corresponding root system is finite.Comment: 21 pages, 7 figures; v2: 23 pages, 8 figures, Remark 3.15 added,
accepted in Journal of Algebra, computational sectio
A Family of Controllable Cellular Automata for Pseudorandom Number Generation
In this paper, we present a family of novel Pseudorandom Number Generators (PRNGs) based on Controllable Cellular Automata (CCA) ─ CCA0, CCA1, CCA2 (NCA), CCA3 (BCA), CCA4 (asymmetric NCA), CCA5, CCA6 and CCA7 PRNGs. The ENT and DIEHARD test suites are used to evaluate the randomness of these CCA PRNGs. The results show that their randomness is better than that of conventional CA and PCA PRNGs while they do not lose the structure simplicity of 1-d CA. Moreover, their randomness can be comparable to that of 2-d CA PRNGs. Furthermore, we integrate six different types of CCA PRNGs to form CCA PRNG groups to see if the randomness quality of such groups could exceed that of any individual CCA PRNG. Genetic Algorithm (GA) is used to evolve the configuration of the CCA PRNG groups. Randomness test results on the evolved CCA PRNG groups show that the randomness of the evolved groups is further improved compared with any individual CCA PRNG
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