By Meyers R.A. (ed.)
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Extra resources for Computational complexity: selected entries from the Encyclopedia of computational complexity and systems science
F. Ioﬀe Physico-Technical Institute St. Petersburg Russia FALOUTSOS, MICHALIS University of California Riverside USA DRAPER, DAVID University of California Santa Cruz USA DUBOIS, DIDIER Universite Paul Sabatier Toulouse Cedex France DUNNE, JENNIFER A. Santa Fe Institute Santa Fe USA Paciﬁc Ecoinformatics and Computational Ecology Lab Berkeley USA DURAND-LOSE, JÉRÔME Université d’Orléans Orléans France DUTTA , PRAJIT K. Columbia University New York USA DŽEROSKI , SAŠO Jožef Stefan Institute Ljubljana Slovenia FOGEDBY, HANS C.
Since then a number of researchers have independently derived similar results [120,121,122,123,124,125,126]. Let A D circ(a0 ; : : : ; be a state in E (A; Zn ). The minimal a n 1 ) and let annihilating polynomial of is the monic polynomial P (z) such that P (A) D 0 mod(p). This polynomial exists since A always satisﬁes its characteristic equation. Let P (z) D z k P (z) with P (0) ¤ 0. The order of P (z), ord(P (z)), is deﬁned as the smallest natural number c such that P (z)j(z c 1). The following theorem is given in [121,122].
14) a rule will be injective on E (A; Zn ) if and only if its diagonalized matrix representation is invertible, hence there can be no zero eigenvalues. Lemma 5 ([81,116]) Let X : E (A; Zn ) 7! E (A; Zn ) be an additive rule represented by A D circ(a0 ; : : : ; a n 1 ). X is injective on Zn Z if and only if PA (! s ) ¤ 0 mod(p) for 0 Ä s Ä n 1. 7 8 Additive Cellular Automata If the rule in Lemma 5 is to be injective on Z then it must be so on Zn for all n. Thus the complex polynomial PA (z) must be irreducible with respect to all nth roots of unity, for all n.