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 Prikl. Diskr. Mat. Suppl., 2015, Issue 8, Pages 115–117 (Mi pdma253)

Applied Theory of Coding, Automata and Graphs

On number of inaccessible states in finite dynamic systems of binary vectors associated with palms orientations

A. V. Zharkova

Saratov State University, Saratov

Abstract: Finite dynamic systems of binary vectors associated with palms orientations are considered. A palm is a tree which is a union of paths having a common end vertex and all these paths, except perhaps one, have the length 1. States of a dynamic system $(P_{s+c},\gamma)$, $s>0$, $c>1$, are all possible orientations of a palm with trunk length $s$ and leafs number $c$, and evolutionary function transforms a given palm orientation by reversing all arcs that enter into sinks. This dynamic system is isomorphic to finite dynamic system ($B^{s+c}$, $\gamma$), $s>0$, $c>1$, where states of this system are all possible binary vectors of dimension $s+c$. Let $v=v_1…v_s.v_{s+1}…v_{s+c}\in B^{s+c}$, then $\gamma(v)=v'$ where $v'$ is obtained by simultaneous application of the following rules: 1) if $v_1=0$, then $v'_1=1$; 2) if $v_i=1$ and $v_{i+1}=0$ for some $i$ where $0<i<s$, then $v'_i=0$ and $v'_{i+1}=1$; 3) if $v_i=1$ for some $i$ where $s<i\leq s+c$, then $v'_i=0$; 4) if $v_s=1$ and $v_i=0$ for all $i$ where $s<i\leq s+c$, then $v'_s=0$ and $v'_i=1$ for all $i$, $s<i\leq s+c$; 5) there are no other differences between $v$ and $\gamma(v)$. A formula for counting the number of inaccessible states in the considered dynamic systems is proposed. The table with the number of inaccessible states in systems $(B^{8+c},\gamma)$ for $1<c<9$ is given.

Keywords: finite dynamic system, inaccessible state, palm, starlike tree.

DOI: https://doi.org/10.17223/2226308X/8/44

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UDC: 519.1

Citation: A. V. Zharkova, “On number of inaccessible states in finite dynamic systems of binary vectors associated with palms orientations”, Prikl. Diskr. Mat. Suppl., 2015, no. 8, 115–117

Citation in format AMSBIB
\Bibitem{Zha15} \by A.~V.~Zharkova \paper On number of inaccessible states in finite dynamic systems of binary vectors associated with palms orientations \jour Prikl. Diskr. Mat. Suppl. \yr 2015 \issue 8 \pages 115--117 \mathnet{http://mi.mathnet.ru/pdma253} \crossref{https://doi.org/10.17223/2226308X/8/44}