Sequential dynamical system について

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Sequential dynamical system ：ウィキペディア英語版

Sequential dynamical system

Sequential dynamical systems (SDSs) are a class of graph dynamical systems. They are discrete dynamical systems which generalize many aspects of for example classical cellular automata, and they provide a framework for studying asynchronous processes over graphs. The analysis of SDSs uses techniques from combinatorics, abstract algebra, graph theory, dynamical systems and probability theory.
==Definition==
An SDS is constructed from the following components:

* A finite ''graph'' ''Y'' with vertex set v() = . Depending on the context the graph can be directed or undirected.
* A state ''x_v'' for each vertex ''i'' of ''Y'' taken from a finite set ''K''. The ''system state'' is the ''n''-tuple ''x'' = (''x''₁, ''x''₂, ... , ''x_n''), and ''x''() is the tuple consisting of the states associated to the vertices in the 1-neighborhood of ''i'' in ''Y'' (in some fixed order).
* A ''vertex function'' ''f_i'' for each vertex ''i''. The vertex function maps the state of vertex ''i'' at time ''t'' to the vertex state at time ''t'' + 1 based on the states associated to the 1-neighborhood of ''i'' in ''Y''.
* A word ''w'' = (''w''₁, ''w''₂, ... , ''w_m'') over ''v''().

It is convenient to introduce the ''Y''-local maps ''F_i'' constructed from the vertex functions by
:

)), x_, \ldots , x_n) \;.

The word ''w'' specifies the sequence in which the ''Y''-local maps are composed to derive the sequential dynamical system map ''F'': ''Kⁿ → Kⁿ'' as
:

) = F_ \circ F_ \circ \cdots \circ F_ \circ F_ \;.

If the update sequence is a permutation one frequently speaks of a ''permutation SDS'' to emphasize this point.
The ''phase space'' associated to a sequential dynamical system with map ''F'': ''Kⁿ → Kⁿ'' is the finite directed graph with vertex set ''Kⁿ'' and directed edges (''x'', ''F''(''x'')). The structure of the phase space is governed by the properties of the graph ''Y'', the vertex functions (''f_i'')''_i'', and the update sequence ''w''. A large part of SDS research seeks to infer phase space properties based on the structure of the system constituents.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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