eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2009-02-19
337
348
10.4230/LIPIcs.STACS.2009.1819
article
Undecidable Properties of Limit Set Dynamics of Cellular Automata
Di Lena, Pietro
Margara, Luciano
Cellular Automata (CA) are discrete dynamical systems and an abstract model of parallel computation. The limit set of a cellular automaton is its maximal topological attractor. A well know result, due to Kari, says that all nontrivial properties of limit sets are undecidable. In this paper we consider properties of limit set dynamics, i.e. properties of the dynamics of Cellular Automata restricted to their limit sets. There can be no equivalent of Kari's Theorem for limit set dynamics. Anyway we show that there is a large class of undecidable properties of limit set dynamics, namely all properties of limit set dynamics which imply stability or the existence of a unique subshift attractor. As a consequence we have that it is undecidable whether the cellular automaton map restricted to the limit set is the identity, closing, injective, expansive, positively expansive, transitive.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol003-stacs2009/LIPIcs.STACS.2009.1819/LIPIcs.STACS.2009.1819.pdf
Cellular automata
Undecidability
Symbolic dynamics