Hardness of conjugacy, embedding and factorization of multidimensional subshifts

Emmanuel Jeandel 1 Pascal Vanier 2
1 CARTE - Theoretical adverse computations, and safety
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
2 LACL - Laboratoire d'Algorithmique Complexité et Logique
Abstract : Subshifts of finite type are sets of colorings of the plane defined by local constraints. They can be seen as a discretization of continuous dynamical systems. We investigate here the hardness of deciding factorization, conjugacy and embedding of subshifts in dimensions $d > 1$ for subshifts of finite type and sofic shifts and in dimensions $d\geq 1$ for effective shifts. In particular, we prove that the factorization and embedding problems are $\Sigma^0_3$ -complete and $\Sigma^0_1$- complete respectively for SFTs, sofic and effective subshifts. Conjugacy on the other side is $\Sigma^0_1$-complete for SFTs and $\Sigma^0_3$-complete for sofic and effective shifts.
Keywords : tilings computability embedding arithmetical hierarchy Subshifts of finite type sofic subshifts effective subshifts factorization conjugacy SFTs subshifts
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Emmanuel Jeandel, Pascal Vanier. Hardness of conjugacy, embedding and factorization of multidimensional subshifts . Journal of Computer and System Sciences, Elsevier, 2015, http://dx.doi.org/10.1016/j.jcss.2015.05.003. ⟨10.1016/j.jcss.2015.05.003⟩. ⟨hal-01150419⟩

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