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Monadic second-order logic and the domino problem on self-similar graphs
arXiv - CS - Logic in Computer Science Pub Date : 2020-11-05 , DOI: arxiv-2011.02735
Laurent Bartholdi

We consider the domino problem on Schreier graphs of self-similar groups, and more generally their monadic second-order logic. On the one hand, we prove that if the group is bounded then the graph's monadic second-order logic is decidable. This covers, for example, the Sierpi\'nski gasket graphs and the Schreier graphs of the Basilica group. On the other hand, we already prove undecidability of the domino problem for a class of self-similar groups, answering a question by Barbieri and Sablik, and some examples including one of linear growth.

中文翻译:

一元二阶逻辑和自相似图上的多米诺问题

我们考虑自相似群的 Schreier 图上的多米诺骨牌问题,更一般地考虑它们的一元二阶逻辑。一方面,我们证明如果群是有界的,那么图的一元二阶逻辑是可判定的。例如,这包括 Sierpi\'nski 垫片图和 Basilica 组的 Schreier 图。另一方面,我们已经证明了一类自相似群的多米诺问题的不可判定性,回答了 Barbieri 和 Sablik 的一个问题,以及一些例子,包括线性增长之一。
更新日期:2020-11-06
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