Structures Without ScatteredAutomatic Presentation
Abstract
Bruyere and Carton lifted the notion of finite automata reading infinite words to finite automata reading words with shape an arbitrary linear order L. Automata on finite words can be used to represent infinite structures, the socalled wordautomatic structures. Analogously, for a linear order L there is the class of Lautomatic structures. In this paper we prove the following limitations on the class of Lautomatic structures for a fixed L of finite condensation rank 1+\alpha. Firstly, no scattered linear order with finite condensation rank above \omega^(\alpha+1) is L\alphaautomatic. In particular, every Lautomatic ordinal is below \omega^\omega^(\alpha+1). Secondly, we provide bounds on the (ordinal) height of wellfounded order trees that are Lautomatic. If \alpha is finite or L is an ordinal, the height of such a tree is bounded by \omega^{\alpha+1}. Finally, we separate the class of treeautomatic structures from that of Lautomatic structures for any ordinal L: the countable atomless boolean algebra is known to be treeautomatic, but we show that it is not Lautomatic.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 DOI:
 10.48550/arXiv.1304.0912
 arXiv:
 arXiv:1304.0912
 Bibcode:
 2013arXiv1304.0912K
 Keywords:

 Computer Science  Formal Languages and Automata Theory;
 Mathematics  Logic
 EPrint:
 10 pages + 20 pages Appendix