Automatic Ordinals
Olivier Finkel (ELM, IMJ), Stevo Todorcevic (ELM, IMJ)

TL;DR
This paper characterizes the class of ordinals that are injectively omega-tree-automatic and omega^n-automatic, establishing their bounds and demonstrating the strict hierarchy of these automatic structures.
Contribution
It proves the bounds of injectively omega-tree-automatic and omega^n-automatic ordinals, extending previous results and showing the hierarchy of such structures is strict.
Findings
Injectively omega-tree-automatic ordinals are smaller than ω^{ω^ω}.
Injectively ω^n-automatic ordinals are smaller than ω^{ω^n}.
The hierarchy of injectively ω^n-automatic structures is strict.
Abstract
We prove that the injectively omega-tree-automatic ordinals are the ordinals smaller than . Then we show that the injectively -automatic ordinals, where is an integer, are the ordinals smaller than . This strengthens a recent result of Schlicht and Stephan who considered in [Schlicht-Stephan11] the subclasses of finite word -automatic ordinals. As a by-product we obtain that the hierarchy of injectively -automatic structures, n>0, which was considered in [Finkel-Todorcevic12], is strict.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsNatural Language Processing Techniques · semigroups and automata theory · Advanced Algebra and Logic
