Automata on ordinals and automaticity of linear orders

2017

COM

Self Cite

An α-automaton (for α some ordinal) is an automaton similar to a Muller automaton that processes words of length α. A structure is called α-automatic if it can be presented by α-automata (completely analogous to the notion of automatic structures which can be presented by the well-known finite automata). We call a structure ordinal-automatic if it is α-automatic for some ordinal α. We continue the study of ordinal-automatic structures initiated by Schlicht and Stephan as well as by Finkel and Todorčević. We develop a pumping lemma for α-automata (processing finite α-words, i.e., words of length α that have one fixed letter at all but finitely many positions). Using this pumping, we provide counterparts for the class of ordinal-automatic structures to several results on automatic structures:• Every finite word α-automatic structure has an injective finite word α-automatic presentation for all α < ω1 + ω ω . This bound is sharp. • We classify completely the finite word ω n -automatic Boolean algebras. Moreover, we show that the countable atomless Boolean algebra does not have an injective finite-word ordinal-automatic presentation. • We separate the class of finite-word ordinal-automatic structures from that of tree-automatic structures by showing that free term algebras with at least 2 generators (and one binary function) are not ordinal-automatic while the free term algebra with countable infinitely many generators is known to be tree-automatic. • For every ordinal α < ω1 + ω ω , we provide a sharp bound on the height of the finite word α-automatic well-founded order forests. • For every ordinal α < ω1 + ω ω , we provide a structure Fα that is complete for the class of finite-word α-automatic structures with respect to first-order interpretations. • As a byproduct, we also lift Schlicht and Stephans's characterisation of the injectively finite-word α-automatic ordinals to the noninjective setting.

Section: Introductionsupporting

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Pumping for ordinal-automatic structures1

Huschenbett

Kartzow

2017

COM

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“…Through this correspondence we obtain by [25,Proposition 16] Corollary 23. The rank of every scattered tree-automatic linear order is below ω ω .…”

Section: Lower and Upper Boundsmentioning

confidence: 99%

“…(b) It follows from [25] and the correspondence in Theorem 22 that maxord(C k ) = ω ω k+1 and minord(C k ) ≤ ω k+1 . To show that minord(C k ) ≥ ω k+1 , it is sufficient to consider a regular tree language B k such that a nodex is a branching node in some tree in B k iffx has at most k 0s and arbitrary many 1s.…”

Section: Lower and Upper Boundsmentioning

confidence: 99%

Tree-automatic scattered linear orders

Jain

Khoussainov

Theoretical Computer Science

et al. 2016

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Abstract. We study tree-automatic linear orders on regular tree languages. We first show that there is no tree-automatic scattered linear order, and in particular no well-order, on the set of all finite labeled trees. This also follows from results of Gurevich-Shelah [8] and Carayol-Löding [4]. We then show that a regular tree language admits a tree-automatic scattered linear order if and only if all trees are included in a subtree of the full binary tree with finite tree-rank. As a consequence of this characterization, we obtain an algorithm which, given a regular tree language, decides if the tree language can be well-ordered by a tree automaton. Finally, we connect tree automata with automata on ordinals and determine sharp lower and upper bounds for tree-automatic well-orders on natural examples of regular tree languages. Our proofs use elementary techniques of automata theory.

Structures without Scattered-Automatic Presentation

Kartzow

Lecture Notes in Computer Science

2013

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Bruyère 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 word-automatic structures. Analogously, for a linear order L there is the class of L-automatic structures. In this paper we prove the following limitations on the class of L-automatic structures for a fixed L of finite condensation rank 1 + α. Firstly, no scattered linear order with finite condensation rank above ω α+1 is Lautomatic. In particular, every L-automatic ordinal is below ω ω α+1. Secondly, we provide bounds on the (ordinal) height of well-founded order trees that are L-automatic. If α is finite or L is an ordinal, the height of such a tree is bounded by ω α+1 . Finally, we separate the class of tree-automatic structures from that of L-automatic structures for any ordinal L: the countable atomless boolean algebra is known to be tree-automatic, but we show that it is not L-automatic.