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Isomorphisms of scattered automatic linear orders

Dietrich Kuske
2014 Theoretical Computer Science  
We prove the undecidability of the existence of an isomorphism between scattered tree-automatic linear orders as well as the existence of automorphisms of scattered word automatic linear orders.  ...  Hence also the isomorphism problem for tree-automatic scattered linear orders is Π 0 1 -hard (Corollary 14). (2) The existence of a non-trivial automorphism of a word-automatic scattered linear order is  ...  Isomorphisms and automorphisms of linear orders on tree languages In this section, we will show that the isomorphism of scattered and tree-automatic linear orders is undecidable.  ... 
doi:10.1016/j.tcs.2014.03.022 fatcat:jb6ed4blzrflpngk2i7ukfypli

Isomorphisms of scattered automatic linear orders [article]

Dietrich Kuske
2012 arXiv   pre-print
We prove that the isomorphism of scattered tree automatic linear orders as well as the existence of automorphisms of scattered word automatic linear orders are undecidable.  ...  For the existence of automatic automorphisms of word automatic linear orders, we determine the exact level of undecidability in the arithmetical hierarchy.  ...  Isomorphisms and automorphisms of linear orders on trees In this section, we will show that the isomorphism of scattered and tree automatic linear orders is undecidable.  ... 
arXiv:1204.5653v1 fatcat:fcfmmwk54zgfverue6k7cmxeve

Tree-automatic scattered linear orders

Sanjay Jain, Bakhadyr Khoussainov, Philipp Schlicht, Frank Stephan
2016 Theoretical Computer Science  
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.  ...  Then there is no tree-automatic scattered linear order on A.  ... 
doi:10.1016/j.tcs.2016.02.008 fatcat:nek7nicerfbmvhbfwwyqdg4t7e

The isomorphism problem on classes of automatic structures with transitive relations

Dietrich Kuske, Jiamou Liu, Markus Lohrey
2013 Transactions of the American Mathematical Society  
(iv) The isomorphism problem for automatic order trees is Σ 1 1 -complete. (v) The isomorphism problem for automatic linear orders is Σ 1 1 -complete.  ...  We also obtain Π 0 1 -completeness of the elementary equivalence problem for several classes of automatic structures and Σ 1 1 -completeness of the isomorphism problem for trees (resp., linear orders)  ...  Hence, the techniques from [26, 34] do not work for automatic order trees, automatic linear orders, and also automatic equivalence relations.  ... 
doi:10.1090/s0002-9947-2013-05766-2 fatcat:ugsemlautrajjnlvmgs47dy7v4

The Rank of Tree-Automatic Linear Orderings [article]

Martin Huschenbett
2012 arXiv   pre-print
We further investigate a restricted form of tree-automaticity and prove that every linear ordering which admits a tree-automatic presentation of branching complexity at most k has FC-rank strictly below  ...  We generalise Delhommé's result that each tree-automatic ordinal is strictly below ω^ω^ω by showing that any tree-automatic linear ordering has FC-rank strictly below ω^ω.  ...  . ◭ Tree-Automatic Linear Orderings The objective of this section is to prove our main result, namely Theorem 4.4, which states that every tree-automatic linear ordering has FC-rank below ω ω .  ... 
arXiv:1204.3048v1 fatcat:sug7g6hikvcyhhuw5gayo7xx2m

The Isomorphism Problem on Classes of Automatic Structures

Dietrich Kuske, Jiamou Liu, Markus Lohrey
2010 2010 25th Annual IEEE Symposium on Logic in Computer Science  
Several undecidability results on isomorphism problems for automatic structures are shown: (i) The isomorphism problem for automatic equivalence relations is The isomorphism problem for automatic linear  ...  orders is not arithmetical.  ...  . ⊓ ⊔ Automatic Linear Orders Our main result for automatic linear orders is: Theorem 4. 1 The isomorphism problem for the class of automatic linear orders is at least as hard as FOTh(N; +, ×).  ... 
doi:10.1109/lics.2010.10 dblp:conf/lics/KuskeLL10 fatcat:hvupoq4nl5cdhcolmfnsef3ice

The Isomorphism Problem On Classes of Automatic Structures [article]

Dietrich Kuske, Jiamou Liu, Markus Lohrey
2010 arXiv   pre-print
(ii) The isomorphism problem for automatic trees of height n ≥ 2 is Π^0_2n-3-complete. (iii) The isomorphism problem for automatic linear orders is not arithmetical.  ...  Automatic structures are finitely presented structures where the universe and all relations can be recognized by finite automata.  ...  For any k ∈ N, there exist two isomorphic automatic trees of finite height (and two automatic linear orders) without any Σ 0 k -isomorphism. Proof.  ... 
arXiv:1001.2086v1 fatcat:ydzxgpdx3zedpfe7vttcur7xs4

Word Automaticity of Tree Automatic Scattered Linear Orderings Is Decidable [article]

Martin Huschenbett
2012 arXiv   pre-print
The question naturally arises whether a given tree automatic structure is already word automatic. We prove that this problem is decidable for tree automatic scattered linear orderings.  ...  Words can be regarded as specific simple trees and a structure is word automatic if it is encodable using only these trees.  ...  Tree automatic structures and tree automatic presentations are defined like in the word automatic case, but based on trees and tree automata. Linear Orderings.  ... 
arXiv:1201.5070v1 fatcat:nujpbiqkunafda4p4e3nlfngtu

Word Automaticity of Tree Automatic Scattered Linear Orderings Is Decidable [chapter]

Martin Huschenbett
2012 Lecture Notes in Computer Science  
The question naturally arises whether a given tree automatic structure is already word automatic. We prove that this problem is decidable for tree automatic scattered linear orderings.  ...  Words can be regarded as specific simple trees and a structure is word automatic if it is encodable using only these trees.  ...  Tree automatic structures and tree automatic presentations are defined like in the word automatic case, but based on trees and tree automata. Linear Orderings.  ... 
doi:10.1007/978-3-642-30870-3_32 fatcat:zwc5ek5fynhg7gbaktpap7b2je

Three Lectures on Automatic Structures [article]

Bakhadyr Khoussainov, Mia Minnes
2008 arXiv   pre-print
We discuss variants of automatic structures related to several models of computation: word automata, tree automata, Buchi automata, and Rabin automata.  ...  The study of properties of automatic structures is a relatively new and very active area of research.  ...  Finally, we consider partial orders as trees and consider the branching complexity of automatic partial order trees.  ... 
arXiv:0809.3430v1 fatcat:s5o3fh45bbf3dmo6ms3h6pjiia

Structures Without Scattered-Automatic Presentation [article]

Alexander Kartzow, Philipp Schlicht
2013 arXiv   pre-print
Bruyere and Carton lifted the notion of finite automata reading infinite words to finite automata reading words with shape an arbitrary linear order L.  ...  Firstly, no scattered linear order with finite condensation rank above ω^(α+1) is L-α-automatic. In particular, every L-automatic ordinal is below ω^ω^(α+1).  ...  . ⊓ ⊔ D.2 Kleene-Brouwer Orders of Trees Let T = (T, ⊑) be a tree and let L = (T, ) be a linear order.  ... 
arXiv:1304.0912v1 fatcat:g7qdutt5tnbgvmofjl2azna33m

Structures without Scattered-Automatic Presentation [chapter]

Alexander Kartzow, Philipp Schlicht
2013 Lecture Notes in Computer Science  
Bruyère and Carton lifted the notion of finite automata reading infinite words to finite automata reading words with shape an arbitrary linear order L.  ...  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 .  ...  T = (T, ⊑) be a tree and let L = (T, ) be a linear order.  ... 
doi:10.1007/978-3-642-39053-1_32 fatcat:hmtbiuz5i5cx7ay6vaywlu65li

Deciding the isomorphism problem in classes of unary automatic structures

Jiamou Liu, Mia Minnes
2011 Theoretical Computer Science  
We solve the isomorphism problem for certain classes of unary automatic structures: unary automatic equivalence relations, unary automatic linear orders, and unary automatic trees.  ...  Minnes). 1 A related paper which focuses on time and space complexity for these unary automatic structures has been submitted for publication by the authors elsewhere. 0304-3975/$ -see front matter  ...  We described algorithms deciding the isomorphism problems for unary automatic equivalence structures, linear orders, and trees. This settled the question of whether such algorithms existed.  ... 
doi:10.1016/j.tcs.2010.12.045 fatcat:6uz3zolnrbbi7mowytvralvo3e

Analysing Complexity in Classes of Unary Automatic Structures [chapter]

Jiamou Liu, Mia Minnes
2009 Lecture Notes in Computer Science  
In particular, we focus on automatic equivalence relations, linear orders, trees, and graphs with finite degree.  ...  This paper addresses the complexity of several families of queries in classes of unary automatic structures. The queries include membership and isomorphism.  ...  A linear order is unary automatic if and only if it is isomorphic to a finite sum of linear orders of type ω, ω * or n. Corollary 1.  ... 
doi:10.1007/978-3-642-00982-2_44 fatcat:kr7ktxo4qjcg5nzk6r5fiepo7q

Ordered Types for Stream Processing of Tree-Structured Data

Ryosuke Sato, Kohei Suenaga, Naoki Kobayashi
2011 Journal of Information Processing  
This paper extends their framework by introducing ordered, non-linear types in addition to ordered linear types.  ...  The key ingredient of the framework was the use of ordered linear types to guarantee that a treeprocessing program traverses an input tree just once in the depth-first, left-to-right order (so that the  ...  trees and ordered linear trees being captured in the closure. • In the rules T-App, T-Plus, T-Node, and T-MNode, the ordered linear and the ordered type environments of M1 and M2 are concatenated in this  ... 
doi:10.2197/ipsjjip.19.74 fatcat:omsrspzvlnhetivvarlc5ysyku
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