70,787 research outputs found

    A Hierarchy of Tree-Automatic Structures

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    We consider ωn\omega^n-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length ωn\omega^n for some integer n1n\geq 1. We show that all these structures are ω\omega-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for ω2\omega^2-automatic (resp. ωn\omega^n-automatic for n>2n>2) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for ωn\omega^n-automatic boolean algebras, n>1n > 1, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a Σ21\Sigma_2^1-set nor a Π21\Pi_2^1-set. We obtain that there exist infinitely many ωn\omega^n-automatic, hence also ω\omega-tree-automatic, atomless boolean algebras BnB_n, n1n\geq 1, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [FT10].Comment: To appear in The Journal of Symbolic Logic. arXiv admin note: substantial text overlap with arXiv:1007.082

    Tree-Automatic Well-Founded Trees

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    We investigate tree-automatic well-founded trees. Using Delhomme's decomposition technique for tree-automatic structures, we show that the (ordinal) rank of a tree-automatic well-founded tree is strictly below omega^omega. Moreover, we make a step towards proving that the ranks of tree-automatic well-founded partial orders are bounded by omega^omega^omega: we prove this bound for what we call upwards linear partial orders. As an application of our result, we show that the isomorphism problem for tree-automatic well-founded trees is complete for level Delta^0_{omega^omega} of the hyperarithmetical hierarchy with respect to Turing-reductions.Comment: Will appear in Logical Methods of Computer Scienc

    Automatic Ordinals

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    We prove that the injectively omega-tree-automatic ordinals are the ordinals smaller than ωωω\omega^{\omega^\omega}. Then we show that the injectively ωn\omega^n-automatic ordinals, where n>0n>0 is an integer, are the ordinals smaller than ωωn\omega^{\omega^n}. This strengthens a recent result of Schlicht and Stephan who considered in [Schlicht-Stephan11] the subclasses of finite word ωn\omega^n-automatic ordinals. As a by-product we obtain that the hierarchy of injectively ωn\omega^n-automatic structures, n>0, which was considered in [Finkel-Todorcevic12], is strict.Comment: To appear in a Special Issue on New Worlds of Computation 2011 of the International Journal of Unconventional Computing. arXiv admin note: text overlap with arXiv:1111.150

    Collapsible Pushdown Graphs of Level 2 are Tree-Automatic

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    We show that graphs generated by collapsible pushdown systems of level 2 are tree-automatic. Even when we allow ϵ\epsilon-contractions and add a reachability predicate (with regular constraints) for pairs of configurations, the structures remain tree-automatic. Hence, their FO theories are decidable, even when expanded by a reachability predicate. As a corollary, we obtain the tree-automaticity of the second level of the Caucal-hierarchy.Comment: 12 pages Accepted for STACS 201

    Transforming structures by set interpretations

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    We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic second-order (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of elements of the orignal structure. The definition of these interpretations directly implies that they send structures with a decidable WMSO theory to structures with a decidable first-order theory. In this paper, we investigate the expressive power of such interpretations applied to infinite deterministic trees. The results can be used in the study of automatic and tree-automatic structures.Comment: 36 page

    Advances and applications of automata on words and trees : abstracts collection

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    From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501 "Advances and Applications of Automata on Words and Trees" was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

    The Isomorphism Relation Between Tree-Automatic Structures

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    An ω\omega-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω\omega-tree-automatic structures. We prove first that the isomorphism relation for ω\omega-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for ω\omega-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is neither a Σ21\Sigma_2^1-set nor a Π21\Pi_2^1-set

    Adaptive content mapping for internet navigation

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    The Internet as the biggest human library ever assembled keeps on growing. Although all kinds of information carriers (e.g. audio/video/hybrid file formats) are available, text based documents dominate. It is estimated that about 80% of all information worldwide stored electronically exists in (or can be converted into) text form. More and more, all kinds of documents are generated by means of a text processing system and are therefore available electronically. Nowadays, many printed journals are also published online and may even discontinue to appear in print form tomorrow. This development has many convincing advantages: the documents are both available faster (cf. prepress services) and cheaper, they can be searched more easily, the physical storage only needs a fraction of the space previously necessary and the medium will not age. For most people, fast and easy access is the most interesting feature of the new age; computer-aided search for specific documents or Web pages becomes the basic tool for information-oriented work. But this tool has problems. The current keyword based search machines available on the Internet are not really appropriate for such a task; either there are (way) too many documents matching the specified keywords are presented or none at all. The problem lies in the fact that it is often very difficult to choose appropriate terms describing the desired topic in the first place. This contribution discusses the current state-of-the-art techniques in content-based searching (along with common visualization/browsing approaches) and proposes a particular adaptive solution for intuitive Internet document navigation, which not only enables the user to provide full texts instead of manually selected keywords (if available), but also allows him/her to explore the whole database
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