110 research outputs found

    Model Theoretic Complexity of Automatic Structures

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    We study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic well- founded partial order is bounded by \omega^\omega ; 2) The ordinal heights of automatic well-founded relations are unbounded below the first non-computable ordinal; 3) For any computable ordinal there is an automatic structure of Scott rank at least that ordinal. Moreover, there are automatic structures of Scott rank the first non-computable ordinal and its successor; 4) For any computable ordinal, there is an automatic successor tree of Cantor-Bendixson rank that ordinal.Comment: 23 pages. Extended abstract appeared in Proceedings of TAMC '08, LNCS 4978 pp 514-52

    Automatic Equivalence Structures of Polynomial Growth

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    In this paper we study the class EqP of automatic equivalence structures of the form ?=(D, E) where the domain D is a regular language of polynomial growth and E is an equivalence relation on D. Our goal is to investigate the following two foundational problems (in the theory of automatic structures) aimed for the class EqP. The first is to find algebraic characterizations of structures from EqP, and the second is to investigate the isomorphism problem for the class EqP. We provide full solutions to these two problems. First, we produce a characterization of structures from EqP through multivariate polynomials. Second, we present two contrasting results. On the one hand, we prove that the isomorphism problem for structures from the class EqP is undecidable. On the other hand, we prove that the isomorphism problem is decidable for structures from EqP with domains of quadratic growth

    On algebraic and logical specifications of classes of regular languages

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    AbstractThe paper studies classes of regular languages based on algebraic constraints imposed on transitions of automata and discusses issues related to specifications of these classes from algebraic, computational and logical points of view

    The isomorphism problem for tree-automatic ordinals with addition

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    This paper studies tree-automatic ordinals (or equivalently, well-founded linearly ordered sets) together with the ordinal addition operation +. Informally, these are ordinals such that their elements are coded by finite trees for which the linear order relation of the ordinal and the ordinal addition operation can be determined by tree automata. We describe an algorithm that, given two tree-automatic ordinals with the ordinal addition operation, decides if the ordinals are isomorphic

    LINEAR ORDERS REALIZED BY CE EQUIVALENCE RELATIONS

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    AbstractLetEbe a computably enumerable (c.e.) equivalence relation on the setωof natural numbers. We say that the quotient setω/E\omega /E(or equivalently, the relationE)realizesa linearly ordered setL{\cal L}if there exists a c.e. relation ⊴ respectingEsuch that the induced structure (ω/E\omega /E; ⊴) is isomorphic toL{\cal L}. Thus, one can consider the class of all linearly ordered sets that are realized byω/E\omega /E; formally,K(E)={L ∣ the order − type L is realized by E}{\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized byE. One can also define the following pre-order≤lo \le _{lo} on the class of all c.e. equivalence relations:E1≤loE2E_1 \le _{lo} E_2 if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order≤lo \le _{lo} . We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees.</jats:p
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