1,421 research outputs found

    Infinite and Bi-infinite Words with Decidable Monadic Theories

    Get PDF
    We study word structures of the form (D,<,P)(D,<,P) where DD is either N\mathbb{N} or Z\mathbb{Z}, << is the natural linear ordering on DD and P⊆DP\subseteq D is a predicate on DD. In particular we show: (a) The set of recursive ω\omega-words with decidable monadic second order theories is Σ3\Sigma_3-complete. (b) Known characterisations of the ω\omega-words with decidable monadic second order theories are transfered to the corresponding question for bi-infinite words. (c) We show that such "tame" predicates PP exist in every Turing degree. (d) We determine, for P⊆ZP\subseteq\mathbb{Z}, the number of predicates Q⊆ZQ\subseteq\mathbb{Z} such that (Z,≤,P)(\mathbb{Z},\le,P) and (Z,≤,Q)(\mathbb{Z},\le,Q) are indistinguishable. Through these results we demonstrate similarities and differences between logical properties of infinite and bi-infinite words

    Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

    Full text link
    We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class Δ21\bm{\Delta}^1_2 in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference, 200

    Model Theoretic Complexity of Automatic Structures

    Get PDF
    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

    Spectra of Monadic Second-Order Formulas with One Unary Function

    Full text link
    We establish the eventual periodicity of the spectrum of any monadic second-order formula where: (i) all relation symbols, except equality, are unary, and (ii) there is only one function symbol and that symbol is unary
    • …
    corecore