1,425 research outputs found
Infinite and Bi-infinite Words with Decidable Monadic Theories
We study word structures of the form where is either
or , is the natural linear ordering on and
is a predicate on . In particular we show:
(a) The set of recursive -words with decidable monadic second order
theories is -complete.
(b) Known characterisations of the -words with decidable monadic
second order theories are transfered to the corresponding question for
bi-infinite words.
(c) We show that such "tame" predicates exist in every Turing degree.
(d) We determine, for , the number of predicates
such that and
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
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
in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference,
200
Model Theoretic Complexity of Automatic Structures
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
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
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