The complexity of quickly decidable ORM-decidable sets

  • J. D. Hamkins, D. Linetsky, and R. Miller, “The Complexity of Quickly Decidable ORM-Decidable Sets,” in Computation and Logic in the Real World – Third Conference of Computability in Europe CiE 2007, Siena, Italy, 2007, pp. 488-496.  
    @INPROCEEDINGS{HamkinsLinetskyMiller2007:ComplexityOfQuicklyDecidableORMSets,
    AUTHOR = "Joel David Hamkins and David Linetsky and Russell Miller",
    TITLE = "The Complexity of Quickly Decidable {ORM}-Decidable Sets",
    BOOKTITLE = "Computation and Logic in the Real World - Third Conference of Computability in Europe CiE 2007",
    YEAR = "2007",
    editor = "Barry Cooper and Benedikt {L\"owe} and Andrea Sorbi",
    volume = "4497",
    number = "",
    series = "Proceedings, Lecture Notes in Computer Science",
    pages = "488--496",
    address = "Siena, Italy",
    month = "",
    organization = "",
    publisher = "",
    note = "",
    abstract = "",
    keywords = "",
    doi = {10.1007/978-3-540-73001-9_51},
    ee = {http://dx.doi.org/10.1007/978-3-540-73001-9_51},
    bibsource = {DBLP, http://dblp.uni-trier.de},
    file = F
    }

The Ordinal Register Machine (ORM) is one of several different machine models for infinitary computability. We classify, by complexity, the sets that can be decided quickly by ORMs. In particular, we show that the arithmetical sets are exactly those sets that can be decided by ORMs in times uniformly less than $\omega^\omega$. Further, we show that the hyperarithmetical sets are exactly those sets that can be decided by an ORM in time uniformly less than $\omega_1^{CK}$.

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