22 research outputs found

    The Complexity of Orbits of Computably Enumerable Sets

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    The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, \E, such that the question of membership in this orbit is Σ11\Sigma^1_1-complete. This result and proof have a number of nice corollaries: the Scott rank of \E is \wock +1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of \E; for all finite α≥9\alpha \geq 9, there is a properly Δα0\Delta^0_\alpha orbit (from the proof). A few small corrections made in this versionComment: To appear in the Bulletion of Symbolic Logi

    Bounded low and high sets

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    Anderson and Csima (Notre Dame J Form Log 55(2):245–264, 2014) defined a jump operator, the bounded jump, with respect to bounded Turing (or weak truth table) reducibility. They showed that the bounded jump is closely related to the Ershov hierarchy and that it satisfies an analogue of Shoenfield jump inversion. We show that there are high bounded low sets and low bounded high sets. Thus, the information coded in the bounded jump is quite different from that of the standard jump. We also consider whether the analogue of the Jump Theorem holds for the bounded jump: do we have A ≤bT B if and only if Ab ≤1 Bb ? We show the forward direction holds but not the reverse

    Degrees of Categoricity and the Isomorphism Problem

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    In this thesis, we study notions of complexity related to computable structures. We first study degrees of categoricity for computable tree structures. We show that, for any computable ordinal α\alpha, there exists a computable tree of rank α+1\alpha+1 with strong degree of categoricity 0(2α){\bf 0}^{(2\alpha)} if α\alpha is finite, and with strong degree of categoricity 0(2α+1){\bf 0}^{(2\alpha+1)} if α\alpha is infinite. For a computable limit ordinal α\alpha, we show that there is a computable tree of rank α\alpha with strong degree of categoricity 0(α){\bf 0}^{(\alpha)} (which equals 0(2α){\bf 0}^{(2\alpha)}). In general, it is not the case that every Turing degree is the degree of categoricity of some structure. However, it is known that every degree that is of a computably enumerable (c.e.) set\ in and above 0(α)\mathbf{0}^{(\alpha)}, for α\alpha a successor ordinal, is a degree of categoricity. In this thesis, we include joint work with Csima, Deveau and Harrison-Trainor which shows that every degree c.e.\ in and above 0(α)\mathbf{0}^{(\alpha)}, for α\alpha a limit ordinal, is a degree of categoricity. We also show that every degree c.e.\ in and above 0(ω)\mathbf{0}^{(\omega)} is the degree of categoricity of a prime model, making progress towards a question of Bazhenov and Marchuk. After that, we study the isomorphism problem for tree structures. It follows from our proofs regarding the degrees of categoricity for these structures that, for every computable ordinal α>0\alpha>0, the isomorphism problem for trees of rank α\alpha is Π2α\Pi_{2\alpha}-complete. We also discuss the isomorphism problem for pregeometries in which dependent elements are dense and the closure operator is relatively intrinsically computably enumerable. We show that, if KK is a class of such pregeometries, then the isomorphism problem for the class KK is Π3\Pi_3-hard. Finally, we study the Turing ordinal. We observed that the definition of the Turing ordinal has two parts each of which alone can define a specific ordinal which we call the upper and lower Turing ordinals. The Turing ordinal exists if and only if these two ordinals exist and are equal. We give examples of classes of computable structures such that the upper Turing ordinal is β\beta and the lower Turing ordinal is α\alpha for all computable ordinals α<β\alpha<\beta

    The Machine as Data: A Computational View of Emergence and Definability

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    Turing’s (Proceedings of the London Mathematical Society 42:230–265, 1936) paper on computable numbers has played its role in underpinning different perspectives on the world of information. On the one hand, it encourages a digital ontology, with a perceived flatness of computational structure comprehensively hosting causality at the physical level and beyond. On the other (the main point of Turing’s paper), it can give an insight into the way in which higher order information arises and leads to loss of computational control—while demonstrating how the control can be re-established, in special circumstances, via suitable type reductions. We examine the classical computational framework more closely than is usual, drawing out lessons for the wider application of information–theoretical approaches to characterizing the real world. The problem which arises across a range of contexts is the characterizing of the balance of power between the complexity of informational structure (with emergence, chaos, randomness and ‘big data’ prominently on the scene) and the means available (simulation, codes, statistical sampling, human intuition, semantic constructs) to bring this information back into the computational fold. We proceed via appropriate mathematical modelling to a more coherent view of the computational structure of information, relevant to a wide spectrum of areas of investigation
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