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Eliminating Unbounded Search in Computable Algebra

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Unveiling Dynamics and Complexity (CiE 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10307))

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Abstract

Klaimullin, Melnikov and Ng [KMNa] have recently suggested a new systematic approach to algorithms in algebra which is intermediate between computationally feasible algebra [CR91, KNRS07] and abstract computable structure theory [AK00, EG00]. In this short survey we discuss some of the key results and ideas of this new topic [KMNa, KMNc, KMNb]. We also suggest several open problems.

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Notes

  1. 1.

    We also agree that all finite structures are fpr.

  2. 2.

    If \(\mathcal {A}\) is fpr-categorical then \(\mathbf {FPR} (\mathcal {A})\) contains a unique degree. Nonetheless, \(A \simeq _{pr} B\) does not necessarily imply that there exists a fpr isomorphism \(A \rightarrow B\) (it is easy to construct a counter-example).

  3. 3.

    This means that all implications that are shown at the diagram above are proper. Furthermore, these implications (and their transitive closures) are the only implications that hold.

  4. 4.

    A total function g is primitively recursively reducible to a function f (\(g\le _{PR}f\)) if \(g=\Phi ^f\) for some f-primitive recursive schema \(\Phi ^f\). This leads to the definitions of \(g\equiv _{PR}f\) and \(g<_{PR}f\) as well as the notion of primitive recursive (PR-) degree. For a total function f let \(\{\Phi ^f_n\}\) be the Gödel numbering of all f-primitive recursive schemata for functions with one variable. Define the primitive recursive jump \(f'_{PR}\) to be the function \(f'_{PR}(n,x)=\Phi ^f_n(x).\) It is easy to check that \(f\le _{PR}g\implies f'_{PR}\le _{PR}g'_{PR}\) and \(f<_{PR}f'_{PR}.\)

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Authors and Affiliations

  1. Massey University, Auckland, New Zealand

    Alexander G. Melnikov

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  1. Alexander G. Melnikov
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Correspondence to Alexander G. Melnikov .

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Editors and Affiliations

  1. University of Turku, Turku, Finland

    Jarkko Kari

  2. Christian-Albrechts-University of Kiel, Kiel, Germany

    Florin Manea

  3. Åbo Akademi University, Turku, Finland

    Ion Petre

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Melnikov, A.G. (2017). Eliminating Unbounded Search in Computable Algebra. In: Kari, J., Manea, F., Petre, I. (eds) Unveiling Dynamics and Complexity. CiE 2017. Lecture Notes in Computer Science(), vol 10307. Springer, Cham. https://doi.org/10.1007/978-3-319-58741-7_8

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  • DOI: https://doi.org/10.1007/978-3-319-58741-7_8

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