Turing Computable Embeddings, Computable Infinitary Equivalence, and Linear Orders

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10307)

Abstract

We study Turing computable embeddings for various classes of linear orders. The concept of a Turing computable embedding (or tc-embedding for short) was developed by Calvert, Cummins, Knight, and Miller as an effective counterpart for Borel embeddings. We are focused on tc-embeddings for classes equipped with computable infinitary\(\varSigma _{\alpha }\)equivalence, denoted by \(\sim ^c_{\alpha }\). In this paper, we isolate a natural subclass of linear orders, denoted by WMB, such that \((WMB,\cong )\) is not universal under tc-embeddings, but for any computable ordinal \(\alpha \ge 5\), \((WMB, \sim ^c_{\alpha })\) is universal under tc-embeddings. Informally speaking, WMB is not tc-universal, but it becomes tc-universal if one imposes some natural restrictions on the effective complexity of the syntax. We also give a complete syntactic characterization for classes \((K,\cong )\) that are Turing computably embeddable into some specific classes \((\mathcal {C},\cong )\) of well-orders. This extends the similar result of Knight, Miller, and Vanden Boom for the class of all finite linear orders \(\mathcal {C}_{fin}\).

Keywords

Turing computable embedding Linear order Ordinal Computable infinitary equivalence Computable structure 

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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