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Theory of Computing Systems

, Volume 61, Issue 4, pp 1254–1287 | Cite as

Semiautomatic Structures

  • Sanjay Jain
  • Bakhadyr Khoussainov
  • Frank Stephan
  • Dan Teng
  • Siyuan Zou
Article
Part of the following topical collections:
  1. Special Issue on Computability, Complexity and Randomness (CCR 2015)

Abstract

Semiautomatic structures generalise automatic structures in the sense that for some of the relations and functions in the structure one only requires the derived relations and functions are automatic when all but one input are filled with constants. One can also permit that this applies to equality in the structure so that only the sets of representatives equal to a given element of the structure are regular while equality itself is not an automatic relation on the domain of representatives. It is shown that one can find semiautomatic representations for the field of rationals and also for finite algebraic field extensions of it. Furthermore, one can show that infinite algebraic extensions of finite fields have semiautomatic representations in which the addition and equality are both automatic. Further prominent examples of semiautomatic structures are term algebras, any relational structure over a countable domain with a countable signature and any permutation algebra with a countable domain. Furthermore, examples of structures which fail to be semiautomatic are provided.

Keywords

Automatic structures Automata theory Basic algebraic structures Groups Rings Semiautomatic structures 

Notes

Acknowledgements

The authors would like to thank Anil Nerode as well as the participants of the IMS Workshop on Automata Theory and Applications in 2011 who discussed the topic and initial results with the authors.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Sanjay Jain
    • 1
  • Bakhadyr Khoussainov
    • 2
  • Frank Stephan
    • 1
    • 3
  • Dan Teng
    • 3
  • Siyuan Zou
    • 3
  1. 1.Department of Computer ScienceNational University of SingaporeSingaporeRepublic of Singapore
  2. 2.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand
  3. 3.Department of MathematicsThe National University of SingaporeSingaporeRepublic of Singapore

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