Representing Hyper-arithmetical Sets by Equations over Sets of Integers

@article{Je2012RepresentingHS,
  title={Representing Hyper-arithmetical Sets by Equations over Sets of Integers},
  author={Artur Jeż and Alexander Okhotin},
  journal={Theory of Computing Systems},
  year={2012},
  volume={51},
  pages={196-228}
}
Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition, defined as S+T={m+n∣m∈S,n∈T}, and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural… 

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