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Computability of Algebraic and Definable Closure

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Book cover Logical Foundations of Computer Science (LFCS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11972))

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Abstract

We consider computability-theoretic aspects of the algebraic and definable closure operations for formulas. We show that for \(\varphi \) a Boolean combination of \(\varSigma _n\)-formulas and in a given computable structure, the set of parameters for which the closure of \(\varphi \) is finite is \(\varSigma ^0_{n+2}\), and the set of parameters for which the closure is a singleton is \(\varDelta ^0_{n+2}\). In addition, we construct examples witnessing that these bounds are tight.

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References

  1. Ackerman, N., Freer, C., Patel, R.: Invariant measures concentrated on countable structures. Forum Math. Sigma 4(e17), 59 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Chen, R., Kechris, A.S.: Structurable equivalence relations. Fund. Math. 242(2), 109–185 (2018)

    Article  MathSciNet  Google Scholar 

  3. Cherlin, G., Shelah, S., Shi, N.: Universal graphs with forbidden subgraphs and algebraic closure. Adv. Appl. Math. 22(4), 454–491 (1999)

    Article  MathSciNet  Google Scholar 

  4. Harizanov, V.S.: Pure computable model theory. In: Handbook of recursive mathematics, Logic and the Foundations of Mathematics, North-Holland, vol. 138, pp. 3–114 (1998)

    Google Scholar 

  5. Hodges, W.: Model theory, Encyclopedia of Mathematics and its Applications, vol. 42. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  6. Soare, R.I.: Turing Computability. Theory and Applications of Computability. Springer, Berlin (2016)

    Book  Google Scholar 

  7. Tent, K., Ziegler, M.: A Course in Model Theory. Lecture Notes in Logic, vol. 40. Cambridge University Press (2012)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Harvard University, Cambridge, MA, 02138, USA

    Nathanael Ackerman

  2. Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Cameron Freer

  3. African Institute for Mathematical Sciences, M’bour–Thiès, Senegal

    Rehana Patel

Authors
  1. Nathanael Ackerman
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  2. Cameron Freer
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  3. Rehana Patel
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Corresponding author

Correspondence to Cameron Freer .

Editor information

Editors and Affiliations

  1. The Graduate Center, CUNY, New York, NY, USA

    Sergei Artemov

  2. Cornell University, Ithaca, NY, USA

    Anil Nerode

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Ackerman, N., Freer, C., Patel, R. (2020). Computability of Algebraic and Definable Closure. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2020. Lecture Notes in Computer Science(), vol 11972. Springer, Cham. https://doi.org/10.1007/978-3-030-36755-8_1

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