Decidability of the AE-theory of the lattice of \({\varPi }_1^0\) classes

Article

Abstract

An AE-sentence is a sentence in prenex normal form with all universal quantifiers preceding all existential quantifiers, and the AE-theory of a structure is the set of all AE-sentences true in the structure. We show that the AE-theory of \((\mathscr {L}({\varPi }_1^0), \cap , \cup , 0, 1)\) is decidable by giving a procedure which, for any AE-sentence in the language, determines the truth or falsity of the sentence in our structure.

Keywords

Lattice Computability Decidability \({\varPi }^0_1\) class 

Mathematics Subject Classification

03D 03G10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cenzer, D., Jockusch, C.: \(\Pi ^0_1\) classes–structure and applications. In: Cholak, P.A., Lempp, S., Lerman, M., Shore, R.A. (eds.) Computability Theory and Its Applications (Boulder, CO, 1999), Contemp. Math. vol. 257, pp. 39–59. American Mathematical Society, Providence (2000)Google Scholar
  2. 2.
    Cholak, P., Coles, R., Downey, R., Herrmann, E.: Automorphisms of the lattice of \(\Pi ^0_1\) classes. Trans. Am. Math. Soc. 353, 4899–4924 (2001) (electronic) Google Scholar
  3. 3.
    Lachlan, A.H.: The elementary theory of recursively enumerable sets. Duke Math. J. 35, 123–146 (1968)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Lawton, L.: Almost complemented \(\Pi _1^0\) classes. Arch. Math. Log. 45, 555–568 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Lempp, S., Lerman, M.: The decidability of the existential theory of the poset of recursively enumerable degrees with jump relations. Adv. Math. 120, 1–142 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Lerman, M.: Degrees of Unsolvability, Perspectives in Mathematical Logic. Springer, Berlin (1983)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Northern Michigan UniversityMarquetteUSA

Personalised recommendations