Abstract
One of the main problems in effective model theory is to find an appropriate information complexity measure of the algebraic structures in the sense of computability. Unlike the commonly used degrees of structures, the structure degree measure is total. We introduce and study the jump operation for structure degrees. We prove that it has all natural jump properties (including jump inversion theorem, theorem of Ash), which show that our definition is relevant. We study the relation between the structure degree jump (in the sense of Soskov) and the jump degrees of a structure (in the sense of Jockusch) and give necessary and sufficient conditions for their existence in the terms of structure degrees. We show some properties, distinguishing the structure degrees from the enumeration degrees.
Similar content being viewed by others
References
Angelov, I.: Enumerable relations of structure degrees coding. Master's thesis, Sofia University, 1999 (in Bulgarian)
Ash, C.J.: Recursive labeling systems and stability of recursive structures in hyperarithmetical degrees. Trans. Amer. Math. Soc. 298, 497–514 (1986)
Ash, C.J.: Generalizations of enumeration reducibility using recursive infinitary propositional sentences. Ann. Pure Appl. Logic 58, 173–184 (1992)
Ash, C.J., Jockusch, C., Knight, J.F.: Jumps of orderings. Trans. Amer. Math. Soc. 319, 573–599 (1990)
Baleva, V.: Regular enumerations for abstract structures. Ann. Univ. Sofia 93, 39–48 (1999)
Baleva, V.: Structure degree jump. PhD thesis, Sofia University, 2001 (in Bulgarian)
Bouchkova, V.: Relative set genericity. To appear in Ann. Univ. Sofia, 2002
Case, J.: Maximal arithmetical reducibilities. Z. Math. Logik Grundlag. Math. 20, 261–270 (1974)
Coles, R., Downey, R., Slaman, T.: Every set has a least jump enumeration. Journal of London Math. Soc. 62 (2), 641–649 (2000)
Cooper, S.B.: Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ2 sets are dense. J. Symbolic Logic 49, 503–513 (1984)
Cooper, S.B.: Enumeration reducibilty, nondeterministic computations and relative computability of partial functions. In: Ambos-Spies, K., Muler, G., Sacks, G.E. (eds), Recursion theory week, Oberwolfach 1989, vol. 1432 of Lecture notes in mathematics, Heidelberg, Springer-Verlag, 1990, pp 57–110,
Downey, R.: Computability, definability and algebraic structures. In: Decheng D., et al., (ed), Proceedings of 7th and 8th Asian logic conference. World Scientific, 2003
Downey, R.G., Knight, J.F.: Orderings with α-th jump degree 0( α ). Proc. Amer. Math. Soc. 114, 545–552 (1992)
Jojgov, G.: Minimal pairs of structure degrees. Master's thesis, Sofia University, 1997 (in Bulgarian)
Knight, J.F.: Degrees coded in jumps of orderings. J. Symbolic Logic 51, 1034–1042 (1986)
Lacombe, D.: Deux généralisations de la notion de recursivité relative. C. R. de l'Academie des Sciences de Paris 258, 3410–3413 (1964)
McEvoy, K.: Jumps of quasi-minimal enumeration degrees. J. Symbolic Logic 50, 839–848 (1985)
Moschovakis, Y.N.: Abstract first order computability I. Trans. Amer. Math. Soc. 138, 427–464 (1969)
Moschovakis, Y.N.: Abstract first order computability II. Trans. Amer. Math. Soc. 138, 465–504 (1969)
Odifreddi, P.G.: Classical recursion theory. volume II. North–Holland, 1999
Richter, L.J.: Degrees of structures. J. Symbolic Logic 46, 723–731 (1981)
Rogers, Jr. H.: Theory of recursive functions and effective computability. McGraw-Hill Book Company, New York, 1967
Selman, A.L.: Arithmetical reducibilities I. Z. Math. Logik Grundlag. Math. 17, 335–350 (1971)
Shoenfield, J.R.: Mathematical logic. Addison-Wesley Publishing Company, 1967
Skordev, D.G.: Computability in combinatory spaces. Kluwer Academic Publishers, Dordrecht – Boston – London, 1992
Soskov, I.N.: A jump inversion theorem for the enumeration jump. Arch. Math. Logic 39, 417–437 (2000)
Soskov, I.N.: Abstract computability and definability. Doctor habil. thesis, Sofia University, 2001 (in Bulgarian)
Soskov, I.N.: Degree spectra and co-spectra of structures. To appear in Ann. Univ. Sofia, 2003
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Baleva, V. The jump operation for structure degrees. Arch. Math. Logic 45, 249–265 (2006). https://doi.org/10.1007/s00153-004-0245-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-004-0245-z