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Definability in the Recursively Enumerable Degrees

  • André Nies (a1), Richard A. Shore (a2) and Theodore A. Slaman (a3)

§1. Introduction. Natural sets that can be enumerated by a computable function (the recursively enumerable or r.e. sets) always seem to be either actually computable (recursive) or of the same complexity (with respect to Turing computability) as the Halting Problem, the complete r.e. set K. The obvious question, first posed in Post [1944] and since then called Post's Problem is then just whether there are r.e. sets which are neither computable nor complete, i.e., neither recursive nor of the same Turing degree as K?

Let be the r.e. degrees, i.e., the r.e. sets modulo the equivalence relation of equicomputable with the partial order induced by Turing computability. This structure is a partial order (indeed, an uppersemilattice or usl)with least element 0, the degree (equivalence class) of the computable sets, and greatest element 1 or 0′, the degree of K. Post's problem then asks if there are any other elements of .

The (positive) solution of Post's problem by Friedberg [1957] and Muchnik [1956] was followed by various algebraic or order theoretic results that were interpreted as saying that the structure was in some way well behaved:

Theorem 1.1 (Embedding theorem; Muchnik [1958], Sacks [1963]). Every countable partial ordering or even uppersemilattice can be embedded into .

Theorem 1.2 (Sacks Splitting Theorem [1963b]). For every nonrecursive r.e. degreeathere are r.e. degreesb, c < asuch thatbc = a.

Theorem 1.3 (Sacks Density Theorem [1964]). For every pair of nonrecursive r.e. degreesa < bthere is an r.e. degreecsuch thata < c < b.

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[1984] K. Ambos-Spies , C. G. Jockusch Jr., R. A. Shore , and R. I. Soare , An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees, Transactions of the American Mathematical Society, vol. 281, pp. 109128.

[1993] K. Ambos-Spies and R. A. Shore , Undecidability and 1-types in the r.e. degrees, Annals of Pure and Applied Logic, vol. 63, pp. 337.

[1990] R. G. Downey , Lattice nonembeddings and initial segments of the recursively enumerable degrees, Annals of Pure and Applied Logic, vol. 49, pp. 97119.

[1957] R. M. Friedberg , Two recursively enumerable sets of incomparable degrees of un-solvability, Proceeding ofthe National Academy ofSciences, vol. 43, pp. 236238.

[1982] L. Harrington and S. Shelah , The undecidability of the recursively enumerable degrees (research announcement), Bulletin of the American Mathematical Society, New Series, vol. 6, pp. 7980.

[1993] W. Hodges , Model theory, Cambridge University Press, Cambridge, England.

[1983] C. G. Jockusch Jr. and R. A. Shore , Pseudo-jump operators I: the r.e. case, Transactions ofthe American Mathematical Society, vol. 275, pp. 599609.

[1972] A. H. Lachlan , Embedding nondistributive lattices in the recursively enumerable degrees, Conference in Mathematical Logic, London, 1970 ( W. Hodges , editor), LNMS, vol. 255, Springer-Verlag, Berlin, pp. 149172.

[1975] A. H. Lachlan , A recursively enumerable degree which will not split over all lesser ones, Annals of Mathematical Logic, vol. 9, pp. 307365.

[1980] A. H. Lachlan and R. I. Soare , Not every finite lattice is embeddable in the recursively enumerable degrees, Advances in Mathematics, vol. 37, pp. 7482.

[1944] E. L. Post , Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50, pp. 284316.

[1971] R. W. Robinson , Interpolation and embedding in the recursively enumerable degrees, Annals of Mathematics. Second Series, vol. 93, pp. 285314.

[1963a] G. E. Sacks , Recursive enumerability and the jump operator, Transactions of the American Mathematical Society, vol. 108, pp. 223239.

[1963b] G. E. Sacks , On the degrees less than 0′, Annals of Mathematics. Second Series, vol. 77, pp. 211231.

[1964] G. E. Sacks , The recursively enumerable degrees are dense, Annals of Mathematics. Second Series, vol. 80, pp. 300312.

[1979] R. A. Shore , The homogeneity conjecture, Proceedings of the National Academy of Sciences, vol. 76, pp. 42184219.

[1982] R. A. Shore , Finitely generated codings and the degrees r.e. in a degree d, Proceedings of the American Mathematical Society, vol. 84, pp. 256263.

[1990] R. A. Shore and T. A. Slaman , Working below a low2recursively enumerable degree, Archive for Mathematical Logic, vol. 29, pp. 201211.

[1987] R. I. Soare , Recursively enumerable sets and degrees, Springer-Verlag, Berlin.

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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
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