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  • Cited by 3
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Downey, R.G. 1989. Intervals and sublattices of the r.e. weak truth table degrees, part II: Nonbounding. Annals of Pure and Applied Logic, Vol. 44, Issue. 3, p. 153.


    Downey, R.G. 1989. Intervals and sublattices of the R.E. weak truth table degrees, part I: Density. Annals of Pure and Applied Logic, Vol. 41, Issue. 1, p. 1.


    Downey, R.G. and Stob, M. 1986. Structural interactions of the recursively enumerable T- and W-degrees. Annals of Pure and Applied Logic, Vol. 31, p. 205.


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Pairs without infimum in the recursively enumerable weak truth table degrees

  • Paul Fischer (a1)
  • DOI: http://dx.doi.org/10.2307/2273948
  • Published online: 12 March 2014
Abstract

wtt-reducibility has become of some importance in the last years through the works of Ladner and Sasso [1975], Stob [1983] and Ambos-Spies [1984]. It differs from Turing reducibility by a recursive bound on the use of the reduction. This makes some proofs easier in the wtt degrees than in the Turing degrees. Certain proofs carry over directly from the Turing to the wtt degrees, especially those based on permitting. But the converse is also possible. There are some r.e. Turing degrees which consist of a single r.e. wtt degree (the so-called contiguous degrees; see Ladner and Sasso [1975]). Thus it suffices to prove a result about contiguous wtt degrees using an easier construction, and it carries over to the corresponding Turing degrees.

In this work we prove some results on pairs of r.e. wtt degrees which have no infimum. The existence of such a pair has been shown by Ladner and Sasso. Here we use a different technique of Jockusch [1981] to prove this result (Theorem 1) together with some stronger ones. We show that a pair without infimum exists above a given incomplete wtt degree (Theorem 5) and below any promptly simple wtt degree (Theorem 12). In Theorem 17 we prove, however, that there are r.e. wtt degrees such that any pair below them has an infimum. This shows that certain initial segments of the wtt degrees are lattices. Thus there is a structural difference between the wtt and Turing degrees where the pairs without infimum are dense (Ambos-Spies [1984]).

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

K. Ambos-Spies [1984], On pairs of recursively enumerable degrees, Transactions of the American Mathematical Society, vol. 283, pp. 507531.

K. Ambos-Spies [1985], Cupping and noncapping in the r.e. weak truth table and Turing degrees, Archiv für Mathematische Logik und Grundlagenforschung (to appear).

K. Ambos-Spies , C. G. Jockusch Jr., R. A. Shore and R. I. Soare [1984], 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.

C. G. Jockusch Jr., [1981], Three easy constructions of recursively enumerable sets, Logic Year 1979–80 (M. Lerman et al., editors), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, pp. 8391.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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