Members of countable π10 classes

Cenzer, Douglas; Clote, Peter; Smith, Rick L.; Soare, Robert I.; Wainer, Stanley S.

doi:10.1016/0168-0072(86)90067-9

Cited by 35 publications

(36 citation statements)

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“…Proof This corollary follows from Theorem 4.6 together with Corollary 3.2 of [5] and Theorem 2.1 of [11]. Given degree b as postulated, there exists (by the cited results) a 0 1 class P of rank two and a real X of degree b such that D(P ) = {X} and furthermore every other element of Y of P is eventually 0, that is, Y = u 0 ω for some u.…”

Section: Corollary 48mentioning

confidence: 84%

“…We note that in several previous articles on effectively closed sets [5,7,8,10] (including the conference version of the present paper), a different definition is given for the CB rank of a countable closed set, namely the least ordinal α such that D α+1 (P ) = ∅. This will always be one less than rk(P ) as defined above.…”

Section: Preliminariesmentioning

confidence: 95%

“…If Y has rank one in Q, then Y = σ e (X) for some e and hence Y has the same Turing degree as X. (Note that due to a different definition of rank in [5,11], P is said to have rank one. )…”

Section: Corollary 48mentioning

confidence: 97%

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Computability of Countable Subshifts in One Dimension

et al. 2011

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We investigate the computability of countable subshifts in one dimension, and their members. Subshifts of Cantor-Bendixson rank two contain only eventually periodic elements. Any rank two subshift in 2 Z is decidable. Subshifts of rank three may contain members of arbitrary Turing degree. In contrast, effectively closed ( 0 1 ) subshifts of rank three contain only computable elements, but 0 1 subshifts of rank four may contain members of arbitrary 0 2 degree. There is no subshift of rank ω + 1.

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Section: Corollary 48mentioning

confidence: 84%

Section: Preliminariesmentioning

confidence: 95%

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Computability of Countable Subshifts in One Dimension

et al. 2011

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“…Lempp gave effective versions of this result in [16]. The complexity of members of classes with a given rank was studied in [5].…”

Section: Definition 3 For All Ordinals α and Allmentioning

confidence: 96%

A connection between the Cantor–Bendixson derivative and the well-founded semantics of finite logic programs

Cenzer

Remmel

2012

Ann Math Artif Intell

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Results of Schlipf (J Comput Syst Sci 51:64-86, 1995) and Fitting (Theor Comput Sci 278:25-51, 2001) show that the well-founded semantics of a finite predicate logic program can be quite complex. In this paper, we show that there is a close connection between the construction of the perfect kernel of a 0 1 class via the iteration of the Cantor-Bendixson derivative through the ordinals and the construction of the well-founded semantics for finite predicate logic programs via Van Gelder's alternating fixpoint construction. This connection allows us to transfer known complexity results for the perfect kernel of 0 1 classes to give new complexity results for various questions about the well-founded semantics wfs(P) of a finite predicate logic program P.

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“…Then by 6D, we know x m s . By Lemma 15.2(ii), at stage t + 1 either (1) Step 6C applies to α and x; (2) Step 1 applies to δ and x for some δ < L α, δ = α(x, t + 1). In case (1), we know f s+1 < L α, so x is γ-ineligible for all stages v ≥ t + 1 and γ ⊇ α, so x cannot re-enter R α .…”

Section: Invariance In Ementioning

confidence: 99%

Invariance in ℰ* and ℰ_{Π}

Weber

2006

Trans. Amer. Math. Soc.

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Abstract. We define G, a substructure of E Π (the lattice of Π 0 1 classes), and show that a quotient structure of G, G ♦ , is isomorphic to E * . The result builds on the ∆ 0 3 isomorphism machinery, and allows us to transfer invariant classes from E * to E Π , though not, in general, orbits. Further properties of G ♦ and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance. IntroductionA Π 0 1 class may be defined as the collection of infinite paths through a computable subtree of 2 <ω , the complete binary-branching tree. Π The lattice of all Π 0 1 classes is called E Π , by analogy with E, the lattice of computably enumerable (c.e.) sets. The properties of E have been extensively studied (for a survey, see [19], chapters X and XV). Research on Π 0 1 classes and E Π is currently quite active, with many open questions (see [3] for a number of examples). However, relatively little is known about the orbits and invariant classes of E Π . The goal of the research presented here is to expand that knowledge, in particular by transferring information to E Π from E * , the lattice of c.e. sets modulo finite difference.A Π 0 1 class P is principal (or clopen) if there is a finite set F of nodes of 2 <ω such that an infinite path of the tree is in P if and only if it extends some σ ∈ F . Cholak, Coles, Downey, and Herrmann [7] showed that there were at most two nonisomorphic intervals of the form [P, 2 ω ] in E Π : those where P is principal and those where it is nonprincipal. Cenzer and Nies [4] showed that these are in fact distinct cases.Nies proceeded to define G = [P, 2 ω ] for P nonprincipal. It is via G that we will transfer information from E * to E Π , and many of Nies' unpublished results are reproduced in § §3-4. Several of the results are directly proved in the setting of Π 0 1 classes. However, although the goal is to transfer information to E Π , it is generally Prior to being investigated in this context, G (as a collection of ideals) arose as part of the study of the lattice of c.e. substructures of a computably presented model, an area suggested by Metakides and Nerode in a 1975 paper [15]. A number of papers emerged studying substructures of particular models, such as vector spaces, algebraically closed fields, and Boolean algebras (see Nerode and Remmel [16] for references). Remmel [17] and later Downey [9,10] generalized the work on specific structures to results about effective closure systems (M, cl), where M is a computable set and cl : P(M ) → P(M ) is an effective closure operator, a map with certain properties. For example, the operator could take a subfield of M to its algebraic closure within M . As part of this work the notion of equivalence modulo finite difference, as in E * , is extended to equivalence modulo "finitely-generated difference." That is, A = * B if there is a finite set X such that cl(A ∪ X) = cl(B ∪ X). Downey in particular gives a long list of examples of effective closure systems which includes the remark that in...

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Members of countable π10 classes

Cited by 35 publications

References 4 publications

Computability of Countable Subshifts in One Dimension

Computability of Countable Subshifts in One Dimension

A connection between the Cantor–Bendixson derivative and the well-founded semantics of finite logic programs

Invariance in ℰ* and ℰ_{Π}

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