Embeddings and extensions of embeddings in the r.e. tt and wtt-degrees

Fejer, Peter A.; Shore, Richard A.

doi:10.1007/bfb0076217

Cited by 16 publications

(20 citation statements)

References 11 publications

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“…Pi which hold in f!lJ, and rows 4,5, and 6 give the interpolants whose existence is asserted by (3). The table maintains (1) and (2) for all of the relationships Pi ~ Pi and Pi V Pi = P k which hold in f!lJ. A representation is called finite if J (the set of indices for the rows) is finite.…”

Section: P4mentioning

confidence: 99%

“…It will then not be necessary to use the difficult result of Pudlak and Tuma. So we assume, without loss of generality, that for some dEw, J = [0, d], n > 0, and for '), As in Fejer and Shore [1], our construction will define a recursive function, f(x, s), where f: w X w ~ [0, d]. We define r.e.…”

Section: P4mentioning

confidence: 99%

“…The decision procedure is the same as the one used in Fejer and Shore [1]. An existential sentence 0 = 3x I , ... , 3x n 'P in .P will be valid in the r.e.…”

Section: (/L)(y) =F-[e Jaja '2(/l)(y)mentioning

confidence: 99%

“…Our notation is standard and follows that of Fejer and Shore [1] very closely. We make implicit use of the convention that if {e L(x) L then x < s. The material in this paper appears as part of my doctoral dissertation [2].…”

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confidence: 99%

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Lattice Embeddings in the Recursively Enumerable Truth Table Degrees

Haught¹

1987

Transactions of the American Mathematical Society

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ABSTRACT. It is shown that every finite lattice, and in fact every recursively presentable lattice, can be embedded in the r.e. tt-degrees by a map preserving least and greatest elements. The decidability of the I-quantifier theory of the Le. ttdegrees in the language with ~, v, /\, 0, and 1 is obtained as a corollary. tt-degrees and about questions relating to the decidability of the theory of the r.e. tt-degrees. They show there that every recursively presentable lattice can be embedded in the r.e. tt-degrees preserving least element. Using this, they show that the 3 theory of the r.e. tt-degrees in the language with ~ , V, 1\,0 is decidable, and ask whether the 3 theory is still decidable when 1 is added to the language. This decidability question can be answered by determining which finite lattices can be embedded in the r.e. tt-degrees preserving least and greatest elements. Jockusch and Mohrherr [3] have shown that the diamond lattice, the pentagon lattice, and the I-n-llattices can be embedded preserving least and greatest elements, but leave open the general question, and even such special cases as the three generator Boolean algebra. The embedding used in their proof requires that the lattice in question have the property that no element which is the inf of a pair of incomparable elements of the lattice can be joined up to the 1 of the lattice (except by 1 itself). We show here that all finite lattices, and in fact all recursively presentable lattices, can be embedded in the r.e. tt-degrees preserving least and greatest elements (provided the lattice has distinct least and greatest elements). Our proof for the general lattices combines a generalization of the coding method used by Jockusch and Mohrherr with the strategy for preserving nonzero infs used by Fejer and Shore. Introduction. A setWe prove first, in §l, that every lattice with a finite representation (and distinct least and greatest elements) can be embedded, and then in §2 outline the modifications needed to embed a recursively presented (possibly infinite) lattice. As a

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Section: P4mentioning

confidence: 99%

Section: P4mentioning

confidence: 99%

“…The decision procedure is the same as the one used in Fejer and Shore [1]. An existential sentence 0 = 3x I , ... , 3x n 'P in .P will be valid in the r.e.…”

Section: (/L)(y) =F-[e Jaja '2(/l)(y)mentioning

confidence: 99%

mentioning

confidence: 99%

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Lattice Embeddings in the Recursively Enumerable Truth Table Degrees

Haught¹

1987

Transactions of the American Mathematical Society

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“…It seems convenient to adopt a variation of the notation of Fejer and Shore [4]. Thus if {e}(:r) j then [e] is the truth table with index {e}(x).…”

Section: Introductionmentioning

confidence: 99%

Two theorems on truth table degrees

Downey¹

1988

Proc. Amer. Math. Soc.

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ABSTRACT. In this article we solve two questions of Odifreddi on the r.e. tt-degrees. First we construct an r.e. tt-degree with anticupping property. In fact, we construct r.e. tt-degrees a,b with 0 < a < b and such that for all (not necessarily r.e.) tt-degrees c if a U c > b then a < c. This result also has ramifications in, for example, the r.e. wtt-degrees. Finally we solve another question of Odifreddi by constructing an r.e. tt-degree with no greatest r.e. m-degree. Introduction.The goal of this paper is to answer two questions of Odifreddi concerning r.e. tt-degrees. For background we refer to Odifreddi [9, 10]

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A nonlow2 R. E. Degree with the Extension of Embeddings Properties of a low2 Degree

Shore

Yue

2002

MLQ - Math. Log. Quart.

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We construct a nonlow 2 r.e. degree d such that every extension of embedding property that holds below every low 2 degree holds below d. Indeed, we can also guarantee the converse so that there is a low r.e. degree c such that that the extension of embedding properties true below c are exactly the ones true below d. Moreover, we can also guarantee that no b ≤ d is the base of a nonsplitting pair.

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Embeddings and extensions of embeddings in the r.e. tt and wtt-degrees

Cited by 16 publications

References 11 publications

Lattice Embeddings in the Recursively Enumerable Truth Table Degrees

Lattice Embeddings in the Recursively Enumerable Truth Table Degrees

Two theorems on truth table degrees

A nonlow2 R. E. Degree with the Extension of Embeddings Properties of a low2 Degree

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