Minimal pairs in initial segments of the recursively enumerable degrees

1997 ◽  
Vol 100 (1) ◽  
pp. 7-27 ◽  
Author(s):  
R. Downey ◽  
M. Stob
Keyword(s):  
Minimal Pairs ◽  
Recursively Enumerable

Bounding minimal pairs

1979 ◽  
Vol 44 (4) ◽  
pp. 626-642 ◽  
Author(s):  
A. H. Lachlan
Keyword(s):  
Minimal Pair ◽  
Minimal Pairs ◽  
Recursively Enumerable ◽  
The Many

A minimal pair of recursively enumerable (r.e.) degrees is a pair of degrees a, b of nonrecursive r.e. sets with the property that if c ≤ a and c ≤ b then c = 0. Lachlan [2] and Yates [4] independently proved the existence of minimal pairs. It was natural to ask whether for an arbitrary nonzero r.e. degree c there is a minimal pair a, b with a ≤ c and b ≤ c. In 1971 Lachlan and Ladner proved that a minimal pair below c cannot be obtained in a uniformly effective way from c for r.e. c ≠ 0. but the result was never published. More recently Cooper [1] showed that if c is r.e. and c′ = 0″ then there is a minimal pair below c.In this paper we prove two results:Theorem 1. There exists a nonzero r.e. degree with no minimal pair below it.Theorem 2. There exists a nonzero r.e. degree c such that, if d is r.e. and 0 < d ≤ c, then there is a minimal pair below d.The second theorem is a straightforward variation on the original minimal pair construction, but the proof of the first theorem has some novel features. After some preliminaries in §1, the first theorem is proved in §2 and the second in §3.I am grateful to Richard Ladner who collaborated with me during the first phase of work on this paper as witnessed by our joint abstract [3]. The many discussions we had about the construction required in Theorem 1 were of great help to me.

On minimal pairs of enumeration degrees

1985 ◽  
Vol 50 (4) ◽  
pp. 983-1001 ◽  
Author(s):  
Kevin McEvoy ◽  
S. Barry Cooper
Keyword(s):  
Minimal Pair ◽  
Natural Numbers ◽  
Enumeration Degrees ◽  
Minimal Pairs ◽  
Recursively Enumerable ◽  
Unpublished Work ◽  
Lower Case ◽  
Low Degree ◽  
Finite Set ◽  
Partial Orderings

For sets of natural numbers A and B, A is enumeration reducible to B if there is some effective algorithm which when given any enumeration of B will produce an enumeration of A. Gutteridge [5] has shown that in the upper semilattice of the enumeration degrees there are no minimal degrees (see Cooper [3]), and in this paper we study those pairs of degrees with gib 0. Case [1] constructed a minimal pair. This minimal pair construction can be relativised to any gib, and following a suggestion of Jockusch we can also fix one of the degrees and still construct the pair. These methods yield an easier proof of Case's exact pair theorem for countable ideals. 0″ is an upper bound for the minimal pair constructed in §1, and in §2 we improve this bound to any Σ2-high Δ2 degree. In contrast to this we show that every low degree c bounds a degree a which is not in any minimal pair bounded by c. The structure of the co-r.e. e-degrees is isomorphic to that of the r.e. Turing degrees, and Gutteridge has constructed co-r.e. degrees which form a minimal pair in the e-degrees. In §3 we show that if a, b is any minimal pair of co-r.e. degrees such that a is low then a, b is a minimal pair in the e-degrees (and so Gutteridge's result follows). As a corollary of this we can embed any countable distributive lattice and the two nondistributive five-element lattices in the e-degrees below 0′. However the lowness assumption is necessary, as we also prove that there is a minimal pair of (high) r.e. degrees which is not a minimal pair in the e-degrees (under the isomorphism). In §4 we present more concise proofs of some unpublished work of Lagemann on bounding incomparable pairs and embedding partial orderings.As usual, {Wi}i ∈ ω is the standard listing of the recursively enumerable sets, Du is the finite set with canonical index u and {‹ m, n ›}m, n ∈ ω is a recursive, one-to-one coding of the pairs of numbers onto the numbers. Capital italic letters will be variables over sets of natural numbers, and lower case boldface letters from the beginning of the alphabet will vary over degrees.

Minimal pairs and high recursively enumerable degrees

1974 ◽  
Vol 39 (4) ◽  
pp. 655-660 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Recursive Function ◽  
Uniform Method ◽  
Minimal Pair ◽  
Minimal Pairs ◽  
Recursively Enumerable ◽  
Finite Approximations ◽  
Degree Bounds ◽  
Recursively Enumerable Degree ◽  
Image Position ◽  
High Set

A. H. Lachlan [2] and C. E. M. Yates [4] independently showed that minimal pairs of recursively enumerable (r.e.) degrees exist. Lachlan and Richard Ladner have shown (unpublished) that there is no uniform method for producing a minimal pair of r.e. degrees below a given nonzero r.e. degree. It is not known whether every nonzero r.e. degree bounds a r.e. minimal pair, but in the present paper it is shown (uniformly) that every high r.e. degree bounds a r.e. minimal pair. (A r.e. degree is said to be high if it contains a high set in the sense of Robert W. Robinson [3].)Theorem. Let a be a recursively enumerable degree for which a′ = 0″. Then there are recursively enumerable degrees b0 and b1 such that0 < bi < a for each i ≤ 1, and b0 ⋂ b1 = 0.The proof is based on the Lachlan minimal r.e. pair construction. For notation see Lachlan [2] or S. B. Cooper [1].By Robinson [3] we can choose a r.e. representative A of the degree a, with uniformly recursive tower {As, ∣ s ≥ 0} of finite approximations to A, such that CA dominates every recursive function whereWe define, stage by stage, finite sets Bi,s, i ≤ 1, s ≥ 0, in such a way that Bi, s + 1 ⊇ Bi,s for each i, s, and {Bi,s ∣ i ≤ 1, s ≥ 0} is uniformly recursive.

scholarly journals Some minimal pairs of α-recursively enumerable degrees

1972 ◽  
Vol 4 (4) ◽  
pp. 415-442 ◽  
Author(s):  
Manuel Lerman ◽  
Gerald E. Sacks
Keyword(s):  
Minimal Pairs ◽  
Recursively Enumerable

Final Consonant Deletion in African American Children Speaking Black English

1993 ◽  
Vol 24 (3) ◽  
pp. 161-166 ◽  
Author(s):  
Michael J. Moran
Keyword(s):  
African American ◽  
Black English ◽  
African American Children ◽  
Vowel Length ◽  
Minimal Pairs ◽  
Speech Evaluation ◽  
American Children

The purpose of this study was to determine whether African American children who delete final consonants mark the presence of those consonants in a manner that might be overlooked in a typical speech evaluation. Using elicited sentences from 10 African American children from 4 to 9 years of age, two studies were conducted. First, vowel length was determined for minimal pairs in which final consonants were deleted. Second, listeners who identified final consonant deletions in the speech of the children were provided training in narrow transcription and reviewed the elicited sentences a second time. Results indicated that the children produced longer vowels preceding "deleted" voiced final consonants, and listeners perceived fewer deletions following training in narrow transcription. The results suggest that these children had knowledge of the final consonants perceived to be deleted. Implications for assessment and intervention are discussed.

scholarly journals DIOPHANTINE SETS OF POLYNOMIALS OVER ALGEBRAIC EXTENSIONS OF THE RATIONALS

2014 ◽  
Vol 79 (3) ◽  
pp. 733-747
Author(s):  
CLAUDIA DEGROOTE ◽  
JEROEN DEMEYER
Keyword(s):  
Minimal Polynomial ◽  
Finite Extension ◽  
Algebraic Extension ◽  
Recursively Enumerable

AbstractLet L be a recursive algebraic extension of ℚ. Assume that, given α ∈ L, we can compute the roots in L of its minimal polynomial over ℚ and we can determine which roots are Aut(L)-conjugate to α. We prove that there exists a pair of polynomials that characterizes the Aut(L)-conjugates of α, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in ℝ, or in a finite extension of ℚp (with p an odd prime). Then we show that subsets of L[X]k that are recursively enumerable for every recursive presentation of L[X], are diophantine over L[X].

scholarly journals When catalytic P systems with one catalyst can be computationally complete

2021 ◽  
Author(s):  
Artiom Alhazov ◽  
Rudolf Freund ◽  
Sergiu Ivanov
Keyword(s):  
Computational Complexity ◽  
P Systems ◽  
Membrane Systems ◽  
Computational Completeness ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets

AbstractCatalytic P systems are among the first variants of membrane systems ever considered in this area. This variant of systems also features some prominent computational complexity questions, and in particular the problem of using only one catalyst in the whole system: is one catalyst enough to allow for generating all recursively enumerable sets of multisets? Several additional ingredients have been shown to be sufficient for obtaining computational completeness even with only one catalyst. In this paper, we show that one catalyst is sufficient for obtaining computational completeness if either catalytic rules have weak priority over non-catalytic rules or else instead of the standard maximally parallel derivation mode, we use the derivation mode maxobjects, i.e., we only take those multisets of rules which affect the maximal number of objects in the underlying configuration.

scholarly journals Regional differences in the evolution of the merger of /ʃ/ and /ç/ in Luxembourgish

2021 ◽  
pp. 1-18
Author(s):  
François Conrad
Keyword(s):  
Language Contact ◽  
Sound Change ◽  
Young Generation ◽  
The South ◽  
Data Set ◽  
West Germanic ◽  
The North ◽  
Minimal Pairs ◽  
Old Men ◽  
Middle Generation

The merger of post-alveolar /ʃ/ and palatal /ç/ into alveolopalatal /ɕ/ has recently gained growing interest in sociophonetic research, especially in the Middle German dialect area. In Luxembourgish, a Continental West Germanic language, the sound change has been linked to age differences, while its origins remain unclear. Two studies with a regional focus are presented in this paper. The first study examines the merger in the Centre and the South of Luxembourg. The acoustic examination of both the spectral peak and the centre of gravity of a spoken data set of five minimal pairs embedded in read and orally translated sentences from 48 speakers (three generations (old generation, 65–91 years; middle generation, 40–64 years; young generation, 20–39 years; each generation, n = 16), men and women) reveals interesting results related to their regional background. In the old generation, the merger is further advanced in the speech of old men from the former mining region in the South compared to their peers in the Centre, the former leading this sound change. On the other hand, young speakers in both regions produce only alveolopalatal /ɕ/, the merger being complete in this generation. The second study presents exploratory data from the East and the North of the country. The analysis of this smaller sample (n = 6 speakers) reveals patterns similar to the central region. Pointing to language contact with Romance in the South as cradle and/or catalyser of the merger, these results not only give further clues as to the development in Luxembourg, but also add to a deeper understanding of sound changes in process in complex sibilant systems.

Scattered Context Grammars with One Non-Context-Free Production are Computationally Complete

2021 ◽  
Vol 179 (4) ◽  
pp. 361-384
Author(s):  
Zbyněk Křivka ◽  
Alexander Meduna
Keyword(s):  
Open Problem ◽  
Recursively Enumerable Language ◽  
Recursively Enumerable ◽  
Enumerable Language ◽  
Context Free

This paper investigates the reduction of scattered context grammars with respect to the number of non-context-free productions. It proves that every recursively enumerable language is generated by a scattered context grammar that has no more than one non-context-free production. An open problem is formulated.

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