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Unpublished Work ◽  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 AL-AKHTHA’ FII IKTISAB AL-TSUNAIYAT AL-SHUGHRA

2011 ◽  
Vol 5 (1) ◽  
Author(s):  
Miftahul Huda
Keyword(s):  
Saudi Arabia ◽  
Language Acquisition ◽  
Foreign Language ◽  
Error Analysis ◽  
Language Learning ◽  
Mother Tongue ◽  
Minimal Pair ◽  
Minimal Pairs ◽  
The Subject ◽  
Child Age

Language acquisition starts from the ability of listening basic letter(iktisab al-ashwat) since child age. The letter of a language is limited in number, and sometimes there is similarity of letters among languages. The similarity of letters in two languages make it easy to learn the language. On the contrary, the obstacle of language learning can be caused by different letters between two languages (mother tongue and second/foreign language). The problem may be caused by minimal pairs (tsunaiyat al-shughro). This research aims at finding out the error of minimal pair acquisition, with the subject of Indonesian students in Jami’ah Malik Saud Saudi Arabia, with the method of error analysis. The study concludes that in iktisab al-ashwat of minimal pairs, the error is around 3,3 %-58,3%. Second: the error on minimal pairs occurs on the letters shift ?? ?? ? to be ? , letter ? to be ? , letter ? to be ? , letter ? to be ? , letter ? to be ?? , and letter ? to be ?.

You sound like Mommy

2014 ◽  
Vol 38 (4) ◽  
pp. 309-316 ◽  
Author(s):  
Christopher Fennell ◽  
Krista Byers-Heinlein
Keyword(s):  
Experimental Design ◽  
Old French ◽  
Task Demands ◽  
Minimal Pair ◽  
Phonological Representations ◽  
Minimal Pairs ◽  
Language Environment ◽  
Target Words

Previous research indicates that monolingual infants have difficulty learning minimal pairs (i.e., words differing by one phoneme) produced by a speaker uncharacteristic of their language environment and that bilinguals might share this difficulty. To clearly reveal infants’ underlying phonological representations, we minimized task demands by embedding target words in naming phrases, using a fully crossed, between-subjects experimental design. We tested 17-month-old French-English bilinguals’ ( N = 30) and English monolinguals’ ( N = 31) learning of a minimal pair (/k∊m/ – /g∊m/) produced by an adult bilingual or monolingual. Infants learned the minimal pair only when the speaker matched their language environment. This vulnerability to subtle changes in word pronunciation reveals that neither monolingual nor bilingual 17-month-olds possess fully generalizable phonological representations.

scholarly journals Mergers in Bardi: contextual probability and predictors of sound change

2018 ◽  
Vol 4 (s2) ◽  
Author(s):  
Sarah Babinski ◽  
Claire Bowern
Keyword(s):  
Recent Work ◽  
Historical Linguistics ◽  
Sound Change ◽  
Minimal Pair ◽  
Functional Load ◽  
Crucial Question ◽  
Minimal Pairs ◽  
Pair Density ◽  
Sound Changes ◽  
Australian Languages

AbstractA crucial question for historical linguistics has been why some sound changes happen but not others. Recent work on the foundations of sound change has argued that subtle distributional facts about segments in a language, such as functional load, play a role in facilitating or impeding change. Thus not only are sound changes not all equally plausible, but their likelihood depends in part on phonotactics and aspects of functional load, such as the density of minimal pairs. Tests of predictability on the chance of phoneme merger suggest that phonemes with low functional load (as defined by minimal pair density) are more likely to merge, but this has been investigated only for a small number of languages with very large corpora and well attested mergers. Here we present work suggesting that the same methods can be applied to much smaller corpora, by presenting the results of a preliminary exploration of nine Australian languages, with a particular focus on Bardi, a Nyulnyulan language from Australia’s northwest. Our results support the hypothesis that the probability of merger is higher when phonemes distinguish few minimal pairs.

Overlap among back vowels before /l/ in Kansas City

2016 ◽  
Vol 28 (3) ◽  
pp. 379-407 ◽  
Author(s):  
Christopher Strelluf
Keyword(s):  
Kansas City ◽  
Minimal Pair ◽  
Minimal Pairs ◽  
Acoustic Profile

AbstractThis research examines pre-/l/ allophones of vowels in five lexical sets—GOOSE, FOOT, GOAT, STRUT, and THOUGHT—in Kansas City. It builds an acoustic profile from 5507 tokens drawn from interviews with 67 Kansas Citians born between 1955 and 1999. Results reveal a variety of overlap patterns among all five vowels, with the most widespread being overlap between the pre-/l/ allophones of FOOT, STRUT, and GOAT. Acoustically, overlap patterns generally do not show a trend of change in apparent time. However, responses to minimal pairs reveal substantial apparent-time increases in judgments of “close” or “same.” Speakers appear to adjust their productions of vowels to match their minimal pair judgments. The interaction of these productions and judgments indicates a different profile for these five vowels than has been observed in other communities and suggests that some of these vowels have become phonemically ambiguous in Kansas City.

An extension of the nondiamond theorem in classical and α-recursion theory

1984 ◽  
Vol 49 (2) ◽  
pp. 586-607 ◽  
Author(s):  
Klaus Ambos-Spies
Keyword(s):  
Boolean Algebra ◽  
Recursion Theory ◽  
Upper Bounds ◽  
Minimal Pair ◽  
Splitting Theorem ◽  
Recursively Enumerable ◽  
Interesting Corollary ◽  
Image Position

Lachlan's nondiamond theorem [7, Theorem 5] asserts that there is no embedding of the four-element Boolean algebra (diamond) in the recursively enumerable degrees which preserves infima, suprema, and least and greatest elements. Lachlan observed that, essentially by relativization, the theorem can be extended toUsing the Sacks splitting theorem he concluded that there exists a pair of r.e. degrees which does not have an infimum, thus showing that the r.e. degrees do not form a lattice.We will first prove the following extension of (1):where an r.e. degree a is non-b-cappable if . From (2) we obtain more information about pairs of r.e. degrees without infima: For every nonzero low r.e. degree there exists an incomparable one such that the two degrees do not have an infimum and there is an r.e. degree which is not half of a pair of incomparable r.e. degrees which has an infimum in the low r.e. degrees. Probably the most interesting corollary of (2) is that the join of any cappable r.e. degree (i.e. half of a minimal pair) and any low r.e. degree is incomplete. Consequently there is an incomplete noncappable degree above every incomplete r.e. degree. Cooper's result [3] that ascending sequences of uniformly r.e. degrees can have minimal upper bounds in the set R of r.e. degrees is another corollary of (2).

A jump class of noncappable degrees

1989 ◽  
Vol 54 (2) ◽  
pp. 324-353 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Minimal Pair ◽  
Recursively Enumerable ◽  
Image Position

Friedberg [3] showed that every degree of unsolvability above 0′ is the jump of some degree, and Sacks [9] showed that the degrees above 0′ which are recursively enumerable (r.e.) in 0′ are the jumps of the r.e. degrees.In this paper we examine the extent to which the Sacks jump theorem can be combined with the minimal pair theorem of Lachlan [4] and Yates [13]. We prove below that there is a degree c > 0′ which is r.e. in 0′ but which is not the jump of half a minimal pair of r.e. degrees.This extends Yates' result [13] proving the existence of noncappable degrees (that is, r.e. degrees a < 0′ for which there is no corresponding r.e. b > 0 with a ∩ b = 0).It also throws more light on the class PS of promptly simple degrees. It was shown by Ambos-Spies, Jockusch, Shore and Soare [1] that PS coincides with the class NC of noncappable degrees, and with the class LC of all low-cuppable degrees, and (using earlier work of Maass, Shore and Stob [5]) that PS splits every class Hn or Ln, n ≥ 0, in the high-low hierarchy of r.e. degrees.If c > 0′, with c r.e. in 0′, letand call c−1 the jump class for c. It is easy to see that every jump class contains members of PS (= NC = LC). By Sacks [8] there exists a low a ∈ LC, where of course [a, 0′] (= {br.e. ∣a ≤ b ≤ 0′}) ⊆ LC = PS. But by Robinson [7] [a, 0′] intersects with every jump class.

scholarly journals Lexical and morphosyntactic minimal pairs: Evidence for different processing and implications in language pathology treatment

2014 ◽  
Author(s):  
Luca Cilibrasi ◽  
Vesna Stojanovik ◽  
Patricia M Riddell
Keyword(s):  
Language Impairment ◽  
Specific Language Impairment ◽  
Minimal Pair ◽  
Language Performance ◽  
Phonological Disorders ◽  
Minimal Pairs ◽  
Peculiar Form ◽  
Language Pathology ◽  
Third Person ◽  
Native Speakers Of English

Minimal pairs are defined as pair of words in a particular language which differ in only one phonological element and have a different meaning (Roach, 2000). Several authors argued their relevance in the treatment of phonological disorders (for instance, Barlow and Gierut, 2002). In this study we investigate the nature of minimal pairs showing that a subtype of them entails a peculiar form of processing. In many languages bound morphemes used to mark inflection generate minimal pairs. In English, the present third person singular morpheme -s and the past tense morpheme -ed generate in most cases minimal pairs, such as “asks / asked”. Several authors (Stemberger and MacWhinney, 1986, Bertram et al, 2000) have argued that inflected forms may be stored in the lexicon as units, i.e. together with the bound morpheme. If inflected forms are stored as units in the lexicon, discriminating lexical minimal pairs and morphosyntactic minimal pairs should not be different processes. Elements should be stored similarly in the lexicon, and then compared phonologically when the subject is presented with a minimal pair. In this study we addressed this question presenting 20 monolingual native speakers of English with lexical and morphosyntactic minimal pairs (30 per condition, frequency differences not significant), and with pairs of identical words (leading, thus, to 120 trials). Participants were asked to press “white” if words were different and “black” if words were identical. Conditions were matched on word length. Results show that subjects are significantly faster in discriminating words generating a lexical minimal pair, such as “back / badge” than words generating a morphosyntactic minimal pair, such as “asks / asked”, t (19) = -4.486, p < .001. A third condition was also present to deepen our understanding of the processing of morphosyntactic minimal pairs. In this condition subjects were presented with morphosyntactic minimal pairs generated by very infrequent verbs. Unexpectedly, minimal pairs generated by infrequent verbs revealed to be faster recognised (19) = 2.120, p < .05 than the other morphosyntactic minimal pairs. Even if this may be interpreted as a consequence of attention arousal for unexpected stimuli, the result is problematic if we assume inflected forms to be stored in the lexicon as units. Together, these results suggest that inflected forms are not stored as units and that the discrimination of morphosyntactic minimal pairs relies on the discrimination of inflectional morphemes. As such, we suggest that increasing the sensibility to morphosyntactic minimal pairs in people with a morphosyntactic disorder, such as children with Specific Language Impairment (SLI), should improve their language performance.

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