Type two partial degrees

1978 ◽  
Vol 43 (4) ◽  
pp. 623-629
Author(s):  
Ko-Wei Lih
Keyword(s):  
Recursive Function ◽  
Natural Extension ◽  
Degree Theory ◽  
Recursion Theory ◽  
Natural Numbers ◽  
Partial Functions ◽  
Reduction Procedure ◽  
Turing Reducibility

Roughly speaking partial degrees are equivalence classes of partial objects under a certain notion of relative recursiveness. To make this notion precise we have to state explicitly (1) what these partial objects are; (2) how to define a suitable reduction procedure. For example, when the type of these objects is restricted to one, we may include all possible partial functions from natural numbers to natural numbers as basic objects and the reduction procedure could be enumeration, weak Turing, or Turing reducibility as expounded in Sasso [4]. As we climb up the ladder of types, we see that the usual definitions of relative recursiveness, equivalent in the context of type-1 total objects and functions, may be extended to partial objects and functions in quite different ways. First such generalization was initiated by Kleene [2]. He considers partial functions with total objects as arguments. However his theory suffers the lack of transitivity, i.e. we may not obtain a recursive function when we substitute a recursive function into a recursive function. Although Kleene's theory provides a nice background for the study of total higher type objects, it would be unsatisfactory when partial higher type objects are being investigated. In this paper we choose the hierarchy of hereditarily consistent objects over ω as our universe of discourse so that Sasso's objects are exactly those at the type-1 level. Following Kleene's fashion we define relative recursiveness via schemes and indices. Yet in our theory, substitution will preserve recursiveness, which makes a degree theory of partial higher type objects possible. The final result will be a natural extension of Sasso's Turing reducibility. Due to the abstract nature of these objects we do not know much about their behaviour except at the very low types. Here we pay our attention mainly to type-2 objects. In §2 we formulate basic notions and give an outline of our recursion theory of partial higher type objects. In §3 we introduce the definitions of singular degrees and ω-consistent degrees which are two important classes of type-2 objects that we are most interested in.

Turing reducibility and Turing degrees

2018 ◽  
Author(s):  
Harold Hodes
Keyword(s):  
Computational Complexity ◽  
Recursion Theory ◽  
Turing Degrees ◽  
Natural Numbers ◽  
Mathematical Objects ◽  
Classical Setting ◽  
Turing Reducibility

A reducibility is a relation of comparative computational complexity (which can be made precise in various non-equivalent ways) between mathematical objects of appropriate sorts. Much of recursion theory concerns such relations, initially between sets of natural numbers (in so-called classical recursion theory), but later between sets of other sorts (in so-called generalized recursion theory). This article considers only the classical setting. Also Turing first defined such a relation, now called Turing- (or just T-) reducibility; probably most logicians regard it as the most important such relation. Turing- (or T-) degrees are the units of computational complexity when comparative complexity is taken to be T-reducibility.

The countably based functionals

1983 ◽  
Vol 48 (2) ◽  
pp. 458-474 ◽  
Author(s):  
John P. Hartley
Keyword(s):  
Finite Type ◽  
Recursion Theory ◽  
Type Structure ◽  
Topological Properties ◽  
Natural Numbers ◽  
Finite Amount ◽  
Amount Of Information ◽  
Natural Concept ◽  
Higher Types

In [5], Kleene extended previous notions of computations to objects of higher finite type in the maximal type-structure of functionals given by:Tp(0) = N = the natural numbers,Tp(n + 1) = NTp(n) = the set of total maps from Tp(n) to N.He gave nine schemata, S1–S9, for describing algorithms for computations from a finite list of functionals, and indices to denote these algorithms. It is generally agreed that S1-S9 give a natural concept of computations in higher types.The type-structure of countable functions, Ct(n) for n ϵ N, was first developed by Kleene [6] and Kreisel [7]. It arises from the notions of ‘constructivity’, and has been extensively studied as a domain for higher type recursion theory. Each countable functional is globally described by a countable amount of information coded in its associate, and it is determined locally by a finite amount of information about its argument. The countable functionals are well summarised in Normann [9], and treated in detail in Normann [8].The purpose of this paper is to discuss a generalisation of the countable functionals, which we shall call the countably based functions, Cb(n) for n ϵ N. It is suggested by the notions of ‘predicativity’, in which we view N as a completed totality, and build higher types on it in a constructive manner. So we shall allow quantification over N and include application of 2E in our schemata. Each functional is determined locally by a countable amount of information about its argument, and so is globally described by a continuum of information coded in its associate, which will now be a type-2 object. This generalisation, obtained via associates, was proposed by Wainer, and seems to reflect topological properties of the countable functionals.

Some formal relative consistency proofs

1953 ◽  
Vol 18 (2) ◽  
pp. 136-144 ◽  
Author(s):  
Robert McNaughton
Keyword(s):  
Number Theory ◽  
Natural Number ◽  
Natural Numbers ◽  
Quantification Theory ◽  
Usual Manner ◽  
Function Calculus ◽  
Image Position ◽  
Positive Integers

These systems are roughly natural number theory in, respectively, nth order function calculus, for all positive integers n. Each of these systems is expressed in the notation of the theory of types, having variables with type subscripts from 1 to n. Variables of type 1 stand for natural numbers, variables of type 2 stand for classes of natural numbers, etc. Primitive atomic wff's (well-formed formulas) of Tn are those of number theory in variables of type 1, and of the following kind for n > 1: xi ϵ yi+1. Other wff's are formed by truth functions and quantifiers in the usual manner. Quantification theory holds for all the variables of Tn. Tn has the axioms Z1 to Z9, which are, respectively, the nine axioms and axiom schemata for the system Z (natural number theory) on p. 371 of [1]. These axioms and axiom schemata contain only variables of type 1, except for the schemata Z2 and Z9, which are as follows:where ‘F(x1)’ can be any wff of Tn. Identity is primitive for variables of type 1; for variables of other types it is defined as follows:

Type 2 Diabetes Overtakes Type 1 in Hispanic Girls

2008 ◽  
Vol 38 (15) ◽  
pp. 18
Author(s):  
SHERRY BOSCHERT
Keyword(s):  
Type 2 Diabetes

Molecular analysis of FVIII gene in severe HA patients of Costa Rica

2010 ◽  
Vol 30 (S 01) ◽  
pp. S150-S152
Author(s):  
G. Jiménez-Cruz ◽  
M. Mendez ◽  
P. Chaverri ◽  
P. Alvarado ◽  
W. Schröder ◽  
...  
Keyword(s):  
Factor Viii ◽  
Coagulation Factor ◽  
Costa Rican ◽  
Factor Viii Gene ◽  
Intron 22 Inversion ◽  
Intron 1 ◽  
Polymerase Chain ◽  
Inversion Analysis

SummaryHaemophilia A (HA) is X-chromosome linked bleeding disorders caused by deficiency of the coagulation factor VIII (FVIII). It is caused by FVIII gene intron 22 inversion (Inv22) in approximately 45% and by intron 1 inversion (Inv1) in 5% of the patients. Both inversions occur as a result of intrachromosomal recombination between homologous regions, in intron 1 or 22 and their extragenic copy located telomeric to the FVIII gene. The aim of this study was to analyze the presence of these mutations in 25 HA Costa Rican families. Patients, methods: We studied 34 HA patients and 110 unrelated obligate members and possible carriers for the presence of Inv22or Inv1. Standard analyses of the factor VIII gene were used incl. Southern blot and long-range polymerase chain reaction for inversion analysis. Results: We found altered Inv22 restriction profiles in 21 patients and 37 carriers. It was found type 1 and type 2 of the inversion of Inv22. During the screening for Inv1 among the HA patient, who were Inv22 negative, we did not found this mutation. Discussion: Our data highlight the importance of the analysis of Inv22 for their association with development of inhibitors in the HA patients and we are continuous searching of Inv1 mutation. This knowledge represents a step for genetic counseling and prevention of the inhibitor development.

Polymorphisms of the Plasminogen Activator Inhibitor- 1 Gene in Type 1 and Type 2 Diabetes, and in Patients with Diabetic Retinopathy

1994 ◽  
Vol 71 (06) ◽  
pp. 731-736 ◽  
Author(s):  
M W Mansfield ◽  
M H Stickland ◽  
A M Carter ◽  
P J Grant
Keyword(s):  
Diabetic Retinopathy ◽  
Plasminogen Activator Inhibitor ◽  
Allelic Frequency ◽  
Repeat Sequence ◽  
Dinucleotide Repeat ◽  
Plasminogen Activator Inhibitor 1 ◽  
The Difference ◽  
Pai 1

SummaryTo identify whether genotype contributes to the difference in PAI-1 levels in type 1 and type 2 diabetic subjects and whether genotype relates to the development of retinopathy, a Hind III restriction fragment length polymorphism and two dinucleotide repeat polymorphisms were studied. In 519 Caucasian diabetic subjects (192 type 1, 327 type 2) and 123 Caucasian control subjects there were no differences in the frequency of the Hind III restriction alleles (type 1 vs type 2 vs control: allele 1 0.397 vs 0.420 vs 0.448; allele 2 0.603 vs 0.580 vs 0.552) nor in the allelic frequency at either dinucleotide repeat sequence. In 86 subjects with no retinopathy at 15 years or more from diagnosis of diabetes and 190 subjects with diabetic retinopathy there was no difference in the frequency of Hind III restriction alleles (retinopathy present vs retinopathy absent: allele 1 0.400 vs 0.467; allele 2 0.600 vs 0.533) nor in the allelic frequencies at either dinucleotide repeat sequence. The results indicate that there is no or minimal influence of the PAI-1 gene on either PAI-1 levels or the development of diabetic retinopathy in patients with diabetes mellitus.

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