A classification of the one-argument primitive recursive functions

1983 ◽  
Vol 23 (1) ◽  
pp. 161-174 ◽  
Author(s):  
Jovan Naumović
Keyword(s):  
Recursive Functions ◽  
The One ◽  
Primitive Recursive

Ordinal recursion, and a refinement of the extended Grzegorczyk hierarchy

1972 ◽  
Vol 37 (2) ◽  
pp. 281-292 ◽  
Author(s):  
S. S. Wainer
Keyword(s):  
Natural Manner ◽  
Recursive Functions ◽  
Class A ◽  
Primitive Recursive ◽  
Theoretic Function ◽  
Number Class

It is well known that iteration of any number-theoretic function f, which grows at least exponentially, produces a new function f′ such that f is elementary-recursive in f′ (in the Csillag-Kalmar sense), but not conversely (since f′ majorizes every function elementary-recursive in f). This device was first used by Grzegorczyk [3] in the construction of a properly expanding hierarchy {ℰn: n = 0, 1, 2, …} which provided a classification of the primitive recursive functions. More recently it was shown in [7] how, by iterating at successor stages and diagonalizing over fundamental sequences at limit stages, the Grzegorczyk hierarchy can be extended through Cantor's second number-class. A problem which immediately arises is that of classifying all recursive functions, and an answer to this problem is to be found in the general results of Feferman [1]. These results show that although hierarchies of various types (including the above extensions of Grzegorczyk's hierarchy) can be produced, which range over initial segments of the constructive ordinals and which do provide complete classifications of the recursive functions, these cannot be regarded as classifications “from below”, since the method of assigning fundamental sequences at limit stages must be highly noneffective. We therefore adopt the more modest aim here (as in [7], [12], [14]) of characterising certain well-known (effectively generated) subclasses of the recursive functions, by means of hierarchies generated in a natural manner, “from below”.

How is it that infinitary methods can be applied to finitary mathematics? Gödel's T: a case study

1998 ◽  
Vol 63 (4) ◽  
pp. 1348-1370 ◽  
Author(s):  
Andreas Weiermann
Keyword(s):  
Recursive Function ◽  
Term Rewriting ◽  
Careful Analysis ◽  
Recursion Theory ◽  
Cut Elimination ◽  
Complexity Bounds ◽  
Recursive Functions ◽  
Collapsing Function ◽  
Primitive Recursive

AbstractInspired by Pohlers' local predicativity approach to Pure Proof Theory and Howard's ordinal analysis of bar recursion of type zero we present a short, technically smooth and constructive strong normalization proof for Gödel's system T of primitive recursive functionals of finite types by constructing an ε0-recursive function []0: T → ω so that a reduces to b implies [a]0 > [b]0. The construction of [ ]0 is based on a careful analysis of the Howard-Schütte treatment of Gödel's T and utilizes the collapsing function ψ: ε0 → ω which has been developed by the author for a local predicativity style proof-theoretic analysis of PA. The construction of [ ]0 is also crucially based on ideas developed in the 1995 paper “A proof of strongly uniform termination for Gödel's T by the method of local predicativity” by the author. The results on complexity bounds for the fragments of T which are obtained in this paper strengthen considerably the results of the 1995 paper.Indeed, for given n let Tn be the subsystem of T in which the recursors have type level less than or equal to n + 2. (By definition, case distinction functionals for every type are also contained in Tn.) As a corollary of the main theorem of this paper we obtain (reobtain?) optimal bounds for the Tn-derivation lengths in terms of ω+2-descent recursive functions. The derivation lengths of type one functionals from Tn (hence also their computational complexities) are classified optimally in terms of <ωn+2 -descent recursive functions.In particular we obtain (reobtain?) that the derivation lengths function of a type one functional a ∈ T0 is primitive recursive, thus any type one functional a in T0 defines a primitive recursive function. Similarly we also obtain (reobtain?) a full classification of T1 in terms of multiple recursion.As proof-theoretic corollaries we reobtain the classification of the IΣn+1-provably recursive functions. Taking advantage from our finitistic and constructive treatment of the terms of Gödel's T we reobtain additionally (without employing continuous cut elimination techniques) that PRA + PRWO(ε0) ⊢ Π20 − Refl(PA) and PRA + PRWO(ωn+2) ⊢ Π20 − Refl(IΣn+1), hence PRA + PRWO(ε0) ⊢ Con(PA) and PRA + PRWO(ωn+2) ⊢ Con(IΣn+1).For programmatic reasons we outline in the introduction a vision of how to apply a certain type of infinitary methods to questions of finitary mathematics and recursion theory. We also indicate some connections between ordinals, term rewriting, recursion theory and computational complexity.

Somatoform disorders: diseases of the civilization

2020 ◽  
pp. 25-30
Author(s):  
I. Kukhtevich
Keyword(s):  
Health Professionals ◽  
Somatoform Disorders ◽  
Neuropsychiatric Disorders ◽  
The Other ◽  
Significant Part ◽  
Organic Basis ◽  
Autonomic Disorders ◽  
The One ◽  
Somatic Diseases

Functional autonomic disorders occupy a significant part in the practice of neurologists and professionals of other specialties as well. However, there is no generally accepted classification of such disorders. In this paper the authors tried to show that functional autonomic pathology corresponds to the concept of somatoform disorders combining syndromes manifested by visceral, borderline psychopathological, neurological symptoms that do not have an organic basis. The relevance of the problem of somatoform disorders is that on the one hand many health professionals are not familiar enough with manifestations of borderline neuropsychiatric disorders, often forming functional autonomic disorders, and on the other hand they overestimate somatoform symptoms that are similar to somatic diseases.

scholarly journals Modes of Interaction in Naturally Occurring Medical Encounters With General Practitioners: The “One in a Million” Study

2021 ◽  
pp. 104973232199379
Author(s):  
Olaug S. Lian ◽  
Sarah Nettleton ◽  
Åge Wifstad ◽  
Christopher Dowrick
Keyword(s):  
General Practitioners ◽  
Narrative Analysis ◽  
Mode Competition ◽  
General Applicability ◽  
Naturally Occurring ◽  
Wide Range ◽  
The One ◽  
Narrative Mode ◽  
Modes Of Interaction

In this article, we qualitatively explore the manner and style in which medical encounters between patients and general practitioners (GPs) are mutually conducted, as exhibited in situ in 10 consultations sourced from the One in a Million: Primary Care Consultations Archive in England. Our main objectives are to identify interactional modes, to develop a classification of these modes, and to uncover how modes emerge and shift both within and between consultations. Deploying an interactional perspective and a thematic and narrative analysis of consultation transcripts, we identified five distinctive interactional modes: question and answer (Q&A) mode, lecture mode, probabilistic mode, competition mode, and narrative mode. Most modes are GP-led. Mode shifts within consultations generally map on to the chronology of the medical encounter. Patient-led narrative modes are initiated by patients themselves, which demonstrates agency. Our classification of modes derives from complete naturally occurring consultations, covering a wide range of symptoms, and may have general applicability.

scholarly journals Testing Todd: family types and development

2021 ◽  
pp. 1-18
Author(s):  
Jerg Gutmann ◽  
Stefan Voigt
Keyword(s):  
Rule Of Law ◽  
Civil Society Organizations ◽  
The Rule Of Law ◽  
State Fragility ◽  
Family Types ◽  
Low Levels ◽  
Parsimonious Model ◽  
The One ◽  
High Level

Abstract Many years ago, Emmanuel Todd came up with a classification of family types and argued that the historically prevalent family types in a society have important consequences for its economic, political, and social development. Here, we evaluate Todd's most important predictions empirically. Relying on a parsimonious model with exogenous covariates, we find mixed results. On the one hand, authoritarian family types are, in stark contrast to Todd's predictions, associated with increased levels of the rule of law and innovation. On the other hand, and in line with Todd's expectations, communitarian family types are linked to racism, low levels of the rule of law, and late industrialization. Countries in which endogamy is frequently practiced also display an expectedly high level of state fragility and weak civil society organizations.

Diverse Dimensions of the Cotard’s Syndrome - an Approach Towards a Clear Classification of the Phenomenon

2009 ◽  
Vol 24 (S1) ◽  
pp. 1-1
Author(s):  
A. Nagy ◽  
V. Voros ◽  
T. Tenyi
Keyword(s):  
Clinical Practice ◽  
Diagnostic Category ◽  
Treatment Strategies ◽  
Differential Diagnostics ◽  
Dimensional System ◽  
Central Feature ◽  
Scientific Databases ◽  
Search Terms ◽  
The One

Aim:The authors present the Cotard's syndrome, a rare psychiatric condition, pointing out the latest results in terms of psychoneurology and classification of the phenomenon. The central feature of the syndrome is a nihilistic delusion, in which the patient denies his or her own existence and that of the external world.Method:We searched electronic scientific databases using the appropriate search terms; relevant articles were carefully reviewed. We also present three cases from our clinical practice.Results:After the overview of the latest biological and neuropsychological findings, the terminology, the nosology, the classification and the differential diagnostics are discussed. To sum up with useful information for the clinical practice, the possible treatment strategies, the course and the prognosis of the disease are also presented.Conclusions:The reported cases together with the reviewed literature suggest that a dimensional system of classifying Cotard's syndrome is preferable. At the one end of the spectrum is the presence of the pure nihilistic delusions, appearing as a symptom of an underlying psychiatric or neurological condition. The full-blown, classical syndrome as a diagnostic category forms the other end of the spectrum. The presented theoretical and practical aspects give a lead on deeper understanding, easier recognition and more adequate therapy of the Cotard's syndrome.

PERCUTANEOUS RENAL BIOPSY IN CHILDREN

PEDIATRICS ◽  
1962 ◽  
Vol 30 (2) ◽  
pp. 287-296
Author(s):  
W. F. Dodge ◽  
C. W. Daeschner ◽  
J. C. Brennan ◽  
H. S. Rosenberg ◽  
L. B. Travis ◽  
...  
Keyword(s):  
Renal Disease ◽  
Renal Biopsy ◽  
Clinical Importance ◽  
Renal Diseases ◽  
Renal Lesion ◽  
Bleeding Tendency ◽  
Percutaneous Renal Biopsy ◽  
The One ◽  
Selection Of

Since 1951, when the percutaneous renal biopsy was introduced as an adjunctive method for study of patients with renal disease, reports of some 4,000 kidney biopsies have appeared in the literature. Only about 250 of these, however, have been performed in children. A biopsy specimen containing 5 to 10 glomeruli has been reported to be adequate for interpretation and to be representative of the total renal parenchyma in 84% of the cases with diffuse renal disease. Using a biopsy technique similar to that described by Kark, we have obtained an adequate specimen in 92% of 205 kidney biopsies performed in 168 children with diffuse renal diseases. Seven deaths have been previously reported in the literature. The circumstances surrounding the death of these seven patients and of the one death that occurred in our series are described. Perirenal hematoma has had a reported incidence of 0.4%. It has been our experience, as well as that of the other investigators, that if blood boss is replaced, the patient has an otherwise uneventful course and the mass subsequently disappears. Gross hematuria has had a reported incidence of 5.2%. Microscopic hematuria, lasting for 6 to 12 hours after biopsy, has been found to be the rule rather than the exception. The complications which have occurred have been associated with bleeding, and therefore a careful history concerning bleeding tendency and a study of the clotting mechanism is essential if the risk of needle renal biopsy is to be minimized. In addition to a bleeding tendency or defect in clotting mechanism, most investigators are agreed that the presence of only one kidney or an uncooperative patient are absolute contraindications to renal biopsy. The renal biopsy is primarily, at present, an additional and most useful investigative tool in the elucidation of the pathogenesis, natural history (by serial studies) and effectiveness of specific therapy upon the various renal diseases. It is of practical clinical importance in the selection of those patients with the nephrotic syndrome in whom glucocorticoid therapy is likely to be beneficial or the patient with anuria whose renal lesion is probably reversible with time; and, as a guide to the effectiveness of therapy in patients with pyelonephritis or lupus nephritis. It is not a technique that can be recommended for general or casual use. A classification of the pathohistobogic findings of diffuse glomerulonephritis, patterned after Ellis, is presented and discussed. This classification will be used in the description and discussion of various renal diseases and systemic diseases with associated nephritis in the three subsequent papers.

N. Belknap's Four-Valued Logic, Belknap Computer and New Proximity Functions for Comparing Discrete Objects

2021 ◽  
Vol 1 (2 (6)) ◽  
Author(s):  
Valerii Dmitrienko ◽  
Sergey Leonov ◽  
Mykola Mezentsev
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
The Other ◽  
Proximity Function ◽  
Input Information ◽  
Proximity Measures ◽  
The One ◽  
True Values

The idea of ​​Belknap's four-valued logic is that modern computers should function normally not only with the true values ​​of the input information, but also under the conditions of inconsistency and incompleteness of true failures. Belknap's logic introduces four true values: T (true - true), F (false - false), N (none - nobody, nothing, none), B (both - the two, not only the one but also the other).  For ease of work with these true values, the following designations are introduced: (1, 0, n, b). Belknap's logic can be used to obtain estimates of proximity measures for discrete objects, for which the functions Jaccard and Needhem, Russel and Rao, Sokal and Michener, Hamming, etc. are used. In this case, it becomes possible to assess the proximity, recognition and classification of objects in conditions of uncertainty when the true values ​​are taken from the set (1, 0, n, b). Based on the architecture of the Hamming neural network, neural networks have been developed that allow calculating the distances between objects described using true values ​​(1, 0, n, b). Keywords: four-valued Belknap logic, Belknap computer, proximity assessment, recognition and classification, proximity function, neural network.

scholarly journals Machine learning for psychiatric patient triaging: an investigation of cascading classifiers

2018 ◽  
Vol 25 (11) ◽  
pp. 1481-1487 ◽  
Author(s):  
Vivek Kumar Singh ◽  
Utkarsh Shrivastava ◽  
Lina Bouayad ◽  
Balaji Padmanabhan ◽  
Anna Ialynytchev ◽  
...  
Keyword(s):  
Machine Learning ◽  
Psychiatric Patient ◽  
Classification Accuracy ◽  
Psychiatric Patients ◽  
Patient Records ◽  
Classification Technique ◽  
Machine Learning Approach ◽  
The One ◽  
Unique Dataset

Abstract Objective Develop an approach, One-class-at-a-time, for triaging psychiatric patients using machine learning on textual patient records. Our approach aims to automate the triaging process and reduce expert effort while providing high classification reliability. Materials and Methods The One-class-at-a-time approach is a multistage cascading classification technique that achieves higher triage classification accuracy compared to traditional multiclass classifiers through 1) classifying one class at a time (or stage), and 2) identification and application of the highest accuracy classifier at each stage. The approach was evaluated using a unique dataset of 433 psychiatric patient records with a triage class label provided by “I2B2 challenge,” a recent competition in the medical informatics community. Results The One-class-at-a-time cascading classifier outperformed state-of-the-art classification techniques with overall classification accuracy of 77% among 4 classes, exceeding accuracies of existing multiclass classifiers. The approach also enabled highly accurate classification of individual classes—the severe and mild with 85% accuracy, moderate with 64% accuracy, and absent with 60% accuracy. Discussion The triaging of psychiatric cases is a challenging problem due to the lack of clear guidelines and protocols. Our work presents a machine learning approach using psychiatric records for triaging patients based on their severity condition. Conclusion The One-class-at-a-time cascading classifier can be used as a decision aid to reduce triaging effort of physicians and nurses, while providing a unique opportunity to involve experts at each stage to reduce false positive and further improve the system’s accuracy.

Classifying Predicates and Languages

1997 ◽  
Vol 08 (01) ◽  
pp. 15-41 ◽  
Author(s):  
Carl H. Smith ◽  
Rolf Wiehagen ◽  
Thomas Zeugmann
Keyword(s):  
Learning Theory ◽  
Classification Problems ◽  
Attention In Learning ◽  
Recursive Functions ◽  
Standard Classification ◽  
Resource Bound ◽  
Multi Classification

The present paper studies a particular collection of classification problems, i.e., the classification of recursive predicates and languages, for arriving at a deeper understanding of what classification really is. In particular, the classification of predicates and languages is compared with the classification of arbitrary recursive functions and with their learnability. The investigation undertaken is refined by introducing classification within a resource bound resulting in a new hierarchy. Furthermore, a formalization of multi-classification is presented and completely characterized in terms of standard classification. Additionally, consistent classification is introduced and compared with both resource bounded classification and standard classification. Finally, the classification of families of languages that have attracted attention in learning theory is studied, too.

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