Universal recursion theoretic properties of r.e. preordered structures

1985 ◽  
Vol 50 (2) ◽  
pp. 397-406 ◽  
Author(s):  
Franco Montagna ◽  
Andrea Sorbi
Keyword(s):  
Equivalence Relation ◽  
Point Of View ◽  
Main Concern ◽  
Natural Numbers ◽  
Axiomatic Theory ◽  
Recursive Functions ◽  
Theoretic Point

When dealing with axiomatic theories from a recursion-theoretic point of view, the notion of r.e. preordering naturally arises. We agree that an r.e. preorder is a pair = 〈P, ≤P〉 such that P is an r.e. subset of the set of natural numbers (denoted by ω), ≤P is a preordering on P and the set {〈;x, y〉: x ≤Py} is r.e.. Indeed, if is an axiomatic theory, the provable implication of yields a preordering on the class of (Gödel numbers of) formulas of .Of course, if ≤P is a preordering on P, then it yields an equivalence relation ~P on P, by simply letting x ~Py iff x ≤Py and y ≤Px. Hence, in the case of P = ω, any preordering yields an equivalence relation on ω and consequently a numeration in the sense of [4]. It is also clear that any equivalence relation on ω (hence any numeration) can be regarded as a preordering on ω. In view of this connection, we sometimes apply to the theory of preorders some of the concepts from the theory of numerations (see also Eršov [6]).Our main concern will be in applications of these concepts to logic, in particular as regards sufficiently strong axiomatic theories (essentially the ones in which recursive functions are representable). From this point of view it seems to be of some interest to study some remarkable prelattices and Boolean prealgebras which arise from such theories. It turns out that these structures enjoy some rather surprising lattice-theoretic and universal recursion-theoretic properties.After making our main definitions in §1, we examine universal recursion-theoretic properties of some r.e. prelattices in §2.

Classifying positive equivalence relations

1983 ◽  
Vol 48 (3) ◽  
pp. 529-538 ◽  
Author(s):  
Claudio Bernardi ◽  
Andrea Sorbi
Keyword(s):  
Topological Space ◽  
Equivalence Relation ◽  
Recursive Function ◽  
Equivalence Classes ◽  
Point Of View ◽  
Classification Theorem ◽  
Equivalence Relations ◽  
Natural Numbers ◽  
Uniformity Property

AbstractGiven two (positive) equivalence relations ~1, ~2 on the set ω of natural numbers, we say that ~1 is m-reducible to ~2 if there exists a total recursive function h such that for every x, y ∈ ω, we have x ~1y iff hx ~2hy. We prove that the equivalence relation induced in ω by a positive precomplete numeration is complete with respect to this reducibility (and, moreover, a “uniformity property” holds). This result allows us to state a classification theorem for positive equivalence relations (Theorem 2). We show that there exist nonisomorphic positive equivalence relations which are complete with respect to the above reducibility; in particular, we discuss the provable equivalence of a strong enough theory: this relation is complete with respect to reducibility but it does not correspond to a precomplete numeration.From this fact we deduce that an equivalence relation on ω can be strongly represented by a formula (see Definition 8) iff it is positive. At last, we interpret the situation from a topological point of view. Among other things, we generalize a result of Visser by showing that the topological space corresponding to a partition in e.i. sets is irreducible and we prove that the set of equivalence classes of true sentences is dense in the Lindenbaum algebra of the theory.

Determinants of Successful Financial Inclusion in Low-Income Rural Population

2020 ◽  
Vol 68 (1) ◽  
pp. 82-100
Author(s):  
Yashwant Kumar Vaid ◽  
Vikram Singh ◽  
Monika Sethi
Keyword(s):  
Low Income ◽  
Rural Population ◽  
Geographical Area ◽  
Developing Nations ◽  
Point Of View ◽  
Financial Inclusion ◽  
Main Concern ◽  
Policy Makers ◽  
Developing Nation ◽  
Financial Products

Finance plays a key role in the growth of developed as well as developing nations. A financially well included society leads to stronger growth. Financial inclusion aims at providing easy and affordable access to financial products and services. The main concern for any developing nation from a growth point of view is advancement of low-income rural population just as much as the high-income population. Taking a note of this, identifying the key determinants that would lead to successful financial inclusion of low-income rural population is equally, if not more, important. The inclusion strategies have to be built around these determinants to promote inclusion and thus, a clear picture of these determinants is a must have for strategy and policy makers. Though the factors may be somewhat similar across the nation, but their significance and impact on financial inclusion varies greatly from one geographical area to other. In line with this, the purpose of this study is to identify the dimensions of successful financial inclusion in the low-income rural segments with special reference to Raipur, Chhattisgarh. The study uses factor analysis to identify the determinants and path analysis to analyse the significance of these factors in financial inclusion.

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

1988 ◽  
Vol 53 (4) ◽  
pp. 1177-1187
Author(s):  
W. A. MacCaull
Keyword(s):  
Intuitionistic Logic ◽  
Main Idea ◽  
Rational Functions ◽  
Completeness Theorem ◽  
Point Of View ◽  
Polynomial Rings ◽  
Von Neumann ◽  
Ordered Fields ◽  
Von Neumann Regular ◽  
Theoretic Point

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

Transcendence of cardinals

1965 ◽  
Vol 30 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
Recursive Functions ◽  
Power Set ◽  
Recursive Predicate

In this paper, by a function of ordinals we understand a function which is defined for all ordinals and each of whose value is an ordinal. In [7] (also cf. [8] or [9]) we defined recursive functions and predicates of ordinals, following Kleene's definition on natural numbers. A predicate will be called arithmetical, if it is obtained from a recursive predicate by prefixing a sequence of alternating quantifiers. A function will be called arithmetical, if its representing predicate is arithmetical.The cardinals are identified with those ordinals a which have larger power than all smaller ordinals than a. For any given ordinal a, we denote by the cardinal of a and by 2a the cardinal which is of the same power as the power set of a. Let χ be the function such that χ(a) is the least cardinal which is greater than a.Now there are functions of ordinals such that they are easily defined in set theory, but it seems impossible to define them as arithmetical ones; χ is such a function. If we define χ in making use of only the language on ordinals, it seems necessary to use the notion of all the functions from ordinals, e.g., as in [6].

Simulation in food catering industry. A dashboard of performance indicators

2018 ◽  
Author(s):  
Alessandro Tufano ◽  
Riccardo Accorsi ◽  
Andrea Gallo ◽  
Riccardo Manzini
Keyword(s):  
Decision Support Tool ◽  
Point Of View ◽  
High Price ◽  
Main Concern ◽  
Support Tool ◽  
Safety Control ◽  
Catering Industry ◽  
Trade Offs ◽  
Single Production

"Contract catering industry is concerned with the production of ready-to-eat meals for schools, hospitals and private companies. The structure of this market is highly competitive, and customers are rarely willing to pay a high price for this catering service. A single production sites may be demanded up to 10.000 meals per day and these operations can hardly be managed via rule of thumbs without any quantitative decision support tool. This situation is common at several stages of a food supply chain and the methodologies presented in this paper are addressed to any food batch production system with similar complexity and trade-offs. This paper proposes an original KPI dashboard, designed to control costs, time and quality efficiency and helping managers to identify criticalities. Special emphasis is given on food safety control which is the management’s main concern and must be carefully monitored in each stage of the production. To calculate the value of KPIs a Montecarlo simulation approach is used to deal with production complexity and uncertainty. A case study showcases the potential of simulation in this complex industrial field. The case study illustrates an application of the methodology on an Italian company suffering local recipe contamination. The company aims at defining the best standard for production, identifying cycles being sustainable from an economic and environmental point of view."

scholarly journals On the Graph of Divisibility of an Integral Domain

2015 ◽  
Vol 58 (3) ◽  
pp. 449-458 ◽  
Author(s):  
Jason Greene Boynton ◽  
Jim Coykendall
Keyword(s):  
Topological Space ◽  
Integral Domain ◽  
Point Of View ◽  
Current Understanding ◽  
Main Theme ◽  
Topological Graph ◽  
Integral Domains ◽  
Graph Theoretic ◽  
Theoretic Point ◽  
Group Of Divisibility

AbstractIt is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.

The Development of African Historical Studies in East Germany; An Outline And Selected Bibliography

1992 ◽  
Vol 19 ◽  
pp. 133-146
Author(s):  
Thea Büttner
Keyword(s):  
World War Ii ◽  
East Germany ◽  
Point Of View ◽  
Sub Saharan Africa ◽  
Main Concern ◽  
African History ◽  
Outward Appearance ◽  
University Of Berlin ◽  
The World ◽  
German Research

My main concern in this paper is to throw some light on the scope of the problem from the view of the development of African historical studies in East Germany after World War II. It is necessary first to discuss some negative and positive sides of German historical African studies before 1945. For several decades German research has demonstrated a startling lack of interest in the research problems of African history. In connection with the colonial conquests of the European powers, special institutes grew in social anthropology, colonial economics, and geography, although the historical development of the peoples of Africa was ignored. As an outward appearance of this development there grew in several German universities, departments for Oriental languages e.g., at the University of Berlin on the direct instruction of Bismarck, and in 1908 the Colonial Institute at Hamburg University.Leading German historians and Africanists of the past demonstrated their theoretical ignorance in relation to African history. They proceeded from the definition of Leopold von Ranke, who classed the African peoples with the “non-history possessing” peoples who have made no contribution to world culture. G. W. F. Hegel uttered only fatalistic and stereotyped ideas—for him Africa was “no historical part of the World, it has no movement or development to exhibit.” These fundamental conceptions penetrated in one degree or another, the majority of publications on Africa up to 1945. Even Dietrich Westerman, one of the best known Africanists, who published one major book on African history in the German language, Geschichte Afrikas, in 1952 made his studies in the old tradition of seeing sub-Saharan Africa predominantly from the European point of view and continuing the image of an African peoples' history that was not accomplished by the world moulding civilized mankind and has not contributed its share to it. In short, the theoretical foundation of colonialism was rooted in German research in a deep racialist ideology. Only a few explorers and scientists swam against the tide.

Multi-Granular Computing through Rough Sets

2014 ◽  
pp. 1-34 ◽  
Author(s):  
B. K. Tripathy
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Intelligent Systems ◽  
Granular Computing ◽  
Rough Set Theory ◽  
Point Of View ◽  
Information Granules ◽  
Computing Platform

Granular Computing has emerged as a framework in which information granules are represented and manipulated by intelligent systems. Granular Computing forms a unified conceptual and computing platform. Rough set theory put forth by Pawlak is based upon single equivalence relation taken at a time. Therefore, from a granular computing point of view, it is single granular computing. In 2006, Qiang et al. introduced a multi-granular computing using rough set, which was called optimistic multigranular rough sets after the introduction of another type of multigranular computing using rough sets called pessimistic multigranular rough sets being introduced by them in 2010. Since then, several properties of multigranulations have been studied. In addition, these basic notions on multigranular rough sets have been introduced. Some of these, called the Neighborhood-Based Multigranular Rough Sets (NMGRS) and the Covering-Based Multigranular Rough Sets (CBMGRS), have been added recently. In this chapter, the authors discuss all these topics on multigranular computing and suggest some problems for further study.

Modularity in Artificial Neural Networks

2011 ◽  
pp. 1095-1101
Author(s):  
Ricardo Téllez ◽  
Cecilio Angulo
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Networks ◽  
Complex Systems ◽  
Intelligent Systems ◽  
Point Of View ◽  
Main Concern ◽  
Complex Structures ◽  
Computational Point ◽  
Points Of View

The concept of modularity is a main concern for the generation of artificially intelligent systems. Modularity is an ubiquitous organization principle found everywhere in natural and artificial complex systems (Callebaut, 2005). Evidences from biological and philosophical points of view (Caelli and Wen, 1999) (Fodor, 1983), indicate that modularity is a requisite for complex intelligent behaviour. Besides, from an engineering point of view, modularity seems to be the only way for the construction of complex structures. Hence, whether complex neural programs for complex agents are desired, modularity is required. This article introduces the concepts of modularity and module from a computational point of view, and how they apply to the generation of neural programs based on modules. Two levels, strategic and tactical, at which modularity can be implemented, are identified. How they work and how they can be combined for the generation of a completely modular controller for a neural network based agent is presented.

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