Recursive Colorings of Highly Recursive Graphs

1981 ◽  
Vol 33 (6) ◽  
pp. 1279-1290 ◽  
Author(s):  
Henry A. Kierstead
Keyword(s):  
Characteristic Function ◽  
Computer Program ◽  
Finite Number ◽  
Natural Numbers ◽  
Recursive Functions ◽  
Explicit Constructions ◽  
Recursive Structures ◽  
Definition Of ◽  
Recursive Relations

One of the attractions of finite combinatorics is its explicit constructions. This paper is part of a program to enlarge the domain of finite combinatorics to certain infinite structures while preserving the explicit constructions of the smaller domain. The larger domain to be considered consists of the recursive structures. While recursive structures may be infinite they are still amenable to explicit constructions. In this paper we shall concentrate on recursive colorings of highly recursive graphs.A function f: Nk → N, where N is the set of natural numbers, is recursive if and only if there exists an algorithm (i.e., a finite computer program) which upon input of a sequence of natural numbers , after a finite number of steps, outputs . A subset of Nk is recursive provided that its characteristic function is recursive. For a more thorough definition of recursive functions and recursive relations see [10].

A note on nominalism and recursive functions

1949 ◽  
Vol 14 (1) ◽  
pp. 27-31 ◽  
Author(s):  
R. M. Martin
Keyword(s):  
General Theory ◽  
Philosophy Of Science ◽  
Recursive Function ◽  
Homogeneous System ◽  
Formal Logic ◽  
Abstract Objects ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursive Functions ◽  
Definition Of

The purpose of this note is (i) to point out an important similarity between the nominalistic system discussed by Quine in his recent paper On universals and the system of logic (the system н) developed by the author in A homogeneous system for formal logic, (ii) to offer certain corrections to the latter, and (iii) to show that that system (н) is adequate for the general theory of ancestrale and for the definition of any general recursive function of natural numbers.Nominalism as a thesis in the philosophy of science, according to Quine, is the view that it is possible to construct a language adequate for the purposes of science, which in no wise admits classes, properties, relations, or other abstract objects as values for variables.

Note on a conjecture of Skolem

1946 ◽  
Vol 11 (3) ◽  
pp. 73-74 ◽  
Author(s):  
Emil L. Post
Keyword(s):  
Recursive Function ◽  
Recursive Relation ◽  
Natural Numbers ◽  
Excellent Review ◽  
Recursive Functions ◽  
Diagonal Argument ◽  
Primitive Recursive ◽  
Recursive Relations

In his excellent review of four notes of Skolem on recursive functions of natural numbers Bernays states: “The question whether every relation y = f(x1,…, xn) with a recursive function ƒ is primitive recursive remains undecided.” Actually, the question is easily answered in the negative by a form of the familiar diagonal argument.We start with the ternary recursive relation R, referred to in the review, such that R(x, y, 0), R(x, y, 1), … is an enumeration of all binary primitive recursive relations.

Preliminaries

2017 ◽  
Author(s):  
David J. Lobina
Keyword(s):  
Data Structures ◽  
Production Systems ◽  
Specific Data ◽  
Recursive Functions ◽  
Recursive Solution ◽  
Partial Recursive Functions ◽  
Recursive Structures ◽  
Recursive Data Structures ◽  
Recursive Data ◽  
The Relationship

Recursion, or the capacity of ‘self-reference’, has played a central role within mathematical approaches to understanding the nature of computation, from the general recursive functions of Alonzo Church to the partial recursive functions of Stephen C. Kleene and the production systems of Emil Post. Recursion has also played a significant role in the analysis and running of certain computational processes within computer science (viz., those with self-calls and deferred operations). Yet the relationship between the mathematical and computer versions of recursion is subtle and intricate. A recursively specified algorithm, for example, may well proceed iteratively if time and space constraints permit; but the nature of specific data structures—viz., recursive data structures—will also return a recursive solution as the most optimal process. In other words, the correspondence between recursive structures and recursive processes is not automatic; it needs to be demonstrated on a case-by-case basis.

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].

Current approaches to comprehensive assessment of child and adolescent health in hygiene and clinical practice

2020 ◽  
pp. 47-52
Author(s):  
Olga Pavlovna Gritsina ◽  
Anna Konstantinovna Yatsenko ◽  
Lidiya Viktorovna Trankovskaya ◽  
Oksana Valerievna Perelomova
Keyword(s):  
Computer Program ◽  
Children And Adolescents ◽  
Integrated Assessment ◽  
Technical Specification ◽  
Health Indicators ◽  
Comprehensive Assessment ◽  
Medical Surveillance ◽  
Human Welfare ◽  
Software Product ◽  
Definition Of

The relevance of improving the quality of preventive medical surveillance of children and adolescents is undeniable, which provides the basis for the search for methodologically sound approaches to an integrated assessment of the health of the child population. The purpose of the study was to develop and create the software product «Computer Program «Comprehensive Assessment of the Health of Children and Adolescents». For realization of the purpose, patent search, compilation of a technical specification on the basis of criteria of assessment of children’s health and assignment to a particular group of health, writing of a software product using modern programming libraries, as well as preparation of accompanying documents for registration of an intellectual property object were performed. The result of the work was the «Computer Program «Comprehensive Assessment of the Health of Children and Adolescents», designed for the integrated assessment of the state of health of children and adolescents during screening and preventive examinations. After filling in all fields, the program processes the received data and displays the final result — assessment of the child’s health status with the definition of the health group. Information about the examined patient is sent to the program database. The built-in database allows you to systematize the data obtained, analyze the health indicators of the surveyed contingents both in one-step and in longitudinal studies. This program product can be used in the work of medical organizations, higher educational institutions of a medical profile, physical education organizations and institutions of the Federal Service for Supervision of Consumer Rights Protection and Human Welfare in the Russian Federation.

On the Dynamic Isotropy of Mechanisms With Two Degrees of Freedom

2005 ◽  
Author(s):  
Raffaele Di Gregorio ◽  
Alessandro Cammarata ◽  
Rosario Sinatra
Keyword(s):  
Finite Number ◽  
Degrees Of Freedom ◽  
Point Of View ◽  
Closed Curves ◽  
Special Cases ◽  
Dynamic Performances ◽  
Definition Of ◽  
Geometric Locus

The comparison of mechanisms with different topology or with different geometry, but with the same topology, is a necessary operation during the design of a machine sized for a given task. Therefore, tools that evaluate the dynamic performances of a mechanism are welcomed. This paper deals with the dynamic isotropy of 2-dof mechanisms starting from the definition introduced in a previous paper. In particular, starting from the condition that identifies the dynamically isotropic configurations, it shows that, provided some special cases are not considered, 2-dof mechanisms have at most a finite number of isotropic configurations. Moreover, it shows that, provided the dynamically isotropic configurations are excluded, the geometric locus of the configuration space that collects the points associated to configurations with the same dynamic isotropy is constituted by closed curves. This results will allow the classification of 2-dof mechanisms from the dynamic-isotropy point of view, and the definition of some methodologies for the characterization of the dynamic isotropy of these mechanisms. Finally, examples of applications of the obtained results will be given.

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.

A Proposed Categorization Method and Category Domain Inequality Development Procedure for Ground Joints of Five Bar Linkages

1998 ◽  
Author(s):  
Chuen-Sen Lin ◽  
Terry Lee ◽  
Bao-Ping Jia
Keyword(s):  
Computer Program ◽  
Single Loop ◽  
Loop Equation ◽  
Development Procedure ◽  
Five Dimension ◽  
Simple Logic ◽  
Logic Operations ◽  
Definition Of ◽  
Dimension Space ◽  
The Given

Abstract This paper presents a method for the development of sets of symbolic inequalities in terms of link lengths for the prediction of the rotation capabilities of ground joints of single-loop five-bar linkages. The inequalities are obtained from the combination of the loop equation of a five-bar linkage and its derivatives and the application of simple logic operations. The rotation capabilities of ground joints are divided into three categories: the incomplete-rotation ground joints, the conditioned complete-rotation ground joints, and the unconditioned complete-rotation ground joints. The derived sets of inequalities define the domain, in a five-dimension space of the five link lengths, for each of the rotation categories. In this paper, the definition of each category is clearly described and the derivations of sets of inequalities are explained in details. A computer program was constructed to examine the completeness and correctness of the categorization method and to analyze the given five-bar linkages to determine the appropriate categories for their ground joints.

A Hierarchy on the Class of Primitive Recursive Ordinal Functions

1976 ◽  
Vol 28 (6) ◽  
pp. 1205-1209
Author(s):  
Stanley H. Stahl
Keyword(s):  
Natural Numbers ◽  
Recursive Functions ◽  
Set Functions ◽  
Primitive Recursive ◽  
Class Of Functions ◽  
Recursive Set

The class of primitive recursive ordinal functions (PR) has been studied recently by numerous recursion theorists and set theorists (see, for example, Platek [3] and Jensen-Karp [2]). These investigations have been part of an inquiry concerning a larger class of functions; in Platek's case, the class of ordinal recursive functions and in the case of Jensen and Karp, the class of primitive recursive set functions. In [4] I began to study PR in depth and this paper is a report on an attractive analogy between PR and its progenitor, the class of primitive recursive functions on the natural numbers (Prim. Rec).

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