Metarecursive sets

1965 ◽  
Vol 30 (3) ◽  
pp. 318-338 ◽  
Author(s):  
G. Kreisel ◽  
Gerald E. Sacks
Keyword(s):  
Mathematical Theory ◽  
Experimental Approach ◽  
Mathematical Structure ◽  
Field Work ◽  
Recursion Theory ◽  
Axiomatic Method ◽  
Axiomatic Treatment ◽  
Abstract Approach ◽  
Priority Method ◽  
Ultimate Purpose

Our ultimate purpose is to give an axiomatic treatment of recursion theory sufficient to develop the priority method. The direct or abstract approach is to keep in mind as clearly as possible the methods actually used in recursion theory, and then to formulate them explicitly. The indirect or experimental approach is to look first for other mathematical theories which seem similar to recursion theory, to formulate the analogies precisely, and then to search for an axiomatic treatment which covers not only recursion theory but also the analogous theories as particular cases.The first approach is more general because it does not depend on the existence of (familiar) analogues. A concrete mathematical theory, it seems, need have no such analogues and still be important, as e.g. classical number theory. In such a case, an axiomatic treatment may still be useful for exhibiting the mathematical structure of the theory considered and the assumptions on which it rests. However, it will lack one of the most heavily advertised advantages of the axiomatic method, namely, the “economy of thought” which results from an uniform theory for several different and interesting cases: we cannot hope for this if, by hypothesis, we know of only one particular case. In contrast, the second approach, if successful at all, is bound to achieve such “economy” because we start out with several interesting particular cases. Another possible virtue of the second approach is that of field work over insight: the abstract pattern that we are looking for and hoping to formalize in axioms, may not be evident in any one mathematical theory, but may spring to the eye if one happens to look simultaneously at several theories which happen to realize the pattern.

Definability of r. e. sets in a class of recursion theoretic structures

1983 ◽  
Vol 48 (3) ◽  
pp. 662-669 ◽  
Author(s):  
Robert E. Byerly
Keyword(s):  
Recursion Theory ◽  
Universal Function ◽  
Independent Interest ◽  
Suitable Model ◽  
Natural Numbers ◽  
Construction Techniques ◽  
Partial Recursive Functions ◽  
Priority Method ◽  
Recursive Isomorphism ◽  
Direct Attack

It is known [4, Theorem 11-X(b)] that there is only one acceptable universal function up to recursive isomorphism. It follows from this that sets definable in terms of a universal function alone are specified uniquely up to recursive isomorphism. (An example is the set K, which consists of all n such that {n}(n) is defined, where λn, m{n}(m) is an acceptable universal function.) Many of the interesting sets constructed and studied by recursion theorists, however, have definitions which involve additional notions, such as a specific enumeration of the graph of a universal function. In particular, many of these definitions make use of the interplay between the purely number-theoretic properties of indices of partial recursive functions and their purely recursion-theoretic properties.This paper concerns r.e. sets that can be defined using only a universal function and some purely number-theoretic concepts. In particular, we would like to know when certain recursion-theoretic properties of r.e. sets definable in this way are independent of the particular choice of universal function (equivalently, independent of the particular way in which godel numbers are identified with natural numbers).We will first develop a suitable model-theoretic framework for discussing this question. This will enable us to classify the formulas defining r.e. sets by their logical complexity. (We use the number of alternations of quantifiers in the prenex form of a formula as a measure of logical complexity.) We will then be able to examine the question at each level.This work is an approach to the question of when the recursion-theoretic properties of an r.e. set are independent of the particular parameters used in its construction. As such, it does not apply directly to the construction techniques most commonly used at this time for defining particular r.e. sets, e.g., the priority method. A more direct attack on this question for these techniques is represented by such works as [3] and [5]. However, the present work should be of independent interest to the logician interested in recursion theory.

scholarly journals The infinite injury priority method

1976 ◽  
Vol 41 (2) ◽  
pp. 513-530 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Recursion Theory ◽  
Powerful Method ◽  
Recursively Enumerable ◽  
Priority Method

One of the most important and distinctive tools in recursion theory has been the priority method whereby a recursively enumerable (r.e.) set A is constructed by stages to satisfy a sequence of conditions {Rn}n∈ω called requirements. If n < m, requirement Rn is given priority over Rm and action taken for Rm at some stage s may at a later stage t > s be undone for the sake of Rn thereby injuring Rm at stage t. The first priority method was invented by Friedberg [2] and Muchnik [11] to solve Post's problem and is characterized by the fact that each requirement is injured at most finitely often.Shoenfield [20, Lemma 1], and then independently Sacks [17] and Yates [25] invented a much more powerful method in which a requirement may be injured infinitely often, and the method was applied and refined by Sacks [15], [16], [17], [18], [19] and Yates [25], [26] to obtain many deep results on r.e. sets and their degrees. In spite of numerous simplifications and variations this infinite injury method has never been as well understood as the finite injury method because of its apparently greater complexity.The purpose of this paper is to reduce the Sacks method to two easily understood lemmas whose proofs are very similar to the finite injury case. Using these lemmas we can derive all the results of Sacks on r.e. degrees, and some by Yates and Robinson as well, in a manner accessible to the nonspecialist. The heart of the method is an ingenious observation of Lachlan [7] which is combined with a further simplification of our own.

An introduction to γ-recursion theory (or what to do in KP – Foundation)

1990 ◽  
Vol 55 (1) ◽  
pp. 194-206 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Recursion Theory ◽  
Reverse Mathematics ◽  
Peano Arithmetic ◽  
Turing Degrees ◽  
The Subject ◽  
Admissible Ordinals ◽  
Limited Induction ◽  
Well Foundedness

The program of reverse mathematics has usually been to find which parts of set theory, often used as a base for other mathematics, are actually necessary for some particular mathematical theory. In recent years, Slaman, Groszek, et al, have given the approach a new twist. The priority arguments of recursion theory do not naturally or necessarily lead to a foundation involving any set theory; rather, Peano Arithmetic (PA) in the language of arithmetic suffices. From this point, the appropriate subsystems to consider are fragments of PA with limited induction. A theorem in this area would then have the form that certain induction axioms are independent of, necessary for, or even equivalent to a theorem about the Turing degrees. (See, for examples, [C], [GS], [M], [MS], and [SW].)As go the integers so go the ordinals. One motivation of α-recursion theory (recursion on admissible ordinals) is to generalize classical recursion theory. Since induction in arithmetic is meant to capture the well-foundedness of ω, the corresponding axiom in set theory is foundation. So reverse mathematics, even in the context of a set theory (admissibility), can be changed by the influence of reverse recursion theory. We ask not which set existence axioms, but which foundation axioms, are necessary for the theorems of α-recursion theory.When working in the theory KP – Foundation Schema (hereinafter called KP−), one should really not call it α-recursion theory, which refers implicitly to the full set of axioms KP. Just as the name β-recursion theory refers to what would be α-recursion theory only it includes also inadmissible ordinals, we call the subject of study here γ-recursion theory. This answers a question by Sacks and S. Friedman, “What is γ-recursion theory?”

scholarly journals De Karisimbi, Een Recente Vulkaan van Het Virungagebied (Rwanda, Zaire)

Afrika Focus ◽  
1986 ◽  
Vol 2 (1) ◽  
pp. 23-54
Author(s):  
Mark de Mulder
Keyword(s):  
Fractional Crystallization ◽  
Field Work ◽  
Geological Structure ◽  
Radiometric Dating ◽  
Chemical Data ◽  
Laboratory Studies ◽  
Crustal Rocks ◽  
Dating Method ◽  
Ultimate Purpose

Karisimbi, A Recent Volcano of the Virunga Area (Rwanda, Zaire) This paper deals with the geological structure and the petrologic evolution of Karisimbi, the highest volcano in the Virunga region. As this paper is intended to be understood by non-geologists, a brief review about the methods used by volcanologists, should make things clear for the reader. The field-work data enabled us to describe the morphology, structure and the evolution of Karisimbi. The results of the laboratory studies are summarized in the section petrography – petrochemistry, where some problems concerning nomenclature and interpretation of chemical data are discussed as well. Petrographical and petrochemical information leads us to the origin and the evolution of magmas, which is the ultimate purpose of every petrologist. In the case of Karisimbi, it is suggested that its petrologic evolution took place by simultaneous fractional crystallization and contamination by crustal rocks. Finally, the ages of some typical Karisimbi lavas have been determined by a radiometric dating method (K-Ar), bearing in mind that large errors on these ages are inevitable.

Representation and Interpretation of Sketches in Mechanical Design: Experimental and Theoretical Approaches

2003 ◽  
Author(s):  
Y. Zeng ◽  
A. Pardasani ◽  
H. Antunes ◽  
Z. Li ◽  
J. Dickinson ◽  
...  
Keyword(s):  
Design Process ◽  
Set Theory ◽  
Experimental Approach ◽  
Theoretical Foundation ◽  
Mechanical Design ◽  
Mathematical Structure ◽  
Assembly System ◽  
Automobile Assembly ◽  
Theoretical Approaches

This paper aims to establish a theoretical foundation for representing and interpreting free-hand design sketches throughout the conceptual design process. Both experimental and theoretical approaches are used. In using the experimental approach, one case study from a book and one case study from an automobile assembly system manufacturer are used to illustrate the characteristics of design sketches. These characteristics provide the requirements for models of sketch representation and interpretation. In using the theoretical approach, a mathematical structure of design sketches is established. This mathematical structure can naturally and logically model the evolving sketches generated in the design process, through integrating the strengths of set theory and mereology. Based on the results of these two approaches, a design sketch language is developed to be a formal foundation of sketch representation and interpretation.

scholarly journals Maximal sets and fragments of Peano arithmetic

1989 ◽  
Vol 115 ◽  
pp. 165-183 ◽  
Author(s):  
C.T. Chong
Keyword(s):  
Recursion Theory ◽  
Peano Arithmetic ◽  
Base Theory ◽  
Abstract Principle ◽  
Priority Method ◽  
General Program ◽  
Fragments Of Peano Arithmetic

This work is inspired by the recent paper of Mytilinaios and Slaman [9] on the infinite injury priority method. It may be considered to fall within the general program of the study of reverse recursion theory: What axioms of Peano arithmetic are required or sufficient to prove theorems in recursion theory? Previous contributions to this program, especially with respect to the finite and infinite injury priority methods, can be found in the works of Groszek and Mytilinaios [4], Groszek and Slaman [5], Mytilinaios [8], Slaman and Woodin [10]. Results of [4] and [9], for example, together pinpoint the existence of an incomplete, nonlow r.e. degree to be provable only within some fragment of Peano arithmetic at least as strong as P- + IΣ2. Indeed an abstract principle on infinite strategies, such as that on the construction of an incomplete high r.e. degree, was introduced in [4] and shown to be equivalent to Σ2 induction over the base theory P- + IΣ0 of Peano arithmetic.

ON A MATHEMATICAL THEORY OF COMPLEX SYSTEMS ON NETWORKS WITH APPLICATION TO OPINION FORMATION

2013 ◽  
Vol 24 (02) ◽  
pp. 405-426 ◽  
Author(s):  
D. KNOPOFF
Keyword(s):  
Kinetic Theory ◽  
Complex Systems ◽  
Mathematical Theory ◽  
Stochastic Games ◽  
Mathematical Structure ◽  
Opinion Formation ◽  
Active Particles ◽  
Mathematical Equations

This paper presents a development of the so-called kinetic theory for active particles to the modeling of living, hence complex, systems localized in networks. The overall system is viewed as a network of interacting nodes, mathematical equations are required to describe the dynamics in each node and in the whole network. These interactions, which are nonlinearly additive, are modeled by evolutive stochastic games. The first conceptual part derives a general mathematical structure, to be regarded as a candidate towards the derivation of models, suitable to capture the main features of the said systems. An application on opinion formation follows to show how the theory can generate specific models.

ON THE DIFFICULT INTERPLAY BETWEEN LIFE, "COMPLEXITY", AND MATHEMATICAL SCIENCES

2013 ◽  
Vol 23 (10) ◽  
pp. 1861-1913 ◽  
Author(s):  
N. BELLOMO ◽  
D. KNOPOFF ◽  
J. SOLER
Keyword(s):  
Kinetic Theory ◽  
Complex Systems ◽  
Case Studies ◽  
Mathematical Theory ◽  
Stochastic Games ◽  
Mathematical Structure ◽  
Living Systems ◽  
Active Particles ◽  
Suitable Generalization ◽  
Mathematical Sciences

This paper presents a revisiting, with developments, of the so-called kinetic theory for active particles, with the main focus on the modeling of nonlinearly additive interactions. The approach is based on a suitable generalization of methods of kinetic theory, where interactions are depicted by stochastic games. The basic idea consists in looking for a general mathematical structure suitable to capture the main features of living, hence complex, systems. Hopefully, this structure is a candidate towards the challenging objective of designing a mathematical theory of living systems. These topics are treated in the first part of the paper, while the second one applies it to specific case studies, namely to the modeling of crowd dynamics and of the immune competition.

scholarly journals Brownfield Management in Greece. The Case of Piraeus

2020 ◽  
Vol 27 (2) ◽  
pp. 5-15
Author(s):  
Tousi Evgenia ◽  
Serraos Konstantinos
Keyword(s):  
Soil Contamination ◽  
Policy Making ◽  
Field Work ◽  
International Literature ◽  
Brownfield Redevelopment ◽  
Brownfield Sites ◽  
Administrative Area ◽  
Ultimate Purpose ◽  
Brownfield Management

The article presents the main findings of a research associated with brownfield redevelopment in Greece. The regional administrative area of Piraeus is used as a pilot case study. Taking into account the international literature, the article presents the contemporary condition in Greece emphasizing on key-obstacles that hinder rehabilitation and reuse of brownfield sites. The research includes field work. Thematic maps depict the current condition of brownfield sites around Piraeus port, using as key-categories the former use, the current use, the system of ownership, the prices of land, the level of soil contamination. This cartographic depiction functions as a necessary tool for policy making on urban regeneration. The conclusions derived from field work provide useful information for further research. The ultimate purpose of the article is to highlight the contemporary problems in Greece, making the appropriate connections with the international experience on the field.  

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