Recursion-theoretic hierarchies

2018 ◽  
Author(s):  
Harold Hodes
Keyword(s):  
Computational Complexity ◽  
Model Theory ◽  
Recursion Theory ◽  
Bottom Up ◽  
Basic Class

In mathematics, a hierarchy is a ‘bottom up’ system classifying entities of some particular sort, a system defined inductively, starting with a ‘basic’ class of such entities, with further (‘higher’) classes of such entities defined in terms of previously defined (‘lower’) classes. Such a classification reflects complexity in some respect, one entity being less complex than another if it appears ‘earlier’ (‘lower’) then that other. Many of the hierarchies studied by logicians construe complexity as complexity of definition, placing such hierarchies within the purview of model theory; but even such notions of complexity are closely tied to species of computational complexity, placing them also in the purview of recursion theory.

scholarly journals FINDING THE LIMIT OF INCOMPLETENESS I

2020 ◽  
Vol 26 (3-4) ◽  
pp. 268-286
Author(s):  
YONG CHENG
Keyword(s):  
Model Theory ◽  
Recursion Theory ◽  
Minimal Degree ◽  
Incompleteness Theorem ◽  
Turing Degree ◽  
Turing Reducibility ◽  
Inseparable Pair

AbstractIn this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds.

J. C. Shepherdson. Algorithmic procedures, generalized Turing algorithms, and elementary recursion theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 285–308. - J. C. Shepherdson. Computational complexity of real functions. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 309–315. - A. J. Kfoury. The pebble game and logics of programs. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 317–329. - R. Statman. Equality between functionals revisited. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 331–338. - Robert E. Byerly. Mathematical aspects of recursive function theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 339–352.

1990 ◽  
Vol 55 (2) ◽  
pp. 876-878
Author(s):  
J. V. Tucker
Keyword(s):  
New York ◽  
Computational Complexity ◽  
Recursive Function ◽  
Function Theory ◽  
Recursion Theory ◽  
Foundations Of Mathematics ◽  
Real Functions ◽  
Pebble Game ◽  
Logics Of Programs

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.

Logic and Philosophical Methodology

2016 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Classical Logic ◽  
Proof Theory ◽  
Recursion Theory ◽  
Theory Model ◽  
Philosophical Methodology ◽  
Paraconsistent Logics ◽  
Philosophical Logic

This article explores the role of logic in philosophical methodology, as well as its application in philosophy. The discussion gives a roughly equal coverage to the seven branches of logic: elementary logic, set theory, model theory, recursion theory, proof theory, extraclassical logics, and anticlassical logics. Mathematical logic comprises set theory, model theory, recursion theory, and proof theory. Philosophical logic in the relevant sense is divided into the study of extensions of classical logic, such as modal or temporal or deontic or conditional logics, and the study of alternatives to classical logic, such as intuitionistic or quantum or partial or paraconsistent logics. The nonclassical consists of the extraclassical and the anticlassical, although the distinction is not clearcut.

scholarly journals Alternating Turing machines and the analytical hierarchy

10.29007/t77g ◽  
2018 ◽  
Author(s):  
Daniel Leivant
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Recursion Theory ◽  
Turing Machines ◽  
Arithmetical Hierarchy ◽  
Definition Of ◽  
Alternating Turing Machines ◽  
Insight Into ◽  
Computably Enumerable

We use notions originating in Computational Complexity to provide insight into the analogies between computational complexity and Higher Recursion Theory. We consider alternating Turing machines, but with a modified, global, definition of acceptance. We show that a language is accepted by such a machine iff it is Pi-1-1. Moreover, total alternating machines, which either accept or reject each input, accept precisely the hyper-arithmetical (Delta-1-1) languages. Also, bounding the permissible number of alternations we obtain a characterization of the levels of the arithmetical hierarchy..The novelty of these characterizations lies primarily in the use of finite computing devices, with finitary, discrete, computation steps. We thereby elucidate the correspondence between the polynomial-time and the arithmetical hierarchies, as well as that between the computably-enumerable, the inductive (Pi-1-1), and the PSpace languages.

One hundred and two problems in mathematical logic

1975 ◽  
Vol 40 (2) ◽  
pp. 113-129 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Model Theory ◽  
Proof Theory ◽  
Recursion Theory ◽  
Logic Model ◽  
Mathematical Statement ◽  
Question Mark ◽  
Truth Value ◽  
Expository Paper

This expository paper contains a list of 102 problems which, at the time of publication, are unsolved. These problems are distributed in four subdivisions of logic: model theory, proof theory and intuitionism, recursion theory, and set theory. They are written in the form of statements which we believe to be at least as likely as their negations. These should not be viewed as conjectures since, in some cases, we had no opinion as to which way the problem would go.In each case where we believe a problem did not originate with us, we made an effort to pinpoint a source. Often this was a difficult matter, based on subjective judgments. When we were unable to pinpoint a source, we left a question mark. No inference should be drawn concerning the beliefs of the originator of a problem as to which way it will go (lest the originator be us).The choice of these problems was based on five criteria. Firstly, we are only including problems which call for the truth value of a particular mathematical statement. A second criterion is the extent to which the concepts involved in the statements are concepts that are well known, well denned, and well understood, as well as having been extensively considered in the literature. A third criterion is the extent to which these problems have natural, simple and attractive formulations. A fourth criterion is the extent to which there is evidence that a real difficulty exists in finding a solution. Lastly and unavoidably, the extent to which these problems are connected with the author's research interests in mathematical logic.

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