Gödel numberings of partial recursive functions

1958 ◽  
Vol 23 (3) ◽  
pp. 331-341 ◽  
Author(s):  
Hartley Rogers
Keyword(s):  
Mathematical Logic ◽  
Recursive Function ◽  
Function Theory ◽  
Recursive Structure ◽  
Partial Functions ◽  
Logical Calculus ◽  
Recursive Functions ◽  
Several Variables ◽  
Partial Recursive Functions ◽  
Made In

In § 1 we present conceptual material concerning the notion of a Gödel numbering of the partial recursive functions. § 2 presents a theorem about these concepts. § 3 gives several applications. The material in § 1 and § 2 grew out of attempts by the author to find routine solutions to some of the problems discussed in § 3. The author wishes to acknowledge his debt in § 2 to the fruitful methods of Myhill in [M] and to thank the referee for an abbreviated and improved version of the proof for Lemma 3 in § 2.In the literature of mathematical logic, “Gödel numbering” usually means an effective correspondence between integers and the well-formed formulas of some logical calculus. In recursive function theory, certain such associations between the non-negative integers and instructions for computing partial recursive functions have been fundamental. In the present paper we shall be concerned only with numberings of the latter, more special, sort. By numbers and integers we shall mean non-negative integers. Our notation is, in general, that of [K]. If ϕ and ψ are two partial functions, ϕ = ψ shall mean that (∀x)[ϕ(x)≃(ψx)], i.e., that ϕ and ψ are defined for the same arguments and are equal on those arguments. We consider partial recursive functions of one variable; applications of the paper to the case of several variables, or to the case of all partial recursive functions in any number of variables, can be made in the usual way using the coordinate functions (a)i of [K, p. 230]. It will furthermore be observed that we consider only concepts that are invariant with respect to general recursive functions; more limited notions of Gödel numbering, taking into account, say, primitive recursive structure, are beyond the scope of the present paper.

scholarly journals Recursive density types and Nerode extensions of arithmetic

1975 ◽  
Vol 20 (2) ◽  
pp. 146-158 ◽  
Author(s):  
P. Aczel
Keyword(s):  
Equivalence Classes ◽  
Partial Functions ◽  
Recursive Functions ◽  
Cardinal Numbers ◽  
Partial Recursive Functions

The notion of a recursive density type (R.D.T.) was introduced by Medvedev and developed by Pavlova (1961). More recently the algebra of R.D.T.'s was initiated by Gonshor and Rice (1969). The R.D.T.'s are equivalence classes of sets of integers, similar in many respects to the R.E.T.'s. They may both be thought of as effective analogues of the cardinal numbers. While the equivalence relationfor R.E.T.'s is defined in terms of partial recursive functions, that for R.D.T.'s may be characterized in terms of recursively bounded partial functions (see 4.22a).

Two recursion theoretic characterizations of proof speed-ups

1989 ◽  
Vol 54 (2) ◽  
pp. 522-526 ◽  
Author(s):  
James S. Royer
Keyword(s):  
Recursive Function ◽  
Formal System ◽  
Complexity Measure ◽  
Incompleteness Theorem ◽  
Similar Treatment ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Speed Up

Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan (improving upon [Kle50] and [Kle52]) showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1is speedable over another system S0on a set of formulaeAiff, for each recursive functionh, there is a formulaαinAsuch that the length of the shortest proof ofαin S0is larger thanhof the shortest proof ofαin S1. (Here we equate the length of a proof with something like the number of characters making it up,notits number of lines.) We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1but unprovable in S0from the collectionA–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below,ϕis an acceptable numbering of the partial recursive functions [Rog87] andΦa (Blum) complexity measure associated withϕ[Blu67], [DW83].

On complexity properties of recursively enumerable sets

1973 ◽  
Vol 38 (4) ◽  
pp. 579-593 ◽  
Author(s):  
M. Blum ◽  
I. Marques
Keyword(s):  
Complexity Theory ◽  
Recursive Function ◽  
Strong Sense ◽  
Important Goal ◽  
Partial Recursive Function ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Partial Recursive Functions ◽  
Do So

An important goal of complexity theory, as we see it, is to characterize those partial recursive functions and recursively enumerable sets having some given complexity properties, and to do so in terms which do not involve the notion of complexity.As a contribution to this goal, we provide characterizations of the effectively speedable, speedable and levelable [2] sets in purely recursive theoretic terms. We introduce the notion of subcreativeness and show that every program for computing a partial recursive function f can be effectively speeded up on infinitely many integers if and only if the graph of f is subcreative.In addition, in order to cast some light on the concepts of effectively speedable, speedable and levelable sets we show that all maximal sets are levelable (and hence speedable) but not effectively speedable and we exhibit a set which is not levelable in a very strong sense but yet is effectively speedable.

Some properties of invariant sets

1984 ◽  
Vol 49 (1) ◽  
pp. 9-21 ◽  
Author(s):  
Robert E. Byerly
Keyword(s):  
Recursive Function ◽  
Invariant Sets ◽  
Turing Degrees ◽  
Partial Recursive Function ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Previous Paragraph

In [1] two interesting invariance notions were introduced: the notions of a set of godel numbers being invariant to automorphisms of the structures (ω, ·) and (ω, E) respectively. Here, · and E are defined by n · m ≃ φn (m) and nEm if and only if n Є Wm, where {φn} and {Wn} are acceptable enumerations of the partial recursive functions and r.e. sets respectively. In this paper we continue the study of the invariant sets, and especially the invariant r.e. sets, of gödel numbers.We start off with an easy result which characterizes the Turing degrees containing invariant sets. We then take a closer look at r.e. sets invariant with respect to automorphisms of (ω,E). Using the characterization [1, Theorem 4.2] of such sets, we will derive a somewhat different characterization (which was stated, but not proved, in [1, Proposition 4.4]) and, using it as a tool for constructing invariant sets, prove that the r.e. sets invariant with respect to automorphisms of (ω, E) cannot be effectively enumerated.We will next discuss representations of r.e. sets invariant with respect to automorphisms of (ω, ·). Although these sets do not have as nice a characterization as the r.e. sets invariant with respect to automorphisms of (ω, E) do, the techniques of [1] can still profitably be used to investigate their structure. In particular, if f is a partial recursive function whose graph is invariant with respect to automorphisms of (ω, ·), then for every a in the domain of f, there is a term t(a) built up from a and · only such that f(a) ≃ t(a). This is an analog to [1, Corollary 4.3]. We will also prove an analog to a result mentioned in the previous paragraph: the r.e. sets invariant with respect to automorphisms of (ω, ·) cannot be effectively enumerated.

Church's thesis without tears

1983 ◽  
Vol 48 (3) ◽  
pp. 797-803 ◽  
Author(s):  
Fred Richman
Keyword(s):  
Function Theory ◽  
Computable Function ◽  
Modern Theory ◽  
Central Feature ◽  
Partial Functions ◽  
Church’S Thesis ◽  
Recursive Functions ◽  
Church's Thesis ◽  
The Church ◽  
Computable Functions

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular ifPis some property, then the statement “for eachmthere isnsuch thatP(m, n)” means that we can construct a (computable) functionθsuch thatP(m, θ(m))for allm. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.

Recursive generation in language

2017 ◽  
Author(s):  
David J. Lobina
Keyword(s):  
Mathematical Logic ◽  
Production System ◽  
General Property ◽  
Recent Literature ◽  
Expressive Power ◽  
The Self ◽  
Recursive Structure ◽  
Embedded Structures ◽  
Mechanical Procedure ◽  
The 1950S

The introduction of recursion into linguistics was the result of applying some of the results of mathematical logic to the study of language. In particular, recursion was introduced in the 1950s as a general property of the mechanical procedure underlying the grammar, in order to account for language’s discrete infinity and expressive power—in the 1950s, this mechanical procedure was a production system, whereas more recently, of course, it is the set-operator merge. Unfortunately, the recent literature has confused the general recursive property of a grammar with specific instances of (recursive) rules/operations within a grammar; more worryingly still, there has been a general conflation of these recursive rules with some of the self-embedded structures these rules can generate, adding to the confusion. The conflation is manifold but always fallacious. Moreover, language manifests a much more generally recursive structure than is usually recognized: bundles of the universal (Specifier)-Head-Complement(s) geometry.

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.

scholarly journals The Level of History Teachers’ Use Active Learning Methods and Technics

2017 ◽  
Vol 10 (12) ◽  
pp. 140
Author(s):  
Sefa YILDIRIM ◽  
Ozkan AKMAN ◽  
Bulent ALAGOZ
Keyword(s):  
Active Learning ◽  
Developed Countries ◽  
Learning Methods ◽  
History Teachers ◽  
Classical Education ◽  
Several Variables ◽  
Kurt Lewin ◽  
The Developed Countries ◽  
New Education ◽  
Made In

An experience theory is required if the education is to be wisely carried out (John Dewey). Education is a discipline that saves lives if it is qualified, but loss of which could not be made up throughout generations if it is not qualified. The roots of society are based on the education, and educated masses and civilizations could either move into the future or could fall behind in the race of becoming civilized. The classical education notion which stays on the level of theory and is carried out, centering the teacher is being left by the developed countries and replaced with the education notion which centers the student and structures information by benefiting from experiences, thus aims to lead civilization race with citizens knowing the ways to reach the information and aware of their duties and responsibilities. While Kurt Lewin says nothing is as practical as a good theory, he also catches attention to the new education notion centering student that has changed and is changing. In this scope, the aim of this study is analyze how often active learning methods are used by history teachers through several variables. In the light of the data, after analysis results and explanations made in accordance with these results are written, the study is concluded with suggestions

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