scholarly journals Computability and Recursion

1996 ◽  
Vol 2 (3) ◽  
pp. 284-321 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Recursive Function ◽  
Conceptual Analysis ◽  
Function Theory ◽  
Computability Theory ◽  
Turing Machines ◽  
Church’S Thesis ◽  
The Future ◽  
Church's Thesis ◽  
The Subject

AbstractWe consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness”. We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas.After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church's Thesis and Turing's Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than “Recursive Function Theory.”

Trial and error predicates and the solution to a problem of Mostowski

1965 ◽  
Vol 30 (1) ◽  
pp. 49-57 ◽  
Author(s):  
Hilary Putnam
Keyword(s):  
Finite Number ◽  
Correct Answer ◽  
Decision Procedure ◽  
Finite Sequence ◽  
Decision Procedures ◽  
Turing Machines ◽  
Trial And Error ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Empirical Means

The purpose of this paper is to present two groups of results which have turned out to have a surprisingly close interconnection. The first two results (Theorems 1 and 2) were inspired by the following question: we know what sets are “decidable” — namely, the recursive sets (according to Church's Thesis). But what happens if we modify the notion of a decision procedure by (1) allowing the procedure to “change its mind” any finite number of times (in terms of Turing Machines: we visualize the machine as being given an integer (or an n-tuple of integers) as input. The machine then “prints out” a finite sequence of “yesses” and “nos”. The last “yes” or “no” is always to be the correct answer.); and (2) we give up the requirement that it be possible to tell (effectively) if the computation has terminated? I.e., if the machine has most recently printed “yes”, then we know that the integer put in as input must be in the set unless the machine is going to change its mind; but we have no procedure for telling whether the machine will change its mind or not.The sets for which there exist decision procedures in this widened sense are decidable by “empirical” means — for, if we always “posit” that the most recently generated answer is correct, we will make a finite number of mistakes, but we will eventually get the correct answer. (Note, however, that even if we have gotten to the correct answer (the end of the finite sequence) we are never sure that we have the correct answer.)

Turing machines

2018 ◽  
Author(s):  
Guglielmo Tamburrini
Keyword(s):  
Finite Number ◽  
Turing Machine ◽  
Finite Alphabet ◽  
Turing Machines ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Internal Configuration ◽  
Class Of Functions ◽  
Alan Mathison Turing ◽  
Algorithmic Procedure

Turing machines are abstract computing devices, named after Alan Mathison Turing. A Turing machine operates on a potentially infinite tape uniformly divided into squares, and is capable of entering only a finite number of distinct internal configurations. Each square may contain a symbol from a finite alphabet. The machine can scan one square at a time and perform, depending on the content of the scanned square and its own internal configuration, one of the following operations: print or erase a symbol on the scanned square or move on to scan either one of the immediately adjacent squares. These elementary operations are possibly accompanied by a change of internal configuration. Turing argued that the class of functions calculable by means of an algorithmic procedure (a mechanical, stepwise, deterministic procedure) is to be identified with the class of functions computable by Turing machines. The epistemological significance of Turing machines and related mathematical results hinges upon this identification, which later became known as Turing’s thesis; an equivalent claim, Church’s thesis, had been advanced independently by Alonzo Church. Most crucially, mathematical results stating that certain functions cannot be computed by any Turing machine are interpreted, by Turing’s thesis, as establishing absolute limitations of computing agents.

Church’s thesis

2018 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Turing Machine ◽  
Recursive Function ◽  
Turing Computability ◽  
Church’S Thesis ◽  
The Third ◽  
Alan Turing ◽  
Rigorous Formulation ◽  
Person Following ◽  
Mechanical Procedure ◽  
Church's Thesis

An algorithm or mechanical procedure A is said to ‘compute’ a function f if, for any n in the domain of f, when given n as input, A eventually produces fn as output. A function is ‘computable’ if there is an algorithm that computes it. A set S is ‘decidable’ if there is an algorithm that decides membership in S: if, given any appropriate n as input, the algorithm would output ‘yes’ if n∈S, and ‘no’ if n∉S. The notions of ‘algorithm’, ‘computable’ and ‘decidable’ are informal (or pre-formal) in that they have meaning independently of, and prior to, attempts at rigorous formulation. Church’s thesis, first proposed by Alonzo Church in a paper published in 1936, is the assertion that a function is computable if and only if it is recursive: ‘We now define the notion…of an effectively calculable function…by identifying it with the notion of a recursive function….’ Independently, Alan Turing argued that a function is computable if and only if there is a Turing machine that computes it; and he showed that a function is Turing-computable if and only if it is recursive. Church’s thesis is widely accepted today. Since an algorithm can be ‘read off’ a recursive derivation, every recursive function is computable. Three types of ‘evidence’ have been cited for the converse. First, every algorithm that has been examined has been shown to compute a recursive function. Second, Turing, Church and others provided analyses of the moves available to a person following a mechanical procedure, arguing that everything can be simulated by a Turing machine, a recursive derivation, and so on. The third consideration is ‘confluence’. Several different characterizations, developed more or less independently, have been shown to be coextensive, suggesting that all of them are on target. The list includes recursiveness, Turing computability, Herbrand–Gödel derivability, λ-definability and Markov algorithm computability.

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 Enumerability and Recursivity

1993 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Recursive Function ◽  
Function Theory ◽  
Closure Properties ◽  
Recursive Enumerability ◽  
Recursively Enumerable ◽  
Special Cases ◽  
Mechanical Procedure ◽  
The Subject ◽  
Definition Of ◽  
Intuitive Sense

Having proved that Peano Arithmetic is incomplete, we can ask another question about the system. Is there any algorithm (mechanical procedure) by which we can determine which sentences are provable in the system and which are not? This brings us to the subject of recursive function theory, to which we now turn. we are denning a relation (or set) to be r.e. (recursively enumerable) iff it is Σ1, and to be recursive iff it and its complement are r.e. An equivalent definition of recursive enumerability is represent ability in some finitely axiomatizable system (as we will prove). Many other characterizations of recursive enumerability and recursivity can be found in the literature (cf., e.g., Kleene [1952], Turing [1936], Post [1944], Smullyan [1961], Markov [1961]), but the Σ1-characterization fits in best with the overall plan of this volume. The fact that so many different and independently formulated definitions turn out to be equivalent adds support to a thesis proposed by Church—namely that any function that is effectively calculable in the intuitive sense is a recursive function. Interesting discussions of Church’s thesis can be found in Kleene [1952] and Rogers [1967]. In this chapter, we establish a few basic properties of recursive enumerability that will be needed in just about all the chapters that follow. §1. Some Closure Properties. It will be convenient to regard sets as special cases of relations (sets are thus relations of one argument or relations of degree 1). It will be convenient to use the l-notation “λx1,...,xn : (...)”, read “the set of all n-tuples (x1,..., xn) such that (...)”. For example, for any relation λ(x1, x2, x3), the relation λx1x2x3: R(x2 x2, x3) is the set of all triples (x1,x2,x3) (of natural numbers) such that R(x2,x2,x1) holds. we sometimes write “x: (. . . )” for “λx: ( . . . ) ”.

The future role of the psychiatrist in Europe – a UEMS perspective

2014 ◽  
Vol 11 (01) ◽  
pp. 35-42
Author(s):  
M. Hermans
Keyword(s):  
Mental Health ◽  
Mental Health Professionals ◽  
Health Professionals ◽  
General Interest ◽  
Future Role ◽  
Personal Opinion ◽  
The Future ◽  
The Subject ◽  
The Many

SummaryThe author presents his personal opinion inviting to discussion on the possible future role of psychiatrists. His view is based upon the many contacts with psychiatrists all over Europe, academicians and everyday professionals, as well as the familiarity with the literature. The list of papers referred to is based upon (1) the general interest concerning the subject when representing ideas also worded elsewhere, (2) the accessibility to psychiatrists and mental health professionals in Germany, (3) being costless downloadable for non-subscribers and (4) for some geographic aspects (e.g. Belgium, Spain, Sweden) and the latest scientific issues, addressing some authors directly.

INVESTIGATION OF CHARACTERISTICS OF THE GRAIN RECEIVING FROM RAILWAY TO THE GRAIN TRANSSHIPMENT TERMINAL

2020 ◽  
Vol 20 (2) ◽  
pp. 7-13
Author(s):  
G. Stankevych ◽  
L. Dmytrenko ◽  
A. Kats ◽  
V. Shpak
Keyword(s):  
Quality Indicators ◽  
Grain Quality ◽  
Total Duration ◽  
Railway Transport ◽  
Oil Crops ◽  
The Future ◽  
Leading Position ◽  
Standard Values ◽  
The Subject ◽  
Sown Area

In the future, in Ukraine it is planned to increase the sown area for cereals, legumes and oilseeds, to increase the gross grain harvest to 80 million tons, and its export abroad was increased twice. Intensive construction in the southern ports of Ukraine of grain transshipment terminals with large metal silos will solve the problem of increasing grain export in the future. At these powerful terminals, the bulk of the grain comes mainly by rail, and is shipped to water. The aim of the work was to study the characteristics of the grain receiving from railway transport to the grain transshipment terminal of LLC “Ukrelevatorprom” in order to improve its works efficiency. The object of the study was the development of technology of grain receiving at the grain transshipment terminal; the subject of research is cereals, legume sand oil crops, as well as data from daily volumes of receiving and dispensing operations at the grain transshipment terminal of LLC “Ukrelevatorprom” for 2015-2016. The studies were carried out on the basis of processing data from the consignment notes for 2015-2016, according to which there was a summed amount of grain (net) daily transported by the railway. Further processing of the obtained data was carried out by a combined graphoanalytical method, for which, on the basis of tabular values for each studied year, the corresponding histograms and graphs were built and the necessary indicators were determined. Analysis of the structure of grain crops supplied by railway to LLC “Ukrelevatorprom” in 2015 and 2016 and their ratio showed that the main share was occupied by cereal crops (78.0 % and 73.1 % respectively), which were mainly represented by corn, share which was significantly dominated by other crops (wheat of various classes and barley) and amounted to 45.8 % and 44.5 %, respectively, which can be explained by its high demand in the international grain market, in which Ukraine occupies a leading position. Oilseeds (rapeseed) were taken in accordance with 19.1 % and 14.9 %, and legumes (soybeans) — 2.9 % and 12.0 %. An analysis of the timing of the unloading of grain wagons (hopper cars) showed that the total duration of this process, depending on the crops, averages 37...59 minutes. The longest steps for unloading wagons are to determine the grain quality indicators, especially rapeseed, and to spill grain from the wagons, therefore, to reduce their duration, it is necessary to form feeds of wagons with grain batches of the same quality and use more modern express analyzers to determine grain quality indicators, which will increase the productivity of the grain receiving line from the railway. According to the research results, the enterprise has the potential to increase by about 30 % the volume of grain intake. It was established that the periods of the grain receipt at the enterprise in 2015-2016 amounted to 349 and 353 days, respectively, the actual coefficients of the daily irregularity Kdaily for the grain receipt from the railway in these years are equal to 1.47 and 1.52, and the monthly irregularity Kmonth, respectively 1.33 and 1.21, does not exceed the standard values Kdaily = 2.5 and Kmonth= 2.0. This made it possible to clarify the database from the actual characteristics of the process of grain receiving by railway and can be used in design and verification calculations of equipment in technological lines for receiving grain from railway transport, and will contribute to increasing the efficiency of grain transshipment terminals. 

Sign in / Sign up

Export Citation Format

Share Document