Who released a paper entitled On computable numbers?
Alan Turing Publishes "On Computable Numbers," Describing What Came to be Called the "Turing Machine" Alan Turing, aged 16. Turing published On Computable Numbers when he was 24 years old.
When did Alan Turing publish?
In 1936, whilst studying for his Ph. D. at Princeton University, the English mathematician Alan Turing published a paper, “On Computable Numbers, with an application to the Entscheidungsproblem,” which became the foundation of computer science.
What did Alan Turing prove in his paper on computable numbers with an application to the Entscheidungsproblem?
In his most famous mathematical paper 'On Computable Numbers, with an Application to the Entscheidungsproblem', Turing set out the idea of a multi-purpose machine, whose function would be changed according to the instructions it was given. This concept of a programmable computer became known as the 'Turing machine'.
Are real numbers computable?
Real numbers used in any explicit way in traditional mathematics are always computable in this sense. But as Turing pointed out, the overwhelming majority of all possible real numbers are not computable. For certainly there can be no more computable real numbers than there are possible Turing machines.
Who broke Enigma code?
Alan Turing was a brilliant mathematician. Born in London in 1912, he studied at both Cambridge and Princeton universities. He was already working part-time for the British Government's Code and Cypher School before the Second World War broke out.
What is the paper Alan Turing published in 1950?
Computing Machinery and Intelligence"Computing Machinery and Intelligence" is a seminal paper written by Alan Turing on the topic of artificial intelligence. The paper, published in 1950 in Mind, was the first to introduce his concept of what is now known as the Turing test to the general public.
What was the problem that Turing set out to solve?
Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program–input pairs cannot exist.
How did Turing break the Enigma code?
While there, Turing built a device known as the Bombe. This machine was able to use logic to decipher the encrypted messages produced by the Enigma. However, it was human understanding that enabled the real breakthroughs. The Bletchley Park team made educated guesses at certain words the message would contain.
What was Alan Turing's theory?
He published just one paper (1952), but it triggered a whole new field of mathematical enquiry into pattern formation. He discovered that a system with just 2 molecules could, at least in theory, create spotty or stripy patterns if they diffused and chemically interacted in just the right way.
How do you know if something is computable?
The definition of computability varies slightly depending on what kind of thing you're talking about, but the basic idea is the same. A function is computable if there's a program that takes as input and produces as output. A real number is computable if there's a program that takes no input and produces as output.
What does the word computable mean?
capable of being computed: capable of being computed.
Is Rayo's number the biggest number?
Rayo's number: The smallest number bigger than any number that can be named by an expression in the language of first order set-theory with less than a googol (10100) symbols.
What did Alan Turing do in 1950?
In 1950, Alan Turing Created a Chess Computer Program That Prefigured A.I. The first chess algorithm didn't even run on a computer. The first chess algorithm didn't even run on a computer. Chess is one of the oldest, and most revered games of strategy and analysis in the world.
What did Alan Turing do in 1940?
Turing pitted machine against machine. The prototype model of his anti-Enigma "bombe", named simply Victory, was installed in the spring of 1940. His bombes turned Bletchley Park into a codebreaking factory.
What did Alan Turing do in 1930?
That was the year that mathematical pioneer Alan Turing imagined a 'universal machine' in his paper 'On Computable Numbers. ' Turing described a machine that could read symbols on a tape and proposed that the tape be used to program the machine.
Who wrote the first AI?
Christopher StracheyThe earliest successful AI program was written in 1951 by Christopher Strachey, later director of the Programming Research Group at the University of Oxford. Strachey's checkers (draughts) program ran on the Ferranti Mark I computer at the University of Manchester, England.
How old was Alan Turing when he published on computable numbers?
Alan Turing, aged 16. Turing published On Computable Numbers when he was 24 years old.
What is the most famous paper in the history of computing?
Undoubtedly the most famous theoretical paper in the history of computing, "On Computable Numbers" is a mathematical description an imaginary computing device designed to replicate the mathematical "states of mind" and symbol-manipulating abilities of a human computer. Turing conceived of the universal machine as a means of answering the last of the three questions about mathematics posed by David Hilbert in 1928: (1) is mathematics complete; (2) is mathematics consistent; and (3) is mathematics decidable.
What did Turing show about mathematics?
To demonstrate this, Turing came up with the concept of "computable numbers," which are numbers defined by some definite rule, and thus calculable on the universal machine.
What is the universal machine?
Turing conceived of the universal machine as a means of answering the last of the three questions about mathematics posed by David Hilbert in 1928: (1) is mathematics complete; (2) is mathematics consistent; and (3) is mathematics decidable. Hilbert's final question, known as the Entscheidungsproblem, concerns whether there exists ...
What is Hilbert's final question?
Hilbert's final question, known as the Entscheidungsproblem, concerns whether there exists a defiinite method— or, in the suggestive words of Turing's teacher Max Newman, a "mechanical process"—that can be applied to any mathematical assertion, and which is guaranteed to produce a correct decision as to whether that assertion is true. The Czech logician Kurt Gödel had already shown that arithmetic (and by extension mathematics) was both inconsistent and incomplete. Turing showed, by means of his universal machine, that mathematics was also undecidable.
Where did Turing report to the government code?
After returning to England, on September 4, 1939, the day after Britain and France declared war on Germany, Turing reported to the Government Code and Cypher School , Bletchley Park, in the town of Bletchley, England.
Who developed the Turing machine?
Independently of Alan Turing, mathematician and logician Emil Post of the City College of New York developed, and published in October 1936, a mathematical model of computation that was essentially equivalent to the Turing machine.
Why do not computable reals form a computable field?
Computable reals however do not form a computable field, because the definition of a computable field requires effective equality.
What are some examples of computer packages that work with computable real numbers?
One example is the RealLib package.
What is a computable number?
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers (vanDerHoeven) or the computable reals or recursive reals.
What is a sequence with this property?
A sequence with this property is known as a Specker sequence, as the first construction is due to E. Specker (1949). Despite the existence of counterexamples such as these, parts of calculus and real analysis can be developed in the field of computable numbers, leading to the study of computable analysis .
What are the arithmetical operations on computable numbers?
The arithmetical operations on computable numbers are themselves computable in the sense that whenever real numbers a and b are computable then the following real numbers are also computable: a + b, a - b, ab, and a/b if b is nonzero.
What are the key notions in the definition of a machine?
The key notions in the definition are (1) that some n is specified at the start, (2) for any n the computation only takes a finite number of steps, after which the machine produces the desired output and terminates.
Is a set of computable real numbers order isomorphic?
The set of computable real numbers (as well as every countable, densely ordered subset of computable reals without ends) is order-isomorphic to the set of rational numbers.
When did the computer come out of the truck?
It didn't have the sleek lines that Steve Jobs would admire and it needed to be carried on a truck, but the computer age had finally arrived in rural England. April 3, 1957.
Who invented the universal Turing machine?
Historical Context. In January 1937 the English mathematician Alan Turing published his paper "On Computable Numbers, with an Application to the Entscheidungsproblem". In it he asserted that decision problems are undecidable using a theoretical machine. By coming up with his universal Turing machine, Turing foreshadowed the invention ...
The sets of integers and rationals are both countably infinite
Intuitively, one might think of the set of integers \ (\mathbb {Z}\) as being “bigger” than the set of the natural numbers \ (\mathbb {N}\).
The set of reals is uncountably infinite
However, real numbers are inherently uncountable. A rephrasing of Cantor’s original proof follows, using a trick that has come to be known as “diagonalization.” No matter what infinite list of real numbers is given, we can generate a new number \ (x\) that cannot possibly be in that list.
Defining computable numbers
We will now move on to Turing’s work on computable numbers, which he defined in his famous paper. We use \ (\mathbb {K}\) to represent this set.
Gödel numbers
Now that we’ve defined our set \ (\mathbb {K}\), we need to show that it is countable. The proof relies on the concept of a Gödel number.
Bringing it together
It remains to show that a Gödel number function \ (\mathrm {g}\) exists for \ (\mathbb {K}\). The trick is to calculate a Gödel number from the program that generates a computable number, rather than from the number itself. Recall that a program consists of a finite block of text.
A counter proof?
Some readers may wonder why we cannot use the same diaganolization trick that Cantor used, and so prove that the computable numbers are uncountable. Let’s try to do it, and we will see where it fails.
Conclusions
What does it mean that the set of computable numbers is countable, but the set of real numbers is not? In a very true sense, the set \ (\mathbb {K}\) contains all the numbers that are knowable.
INDEX
The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means.
APPENDIX
The theorem that all effectively calculable ( V -definable) sequences are computable and its converse are proved below in outline. It is assumed that the terms “well-formed formula” (W.F.F.) and “conversion” as used by Church and Kleene are understood.
Endnotes
Gödel, “Uber formal unentscheidbare Satze der Principia Mathernatica und verwant der Systeme, I”, Monatshefte Math. Phys ., 38 (1931). 173-198.
How to do adlrd in a tally?
H increments N = 13473 and converts "13473" to symbol string ADRLD. If sub-machine D deems ADLRD unsatisfactory, then H leaves the tally-record R at 5. H will increment the number N to 13474 and proceed onward. On the other hand, if D deems ADRLD satisfactory then H will increment R to 6. H will convert N (again) into ADLRD [this is just an example, ADLRD is probably useless] and “run” it using the universal machine U until this machine-under-test (U "running" ADRLD) prints its 6th “figure” i.e. 1 or 0. H will print this 6th number (e.g. “0”) in the “output” region of its tape (e.g. B’ = “.100110”).
What is Turing's proof?
Turing's proof is a proof by Alan Turing, first published in January 1937 with the title " On Computable Numbers, with an Application to the Entscheidungsproblem ." It was the second proof (after Church's theorem) of the conjecture that some purely mathematical yes–no questions can never be answered by computation; more technically, that some decision problems are " undecidable " in the sense that there is no single algorithm that infallibly gives a correct "yes" or "no" answer to each instance of the problem. In Turing's own words: "...what I shall prove is quite different from the well-known results of Gödel ... I shall now show that there is no general method which tells whether a given formula U is provable in K [ Principia Mathematica ]..." ( Undecidable, p. 145).
How does Turing proceed?
Turing proceeds by reductio ad absurdum. He asserts the existence of a machine E, which when given the S.D (Standard Description, i.e. "program") of an arbitrary machine M, will determine whether M ever prints a given symbol (0 say). He does not assert that this M is a "computing machine".
What is Turing's construction of machines?
We must emphasize the "constructive" nature of this proof. Turing describes what could be a real machine, really buildable. The only questionable element is the existence of machine D, which this proof will eventually show to be impossible.
How does the process unravel?
The whole process unravels when H arrives at its own number K. We will proceed with our example. Suppose the successful-tally/record R stands at 12. H finally arrives at its own number minus 1, i.e. N = K-1 = 4335...321 4, and this number is unsuccessful. Then H increments N to produce K = 4355...321 5, i.e. its own number. H converts this to “LDDR...DCAR” and passes it to decision-machine D. Decision-machine D must return “satisfactory” (that is: H must by definition go on and on testing, ad infinitum, because it is "circle-free"). So H now increments tally R from 12 to 13 and then re-converts the number-under-test K into its S.D and uses U to simulate it. But this means that H will be simulating its own motions. What is the first thing the simulation will do? This simulation K-aka-H either creates a new N or “resets” the “old” N to 1. This "K-aka-H" either creates a new R or “resets” the “old” R to 0. Old-H “runs” new "K-aka-H" until it arrives at its 12th figure.
What is the second proof of Rice's theorem?
Second proof: This one is perhaps more familiar to readers as Rice's theorem: "We can show further that there can be no machine E which, when supplied with the S.D ["program"] of an arbitrary machine M, will determine whether M ever prints a given symbol (0 say) " (his italics, Undecidable, p. 134).
Which theorem is most relevant to the halting problem?
His first theorem is most relevant to the halting problem, the second is more relevant to Rice's theorem .
What is a computable number?
In other words, we define a real number as computable if there is an algorithm which, given n, returns the first n digits of the number . Informally, we can think of a computable number as Marvin Minsky did, namely as “A number for which there is a Turing machine which, given n on its initial tape, terminates with the nth digit of that number [encoded on its tape] ”.
Who came to Princeton to work on Hilbert's Entscheidungsproblem?
Simultaneously Alan Turing came to Princeton to work on the same problems of both attempting to settle Hilbert’s Entscheidungsproblem and in the process provide a complete and final definition of computability. When published in the Proceedings of the London Mathematical Society 42 on November 30th 1936, Turing’s paper defined computability as (Soare, 2013):
Who was Turing's thesis advisor?
Turing’s thesis advisor Alonzo Church first began working on the problem in 1933. He presented his proposed solution featuring what he called effectively calculable functions to Gödel around March 1934. Supposedly, Gödel rejected Church’s approach as “thoroughly unsatisfactory” (Soare, 2013). Based on λ - calculus, Church’s first proposition said:
Who proved the incompleteness of formal systems?
As Robert I. Soare recounts (Copeland et al, 2013 p. 207), by 1931 logic as applied to computability was a young man’s game. Gödel, born in 1906, was 25 years old when he proved the incompleteness of formal systems. Church was 33 when he proved the undecidability of the Entscheidungsproblem with his invention of the concept of “effective calculability” based on λ - calculus. Turing, independently proved the same result in the same year with the invention of Turing machines. Born in 1912, he was 24 at the time.
Is penrose tiling fractal?
Because such patterns are non-periodic, they lack translational symmetry. They are also self-similar (“fractal”), as they appear the same at larger and larger scales.
Who cracked the Enigma encryption device?
The now famed Alan Turing (1912–1954) was an English mathematician most well known as the tour de force behind the successful Allied effort to crack the Axis communication encryption device Enigma. For mathematicians, Turing’s name is synonymous with genius not only for this applied work but rather instead for his exceptionally visionary work in pure mathematics and logic.
Is a set of real numbers countable?
While we know that the set of real numbers is uncountable, the set of computable numbers is countable, and thus we know that most real numbers are not computable. The proof that the computable numbers is countable arises intuitively from the fact that they may all be produced by Turing machines, of which there are only countably many variations (i.e. they can be put into one-to-one correspondance with the natural numbers). Formally:
Overview
Properties
Assigning a Gödel number to each Turing machine definition produces a subset of the natural numbers corresponding to the computable numbers and identifies a surjection from to the computable numbers. There are only countably many Turing machines, showing that the computable numbers are subcountable. The set of these Gödel numbers, however, is not computably enumerable (and consequently, neither are subsets of that are defined in terms of it). …
Informal definition using a Turing machine as example
In the following, Marvin Minsky defines the numbers to be computed in a manner similar to those defined by Alan Turing in 1936; i.e., as "sequences of digits interpreted as decimal fractions" between 0 and 1:
A computable number [is] one for which there is a Turing machine which, given n on its initial tape, terminates with the nth digit of that number [encoded on its tape].
Formal definition
A real number a is computable if it can be approximated by some computable function in the following manner: given any positive integer n, the function produces an integer f(n) such that:
There are two similar definitions that are equivalent:
• There exists a computable function which, given any positive rational error bound , produces a rational number r such that
Digit strings and the Cantor and Baire spaces
Turing's original paper defined computable numbers as follows:
A real number is computable if its digit sequence can be produced by some algorithm or Turing machine. The algorithm takes an integer as input and produces the -th digit of the real number's decimal expansion as output.
(The decimal expansion of a only refers to the digits following the decimal point.)
Use in place of the reals
The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers. Though the computable reals exhaust those reals we can calculate or approximate, the assumption that all reals are computable leads to substantially different conclusions about the real numbers. The question naturally arises of whether it is possible to dispose of the full set of reals and use comp…
Implementations of exact arithmetic
Computer packages representing real numbers as programs computing approximations have been proposed as early as 1985, under the name "exact arithmetic". Modern examples include the CoRN library (Coq), and the RealLib package (C++). A related line of work is based on taking a real RAM program and running it with rational or floating-point numbers of sufficient precision, such as the iRRAM package.
See also
• Constructible number
• Definable number
• Semicomputable function
• Transcomputational problem
The Sets of Integers and Rationals Are Both Countably Infinite
The Set of Reals Is Uncountably Infinite
- However, real numbers are inherently uncountable. A rephrasing of Cantor's original proof follows, using a trick that has come to be known as "diagonalization." No matter what infinite list of real numbers is given, we can generate a new number xxthat cannot possibly be in that list. To prove the real numbers are uncountable, let's assume they are not. Then there would exist some one-t…
Defining Computable Numbers
- We will now move on to Turing's work on computable numbers, which he defined in his famous paper. We use KK to represent this set. (CC is taken, as it usually refers to the set of complexnumbers.) Turing defined a computable number as one that can be calculated to within an arbitrary precision, within a finite amount of time. These days, the easiest way to do that is oft…
Gödel Numbers
- Now that we've defined our set KK, we need to show that it is countable. The proof relies on the concept of a Gödel number. Gödel numbers were developed by Kurt Gödel himself, in order to prove his "incompleteness theorems." (If you are new to German, Gödel's name is pronounced similarly to the English word "girdle", but without the "r" sound.) Definition: Gödel Number A Göd…
Bringing It Together
- It remains to show that a Gödel number function gg exists for KK. The trick is to calculate a Gödel number from the program that generates a computable number, rather than from the number itself. Recall that a program consists of a finite block of text. Most computers use the ASCII code or Unicode to represent a text file, in which every valid character is a encoded as a number betw…
A Counter Proof?
- Some readers may wonder why we cannot use the same diaganolization trick that Cantor used, and so prove that the computable numbers are uncountable. Let's try to do it, and we will see where it fails. Let us say that we have an allegedly complete (though infinite) enumeration of every computable number. We will now attempt to use diagonalization in order to produce a new comp…
Conclusions
- What does it mean that the set of computable numbers is countable, but the set of real numbers is not? In a very true sense, the set KK contains all the numbers that are knowable. Real numbers that aren't computable can't be calculated, or even thought about except in the most abstract manner (such as xx from the last section, the uncalculable result of using the diagonalization pro…
Acknowledgements
- Thank you to David Bau for permission to use his implementation of the ππspigot algorithm. Thank you also to Ursula Whitcher for encouragement and assistance with one of the proofs.
References
- D. Bau. A dabbler's weblog: Python pi.py spigot. http://davidbau.com/archives/2010/03/14/python_pipy_spigot.html, March 2010.
- G. Cantor. Über eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen (On a property of the collection of all real algebraic numbers). Journal für die reine und angewandte Mathematik,...
- D. Bau. A dabbler's weblog: Python pi.py spigot. http://davidbau.com/archives/2010/03/14/python_pipy_spigot.html, March 2010.
- G. Cantor. Über eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen (On a property of the collection of all real algebraic numbers). Journal für die reine und angewandte Mathematik,...
- G. Cantor. Über eine elementare Frage der Mannigfaltigkeitslehre (On an elementary question concerning the theory of manifolds). Jahresbericht der Deutschen Mathematiker-Vereinigung, 1, 1892.
- A. Church. A note on the Entscheidungsproblem. The Journal of Symbolic Logic, 1(01):40–41, 1936.
