Church-Turing thesis - Church-Turing thesis In computer science, the Church-Turing thesis states in its most common form that every effective computation or algorithm can be carried out by a Turing machine. Any computer program in any of the conventional programming languages can be translated into a Turing machine, and any Turing machine can be translated into most programming languages, so the thesis is equivalent to saying that the conventional programming languages are sufficient to express any algorithm. The thesis, which is now generally assumed to be true, is also known as Church's thesis or Church's conjecture (named after Alonzo Church) and Turing's thesis (named after Alan Turing). The thesis might be rephrased as saying that the notion of effective or mechanical method in logic and mathematics is captured by.
Alan Turing - Alan Turing simple:Alan Turing Alan Turing Alan Mathison Turing (June 23, 1912 - June 7, 1954) was a British mathematician and is considered to be one of the fathers of modern computer science. He provided an influential formalisation of the concept of algorithm and computation: the Turing machine. He formulated the now widely accepted Church-Turing thesis, namely that every other practical computing model had either the equivalent or a subset of the capabilities of a Turing machine. During World War II he headed a successful effort of breaking the German secret code. After the war, he worked with one of the earliest digital computers, and later he provided a provocative contribution to the discussion "Can machines think?" Table of contents showTocToggle("show","hide") 1 Childhood and youth 2 College.
Alonzo Church - Alonzo Church Alonzo Church (June 14, 1903 - August 11, 1995) was an American mathematician who was responsible for some of the foundations of theoretical computer science. Born in Washington, DC, he attended Princeton University as an undergraduate and continued there, completing his PhD in 1927. He became a professor of mathematics at Princeton in 1929. He is best known for the development of the lambda calculus, his 1936 paper that showed the existence of an "undecidable problem" in it. This result preempted Alan Turing's famous work on the halting problem which also demonstrated the existence of a problem unsolvable by mechanical means. Supervising Turing's doctoral thesis, they then showed that the lambda calculus and the Turing machine used in Turing's halting problem were equivalent in capabilities,.
Church - Church A church can be a Christian building of worship. Here Lärbro church at Gotland, Sweden The word church has several meanings, including: A Christian building of worship. See altar, altar rails, confessional, dome, nave, pew, pulpit, sanctuary, lych gate. An assembly of Christian believers who worship together. This is one translation of the Greek Koine word "Ecclesia," used in the New Testament, and is the sense used by many Christians. In Christian theology, the Body of Christ composed of Jesus Christ and all Christians, living and dead. This is another sense of the word used in the New Testament, also used by the Nicene and Apostles' Creeds ("... one holy catholic and apostolic church ..."), and the sense used by many Christians. A religious organization.
Thesis - Thesis A thesis is either (1) an academic treatise advancing a new point of view resulting from research, or (2) a research paper that students write in order to complete the requirements for a graduate degree. At some universities, the doctoral thesis is officially called dissertation. See also thesis committee. Famous thesis Church-Turing thesis.
Turing machine - Turing machine The Turing machine is an abstract model of computer execution and storage introduced in 1936 by Alan Turing to give a mathematically precise definition of algorithm or 'mechanical procedure'. As such it is still widely used in theoretical computer science, especially in complexity theory and the theory of computation. The thesis that states that Turing machines indeed capture the informal notion of effective or mechanical method in logic and mathematics is known as the Church-Turing thesis. Turing machines shouldn't be confused with the Turing test, Turing's attempt to capture the notion of artificial intelligence. A Turing machine that is able to simulate any other Turing machine is called a universal Turing machine. Table of contents showTocToggle("show","hide") 1 Definition 2 Example 3 Universal Turing machines.
Turing-complete - Turing-complete In the theory of computers both imagined and real, of programming languages, and of other logical systems, a Turing-complete system is one which has computational power equivalent to a universal Turing machine. The concept is named in honor of Alan Turing. In other words, the system and the universal Turing machine can emulate each other. No computers completely meet this requirement, as a Turing machine has unlimited storage capacity, impossible to emulate on a real device. With this proviso, however, all modern computers are Turing-complete, as are all general-purpose programming languages. Turing-completeness is significant in that every plausible design for a computing device so far advanced (even quantum computers) can be emulated by a universal Turing machine. Thus, a machine that can act as a.
Sapir-Whorf Hypothesis - themselves. Most people do some of their thinking by imagining images and other sensory phantasms. To the extent that people think by talking to themselves they are limited by their vocabulary and the structure of their language and their linguistic habits. (However it should also be noted that individuals have idiolects.) John Grinder, a founder of NLP, was a linguistics professor who perhaps unconsciously combined the ideas of Chomsky with the Sapir-Whorf hypothesis. A seminal NLP insight came from a challenge he gave to his students: coin a neologism to describe a distinction for which you have no words. Student Robert Dilts coined a word for the way people stare into space when they are thinking, and for the different directions they stare. These new words enabled users to describe patterns.
Hypercomputation - they can compute is studied in the field known as recursion theory. A similar recent term is super-Turing computation, which has been used in the neural network literature to describe machines with various expanded abilities, including the ability to compute directly on real numbers, the ability to carry out infinitely many computations simultaneously, or the ability to carry out computations with exponentially lower complexity than standard Turing machines. Hypercomputation was first introduced by Alan Turing in his 1939 paper Systems of logic based on ordinals, which investigated mathematical systems in which an oracle was available to compute a single arbitrary (non-recursive) function from naturalss to naturals. Other posited kinds of hypercomputer include: A quantum mechanical system which somehow uses (for example) an infinite superposition of states to compute a non-recursive function1..
Functionalism (philosophy of mind) - or the mental, is best understood when considering the analogy made by functionalists between the mind and the modern digital computer. More specifically, the analogy is made to the machine that the Church-Turing thesis posits which is capable of, in principle, computing almost any given algorithm (the algorithm must have certain limitations); namely the Turing machine. A Turing machine must possess certain characteristics: Data input (analogous to the senses in humans). Data output (analogous to both behaviour and memory). Functional states (analogous to mental states), The ability to move from one functional state into another. The definition of functional states with reference to the part they play in the operation of the entire entity - ie. in reference to the other functional states. This variety of functionalism was developed by Hilary.
Emulator - on a platform (computer architecture and/or operating system) other than the one they were originally written for. It does this by "emulating", or reproducing, the behavior of one type of computer on another by accepting the same data, executing the same programs, and achieving the same results. In a technical sense, the Church-Turing thesis implies that any operating environment can be emulated within any other. In practice, it can be quite difficult, particularly when the exact behaviour of the system to be emulated is not documented and has to be deduced through reverse engineering. It also says nothing about timing constraints; if the emulator does not perform as quickly as the original hardware, the emulated software may run much more slowly than it would have on the original hardware. Most emulators.
Entscheidungsproblem - decides for given first-order statements whether they are universally valid or not. Alonzo Church and independently Alan Turing showed in 1936 that this is impossible. As a consequence, it is in particular impossible to algorithmically decide whether statements in arithmetic are true or false. The question goes back to Gottfried Leibniz, who in the seventeenth century, after having constructed a successful mechanical calculating machine, dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements. He realized that the first step would have to be a clean formal language, and much of his subsequent work was directed towards that goal. In 1928, David Hilbert and Wilhelm Ackermann posed the question in the form outlined above. A first-order statement is called "universally valid" or.
Algorithm - decisions (logic and comparison) until the task is completed. In formal mathematical terms, an algorithm is considered to be any sequence of operations which can be performed by a Turing-complete system. Different algorithms may complete the same task with a different set of instructions in more or less time, space, or effort than others. A cooking recipe is an example of an algorithm. Given two different recipes for making potato salad, one may have "peel the potato" before "boil the potato" while the other presents the steps in the reverse order, yet they both call for these steps to be repeated for all potatoes and end when the potato salad is ready to be eaten. Correctly performing an algorithm will not solve a problem if the algorithm is flawed, or not.
Algorithmic learning theory - hypothesis must be correct for all the sentences so far. A particular learner is said to be able to "learn a language in the limit" if there is a certain number of steps beyond which its hypothesis no longer changes. At this point it has indeed learned the language, because every possible sentence appears somewhere in the sequence of inputs (past or future), and the hypothesis is correct for all inputs (past or future), so the hypothesis is correct for every sentence. The learner is not required to be able to tell when it has reached a correct hypothesis, all that is required is that it be true. Gold showed that any language which is defined by a Turing machine program can be learned in the limit by another Turing-complete machine.
Busy beaver - a busy beaver (from the colloquial expression for "industrious person") is a Turing machine that, when given an initially empty (binary?) tape (a string of only 0's), does a lot of work, then halts. The functions defined below, called busy beaver functions, were introduced and their properties proved in 1952, by Tibor Rado. There are two main 'categories': Σ(n): the largest number of 1's printable by an n-state machine before halting, and S(n): the largest number of steps taken by an n-state machine before halting. All machines are started on initially blank tapes and those that do not halt are not candidates. Both of these functions are uncomputable in general. That they grow faster than any computable functions can be easily shown. Take any computable function f of x. Take a.
Computer science - couple of hundred years. However, this does not mean that there is significantly less on the computer scientist's plate than on the physicist's: younger it may be, but it has had a far more intense upbringing! - Richard Feynman The Church-Turing thesis states that all known kinds of general computing devices are essentially equivalent in what they can do, although they vary in time and space efficiency. This thesis is sometimes treated as the fundamental principle of computer science. Most research in computer science has been related to von Neumann computerss or Turing machines (computers that do one small, deterministic task at a time), because they resemble most real computers in use today. Computer scientists also study other kinds of machines, some practical (like parallel and quantum machines) and some theoretical.
Computation - mathematicians were trying to find which math problems can be solved by simple methods and which cannot. The first step was to define what they meant by a "simple method" for solving a problem. In other words, they needed a formal model of computation. Several different computational models were devised by these early researchers. One model, the Turing machine, stores characters on an infinitely long tape, with one square at any given time being scanned by a read/write head. Another model, recursive functions, uses functions and function composition to operate on numbers. The lambda calculus uses a similar approach. Still others, including Markov algorithms and Post systems, use grammar-like rules to operate on strings. All of these formalisms were shown to be equivalent in computational power -- that is, any computation.
Computability theory - the theory of computation dealing with which problems are solvable by algorithms (equivalently, by Turing machines), with various restrictions and extensions. Computability theory addresses four main questions: What problems can Turing machines solve? What other systems are equivalent to Turing machines? What problems require more powerful machines? What problems can be solved by less powerful machines? See the article on theory of computation for a chart showing which classes of problems are subsets of other classes. Table of contents showTocToggle("show","hide") 1 What problems can Turing machines solve? 2 What other systems are equivalent to Turing machines? 3 What problems require more powerful machines? 4 What problems can be solved by less powerful machines? What problems can Turing machines solve? Not all problems can be solved. An undecidable problem is one that.
Combinatory logic - of the formal parameter of E1, and the result is a new lambda term which is equivalent to the old one. If a lambda term contains no subterms of the form (λv.E1 E2) then it cannot be reduced, and is said to be in normal form. The expression E[a/v] represents the result of taking the term E and replacing all free occurrences of v with a. Thus we write (λv.E a) => E[a/v] By convention, we take (a b c d ... z) as short for ''(...(((a b) c) d) ... z)''. The motivation for this definition of reduction is that it captures the essential behavior of all mathematical functions. For example, consider the function that computes the square of a number. We might write The square of x is x*x.
Stephen Cole Kleene - theoretical computer science. Kleene was best known for founding the branch of mathematical logic known as recursion theory together with Alonzo Church, Kurt Gödel, Alan Turing and others; and for inventing regular expressions. By providing methods of determining which problems are soluble, Kleene's work led to the study of which functions are computable. The Kleene star, Kleene's recursion theorem and the Ascending Kleene Chain are named after him. He also contributed to mathematical intuitionism as founded by Luitzen Egbertus Jan Brouwer. Kleene was born in Hartford, Connecticut, USA. He received his bachelor of arts degree from Amherst College in 1930. From 1930 to 1935, he was a graduate student and research assistant at Princeton University, where he received his doctorate in mathematics in 1934, supervised by Alonzo Church, for a thesis.
