Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions (general recursion Recursion, in mathematics and computer science, is a method of defining functions in which the function being defined is applied within its own definition; specifically it is defining an infinite statement using finite components. The term is also used more generally to describe a process of repeating objects in a self-similar way. For instance,, Turing machines A Turing machine is a theoretical device that manipulates symbols contained on a strip of tape. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a computer. The "Turing" machine was described by Alan Turing in 193, λ-calculus In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. It was introduced by Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics. After the original system was shown to be logically inconsistent ,) that later were shown to be equivalent; the notion captured by these definitions is known as (recursive) computability Computable functions are the basic objects of study in computability theory. The set of computable functions is equivalent to the set of Turing-computable functions and partial recursive functions. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any.
Church's thesis In computability theory the Church–Turing thesis is a combined hypothesis ("thesis") about the nature of effectively calculable (computable) functions by recursion (Church's Thesis), by mechanical device equivalent to a Turing machine (Turing's Thesis) or by use of Church's λ-calculus. The three computational processes (recursion, λ- states that the two notions coincide: any number-theoretic function that is effectively calculable is recursively computable Computable functions are the basic objects of study in computability theory. The set of computable functions is equivalent to the set of Turing-computable functions and partial recursive functions. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any. Church's thesis is not a mathematical statement and cannot be proved by a mathematical proof In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproved proposition that is believed to.
A further elucidation of the term "effective method" may include the requirement that, when given a problem from outside the class for which the method is effective, the method may halt or loop forever without halting, but must not return a result as if it were the answer to the problem.
An essential feature of an effective method is that it does not require any ingenuity Ingenuity refers to the process of applying ideas to solve problems or meet challenges. The process of figuring out how to cross a mountain stream using a fallen log, build an airplane from a sheet of paper, or start a new company in a foreign culture all involve the exercising of ingenuity. Human ingenuity has led to technological developments from any person or machine executing it.[1]
See also
- Decision problem In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem. The answer can be either 'yes' or 'no', and
- Function problem In computational complexity theory, a function problem is a problem other than a decision problem, that is, a problem requiring a more complex answer than just YES or NO
- Effective results in number theory
- Recursive set In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set. A set which is not computable is called noncomputable or undecidable
References
- ^ The Cambridge Dictionary of Philosophy, effective procedure
- S. C. Kleene (1967), Mathematical logic. Reprinted, Dover, 2002, ISBN 0-486-42533-9, pp. 233 ff., esp. p. 231.
Categories: Metalogic Metalogic is the study of the metatheory of logic. While logic is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves. According to Geoffrey Hunter, while logic concerns itself with the "truths of logic," metalogic concerns | Recursion theory | Theory of computation |
