Computation

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A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. [1] [2] Common examples of computation are mathematical equation solving and the execution of computer algorithms.

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Mechanical or electronic devices (or, historically, people) that perform computations are known as computers .

Computer science is an academic field that involves the study of computation.

Introduction

The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s, [3] but agreement on a suitable definition proved elusive. [4] A candidate definition was proposed independently by several mathematicians in the 1930s. [5] The best-known variant was formalised by the mathematician Alan Turing, who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a Turing machine. [6] Other (mathematically equivalent) definitions include Alonzo Church's lambda-definability , Herbrand-Gödel-Kleene's general recursiveness and Emil Post's 1-definability. [5]

Today, any formal statement or calculation that exhibits this quality of well-definedness is termed computable, while the statement or calculation itself is referred to as a computation.

Turing's definition apportioned "well-definedness" to a very large class of mathematical statements, including all well-formed algebraic statements, and all statements written in modern computer programming languages. [7]

Despite the widespread uptake of this definition, there are some mathematical concepts that have no well-defined characterisation under this definition. This includes the halting problem and the busy beaver game. It remains an open question as to whether there exists a more powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements. [note 1] [8]

Some examples of mathematical statements that are computable include:

Some examples of mathematical statements that are not computable include:

The Physical process of computation

Computation can be seen as a purely physical process occurring inside a closed physical system called a computer. Turing's 1937 proof, On Computable Numbers, with an Application to the Entscheidungsproblem , demonstrated that there is a formal equivalence between computable statements and particular physical systems, commonly called computers. Examples of such physical systems are: Turing machines, human mathematicians following strict rules, digital computers, mechanical computers, analog computers and others.

Alternative accounts of computation

The mapping account

An alternative account of computation is found throughout the works of Hilary Putnam and others. Peter Godfrey-Smith has dubbed this the "simple mapping account." [9] Gualtiero Piccinini's summary of this account states that a physical system can be said to perform a specific computation when there is a mapping between the state of that system and the computation such that the "microphysical states [of the system] mirror the state transitions between the computational states." [10]

The semantic account

Philosophers such as Jerry Fodor [11] have suggested various accounts of computation with the restriction that semantic content be a necessary condition for computation (that is, what differentiates an arbitrary physical system from a computing system is that the operands of the computation represent something). This notion attempts to prevent the logical abstraction of the mapping account of pancomputationalism, the idea that everything can be said to be computing everything.

The mechanistic account

Gualtiero Piccinini proposes an account of computation based on mechanical philosophy. It states that physical computing systems are types of mechanisms that, by design, perform physical computation, or the manipulation (by a functional mechanism) of a "medium-independent" vehicle according to a rule. "Medium-independence" requires that the property can be instantiated[ clarification needed ] by multiple realizers[ clarification needed ] and multiple mechanisms, and that the inputs and outputs of the mechanism also be multiply realizable. In short, medium-independence allows for the use of physical variables with properties other than voltage (as in typical digital computers); this is imperative in considering other types of computation, such as that which occurs in the brain or in a quantum computer. A rule, in this sense, provides a mapping among inputs, outputs, and internal states of the physical computing system. [12]

Mathematical models

In the theory of computation, a diversity of mathematical models of computation has been developed. Typical mathematical models of computers are the following:

Giunti calls the models studied by computation theory computational systems, and he argues that all of them are mathematical dynamical systems with discrete time and discrete state space. [13] : ch.1  He maintains that a computational system is a complex object which consists of three parts. First, a mathematical dynamical system with discrete time and discrete state space; second, a computational setup , which is made up of a theoretical part , and a real part ; third, an interpretation , which links the dynamical system with the setup . [14] : pp.179–80 

See also

Notes

  1. ↑ The study of non-computable statements is the field of hypercomputation.

Related Research Articles

<span class="mw-page-title-main">Computable number</span> Real number that can be computed within arbitrary precision

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, computable reals, or recursive reals. The concept of a computable real number was introduced by Émile Borel in 1912, using the intuitive notion of computability available at the time.

In computability theory, the Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formalize the notion of computability:

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<span class="mw-page-title-main">Turing machine</span> Computation model defining an abstract machine

A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algorithm.

In computer science, a universal Turing machine (UTM) is a Turing machine capable of computing any computable sequence, as described by Alan Turing in his seminal paper "On Computable Numbers, with an Application to the Entscheidungsproblem". Common sense might say that a universal machine is impossible, but Turing proves that it is possible. He suggested that we may compare a human in the process of computing a real number to a machine which is only capable of a finite number of conditions ⁠⁠; which will be called "m-configurations". He then described the operation of such machine, as described below, and argued:

It is my contention that these operations include all those which are used in the computation of a number.

Hypercomputation or super-Turing computation is a set of hypothetical models of computation that can provide outputs that are not Turing-computable. For example, a machine that could solve the halting problem would be a hypercomputer; so too would one that could correctly evaluate every statement in Peano arithmetic.

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<span class="mw-page-title-main">Expression (mathematics)</span> Symbolic description of a mathematical object

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Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the general recursive functions.

In philosophy of mind, the computational theory of mind (CTM), also known as computationalism, is a family of views that hold that the human mind is an information processing system and that cognition and consciousness together are a form of computation. It is closely related to functionalism, a broader theory that defines mental states by what they do rather than what they're made of.

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The Penrose–Lucas argument is a logical argument partially based on a theory developed by mathematician and logician Kurt Gödel. In 1931, he proved that every effectively generated theory capable of proving basic arithmetic either fails to be consistent or fails to be complete. Due to human ability to see the truth of formal system's Gödel sentences, it is argued that the human mind cannot be computed on a Turing Machine that works on Peano arithmetic because the latter can't see the truth value of its Gödel sentence, while human minds can. Mathematician Roger Penrose modified the argument in his first book on consciousness, The Emperor's New Mind (1989), where he used it to provide the basis of his theory of consciousness: orchestrated objective reduction.

References

  1. ↑ "Definition of COMPUTATION". www.merriam-webster.com. 2024-10-11. Retrieved 2024-10-12.
  2. ↑ "Computation: Definition and Synonyms from Answers.com". Answers.com. Archived from the original on 22 February 2009. Retrieved 26 April 2017.
  3. ↑ Couturat, Louis (1901). la Logique de Leibniz a'Après des Documents Inédits. Paris. ISBN   978-0343895099.
  4. ↑ Davis, Martin; Davis, Martin D. (2000). The Universal Computer. W. W. Norton & Company. ISBN   978-0-393-04785-1.
  5. 1 2 Davis, Martin (1982-01-01). Computability & Unsolvability. Courier Corporation. ISBN   978-0-486-61471-7.
  6. ↑ Turing, A.M. (1937) [Delivered to the Society November 1936]. "On Computable Numbers, with an Application to the Entscheidungsproblem" (PDF). Proceedings of the London Mathematical Society. 2. Vol. 42. pp. 230–65. doi:10.1112/plms/s2-42.1.230.
  7. 1 2 Davis, Martin; Davis, Martin D. (2000). The Universal Computer. W. W. Norton & Company. ISBN   978-0-393-04785-1.
  8. ↑ Davis, Martin (2006). "Why there is no such discipline as hypercomputation". Applied Mathematics and Computation. 178 (1): 4–7. doi:10.1016/j.amc.2005.09.066.
  9. ↑ Godfrey-Smith, P. (2009), "Triviality Arguments against Functionalism", Philosophical Studies, 145 (2): 273–95, doi:10.1007/s11098-008-9231-3, S2CID   73619367
  10. ↑ Piccinini, Gualtiero (2015). Physical Computation: A Mechanistic Account. Oxford: Oxford University Press. p. 18. ISBN   9780199658855.
  11. ↑ Fodor, J. A. (1986), "The Mind-Body Problem", Scientific American, 244 (January 1986)
  12. ↑ Piccinini, Gualtiero (2015). Physical Computation: A Mechanistic Account. Oxford: Oxford University Press. p. 10. ISBN   9780199658855.
  13. ↑ Giunti, Marco (1997). Computation, Dynamics, and Cognition. New York: Oxford University Press. ISBN   978-0-19-509009-3.
  14. ↑ Giunti, Marco (2017), "What is a Physical Realization of a Computational System?", Isonomia -- Epistemologica, 9: 177–92, ISSN   2037-4348
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