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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.

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:

All statements characterised in modern programming languages, including C++, Python, and Java.^[7]
All calculations carried by an electronic computer, calculator or abacus.
All calculations carried out on an analytical engine.
All calculations carried out on a Turing Machine.
The majority of mathematical statements and calculations given in maths textbooks.

Some examples of mathematical statements that are not computable include:

Calculations or statements which are ill-defined, such that they cannot be unambiguously encoded into a Turing machine: ("Paul loves me twice as much as Joe").
Problem statements which do appear to be well-defined, but for which it can be proved that no Turing machine exists to solve them (such as the halting problem).

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:

State models including Turing machine, pushdown automaton, finite state automaton, and PRAM
Functional models including lambda calculus
Logical models including logic programming
Concurrent models including actor model and process calculi

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 $DS$ with discrete time and discrete state space; second, a computational setup $H=\left(F,B_{F}\right)$ , which is made up of a theoretical part $F$ , and a real part $B_{F}$ ; third, an interpretation $I_{DS,H}$ , which links the dynamical system $DS$ with the setup $H$ .^[14]^{: pp.179–80}

Notes

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

References

^ "Definition of COMPUTATION". www.merriam-webster.com. 2024-10-11. Retrieved 2024-10-12.
^ "Computation: Definition and Synonyms from Answers.com". Answers.com. Archived from the original on 22 February 2009. Retrieved 26 April 2017.
^ Couturat, Louis (1901). la Logique de Leibniz a'Après des Documents Inédits. Paris. ISBN 978-0343895099.
^ Davis, Martin; Davis, Martin D. (2000). The Universal Computer. W. W. Norton & Company. ISBN 978-0-393-04785-1.
^ ^a ^b Davis, Martin (1982-01-01). Computability & Unsolvability. Courier Corporation. ISBN 978-0-486-61471-7.
^ 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.
^ ^a ^b Davis, Martin; Davis, Martin D. (2000). The Universal Computer. W. W. Norton & Company. ISBN 978-0-393-04785-1.
^ 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.
^ Godfrey-Smith, P. (2009), "Triviality Arguments against Functionalism", Philosophical Studies, 145 (2): 273–95, doi:10.1007/s11098-008-9231-3, S2CID 73619367
^ Piccinini, Gualtiero (2015). Physical Computation: A Mechanistic Account. Oxford: Oxford University Press. p. 18. ISBN 9780199658855.
^ Fodor, J. A. (1986), "The Mind-Body Problem", Scientific American, 244 (January 1986)
^ Piccinini, Gualtiero (2015). Physical Computation: A Mechanistic Account. Oxford: Oxford University Press. p. 10. ISBN 9780199658855.
^ Giunti, Marco (1997). Computation, Dynamics, and Cognition. New York: Oxford University Press. ISBN 978-0-19-509009-3.
^ Giunti, Marco (2017), "What is a Physical Realization of a Computational System?", Isonomia -- Epistemologica, 9: 177–92, ISSN 2037-4348

[8] The study of non-computable statements is the field of hypercomputation.

[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.

[Davis_Davis_2000-4] Davis, Martin; Davis, Martin D. (2000). The Universal Computer. W. W. Norton & Company. ISBN 978-0-393-04785-1.

[Davis-5] 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.

[Davis_Davis_2000_p.-7] Davis, Martin; Davis, Martin D. (2000). The Universal Computer. W. W. Norton & Company. ISBN 978-0-393-04785-1.

[9] 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.

[10] Godfrey-Smith, P. (2009), "Triviality Arguments against Functionalism", Philosophical Studies, 145 (2): 273–95, doi:10.1007/s11098-008-9231-3, S2CID 73619367

[11] Piccinini, Gualtiero (2015). Physical Computation: A Mechanistic Account. Oxford: Oxford University Press. p. 18. ISBN 9780199658855.

[12] Fodor, J. A. (1986), "The Mind-Body Problem", Scientific American, 244 (January 1986)

[13] Piccinini, Gualtiero (2015). Physical Computation: A Mechanistic Account. Oxford: Oxford University Press. p. 10. ISBN 9780199658855.

[14] Giunti, Marco (1997). Computation, Dynamics, and Cognition. New York: Oxford University Press. ISBN 978-0-19-509009-3.

[15] Giunti, Marco (2017), "What is a Physical Realization of a Computational System?", Isonomia -- Epistemologica, 9: 177–92, ISSN 2037-4348

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@@ Line 1: / Line 1: @@
 {{short description|Any type of calculation}}
+A '''computation''' is any type of [[arithmetic]] or non-arithmetic [[calculation]] that is well-defined.<ref>{{Cite web |date=2024-10-11 |title=Definition of COMPUTATION |url=https://www.merriam-webster.com/dictionary/computation |access-date=2024-10-12 |website=www.merriam-webster.com |language=en}}</ref><ref>{{cite web|title=Computation: Definition and Synonyms from Answers.com|url=http://www.answers.com:80/topic/computation|website=Answers.com|access-date=26 April 2017|archive-url=https://web.archive.org/web/20090222005439/http://www.answers.com/topic/computation|archive-date=22 February 2009|url-status=dead}}</ref> Common examples of computation are [[mathematical equation]] solving and the [[Execution (computing)|execution]] of computer [[algorithms]].
-{{original research|date/^May 2022|date=July 2022}}
-'''Computation''' is any type of [[arithmetic]] or non-arithmetic [[calculation]] that follows a well-defined model (e.g., an [[algorithm]]).<ref>[http://www.merriam-webster.com/dictionary/computation Computation] from the Free Merriam-Webster Dictionary</ref><ref>{{cite web|title=Computation: Definition and Synonyms from Answers.com|url=http://www.answers.com:80/topic/computation|website=Answers.com|access-date=26 April 2017|archive-url=https://web.archive.org/web/20090222005439/http://www.answers.com/topic/computation|archive-date=22 February 2009|url-status=dead}}</ref>
-Mechanical or electronic devices (or, [[History of computing hardware|historically]], people) that perform computations are known as ''[[computer]]s''. An especially well-known discipline of the study of computation is [[computer science]].
+Mechanical or electronic devices (or, [[History of computing hardware|historically]], people) that perform computations are known as ''[[computer]]s''.
+[[Computer science]] is an academic field that involves the study of computation.
-== Physical process ==
-Computation can be seen as a purely physical process occurring inside a closed [[physical system]] called a [[computer]]. Examples of such physical systems are [[digital computer]]s, [[mechanical computer]]s, [[quantum computer]]s, [[DNA computer]]s, [[molecular computer]]s, microfluidics-based computers, [[analog computer]]s, and [[wetware computer]]s.
+== Introduction ==
-This point of view has been adopted by the [[physics of computation]], a branch of theoretical physics, as well as the field of [[natural computing]].
+The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the [[1600s (decade)|1600s]],<ref>{{cite book | last=Couturat | first=Louis | title=la Logique de Leibniz a'Après des Documents Inédits | publisher=Paris | date=1901 | isbn=978-0343895099}}</ref> but agreement on a suitable definition proved elusive.<ref name="Davis Davis 2000">{{cite book | last1=Davis | first1=Martin | last2=Davis | first2=Martin D. | title=The Universal Computer | publisher=W. W. Norton & Company | date=2000 | isbn=978-0-393-04785-1}}</ref> A candidate definition was proposed independently by several mathematicians in the 1930s.<ref name="Davis">{{cite book | last=Davis | first=Martin | title=Computability & Unsolvability | publisher=Courier Corporation | date=1982-01-01 | isbn=978-0-486-61471-7}}</ref> 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]].<ref>{{Cite news | last= Turing | first= A.M. |year = 1937 | title = On Computable Numbers, with an Application to the Entscheidungsproblem | orig-year = Delivered to the Society November 1936 | periodical = Proceedings of the London Mathematical Society | series = 2 | volume = 42 | pages = 230–65 | doi= 10.1112/plms/s2-42.1.230 | url = http://www.comlab.ox.ac.uk/activities/ieg/e-library/sources/tp2-ie.pdf }}</ref> Other (mathematically equivalent) definitions include [[Alonzo Church]]'s ''[[Lambda calculus|lambda-definability]]'', [[Herbrand]]-[[Gödel]]-[[Kleene]]'s ''[[General recursive function|general recursiveness]]'' and [[Emil Post]]'s ''1-definability''.<ref name="Davis"/>
+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'''.
-An even more radical point of view, [[pancomputationalism]], is the postulate of [[digital physics]] that argues that the evolution of the universe is itself a computation.
+Turing's definition apportioned "well-definedness" to a very large class of mathematical statements, including all well-formed [[equations|algebraic statements]], and all statements written in modern computer programming languages.<ref name="Davis Davis 2000 p. ">{{cite book | last1=Davis | first1=Martin | last2=Davis | first2=Martin D. | title=The Universal Computer | publisher=W. W. Norton & Company | date=2000 | isbn=978-0-393-04785-1 | page=}}</ref>
+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 [[busy beaver|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.<ref group=note>The study of non-computable statements is the field of [[hypercomputation]].</ref><ref>{{cite journal | author=Davis, Martin | title = Why there is no such discipline as hypercomputation | journal = Applied Mathematics and Computation | volume = 178 | issue = 1 <!-- Special Issue on Hypercomputation --> | year = 2006 | pages = 4–7 | doi = 10.1016/j.amc.2005.09.066}}</ref>
+Some examples of mathematical statements that are computable include:
+* All statements characterised in modern programming languages, including [[C++]], [[Python (programming language)|Python]], and [[Java (programming language)|Java]].<ref name="Davis Davis 2000 p. "/>
+* All calculations carried by an electronic [[computer]], [[calculator]] or [[abacus]].
+* All calculations carried out on an [[analytical engine]].
+* All calculations carried out on a [[Turing Machine]].
+* The majority of mathematical statements and calculations given in maths textbooks.
+Some examples of mathematical statements that are ''not'' computable include:
+* Calculations or statements which are ill-defined, such that they cannot be unambiguously encoded into a Turing machine: ("Paul loves me twice as much as Joe").
+* Problem statements which do appear to be well-defined, but for which it can be proved that no Turing machine exists to solve them (such as [[the halting problem]]).
+=== 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 computer]]s, [[mechanical computer]]s, [[analog computer]]s and others.
+== Alternative accounts of computation ==
 === The mapping account ===
-The classic account of computation is found throughout the works of [[Hilary Putnam]] and others. [[Peter Godfrey-Smith]] has dubbed this the "simple mapping account."<ref>{{Citation|last=Godfrey-Smith|first=P.|year=2009|title=Triviality Arguments against Functionalism|journal=Philosophical Studies|volume=145|issue=2|pages=273–95|doi=10.1007/s11098-008-9231-3}}</ref> [[Gualtiero Piccinini|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."<ref>{{cite book|last=Piccinini|first=Gualtiero|title=Physical Computation: A Mechanistic Account|place=Oxford|publisher=Oxford University Press|year=2015|page=18|isbn=9780199658855}}</ref>
+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."<ref>{{Citation|last=Godfrey-Smith|first=P.|year=2009|title=Triviality Arguments against Functionalism|journal=Philosophical Studies|volume=145|issue=2|pages=273–95|doi=10.1007/s11098-008-9231-3|s2cid=73619367 }}</ref> [[Gualtiero Piccinini|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."<ref>{{cite book|last=Piccinini|first=Gualtiero|title=Physical Computation: A Mechanistic Account|place=Oxford|publisher=Oxford University Press|year=2015|page=18|isbn=9780199658855}}</ref>
 === The semantic account ===
@@ Line 22: / Line 42: @@
 == Mathematical models ==
-{{main|Model of computation}}
+{{Main|Model of computation}}
-In the [[theory of computation]], a diversity of mathematical models of computers has been developed.
+In the [[theory of computation]], a diversity of mathematical models of computation has been developed.
 Typical mathematical [[Model of computation|models of computers]] are the following:
 * State models including [[Turing machine]], [[pushdown automaton]], [[finite state automaton]], and [[Parallel random access machine|PRAM]]
@@ Line 29: / Line 49: @@
 * Logical models including [[logic programming]]
 * Concurrent models including [[actor model]] and [[process calculi]]
-Giunti calls the models studied by computation theory ''computational systems,'' and he argues that all of them are mathematical [[dynamical system]]s with discrete time and discrete state space.<ref>{{cite book|last=Giunti|first=Marco|title=Computation, Dynamics, and Cognition|place=New York|publisher=Oxford University Press|year=1997|isbn=978-0-19-509009-3}}</ref>{{rp|ch.1}} He maintains that a computational system is a complex object which consists of three parts. First, a mathematical dynamical system <math>DS</math> with discrete time and discrete state space; second, a computational setup <math>H=\left(F, B_F\right)</math>, which is made up of a theoretical part <math>F</math>, and a real part <math>B_F</math>; third, an  interpretation <math>I_{DS,H}</math>, which links the dynamical system <math>DS</math> with the setup <math>H</math>.<ref>{{Citation|last=Giunti|first=Marco|year=2017|title=What is a Physical Realization of a Computational System?|journal=Isonomia -- Epistemologica|volume=9|pages=177–92|ISSN=2037-4348|URL=https://www.researchgate.net/publication/319631506_What_is_a_Physical_Realization_of_a_Computational_System}}</ref>{{rp|pp.179–80}}
+Giunti calls the models studied by computation theory ''computational systems,'' and he argues that all of them are mathematical [[dynamical system]]s with discrete time and discrete state space.<ref>{{cite book|last=Giunti|first=Marco|title=Computation, Dynamics, and Cognition|place=New York|publisher=Oxford University Press|year=1997|isbn=978-0-19-509009-3}}</ref>{{rp|ch.1}} He maintains that a computational system is a complex object which consists of three parts. First, a mathematical dynamical system <math>DS</math> with discrete time and discrete state space; second, a computational setup <math>H=\left(F, B_F\right)</math>, which is made up of a theoretical part <math>F</math>, and a real part <math>B_F</math>; third, an interpretation <math>I_{DS,H}</math>, which links the dynamical system <math>DS</math> with the setup <math>H</math>.<ref>{{Citation|last=Giunti|first=Marco|year=2017|title=What is a Physical Realization of a Computational System?|journal=Isonomia -- Epistemologica|volume=9|pages=177–92|issn=2037-4348|url=https://www.researchgate.net/publication/319631506}}</ref>{{rp|pp.179–80}}
 == See also ==
-* [[Computationalism]]
+* [[Computability theory]]
-* [[Real computation]]
-* [[Reversible computation]]
 * [[Hypercomputation]]
-* [[Lateral computing]]
 * [[Computational problem]]
-* [[Multiple realizability]]
 * [[Limits of computation]]
+* [[Computationalism]]
+== Notes ==
+{{reflist|group=note}}
-==References==
+== References ==
 {{Reflist}}