Difference between revisions of "Monoid"
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[[wikipedia:Monoid|Monoid]] ([[wikipedia:zh:幺半群|幺半群]]), according to Wikipedia, it is a set equipped with an associative binary operation and an identity element. It is a foundational algebraic structure that can be applied to many fields. | [[wikipedia:Monoid|Monoid]] ([[wikipedia:zh:幺半群|幺半群]]), according to Wikipedia, it is a set equipped with an associative binary operation and an identity element. It is a foundational algebraic structure that can be applied to many fields. | ||
For instance, in a Turing Machine, all operations can be thought of as some sequences of stack operations<ref>{{:Paper/mov is Turing-complete}}</ref><ref>{{:Video/What is a monoid?}}</ref>. This idea has been extended to Bayesian inferencing<ref>{{:Thesis/Causal Theories: A Categorical Perspective on Bayesian Networks}}</ref>, and other ideas, such as [[:Book/The Master Algorithm|The Master Algorithm]]<ref>{{:Book/The Master Algorithm}}</ref>. | For instance, in a Turing Machine, all operations can be thought of as some sequences of stack operations<ref>{{:Paper/mov is Turing-complete}}</ref><ref>{{:Video/What is a monoid?}}</ref>. This idea has been extended to Bayesian inferencing<ref>{{:Thesis/Causal Theories: A Categorical Perspective on Bayesian Networks}}</ref>, and other ideas, such as [[:Book/The Master Algorithm|The Master Algorithm]]<ref>{{:Book/The Master Algorithm}}</ref>. | ||
=Implementation and Applications= | |||
One may consider the idea of using a single data type as a programming language to be associated with the idea of Monoid. That is also the basis of functional programming languages are nothing but the manipulation of [[S-expression]]s. Therefore, one may consider Lambda-Calculus as an implementation of this Monoid-like language specification. Later, the idea of all languages can be treated as some textual stream of data content, and processed by [[Large Language Model]]s, or [[Generative Pre-trained Transfomer]]s, would be another example of Monoid. | |||
Revision as of 04:40, 6 July 2023
Monoid (幺半群), according to Wikipedia, it is a set equipped with an associative binary operation and an identity element. It is a foundational algebraic structure that can be applied to many fields.
For instance, in a Turing Machine, all operations can be thought of as some sequences of stack operations[1][2]. This idea has been extended to Bayesian inferencing[3], and other ideas, such as The Master Algorithm[4].
Implementation and Applications
One may consider the idea of using a single data type as a programming language to be associated with the idea of Monoid. That is also the basis of functional programming languages are nothing but the manipulation of S-expressions. Therefore, one may consider Lambda-Calculus as an implementation of this Monoid-like language specification. Later, the idea of all languages can be treated as some textual stream of data content, and processed by Large Language Models, or Generative Pre-trained Transfomers, would be another example of Monoid.
References
- ↑ Dolan, Stephen (July 19, 2013). "mov is Turing-complete" (PDF). local page: Computer Laboratory, University of Cambridge.
- ↑ All Angles, ed. (Aug 14, 2021). What is a monoid?. local page: All Angles.
- ↑ Fong, Brendan (2013). Causal Theories: A Categorical Perspective on Bayesian Networks (Master). local page: University of Oxford. Retrieved Jan 26, 2013.
- ↑ Domingos, Pedro (February 13, 2018). The Master Algorithm: How the Quest for the Ultimate Learning Machine Will Remake Our World (Revised ed.). local page: Basic Books. ISBN 978-0465094271.
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