Difference between revisions of "Monoid"
| (23 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
[[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>. | ||
[[ | |||
=Implementation= | |||
In abstract algebra, a [[lattice]] can be considered as a [[monoid]]. 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 Transformer]]s, would be another example of Monoid. Most importantly, the software pattern, or the mechanism of manipulating text-based expressions in an iterative loop, called ([[Read-eval-print loop]])[[REPL]], would be a kind of [[Monoid]]. | |||
=Applications= | |||
Based on the idea of singular data type, the creation of [[IPFS]] as a unifying namespace for all data content can be thought of as a Monoidal data space. Therefore, it is also possible to create a system, that universally treats all data objects as files, and therefore, the whole system becomes a monoidal structure. That would be very close to the approach that [[Unix]] took in its design intent. | |||
==Applying Monoidal Idea to Personalized Data Management== | |||
One possible way to utilize Monoid for personal data management, is to create a network-based operating system that uniformly indexes and organizes one's data asset in a singular namespace, and therefore can be accessed anywhere using some unifying authentication system. This way, one just need to throw personal data assets into the system, and all the data navigation mechanisms would be applied to the data content, and enhance the efficiency of getting access to that piece of data. This is also the purpose of [[PKC]]. | |||
<noinclude> | |||
{{PagePostfix | |||
|category_csd=Category Theory,Monoid,Lattice,Large Language Model,GPT,REPL | |||
|semantic_labels= | |||
}} | |||
</noinclude> | |||
Latest revision as of 05:23, 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
In abstract algebra, a lattice can be considered as a monoid. 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 Transformers, would be another example of Monoid. Most importantly, the software pattern, or the mechanism of manipulating text-based expressions in an iterative loop, called (Read-eval-print loop)REPL, would be a kind of Monoid.
Applications
Based on the idea of singular data type, the creation of IPFS as a unifying namespace for all data content can be thought of as a Monoidal data space. Therefore, it is also possible to create a system, that universally treats all data objects as files, and therefore, the whole system becomes a monoidal structure. That would be very close to the approach that Unix took in its design intent.
Applying Monoidal Idea to Personalized Data Management
One possible way to utilize Monoid for personal data management, is to create a network-based operating system that uniformly indexes and organizes one's data asset in a singular namespace, and therefore can be accessed anywhere using some unifying authentication system. This way, one just need to throw personal data assets into the system, and all the data navigation mechanisms would be applied to the data content, and enhance the efficiency of getting access to that piece of data. This is also the purpose of PKC.
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.
Related Pages