Difference between revisions of "Algebra of Systems"
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Algebra of Systems<ref>{{:Paper/Algebra of Systems}}</ref> is a paper based on [[User:Benkoo|Koo]]'s thesis<ref>{{:Thesis/A Meta-language for Systems Architecting}}</ref>. | Algebra of Systems<ref>{{:Paper/Algebra of Systems}}</ref> is a [[many-sorted algebra]], a computable data format that is summarized in a [[Paper/Algebra of Systems|2009 paper]] based on [[User:Benkoo|Koo]]'s 2005 doctoral thesis<ref>{{:Thesis/A Meta-language for Systems Architecting}}</ref>. | ||
=A | ==A Concise Algebra for automating engineering tasks== | ||
This year 2009 paper summarized the following statement in the conclusion: | This year 2009 paper summarized the following statement in the conclusion: | ||
In [[Laws of programming]]<ref> | In [[Laws of programming]]<ref>{{:Paper/Laws of programming}}</ref>, Hoare et al. questioned whether a small set of algebraic laws can be directly useful in a practical engineering design problem. The absence of a tool that can bridge the cognitive gap between mathematical abstractions and engineering problems may have been the main reason for their conservative attitude. | ||
The above statement echos who [[Dana Scott]]<ref>[ | The above statement echos who [[Dana Scott]]<ref>[https://youtu.be/8zk0yS8Jp5w?t=1300 Scott Commenting on a small algebra for combinators]</ref> was saying in the 2018 Lambda Conference. | ||
[[Brendan Fong]]'s doctorial thesis<ref>{{:Thesis/The Algebra of Open and Interconnected Systems}}</ref> on [[Thesis/The Algebra of Open and Interconnected Systems|The Algebra of Open and Interconnected Systems]] is a rigorous treatment to the subject matter on [[AoS]]. | |||
==A lookup table as a enumerable function== | |||
He said in this lecture<ref>[https://youtu.be/8zk0yS8Jp5w?t=1468 If the giant lookup table is enumerable ...]</ref>: | |||
If that lookup table is enumerable, that is a good definition of say that the enumeration operation is computable. ... So that computability here is on the same plane with enumerability. | |||
<noinclude> | <noinclude> | ||
==Always think positively== | |||
He said the following<ref>[https://youtu.be/8zk0yS8Jp5w?t=1860 Always think positively]</ref>: | |||
Don't think of divergence and all of that. ... You can only achieve what can possibly achieve. Don't think of things that cannot be done. This is the way how enumeration works. Working with the positives. Of course, you can think of complementary sets. | |||
This is also where one can starting relating to the laws of [[composition]]. | |||
==Lambda Calculus allows you to notate Least Fixed Points== | |||
There is a '''well-foundedness''' as he mentioned at about [https://youtu.be/8zk0yS8Jp5w?t=2340 39 minutes] into this video. He started talking about <ref>[https://youtu.be/8zk0yS8Jp5w?t=2380 Least Fixed Point and Lambda Calculus]</ref>. There is a connection between <math>lfp(x)</math> or Least Fixed Point of <math>x</math>, and lambda calculus. | |||
==All Computable/Continuous Functions can be composed using Lambda Calculus== | |||
This where things get related to [[composition]]<ref>[https://youtu.be/8zk0yS8Jp5w?t=2485 All continuous functions can be composed of Lambda Calculus]</ref>. | |||
All Computable/Continuous Functions can be composed using Lambda Calculus and arithmetics. Arithmetics gives you the power of analyzing Gödel numbers and other kinds of structures. | |||
==Number 0,1 is related to K and S respectively== | |||
The arithmetic mechanism can be represented using [[SK Combinators]], therefore, each can be related to a specific kind of number<ref>[https://youtu.be/8zk0yS8Jp5w?t=2555 S and K as numbers 1 and 0 respectively]</ref>. The smaller numbers<code>2, 3, 4</code> are assigned to do arithmetics. | |||
==Sub models with Closure Conditions within Computable Recursive Operators== | |||
There are ways to take closures in sub [https://youtu.be/8zk0yS8Jp5w?t=2978 Lambad Calculus and iteration gives you recursive theory]. It also allows one to filter out certain elements. | |||
Closure is something can be expressed in Lambda Calculus<ref>[https://youtu.be/8zk0yS8Jp5w?t=2973 Closure expressed in Lambda Calculus]</ref>. | |||
==Closed elements in the closed ... forms an algebra== | |||
[https://youtu.be/8zk0yS8Jp5w?t=3310 Closed elements forms ...] | |||
The last few minutes of lecture 3, Scott showed that there is a strong connection between algebra, complete [[lattice]], upper/lower bounds as [[closure]], and finally, [[fixed point]]. | |||
=References= | =References= | ||
Latest revision as of 09:34, 23 February 2022
Algebra of Systems[1] is a many-sorted algebra, a computable data format that is summarized in a 2009 paper based on Koo's 2005 doctoral thesis[2].
A Concise Algebra for automating engineering tasks
This year 2009 paper summarized the following statement in the conclusion:
In Laws of programming[3], Hoare et al. questioned whether a small set of algebraic laws can be directly useful in a practical engineering design problem. The absence of a tool that can bridge the cognitive gap between mathematical abstractions and engineering problems may have been the main reason for their conservative attitude.
The above statement echos who Dana Scott[4] was saying in the 2018 Lambda Conference.
Brendan Fong's doctorial thesis[5] on The Algebra of Open and Interconnected Systems is a rigorous treatment to the subject matter on AoS.
A lookup table as a enumerable function
He said in this lecture[6]:
If that lookup table is enumerable, that is a good definition of say that the enumeration operation is computable. ... So that computability here is on the same plane with enumerability.
Always think positively
He said the following[7]:
Don't think of divergence and all of that. ... You can only achieve what can possibly achieve. Don't think of things that cannot be done. This is the way how enumeration works. Working with the positives. Of course, you can think of complementary sets.
This is also where one can starting relating to the laws of composition.
Lambda Calculus allows you to notate Least Fixed Points
There is a well-foundedness as he mentioned at about 39 minutes into this video. He started talking about [8]. There is a connection between or Least Fixed Point of , and lambda calculus.
All Computable/Continuous Functions can be composed using Lambda Calculus
This where things get related to composition[9].
All Computable/Continuous Functions can be composed using Lambda Calculus and arithmetics. Arithmetics gives you the power of analyzing Gödel numbers and other kinds of structures.
The arithmetic mechanism can be represented using SK Combinators, therefore, each can be related to a specific kind of number[10]. The smaller numbers2, 3, 4 are assigned to do arithmetics.
Sub models with Closure Conditions within Computable Recursive Operators
There are ways to take closures in sub Lambad Calculus and iteration gives you recursive theory. It also allows one to filter out certain elements.
Closure is something can be expressed in Lambda Calculus[11].
Closed elements in the closed ... forms an algebra
Closed elements forms ... The last few minutes of lecture 3, Scott showed that there is a strong connection between algebra, complete lattice, upper/lower bounds as closure, and finally, fixed point.
References
- ↑ Koo, Hsueh-Yung Benjamin; Simmons, Willard; Crawley, Edward (Nov 16, 2021). "Algebra of Systems as a Meta Language for Model Synthesis and Analysis" (PDF). local page: IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS.
- ↑ Koo, Hsueh-Yung Benjamin (31 Jan 2005). A Meta-language for Systems Architecting (PDF) (Ph.D.). local page: MIT. Retrieved July 18, 2021.
- ↑ Hoare, C. A. R.; Hayes, I. J.; He, Jifeng; Morgan, C. C.; Roscoe, A. W.; Sanders, J. W.; Sorensen, I. H.; Spivey, J. M.; Sufrin, B. A. (Aug 1987). "Laws of Programming" (PDF). 30 (8). local page: ACM.
- ↑ Scott Commenting on a small algebra for combinators
- ↑ Fong, Brendan (2016). The Algebra of Open and Interconnected Systems (PDF) (Ph.D.). local page: University of Oxford. Retrieved October 15, 2021.
- ↑ If the giant lookup table is enumerable ...
- ↑ Always think positively
- ↑ Least Fixed Point and Lambda Calculus
- ↑ All continuous functions can be composed of Lambda Calculus
- ↑ S and K as numbers 1 and 0 respectively
- ↑ Closure expressed in Lambda Calculus