View source for Algebra of Systems

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>.

==A Concise Algebra for automating engineering tasks==
This year 2009 paper summarized the following statement in the conclusion:
 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>[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>

==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/>

=Related Pages=
[[Category:AoS]]
[[Category:Meta Mathematics]]
</noinclude>