Dana Scott on Lambda Calculus
Prof. Dana Scott gave a few talks on Lambda Calculus, and some of them are available on Youtube.
A list of them can be found here:
This video series seems to be taken in the same day, a total of 5 hours. Prof. Scott offered many anecdotal insights on how calculus was invented and formed. It directly relates to the notion of function and combinators. Particularly, the SK Combinators.
Lecture 1
This starting lecture talks about the name of Lambda came from[1].
Lecture 2
Godel Numbering
Think about variables in terms of special numbers. This is an insight from Godel[2]Cite error: Invalid <ref> tag; invalid names, e.g. too many, and later utilized to created Universal computation.
We don't need Turing Machine
In this lectureCite error: Invalid <ref> tag; invalid names, e.g. too many, Scott explicitly stated that:
"You don't need Turing Machine to understand it, I hope I can convince you of that."
Scott's Universe is the Powerset of Integers
In this lectureCite error: Invalid <ref> tag; invalid names, e.g. too many, Scott explicitly stated that:
"The Universe if the Powerset of Integers."
Sophomores or Juniors should learn some Topology
Sophomores or juniors should have some topology from calculus...
A neighborhood of a possibly infinite set...
- The neighborhood of a possibly infinite set is just determined by a finite subset... and its complement
- A stronger topology, called product topology, where its complement can also be expressed with finite information... Hausdorf set taking half the topology
Once you define Topology, you may define continuous functions
- Define Continuous Functions
- The main difficulty is that there are two quantifiers, forming a rational number
- Finite amount of information can only be represented by a finite amount of rational numbers
Lecture 3
This lecture[3] starts to mention the notion of algebraic closure and fixed points.
Lecture 4
This lecture[4]Cite error: Invalid <ref> tag; invalid names, e.g. too many mentioned three important persons in logic.
- John Myhill
- John Sheperdson
- Hartley Rogers Jr.
Why he kicks himself in the middle of the night
This is also the place where he starts talking about the recursive combinators, and how this enumerative device can be used to make one rich and famous.
- Here is where he wants to kick himself at the night
- No body said: "do this operators have any algebra to them...", if only, if only ...
This statement relates to the paper[5] on Algebra of Systems and this statement in particular.[6]. Algebra of Systems[7] is a many-sorted algebra, a computable data format that is summarized in a 2009 paper based on Koo's 2005 doctoral thesis[8].
A Concise Algebra for automating engineering tasks
This year 2009 paper summarized the following statement in the conclusion:
In Laws of programming[9], 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[10] was saying in the 2018 Lambda Conference.
Brendan Fong's doctorial thesis[11] 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[12]:
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.
References
- ↑ Scott, Dana (Oct 12, 2017). Dana Scott - Theory and Models of Lambda Calculus Untyped and Typed - Part 1 of 5 - λC 2017. local page: LambdaConf.
- ↑ Scott, Dana (Oct 12, 2017). Dana Scott - Theory and Models of Lambda Calculus Untyped and Typed - Part 2 of 5 - λC 2017. local page: LambdaConf.
- ↑ Scott, Dana (Oct 12, 2017). Dana Scott - Theory and Models of Lambda Calculus Untyped and Typed - Part 3 of 5 - λC 2017. local page: LambdaConf.
- ↑ Scott, Dana (Oct 12, 2017). Dana Scott - Theory and Models of Lambda Calculus Untyped and Typed - Part 4 of 5 - λC 2017. local page: LambdaConf.
- ↑ 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.
- ↑ A Concise Algebra for automating engineering tasks
- ↑ 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 ...