Initial notes on lambda-calculus.

2024-05-23 14:02:55 -06:00 · 2024-05-23 14:02:55 -06:00 · 8fda87c218
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    "Basic": [
--- a/notes/_journal/2024-05-23.md
+++ b/notes/_journal/2024-05-23.md
@ -8,4 +8,5 @@ title: "2024-05-23"
 - [ ] Go (1 Life & Death Problem)
 - [ ] Korean (Read 1 Story)

-* Watched [Lecture #10 - Sorting & Aggregation Algorithms](https://www.youtube.com/watch?v=CMzf9Az1vl4) on databases.
+* Watched [Lecture #10 - Sorting & Aggregation Algorithms](https://www.youtube.com/watch?v=CMzf9Az1vl4) on databases.
+* Work through chapter 1 of "Lambda-Calculus and Combinators, an Introduction".
--- a/notes/programming/λ-Calculus.md
+++ b/notes/programming/λ-Calculus.md
@ -0,0 +1,194 @@
+---
+title: λ-Calculus
+TARGET DECK: Obsidian::STEM
+FILE TAGS: λ-calculus
+tags:
+  - λ-calculus
+---
+
+## Overview
+
+Assume that there is given an infinite sequence of expressions called **variables** and a finite or infinite sequence of expressions called **atomic constants**, different from the variables. The set of expressions called $\lambda$-terms is defined inductively as follows:
+
+* all variables and atomic constants are $\lambda$-terms (called **atoms**)
+* if $M$ and $N$ are $\lambda$-terms, then $(MN)$ is a $\lambda$-term (called **application**)
+* if $M$ is a $\lambda$-term and $x$ is a variable, then $(\lambda x. M)$ is a $\lambda$-term (called **abstraction**)
+
+If the sequence of atomic constants is empty, the system is called **pure**. Otherwise it is called **applied**.
+
+%%ANKI
+Basic
+What does a "higher-order function" refer to?
+Back: A function that acts on other functions.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526287-->
+END%%
+
+%%ANKI
+Basic
+How is $f(x) = x - y$ written using $\lambda$-calculus?
+Back: $\lambda x. x - y$
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526290-->
+END%%
+
+%%ANKI
+Basic
+How is $f(x, y) = x - y$ written using (uncurried) $\lambda$-calculus?
+Back: $\lambda x y. x - y$
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526293-->
+END%%
+
+%%ANKI
+Basic
+How do you curry expression $\lambda x y. x - y$?
+Back: $\lambda x. \lambda y. x - y$
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526297-->
+END%%
+
+%%ANKI
+Basic
+How do you uncurry expression $\lambda x. \lambda y. x - y$?
+Back: $\lambda x y. x - y$
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526300-->
+END%%
+
+%%ANKI
+Basic
+What does $(\lambda x. x - y)(0)$ evaluate to?
+Back: $0 - y$
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526303-->
+END%%
+
+%%ANKI
+Basic
+How many variables exist in a $\lambda$-calculus formal system?
+Back: An infinite number.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526306-->
+END%%
+
+%%ANKI
+Basic
+How many atomic constants exist in a $\lambda$-calculus formal system?
+Back: Zero or more.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526309-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes variables and atomic constants?
+Back: The latter is meant to refer to constants outside the formal system.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526312-->
+END%%
+
+%%ANKI
+Basic
+What two classes of expressions does an "atom" potentially refer to?
+Back: Variables and atomic constants.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526316-->
+END%%
+
+%%ANKI
+Basic
+What general term describes both variables and atomic constants?
+Back: Atoms.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526319-->
+END%%
+
+%%ANKI
+Basic
+Why are variables and atomic constants called "atoms"?
+Back: They are not composed of smaller $\lambda$-terms.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526322-->
+END%%
+
+%%ANKI
+Basic
+When is a $\lambda$-calculus considered pure?
+Back: When there exist no atomic constants in the system.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526325-->
+END%%
+
+%%ANKI
+Basic
+When is a $\lambda$-calculus considered applied?
+Back: When there exists at least one atomic constant in the system.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526328-->
+END%%
+
+%%ANKI
+Cloze
+A $\lambda$-calculus is either {pure} or {applied}.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526331-->
+END%%
+
+%%ANKI
+Basic
+What term refers to the base case of the $\lambda$-term definition?
+Back: The atoms.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526334-->
+END%%
+
+%%ANKI
+Basic
+What term refers to the inductive cases of the $\lambda$-term definition?
+Back: Application and abstraction.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526337-->
+END%%
+
+%%ANKI
+Cloze
+Given $\lambda$-terms $M$ and $N$, {$(MN)$} is referred to as {application}.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526340-->
+END%%
+
+%%ANKI
+Cloze
+Given $\lambda$-term $M$ and variable $x$, {$(\lambda x. M)$} is referred to as {abstraction}.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526343-->
+END%%
+
+%%ANKI
+Basic
+Consider term $(\lambda x. x)(0)$. Is our $\lambda$-calculus pure or applied?
+Back: Applied.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526346-->
+END%%
+
+%%ANKI
+Basic
+Consider term $(\lambda x. x)(y)$. Is our $\lambda$-calculus pure or applied?
+Back: Indeterminate.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526349-->
+END%%
+
+%%ANKI
+Basic
+What three terms categorize all $\lambda$-terms?
+Back: Atoms, applications, and abstractions.
+Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
+<!--ID: 1716494526352-->
+END%%
+
+## Bibliography
+
+* Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
--- a/notes/set/index.md
+++ b/notes/set/index.md
@ -134,6 +134,45 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1715786028667-->
 END%%

+%%ANKI
+Basic
+What does an atom refer to in set theory?
+Back: Any entity that is not a set but can exist in one.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1716494526269-->
+END%%
+
+%%ANKI
+Cloze
+An {1:atom} in set theory is to {2:atomic} logical statements whereas {2:sets} are to {1:molecular} statements.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1716494526273-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes a set from an atom?
+Back: An atom cannot contain other entitites.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1716494526277-->
+END%%
+
+%%ANKI
+Basic
+What intuition is broken when a box is viewed as an atom?
+Back: When viewed as an atom, the box is no longer a container.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1716494526280-->
+END%%
+
+%%ANKI
+Basic
+Enderton's exposition makes what assumption about atoms?
+Back: The set of all atoms is the empty set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1716494526284-->
+END%%
+
 ## Extensionality

 If two sets have exactly the same members, then they are equal: $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$