beta-normal forms and ARSs.

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notes/_journal/2024-06-22.md

@@ -0,0 +1,12 @@
+---
+title: "2024-06-22"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on [[beta-reduction#Normal Form|β-normal forms]].
+* Very basic notes on [[ars|abstract rewriting systems]].

notes/c17/declarations.md

@@ -212,7 +212,7 @@ END%%
 
 ## Integer Literals
 
-Negative integer literals are typed in a counterintuitive way. When the compiler sees a number of form `-X`, the type of `X` is first determined *before* then being negated. Promotion rules are as follows:
+Negative integer literals are typed in a counterintuitive way. When the compiler sees a number of form `-X`, the type of `X` is determined *before* being negated. Promotion rules are as follows:
 
 Decimal     | Other Bases
 ----------- | --------------------

notes/lambda-calculus/alpha-conversion.md

@@ -20,6 +20,13 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
 <!--ID: 1717687744134-->
 END%%
 
+%%ANKI
+Cloze
+$P \equiv Q$ is to {equivalent} whereas $P \equiv_\alpha Q$ is to {congruent}.
+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: 1719065602220-->
+END%%
+
 %%ANKI
 Basic
 What two ways can we pronounce $P \equiv_\alpha Q$?

notes/lambda-calculus/beta-reduction.md

@@ -167,6 +167,105 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
 <!--ID: 1718475424868-->
 END%%
 
+## Normal Form
+
+A term $Q$ which contains no $\beta$-redexes is called a **$\beta$-normal form** (or a **term in $\beta$-normal form** or just a **$\beta$-nf**). The class of all $\beta$-normal forms is called $\beta$-nf or $\lambda\beta$-nf. If a term $P$ $\beta$-reduces to a term $Q$ in $\beta$-nf, then $Q$ is called a **$\beta$-normal form of $P$**.
+
+%%ANKI
+Basic
+$\beta$-reduction terminates if and only if what?
+Back: We reduce to a term in $\beta$-normal form.
+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: 1719065185793-->
+END%%
+
+%%ANKI
+Basic
+Is $\beta$-reduction guaranteed to terminate?
+Back: No.
+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: 1719065185796-->
+END%%
+
+%%ANKI
+Basic
+Is $\beta$-reduction guaranteed to simplify?
+Back: No.
+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: 1719065185803-->
+END%%
+
+%%ANKI
+Basic
+How is a $\beta$-normal form defined?
+Back: As a $\lambda$-term that contains no $\beta$-redexes.
+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: 1719065185808-->
+END%%
+
+%%ANKI
+Cloze
+The class of {all $\beta$-normal forms} is called {$\beta$-nf/$\lambda\beta$-nf}.
+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: 1719065185812-->
+END%%
+
+%%ANKI
+Basic
+What ambiguity does term "$\beta$-nf" introduce?
+Back: It refers to a specific $\beta$-normal form or the class of $\beta$-normal forms.
+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: 1719065185815-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for term $Q$ to be a $\beta$-normal form of term $P$?
+Back: $P$ $\beta$-reduces to a term $Q$ in $\beta$-nf.
+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: 1719065185819-->
+END%%
+
+%%ANKI
+Basic
+How is the class $\beta$-nf alternatively denoted?
+Back: As $\lambda\beta$-nf.
+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: 1719065185823-->
+END%%
+
+%%ANKI
+Basic
+How is the class $\lambda\beta$-nf alternatively denoted?
+Back: As $\beta$-nf.
+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: 1719065185799-->
+END%%
+
+%%ANKI
+Basic
+What is the $\beta$-normal form of $(\lambda x. x(xy))N$?
+Back: $N(Ny)$
+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: 1719065602204-->
+END%%
+
+%%ANKI
+Basic
+What is the $\beta$-normal form of $(\lambda x. xx)(\lambda x. xx)$?
+Back: N/A.
+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: 1719065602211-->
+END%%
+
+%%ANKI
+Basic
+How many $\beta$-reductions can a term have?
+Back: One 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: 1719065602215-->
+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).

notes/lambda-calculus/index.md

@@ -777,6 +777,22 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
 <!--ID: 1717036717102-->
 END%%
 
+%%ANKI
+Basic
+Is $(\lambda x. xy)N \equiv Ny$?
+Back: No.
+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: 1719065185785-->
+END%%
+
+%%ANKI
+Basic
+Is $[N/x]xy \equiv Ny$?
+Back: Yes.
+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: 1719065185789-->
+END%%
+
 For all $\lambda$-terms $M$, $N$, and variables $x$:
 
 * $[x/x]M \equiv M$

notes/logic/equiv-trans.md

@@ -830,22 +830,6 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1714395640893-->
 END%%
 
-%%ANKI
-Basic
-Given valid expression $(b; [i]{\circ}s{:}e)$, what is the type of $b$?
-Back: A function (an array).
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1714395640896-->
-END%%
-
-%%ANKI
-Basic
-Given valid expression $(b; \epsilon{:}e)$, what is the type of $b$?
-Back: A function or scalar.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1714395640898-->
-END%%
-
 %%ANKI
 Basic
 What is the base case of selector update syntax?

notes/programming/ars.md

@@ -0,0 +1,65 @@
+---
+title: Abstract Rewriting Systems
+TARGET DECK: Obsidian::STEM
+FILE TAGS: ars
+tags:
+  - ars
+---
+
+## Overview
+
+In an **abstract rewriting system** (ARS), an object is said to be in **normal form** if it cannot be rewritten any further, i.e. it is irreducible. An object is said to be in **canonical form** if it is presented in the "standard" representation (where "standard" is defined per field).
+
+In most fields, a canoncial form specifies a *unique* representation.
+
+%%ANKI
+Basic
+What is ARS an acronym for?
+Back: **A**bstract **R**ewriting **S**ystem.
+Reference: “Normal Form,” in _Wikipedia_, April 27, 2024, [https://en.wikipedia.org/w/index.php?title=Normal_form](https://en.wikipedia.org/w/index.php?title=Normal_form_(abstract_rewriting)&oldid=1221094193).
+<!--ID: 1719067812812-->
+END%%
+
+%%ANKI
+Basic
+Canonical/normal forms are associated with what general class of abstract systems?
+Back: Abstract rewriting systems.
+Reference: “Canonical Form,” in _Wikipedia_, January 7, 2024, [https://en.wikipedia.org/w/index.php?title=Canonical_form](https://en.wikipedia.org/w/index.php?title=Canonical_form&oldid=1194093963).
+<!--ID: 1719067812820-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for an object to be in normal form?
+Back: It cannot be rewritten any further, i.e. it is irreducible.
+Reference: Normal Form,” in _Wikipedia_, April 27, 2024, [https://en.wikipedia.org/w/index.php?title=Normal_form](https://en.wikipedia.org/w/index.php?title=Normal_form_(abstract_rewriting)&oldid=1221094193).
+END%%
+
+%%ANKI
+Basic
+What does it mean for an object to be in canonical form?
+Back: It is presented in the "standard" way with respect to the field its presented in.
+Reference: “Canonical Form,” in _Wikipedia_, January 7, 2024, [https://en.wikipedia.org/w/index.php?title=Canonical_form](https://en.wikipedia.org/w/index.php?title=Canonical_form&oldid=1194093963).
+<!--ID: 1719067812824-->
+END%%
+
+%%ANKI
+Basic
+What two assumptions are usually associated with an object in canonical form?
+Back: It is as simply represented as possible and it is uniquely identifying.
+Reference: “Canonical Form,” in _Wikipedia_, January 7, 2024, [https://en.wikipedia.org/w/index.php?title=Canonical_form](https://en.wikipedia.org/w/index.php?title=Canonical_form&oldid=1194093963).
+<!--ID: 1719067812829-->
+END%%
+
+%%ANKI
+Basic
+How do "canonical" and "normal" forms differ in most fields?
+Back: The former usually specifies a unique representation for the object.
+Reference: “Canonical Form,” in _Wikipedia_, January 7, 2024, [https://en.wikipedia.org/w/index.php?title=Canonical_form](https://en.wikipedia.org/w/index.php?title=Canonical_form&oldid=1194093963).
+<!--ID: 1719067812833-->
+END%%
+
+## Bibliography
+
+* “Canonical Form,” in _Wikipedia_, January 7, 2024, [https://en.wikipedia.org/w/index.php?title=Canonical_form](https://en.wikipedia.org/w/index.php?title=Canonical_form&oldid=1194093963).
+* Normal Form,” in _Wikipedia_, April 27, 2024, [https://en.wikipedia.org/w/index.php?title=Normal_form](https://en.wikipedia.org/w/index.php?title=Normal_form_(abstract_rewriting)&oldid=1221094193).

notes/proofs/induction.md

@@ -119,7 +119,7 @@ END%%
 
 %%ANKI
 Basic
-What contradiction is introduced to explain why the strong induction assumption is valid?
+What negation is introduced to explain why the strong induction assumption is valid?
 Back: If $P(n)$ is not true for all $n$, there exists a *first* $n_0$ for which $\neg P(n_0)$.
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
 <!--ID: 1714574131963-->