beta-normal forms and ARSs.
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},
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"fields_dict": {
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"Basic": [
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---
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title: "2024-06-22"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [x] OGS
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- [ ] Sheet Music (10 min.)
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- [ ] Korean (Read 1 Story)
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* Notes on [[beta-reduction#Normal Form|β-normal forms]].
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* Very basic notes on [[ars|abstract rewriting systems]].
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@ -212,7 +212,7 @@ END%%
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## Integer Literals
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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:
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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:
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Decimal | Other Bases
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----------- | --------------------
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@ -20,6 +20,13 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
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<!--ID: 1717687744134-->
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END%%
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%%ANKI
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Cloze
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$P \equiv Q$ is to {equivalent} whereas $P \equiv_\alpha Q$ is to {congruent}.
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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).
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<!--ID: 1719065602220-->
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END%%
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%%ANKI
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Basic
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What two ways can we pronounce $P \equiv_\alpha Q$?
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@ -167,6 +167,105 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
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<!--ID: 1718475424868-->
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END%%
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## Normal Form
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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 $P$**.
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%%ANKI
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Basic
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$\beta$-reduction terminates if and only if what?
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Back: We reduce to a term in $\beta$-normal form.
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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).
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<!--ID: 1719065185793-->
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END%%
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%%ANKI
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Basic
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Is $\beta$-reduction guaranteed to terminate?
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Back: No.
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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).
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<!--ID: 1719065185796-->
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END%%
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%%ANKI
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Basic
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Is $\beta$-reduction guaranteed to simplify?
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Back: No.
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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).
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<!--ID: 1719065185803-->
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END%%
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%%ANKI
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Basic
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How is a $\beta$-normal form defined?
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Back: As a $\lambda$-term that contains no $\beta$-redexes.
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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).
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<!--ID: 1719065185808-->
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END%%
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%%ANKI
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Cloze
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The class of {all $\beta$-normal forms} is called {$\beta$-nf/$\lambda\beta$-nf}.
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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).
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<!--ID: 1719065185812-->
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END%%
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%%ANKI
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Basic
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What ambiguity does term "$\beta$-nf" introduce?
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Back: It refers to a specific $\beta$-normal form or the class of $\beta$-normal forms.
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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).
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<!--ID: 1719065185815-->
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END%%
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%%ANKI
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Basic
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What does it mean for term $Q$ to be a $\beta$-normal form of term $P$?
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Back: $P$ $\beta$-reduces to a term $Q$ in $\beta$-nf.
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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).
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<!--ID: 1719065185819-->
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END%%
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%%ANKI
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Basic
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How is the class $\beta$-nf alternatively denoted?
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Back: As $\lambda\beta$-nf.
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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).
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<!--ID: 1719065185823-->
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END%%
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%%ANKI
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Basic
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How is the class $\lambda\beta$-nf alternatively denoted?
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Back: As $\beta$-nf.
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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).
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<!--ID: 1719065185799-->
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END%%
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%%ANKI
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Basic
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What is the $\beta$-normal form of $(\lambda x. x(xy))N$?
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Back: $N(Ny)$
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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).
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<!--ID: 1719065602204-->
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END%%
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%%ANKI
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Basic
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What is the $\beta$-normal form of $(\lambda x. xx)(\lambda x. xx)$?
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Back: N/A.
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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).
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<!--ID: 1719065602211-->
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END%%
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%%ANKI
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Basic
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How many $\beta$-reductions can a term have?
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Back: One or more.
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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).
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<!--ID: 1719065602215-->
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END%%
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## Bibliography
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* 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).
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@ -777,6 +777,22 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
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<!--ID: 1717036717102-->
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END%%
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%%ANKI
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Basic
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Is $(\lambda x. xy)N \equiv Ny$?
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Back: No.
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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).
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<!--ID: 1719065185785-->
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END%%
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%%ANKI
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Basic
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Is $[N/x]xy \equiv Ny$?
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Back: Yes.
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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).
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<!--ID: 1719065185789-->
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END%%
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For all $\lambda$-terms $M$, $N$, and variables $x$:
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* $[x/x]M \equiv M$
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@ -830,22 +830,6 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
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<!--ID: 1714395640893-->
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END%%
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%%ANKI
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Basic
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Given valid expression $(b; [i]{\circ}s{:}e)$, what is the type of $b$?
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Back: A function (an array).
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1714395640896-->
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END%%
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%%ANKI
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Basic
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Given valid expression $(b; \epsilon{:}e)$, what is the type of $b$?
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Back: A function or scalar.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1714395640898-->
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END%%
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%%ANKI
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Basic
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What is the base case of selector update syntax?
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---
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title: Abstract Rewriting Systems
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TARGET DECK: Obsidian::STEM
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FILE TAGS: ars
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tags:
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- ars
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---
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## Overview
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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).
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In most fields, a canoncial form specifies a *unique* representation.
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%%ANKI
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Basic
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What is ARS an acronym for?
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Back: **A**bstract **R**ewriting **S**ystem.
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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).
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<!--ID: 1719067812812-->
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END%%
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%%ANKI
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Basic
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Canonical/normal forms are associated with what general class of abstract systems?
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Back: Abstract rewriting systems.
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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).
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<!--ID: 1719067812820-->
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END%%
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%%ANKI
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Basic
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What does it mean for an object to be in normal form?
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Back: It cannot be rewritten any further, i.e. it is irreducible.
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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).
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END%%
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%%ANKI
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Basic
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What does it mean for an object to be in canonical form?
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Back: It is presented in the "standard" way with respect to the field its presented in.
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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).
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<!--ID: 1719067812824-->
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END%%
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%%ANKI
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Basic
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What two assumptions are usually associated with an object in canonical form?
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Back: It is as simply represented as possible and it is uniquely identifying.
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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).
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<!--ID: 1719067812829-->
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END%%
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%%ANKI
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Basic
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How do "canonical" and "normal" forms differ in most fields?
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Back: The former usually specifies a unique representation for the object.
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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).
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<!--ID: 1719067812833-->
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END%%
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## Bibliography
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* “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).
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* 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).
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@ -119,7 +119,7 @@ END%%
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%%ANKI
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Basic
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What contradiction is introduced to explain why the strong induction assumption is valid?
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What negation is introduced to explain why the strong induction assumption is valid?
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Back: If $P(n)$ is not true for all $n$, there exists a *first* $n_0$ for which $\neg P(n_0)$.
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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).
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<!--ID: 1714574131963-->
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Reference in New Issue