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Notes on alpha-conversion and beta-reduction.

c-declarations
Joshua Potter 2024-06-15 12:20:06 -06:00
parent 6c9e34e192
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notes/_journal/2024-06-15.md
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@ -8,4 +8,5 @@ title: "2024-06-15"
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* [[functions|Notes]] on injections, surjections, and bijections.
* [[functions|Notes]] on injections, surjections, and bijections.
* Start defining [[beta-reduction|β-reduction]].

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@ -20,6 +20,14 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
<!--ID: 1717687744134-->
END%%
%%ANKI
Basic
What two ways can we pronounce $P \equiv_\alpha Q$?
Back: "$P$ is congruent to $Q$" and "$P$ $\alpha$-converts to $Q$".
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: 1718475477173-->
END%%
%%ANKI
Basic
If $P \equiv_\alpha Q$, does $P \equiv Q$?
@ -28,6 +36,22 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
<!--ID: 1717687744141-->
END%%
%%ANKI
Basic
What does an $\alpha$-conversion refer to?
Back: The act of replacing an occurrence of $(\lambda x. M)$ with $\lambda y. [y/x]M$.
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: 1718475424870-->
END%%
%%ANKI
Basic
What distinguishes terms "$\alpha$-conversion" and "$\alpha$-converts"?
Back: The latter refers to 0 or more applications of the former.
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: 1718475424871-->
END%%
%%ANKI
Basic
$\alpha$-conversion is most related to what kind of $\lambda$-term?
@ -44,6 +68,20 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
<!--ID: 1717687744147-->
END%%
%%ANKI
Cloze
"$\alpha$-{conversion}" refers to exactly one change of bound variable.
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: 1718475424873-->
END%%
%%ANKI
Cloze
"$\alpha$-{converts}" refers to zero or more change of bound variables.
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: 1718475424874-->
END%%
%%ANKI
Basic
What *kind* of conversion is a change of bound variable?
@ -102,11 +140,19 @@ END%%
%%ANKI
Cloze
$\alpha$-conversion is known as a change of {bound variables}.
$\alpha$-conversion is known as a change of {bound variable}.
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: 1717687744176-->
END%%
%%ANKI
Basic
What greek-prefixed term is a change of bound variable known as?
Back: An $\alpha$-conversion.
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: 1718475424876-->
END%%
%%ANKI
Basic
If $P \equiv_\alpha Q$, what can be said about the free variables of $P$ and $Q$?
@ -384,6 +430,50 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
<!--ID: 1718422981125-->
END%%
## Simultaneous Substitution
Substitution can be generalized in the natural way to define simultaneous substitution $$[N_1/x_1, N_2/x_2, \ldots, N_n/x_n]M$$ for $n \geq 2$. As in [[equiv-trans#Substitution|equivalence-transformation]], simultaneous substitution is different from sequential substitution.
%%ANKI
Basic
How is simultaneous substitution of $N_1$ for $x_1$ and $N_2$ for $x_2$ in $M$ denoted?
Back: $[N_1/x_1, N_2/x]M$
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: 1718473252304-->
END%%
%%ANKI
Basic
How is $[N_1/x_1, N_2/x_2]M$ denoted in the equivalence-transformation system?
Back: $M_{N_1, N_2}^{x_1, x_2}$
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: 1718473252307-->
END%%
%%ANKI
Basic
How is $M_{N_1, N_2}^{x_1, x_2}$ denoted in $\lambda$-calculus?
Back: $[N_1/x_1, N_2/x_2]M$
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: 1718473252312-->
END%%
%%ANKI
Basic
Suppose $M \equiv x_1x_2$. What is the result of $[u/x_1]([x_1/x_2]M)$?
Back: $uu$
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: 1718473252309-->
END%%
%%ANKI
Basic
Suppose $M \equiv x_1x_2$. What is the result of $[u/x_1, x_1/x_2]M$?
Back: $ux_1$
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: 1718473252311-->
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).

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---
title: β-reduction
TARGET DECK: Obsidian::STEM
FILE TAGS: λ-calculus
tags:
- λ-calculus
---
## Overview
Any term of form $(\lambda x. M)N$ is called a **$\beta$-redex**. The corresponding term $[N/x]M$ is its **contractum**. If and only if a term $P$ contains an occurrence of $(\lambda x. M)N$ and we replace that occurrence by $[N/x]M$, and the result is $P'$, we say we have **contracted** the redex-occurrence in $P$, and $P$ $\beta$-contracts to $P'$ or $P \,\triangleright_{1\beta}\, P'$.
If and only if $P$ can be changed to a term $Q$ by a finite series of $\beta$-contractions and changes of bound variables, we say $P$ $\beta$-reduces to $Q$, or $P \,\triangleright_{\beta}\, Q$.
%%ANKI
Cloze
$\alpha$-{converts} is to $\beta$-{reduces}.
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: 1718475424836-->
END%%
%%ANKI
Cloze
$\alpha$-{conversion} is to $\beta$-{contraction}.
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: 1718475424840-->
END%%
%%ANKI
Cloze
"$\beta$-{contracts}" refers to exactly one contraction of a redex-occurrence.
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: 1718475424841-->
END%%
%%ANKI
Cloze
"$\beta$-{reduces}" refers to zero or more contractions of redex-occurrences.
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: 1718475424843-->
END%%
%%ANKI
Cloze
{1:$(\lambda x.M)N$} is to a {2:$\beta$-redex} whereas {2:$[N/x]M$} is to a {1:contractum}.
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: 1718475424844-->
END%%
%%ANKI
Basic
What is a $\lambda$-term of $(\lambda x.M)N$ called?
Back: A $\beta$-redex.
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: 1718475424846-->
END%%
%%ANKI
Basic
What contractum corresponds to $\beta$-redex $(\lambda x. M)N$?
Back: $[N/x]M$
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: 1718475424848-->
END%%
%%ANKI
Basic
What $\beta$-redex corresponds to contractum $[N/x]M$?
Back: $(\lambda x. M)N$
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: 1718475424849-->
END%%
%%ANKI
Basic
What does it mean to contract a redex-occurrence of $P$?
Back: A $\beta$-redex in $P$ was replaced by its contractum.
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: 1718475424850-->
END%%
%%ANKI
Basic
How do we denote "$P$ $\beta$-contracts to $Q$"?
Back: $P \,\triangleright_{1\beta}\, Q$
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: 1718475424852-->
END%%
%%ANKI
Basic
How do we denote "$P$ $\beta$-reduces to $Q$"?
Back: $P \,\triangleright_{\beta}\, Q$
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: 1718475424853-->
END%%
%%ANKI
Basic
Given $\lambda$-term $P$, is $P \,\triangleright_{1\beta}\, P$ true?
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: 1718475424855-->
END%%
%%ANKI
Basic
Given $\lambda$-term $P$, *why* isn't $P \,\triangleright_{1\beta}\, P$ true?
Back: Replacing a $\beta$-redex in $P$ with its contractum cannot again yield $P$ again.
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: 1718475424857-->
END%%
%%ANKI
Basic
Given $\lambda$-term $P$, is $P \,\triangleright_{\beta}\, P$ true?
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: 1718475424859-->
END%%
%%ANKI
Basic
Is $(\lambda x. x) \,\triangleright_{1\beta}\, (\lambda y. y)$ true?
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: 1718475424860-->
END%%
%%ANKI
Basic
Is $(\lambda x. x) \,\triangleright_{\beta}\, (\lambda y. y)$ true?
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: 1718475424862-->
END%%
%%ANKI
Basic
In what way is $\beta$-contraction a stricter operation than $\beta$-reduction?
Back: The former *requires* replacing a $\beta$-redex occurrence with its contractum.
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: 1718475424864-->
END%%
%%ANKI
Basic
In what way is $\beta$-reduction more general than $\alpha$-conversion?
Back: $\beta$-reduction involves a finite sequence of $\beta$-contractions *and* $\alpha$-conversions.
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: 1718475424865-->
END%%
%%ANKI
Basic
How do we pronounce $P \,\triangleright_{1\beta}\, Q$?
Back: $P$ $\beta$-contracts to $Q$.
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: 1718475424867-->
END%%
%%ANKI
Basic
How do we pronounce $P \,\triangleright_{\beta}\, Q$?
Back: $P$ $\beta$-reduces to $Q$.
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: 1718475424868-->
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).