More notes on the recursion theorem and C loop constructs.

2024-09-29 12:07:51 -06:00 · 2024-09-29 12:07:51 -06:00 · f2781330db
parent baf5e52c0d
commit f2781330db
14 changed files with 185 additions and 50 deletions
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--- a/notes/_journal/2024-09-29.md
+++ b/notes/_journal/2024-09-29.md
@ -2,8 +2,11 @@
 title: "2024-09-29"
 ---

- [ ] Anki Flashcards
+- [x] Anki Flashcards
 - [x] KoL
 - [x] OGS
 - [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
+- [ ] Korean (Read 1 Story)
+
+* Flashcards on the proof of the recursion theorem on $\omega$.
+* Read through how different C loop constructs are compiled into assembly.
--- a/notes/algebra/sequences/delta-constant.md
+++ b/notes/algebra/sequences/delta-constant.md
@ -46,8 +46,8 @@ END%%

 %%ANKI
 Basic
-What is the recurrence of the recursive definition of the $(k + 1)$st differences of $(a_n)$?
-Back: The $(k + 1)$st differences is the differences of the $k$th differences.
+How are the $(k + 1)$st differences of $(a_n)$ defined recursively?
+Back: As the differences of the $k$th differences of $(a_n)$.
 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: 1713580109153-->
 END%%
--- a/notes/combinatorics/permutations.md
+++ b/notes/combinatorics/permutations.md
@ -104,7 +104,7 @@ END%%

 %%ANKI
 Basic
-$n!$ is shorthand for what other "big operator" formula?
+$n!$ is an abbreviation of what "big operator" formula?
 Back: $\Pi_{k=1}^n k$
 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: 1708366788594-->
--- a/notes/formal-system/logical-system/pred-logic.md
+++ b/notes/formal-system/logical-system/pred-logic.md
@ -92,7 +92,7 @@ END%%

 %%ANKI
 Basic
-$\exists x : S, P(x)$ is shorthand for what?
+$\exists x : S, P(x)$ is an abbreviation for what?
 Back: $\exists x, x \in S \land P(x)$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1707494819968-->
@ -261,7 +261,7 @@ END%%

 %%ANKI
 Basic
-$\forall x : S, P(x)$ is shorthand for what?
+$\forall x : S, P(x)$ is an abbreviation for what?
 Back: $\forall x, x \in S \Rightarrow P(x)$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1707494819976-->
--- a/notes/formal-system/proof-system/equiv-trans.md
+++ b/notes/formal-system/proof-system/equiv-trans.md
@ -1393,7 +1393,7 @@ END%%

 %%ANKI
 Basic
-What proposition represents states $\{(b, T), (c, T)\}$ and $\{(b, F), (c, F)\}$?
+What DNF proposition represents states $\{(b, T), (c, T)\}$ and $\{(b, F), (c, F)\}$?
 Back: $(b \land c) \lor (\neg b \land \neg c)$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1706994861337-->
--- a/notes/git/remotes.md
+++ b/notes/git/remotes.md
@ -30,14 +30,6 @@ Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Soft
 <!--ID: 1715623927187-->
 END%%

-%%ANKI
-Basic
-Where are git remotes specified within the `.git` directory?
-Back: In `.git/config`
-Reference: Scott Chacon, *Pro Git*, Second edition, The Expert’s Voice in Software Development (New York, NY: Apress, 2014).
-<!--ID: 1715623927191-->
-END%%
-
 %%ANKI
 Basic
 What kind of git refs are associated with remotes?
--- a/notes/lambda-calculus/index.md
+++ b/notes/lambda-calculus/index.md
@ -370,7 +370,7 @@ END%%

 %%ANKI
 Cloze
-The phrase "{induction on $M$}" is shorthand for phrase "{induction on $lgh(M)$}".
+The phrase "{induction on $M$}" is an abbrevation of phrase "{induction on $lgh(M)$}".
 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: 1716743248113-->
 END%%
--- a/notes/set/functions.md
+++ b/notes/set/functions.md
@ -1747,7 +1747,7 @@ END%%

 %%ANKI
 Basic
-Let $F \colon A \rightarrow B$. Term "$\mathop{\text{coim}}F$" is shorthand for what?
+Let $F \colon A \rightarrow B$. Term "$\mathop{\text{coim}}F$" is an abbreviation for what?
 Back: The **coim**age of $F$.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1721223015596-->
--- a/notes/set/natural-numbers.md
+++ b/notes/set/natural-numbers.md
@ -850,6 +850,14 @@ Reference: “Recursion,” in _Wikipedia_, September 23, 2024, [https://en.wiki
 <!--ID: 1727492422625-->
 END%%

+%%ANKI
+Basic
+The recursion theorem assumes existence of what Peano system?
+Back: $\langle \omega, \sigma, 0 \rangle$ where $\sigma$ is the successor operation restricted to the natural numbers.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727629020357-->
+END%%
+
 %%ANKI
 Basic
 What entities does the recursion theorem presume the existence of?
@ -906,14 +914,6 @@ Reference: “Recursion,” in _Wikipedia_, September 23, 2024, [https://en.wiki
 <!--ID: 1727492422693-->
 END%%

-%%ANKI
-Basic
-The recursion theorem proves $h$ exists. What kind of mathematical entity is $h$?
-Back: A function.
-Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
-<!--ID: 1727492422702-->
-END%%
-
 %%ANKI
 Basic
 The recursion theorem proves function $h$ exists. What is the domain of $h$?
@ -938,6 +938,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1727492422716-->
 END%%

+%%ANKI
+Basic
+The recursion theorem shows existence of $h \colon \omega \rightarrow A$. What is $A$?
+Back: A set fixed before application of the recursion theorem.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727629020364-->
+END%%
+
 %%ANKI
 Basic
 Let $a \in A$ and $F \colon A \rightarrow A$. Using the recursion theorem, how else is $F(F(F(F(a))))$ expressed?
@ -954,6 +962,107 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1727492422724-->
 END%%

+%%ANKI
+Basic
+In Enderton's recursion theorem proof, function $h \colon \omega \rightarrow A$ is defined as the union of what?
+Back: All "acceptable" functions.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727627702457-->
+END%%
+
+%%ANKI
+Basic
+In Enderton's recursion theorem proof, what is the domain of an acceptable function?
+Back: A subset of $\omega$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727627702459-->
+END%%
+
+%%ANKI
+Basic
+In Enderton's recursion theorem proof, what is the codomain of an acceptable function?
+Back: A subset of some fixed set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727627702461-->
+END%%
+
+%%ANKI
+Basic
+In Enderton's recursion theorem proof, what follows if $0 \in \mathop{\text{dom}} v$ for acceptable function $v$?
+Back: $v(0) = a$ for some fixed $a \in A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727627702462-->
+END%%
+
+%%ANKI
+Basic
+In Enderton's recursion theorem proof, what follows if $n^+ \in \mathop{\text{dom}} v$ for acceptable function $v$?
+Back: $n \in \mathop{\text{dom}} v$ and $v(n^+) = F(v(n))$ for some fixed $F \colon A \rightarrow A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727627702464-->
+END%%
+\
+%%ANKI
+Basic
+In Enderton's recursion theorem proof, what term refers to the "approximating" functions?
+Back: They are called "acceptable".
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727627702465-->
+END%%
+
+%%ANKI
+Cloze
+In Enderton's recursion theorem proof, desired $h \colon \omega \rightarrow A$ is defined as $\bigcup$ {$\{ v \mid v \text{ is acceptable} \}$}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727627702466-->
+END%%
+
+%%ANKI
+Cloze
+In Enderton's recursion theorem proof, desired $h \colon \omega \rightarrow A$ is {a function} because {$\{ n \in \omega \mid \text{ at most one } y \text{ such that } \langle n, y \rangle \in h \}$} is {an inductive set}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727627702468-->
+END%%
+
+%%ANKI
+Basic
+In Enderton's recursion theorem proof, how is it shown the domain of desired $h \colon \omega \rightarrow A$ equals $\omega$?
+Back: By proving $\mathop{\text{dom}} h$ is an inductive set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727627702469-->
+END%%
+
+%%ANKI
+Cloze
+In Enderton's recursion theorem proof,desired $h \colon \omega \rightarrow A$ is {unique} because {$\{ n \in \omega \mid h_1(n) = h_2(n) \}$} is {an inductive set}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727627702470-->
+END%%
+
+%%ANKI
+Basic
+*Why* is there no function $h \colon \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all $n \in \mathbb{Z}$, $$\begin{align*} h(0) & = 0 \\ h(n + 1) & = h(n) + 1 \end{align*}$$
+Back: Because $\mathbb{Z}$ has no "starting point" to ground the recursive definition.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727629020369-->
+END%%
+
+%%ANKI
+Basic
+*Why* is there no function $h \colon \mathbb{N} \rightarrow \mathbb{N}$ such that for all $n \in \mathbb{N}$$, $$\begin{align*} h(0) & = 0 \\ h(n + 1) & = h(n) + 1 \end{align*}$$
+Back: N/A. The resursive theorem of $\omega$ states such an $h$ exists.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727629020375-->
+END%%
+
+%%ANKI
+Basic
+In what natural way could we generalize the recursion theorem on $\omega$?
+Back: By stating the theorem in terms of arbitrary Peano systems.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727629232445-->
+END%%
+
 ## Bibliography

 * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
--- a/notes/set/order.md
+++ b/notes/set/order.md
@ -350,8 +350,8 @@ A binary relation $R$ on set $A$ is an **equivalence relation on $A$** iff it is

 %%ANKI
 Basic
-Given $R = \{\langle a, a \rangle, \langle b, b \rangle\}$, which of reflexivity (on $\{a, b\}$), symmetry, and transitivity does $R$ exhibit?
-Back: Reflexivity on $\{a, b\}$ and symmetry.
+Given $R = \{\langle a, a \rangle, \langle b, b \rangle\}$, which of reflexivity, symmetry, and/or transitivity does $R$ exhibit?
+Back: All three.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1720967429839-->
 END%%
@ -562,8 +562,8 @@ END%%

 %%ANKI
 Basic
-Let $\Pi$ be a partition of set $A$. What property must each *individual* member of $\Pi$ exhibit?
-Back: Each member is nonempty.
+Let $\Pi$ be a partition of set $A$. What two properties must each *individual* member of $\Pi$ exhibit?
+Back: Each member is a nonempty subset of $A$.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
 <!--ID: 1721098094065-->
 END%%
--- a/notes/startups/financing-rounds.md
+++ b/notes/startups/financing-rounds.md
@ -377,7 +377,7 @@ END%%

 %%ANKI
 Cloze
-An {angel} is a shorthand term for an {angel investor}.
+An {angel} is an abbreviated term for an {angel investor}.
 Reference: Brad Feld and Jason Mendelson, _Venture Deals_, 3rd ed., n.d.
 <!--ID: 1719360934700-->
 END%%
--- a/notes/x86-64/instructions/arithmetic.md
+++ b/notes/x86-64/instructions/arithmetic.md
@ -98,20 +98,20 @@ Assume `%rbx` holds $p$. What is the value of `%rax` in the following?
 ```asm
 leaq 2(%rbx, %rbx, 7),%rax
 ```
-Back: $2 + 8p$
+Back: N/A. A scaling factor of $7$ is not allowed.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1715781031938-->
+<!--ID: 1727629670917-->
 END%%

 %%ANKI
 Basic
 Assume `%rdx` holds $q$. What is the value of `%rax` in the following?
 ```asm
-leaq 0xE(, %rdx, 3),%rax
+leaq 0xE(, %rdx, 4),%rax
 ```
-Back: $14 + 3q$
+Back: $14 + 4q$
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1715781031941-->
+<!--ID: 1727629670921-->
 END%%

 ### Unary Operations
--- a/notes/x86-64/instructions/conditions.md
+++ b/notes/x86-64/instructions/conditions.md
@ -898,6 +898,37 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1727357091109-->
 END%%

+%%ANKI
+Basic
+In most cases, how can the following `for` loop be rewritten as a `while` loop?
+```c
+for (init; test; update) { body; }
+```
+Back:
+```c
+init;
+while (test) { body; update }
+```
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1727632288464-->
+END%%
+
+%%ANKI
+Basic
+In what situation are the following two blocks of code not equivalent?
+```c
+for (init; test; update) { body; }
+
+init;
+while (test) { body; update }
+```
+Back: Situations in which the `for`-loop has a `continue` statement in the `body`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1727632288468-->
+END%%
+
 ### CMOV

 Like [[access#MOV|MOV]] instructions, but with the data transfer only happening if the move condition is satisfied.