DFS and top-down/bottom-up closure equivalence.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

@@ -214,7 +214,8 @@
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     "b-tree-initial.png",
     "b-tree-inserted-b.png",
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-    "set/functions.md": "3d08bbd3fb31eba419058264ed804e22",
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     "lambda-calculus/beta-reduction.md": "0935987f2bac0e6298735f2b26fd5885",
@@ -698,7 +699,7 @@
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-    "formal-system/logical-system/prop-logic.md": "b61ce051795d5a951c763b928ec5cea8",
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@@ -791,7 +792,7 @@
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-    "c17/types/simple.md": "48e455d1cb6a631fe74e419e52e0a695",
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-    "algorithms/bfs.md": "bc322d510fe25389b3996344b126e497",
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-    "x86-64/procedures.md": "c36268b99c0681bfca245e0538a302e0",
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+    "_journal/2024-10-22.md": "4af65962007cfecdb2c679b44b56d25f",
+    "algorithms/dfs.md": "5fe40544db71b5cc79f4f8149478fc8e",
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   "fields_dict": {
     "Basic": [

notes/_journal/2024-10-23.md

@@ -0,0 +1,11 @@
+---
+title: "2024-10-23"
+---
+
+- [ ] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on closures and proof bottom-up and top-down are equal.

notes/_journal/2024-10/2024-10-20.md

@@ -2,7 +2,7 @@
 title: "2024-10-20"
 ---
 
-- [ ] Anki Flashcards
+- [x] Anki Flashcards
 - [x] KoL
 - [ ] OGS
 - [ ] Sheet Music (10 min.)

notes/_journal/2024-10/2024-10-21.md

@@ -0,0 +1,11 @@
+---
+title: "2024-10-21"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* More notes on the third integral argument for x86-64 procedures.

notes/_journal/2024-10/2024-10-22.md

@@ -0,0 +1,12 @@
+---
+title: "2024-10-22"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Added notes on [[dfs|DFS]].
+* Proof on isomorphism between Peano systems and $\langle \omega, \sigma, 0 \rangle$.

notes/algebra/floor-ceiling.md

@@ -44,7 +44,7 @@ END%%
 
 %%ANKI
 Basic
-When does $\lfloor x / 2 \rfloor = \lceil x / 2 \rceil$?
+Given integer $x$, when does $\lfloor x / 2 \rfloor = \lceil x / 2 \rceil$?
 Back: When $x$ is even.
 Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
 <!--ID: 1708110779668-->
@@ -52,7 +52,7 @@ END%%
 
 %%ANKI
 Basic
-When does $\lfloor x / 2 \rfloor \neq \lceil x / 2 \rceil$?
+Given integer $x$, when does $\lfloor x / 2 \rfloor \neq \lceil x / 2 \rceil$?
 Back: When $x$ is odd.
 Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
 <!--ID: 1708110779674-->

notes/algorithms/bfs.md

@@ -9,7 +9,7 @@ tags:
 
 ## Overview
 
-Bread-first search operates on a graph $G = \langle V, E \rangle$ and a **source** vertex $s$. It works by distinguishing between discovered and undiscovered nodes, incrementally marking nodes adjacent to discovered nodes from undiscovered to discovered.
+Bread-first search operates on a graph $G = \langle V, E \rangle$ and a **source** vertex $s$.
 
 ![[bfs.gif]]
 
@@ -167,7 +167,7 @@ END%%
 %%ANKI
 Basic
 *Why* is BFS of an adjacency-list representation $O(\lvert V \rvert + \lvert E \rvert)$?
-Back: For each vertex being analyzed, we only examine its immediately adjacent vertices.
+Back: For each vertex being analyzed, we examine all of its adjacent vertices.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1727044184060-->
 END%%

notes/algorithms/dfs.md

@@ -0,0 +1,129 @@
+---
+title: Depth-First Search
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algorithm data_structure::graph
+tags:
+  - dfs
+  - graph
+---
+
+## Overview
+
+Depth-first search operates on a graph $G = \langle V, E \rangle$ and a **source** vertex $s$.
+
+![[dfs.gif]]
+
+%%ANKI
+Basic
+What is DFS an acronym for?
+Back: **D**epth-**f**irst **s**earch.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729224-->
+END%%
+
+%%ANKI
+Cloze
+Depth-first search is characterized by a graph and a {source vertex}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729228-->
+END%%
+
+%%ANKI
+Basic
+Which of undirected and directed graphs is DFS applicable to?
+Back: Both.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729231-->
+END%%
+
+%%ANKI
+Basic
+With respect to depth-first trees, what does the predecessor of a node $N$ refer to?
+Back: The node from which $N$ was discovered.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729235-->
+END%%
+
+%%ANKI
+Basic
+What ADT is typically used to manage the set of most recently discovered DFS vertices?
+Back: A stack.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729238-->
+END%%
+
+%%ANKI
+Cloze
+A {1:queue} is to {2:BFS} whereas a {2:stack} is to {1:DFS}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729242-->
+END%%
+
+%%ANKI
+Basic
+Which vertices are not discovered during a graph DFS?
+Back: Those not reachable from the source vertex.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729245-->
+END%%
+
+%%ANKI
+Basic
+What basic graph algorithm is the following a demonstration of?
+![[dfs.gif]]
+Back: Depth-first search.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729249-->
+END%%
+
+%%ANKI
+Basic
+Which standard graph representation has worst-case DFS running time of $O(\lvert V \rvert + \lvert E \rvert)$?
+Back: The adjacency-list representation.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729252-->
+END%%
+
+%%ANKI
+Basic
+Given graph $\langle V, E \rangle$ with adjacency-list representation, what is the worst-case run time of DFS?
+Back: $O(\lvert V \rvert + \lvert E \rvert)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729256-->
+END%%
+
+%%ANKI
+Basic
+Which standard graph representation has worst-case DFS running time of $O(\lvert V \rvert^2)$?
+Back: The adjacency-matrix representation.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729260-->
+END%%
+
+%%ANKI
+Basic
+Given graph $\langle V, E \rangle$ with adjacency-matrix representation, what is the worst-case run time of DFS?
+Back: $O(\lvert V \rvert^2)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729264-->
+END%%
+
+%%ANKI
+Basic
+*Why* is DFS of an adjacency-list representation $O(\lvert V \rvert + \lvert E \rvert)$?
+Back: For each vertex being analyzed, we examine all of its adjacent vertices.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729268-->
+END%%
+
+%%ANKI
+Basic
+*Why* is DFS of an adjacency-matrix representation $O(\lvert V \rvert^2)$?
+Back: For each vertex being analyzed, we must examine $\lvert V \rvert$ entries for adjacent vertices.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1729641729272-->
+END%%
+
+## Bibliography
+
+* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

notes/c17/types/simple.md

@@ -1211,7 +1211,7 @@ END%%
 %%ANKI
 Basic
 Suppose `uintN_t` exists. What is its precision?
-Back: `N`
+Back: `N` bits.
 Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
 <!--ID: 1727551341574-->
 END%%

notes/formal-system/logical-system/prop-logic.md

@@ -34,14 +34,6 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
 <!--ID: 1708199272076-->
 END%%
 
-%%ANKI
-Basic
-What two categories do propositions fall within?
-Back: Atomic and molecular propositions.
-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: 1708199272083-->
-END%%
-
 %%ANKI
 Basic
 What is an atomic proposition?

notes/set/functions.md

@@ -1708,6 +1708,164 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1726363070015-->
 END%%
 
+Let $f$ be a function from $B$ into $B$ and assume $A \subseteq B$. There are two possible methods for constructing the **closure** $C$ of $A$ under $f$. The top-down approach defines $C^*$ to be the intersection of all closed supersets of $A$: $$C^* = \bigcap\, \{X \mid A \subseteq X \subseteq B \land f[\![X]\!] \subseteq X \}$$
+
+The bottom-up approach defines $C_*$ to be $$C_* = \bigcup_{i \in \omega} h(i)$$
+where $h \colon \omega \rightarrow \mathscr{P}(B)$ is recursively defined as: $$\begin{align*} h(0) & = A, \\ h(n^+) &= h(n) \cup f[\![h(n)]\!]. \end{align*}$$
+
+Note that the [[natural-numbers#Recursion Theorem|recursion theorem]] proves $h$ is indeed a function.
+
+%%ANKI
+Basic
+Let $f \colon B \rightarrow B$ and $A \subseteq B$. How is the top-down closure $C^*$ of $A$ under $f$ defined?
+Back: $\bigcap\, \{ X \mid A \subseteq X \subseteq B \land f[\![X]\!] \subseteq X \}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379328-->
+END%%
+
+%%ANKI
+Basic
+Let $f \colon B \rightarrow B$ and $A \subseteq B$. What is the smallest set the closure $C^*$ of $A$ under $f$ can be?
+Back: $A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379332-->
+END%%
+
+%%ANKI
+Basic
+Let $f \colon B \rightarrow B$ and $A \subseteq B$. What is the largest set the closure $C^*$ of $A$ under $f$ can be?
+Back: $B$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379336-->
+END%%
+
+%%ANKI
+Basic
+Let $f \colon B \rightarrow B$ and $A \subseteq B$. How is the bottom-up closure $C_*$ of $A$ under $f$ defined assuming appropriate $h \colon \omega \rightarrow \mathscr{P}(B)$?
+Back: $\bigcup \mathop{\text{ran}} h$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379339-->
+END%%
+
+%%ANKI
+Basic
+Let $f \colon B \rightarrow B$ and $A \subseteq B$. What is the smallest set the closure $C_*$ of $A$ under $f$ can be?
+Back: $A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379342-->
+END%%
+
+%%ANKI
+Basic
+Let $f \colon B \rightarrow B$ and $A \subseteq B$. What is the largest set the closure $C_*$ of $A$ under $f$ can be?
+Back: $B$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379345-->
+END%%
+
+%%ANKI
+Basic
+Let $C$ be the closure of $A$ under $f$. What kind of mathematical entity is $A$?
+Back: A set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379348-->
+END%%
+
+%%ANKI
+Basic
+Let $C$ be the closure of $A$ under $f$. What kind of mathematical entity is $f$?
+Back: A function.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379352-->
+END%%
+
+%%ANKI
+Basic
+Let $C$ be the closure of $A$ under $f$. What kind of mathematical entity is $C$?
+Back: A set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379355-->
+END%%
+
+%%ANKI
+Basic
+Let $C$ be the closure of $A$ under $f$. What two ways can $C$ be defined?
+Back: Bottom-up or top-down.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379358-->
+END%%
+
+%%ANKI
+Basic
+Let $C$ be the closure of $A$ under $f$. How is the top-down closure denoted?
+Back: As $C^*$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379361-->
+END%%
+
+%%ANKI
+Basic
+Let $C$ be the closure of $A$ under $f$. How is the bottom-up closure denoted?
+Back: As $C_*$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379364-->
+END%%
+
+%%ANKI
+Basic
+Let $C$ be the closure of $A$ under $f$. What is the "signature" of $f$?
+Back: $f \colon B \rightarrow B$ for some $B \supseteq A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379367-->
+END%%
+
+%%ANKI
+Basic
+Let $C_*$ be the closure of $A$ under $f$ defined in terms of function $h$. What is $h$'s domain?
+Back: $\omega$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379370-->
+END%%
+
+%%ANKI
+Basic
+Let $C_*$ be the closure of $A$ under $f$ defined in terms of function $h$. What is $h$'s codomain?
+Back: Assume $f$ maps $B$ into $B$, Then $h$'s codomain is $B$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379373-->
+END%%
+
+%%ANKI
+Basic
+Let $C_*$ be the closure of $A$ under $f$ defined in terms of function $h$. What does $h(0)$ evaluate to?
+Back: $A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379378-->
+END%%
+
+%%ANKI
+Basic
+Let $C_*$ be the closure of $A$ under $f$ defined in terms of function $h$. What does $h(n^+)$ evaluate to?
+Back: $h(n) \cup f[\![h(n)]\!]$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379385-->
+END%%
+
+%%ANKI
+Basic
+Let $C_*$ be the closure of $A$ under $f$ defined in terms of function $h$. What theorem proves $h$'s existence?
+Back: The recursion theorem.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379392-->
+END%%
+
+%%ANKI
+Cloze
+The top-down closure $C^*$ of $A$ under $f$ is the {intersection} of all {closed supersets} of $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684379398-->
+END%%
+
 ## Kernels
 
 Let $F \colon A \rightarrow B$. Define [[relations#Equivalence Relations|equivalence relation]] $\sim$ as $$x \sim y \Leftrightarrow f(x) = f(y)$$

notes/set/index.md

@@ -735,6 +735,13 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1716397645564-->
 END%%
 
+%%ANKI
+Cloze
+Let $A$ be a set and $C = \bigcup\, \{ x \mid \_\_\_ \}$. Then $C$ {$\supseteq$} $A$ if $A$ satisfies the {entrance requirement}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684927439-->
+END%%
+
 ## Power Set Axiom
 
 For any set $a$, there is a set whose members are exactly the subsets of $a$: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$
@@ -988,6 +995,13 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1716309007881-->
 END%%
 
+%%ANKI
+Cloze
+Let $A$ be a set and $C = \bigcap\, \{ x \mid \_\_\_ \}$. Then $C$ {$\subseteq$} $A$ if $A$ satisfies the {entrance requirement}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1729684927446-->
+END%%
+
 %%ANKI
 Basic
 What set operation is shaded green in the following venn diagram?

notes/x86-64/procedures.md

@@ -401,6 +401,206 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1728559336879-->
 END%%
 
+%%ANKI
+Basic
+How many bytes make up the `%rdx` register?
+Back: $8$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729533668317-->
+END%%
+
+%%ANKI
+Basic
+How many bytes make up the `%dx` register?
+Back: $2$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729533668324-->
+END%%
+
+%%ANKI
+Basic
+How many bytes make up the `%dl` register?
+Back: $1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729533668328-->
+END%%
+
+%%ANKI
+Basic
+How many bytes make up the `%edx` register?
+Back: $4$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729533668334-->
+END%%
+
+%%ANKI
+Cloze
+By convention, register {`%rdx`} is used for {the third integral argument}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729533668338-->
+END%%
+
+%%ANKI
+Cloze
+{1:Words} are to {2:`%dx`} whereas {2:double words} are to {1:`%edx`}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729533668341-->
+END%%
+
+%%ANKI
+Cloze
+{1:Bytes} are to {2:`%dl`} whereas {2:quad words} are to {1:`%rdx`}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729533668345-->
+END%%
+
+%%ANKI
+Basic
+How do you access the low-order 2 bytes of `%rdx`?
+Back: By using `%dx`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729174-->
+END%%
+
+%%ANKI
+Basic
+How do you access the low-order 4 bytes of `%rdx`?
+Back: By using `%edx`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729533668349-->
+END%%
+
+%%ANKI
+Basic
+How do you access the low-order byte of `%rdx`?
+Back: By using `%dl`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729533668352-->
+END%%
+
+%%ANKI
+Basic
+Which register should the third integral argument of a procedure be placed in?
+Back: `%rdx`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729533668358-->
+END%%
+
+%%ANKI
+Basic
+From smallest to largest, list the four "third integral argument" registers.
+Back: `%dl`, `%dx`, `%edx`, and `%rdx`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729533668361-->
+END%%
+
+%%ANKI
+Cloze
+{1:`%dil`} is to the {2:first} integral argument whereas {2:`%dl`} is to the {1:third} integral argument.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729612004982-->
+END%%
+
+%%ANKI
+Basic
+How many bytes make up the `%rcx` register?
+Back: $8$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729180-->
+END%%
+
+%%ANKI
+Basic
+How many bytes make up the `%ecx` register?
+Back: $4$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729183-->
+END%%
+
+%%ANKI
+Basic
+How many bytes make up the `%cx` register?
+Back: $2$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729186-->
+END%%
+
+%%ANKI
+Basic
+How many bytes make up the `%cl` register?
+Back: $1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729189-->
+END%%
+
+%%ANKI
+Cloze
+By convention, register {`%rcx`} is used for {the fourth integral argument}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729193-->
+END%%
+
+%%ANKI
+Cloze
+{1:Words} are to {2:`%cx`} whereas {2:quad words} are to {1:`%rcx`}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729196-->
+END%%
+
+%%ANKI
+Cloze
+{1:Bytes} are to {2:`%cl`} whereas {2:double words} are to {1:`%ecx`}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729200-->
+END%%
+
+%%ANKI
+Basic
+How do you access the low-order 2 bytes of `%rcx`?
+Back: By using `%cx`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729203-->
+END%%
+
+%%ANKI
+Basic
+How do you access the low-order 4 bytes of `%rcx`?
+Back: By using `%ecx`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729207-->
+END%%
+
+%%ANKI
+Basic
+How do you access the low-order byte of `%rcx`?
+Back: By using `%cl`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729210-->
+END%%
+
+%%ANKI
+Basic
+Which register should the fourth integral argument of a procedure be placed in?
+Back: `%rcx`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729214-->
+END%%
+
+%%ANKI
+Basic
+From smallest to largest, list the four "fourth integral argument" registers.
+Back: `%cl`, `%cx`, `%ecx`, and `%rcx`.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729217-->
+END%%
+
+%%ANKI
+Cloze
+{1:`%di`} is to the {2:first} integral argument whereas {2:`%cx`} is to the {1:fourth} integral argument.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1729641729221-->
+END%%
+
 ## Bibliography
 
 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.