Macros and the recursion theorem.

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     "Basic": [

notes/_journal/2024-09-27.md

@@ -0,0 +1,12 @@
+---
+title: "2024-09-27"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on [[macros]]. Read Gustedt's take on signed/unsigned integer binary representations.
+* Initial notes of the [[natural-numbers#Recursion Theorem|recursion theorem]] (on $\omega$).

notes/_journal/2024-09/2024-09-26.md

@@ -2,7 +2,7 @@
 title: "2024-09-26"
 ---
 
-- [ ] Anki Flashcards
+- [x] Anki Flashcards
 - [x] KoL
 - [x] OGS
 - [ ] Sheet Music (10 min.)

notes/c17/index.md

@@ -356,14 +356,6 @@ Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co
 <!--ID: 1723856661371-->
 END%%
 
-%%ANKI
-Basic
-What is the binary representation of a value?
-Back: N/A. Binary representations describe types not values.
-Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
-<!--ID: 1723856661379-->
-END%%
-
 %%ANKI
 Basic
 What is the object representation of a type?
@@ -394,6 +386,14 @@ Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co
 <!--ID: 1723856661405-->
 END%%
 
+%%ANKI
+Basic
+Why might the same value have different binary representations?
+Back: Because the binary representation corresponds to the type of the value.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432711873-->
+END%%
+
 ## Bibliography
 
 * “ISO: Programming Languages - C,” April 12, 2011, [https://port70.net/~nsz/c/c11/n1570.pdf](https://port70.net/~nsz/c/c11/n1570.pdf).

notes/c17/macros.md

@@ -0,0 +1,159 @@
+---
+title: Macros
+TARGET DECK: Obsidian::STEM
+FILE TAGS: c17::macro
+tags:
+  - c17
+---
+
+## Overview
+
+Macros refer to `#define` directives that specify terms that should be textually replaced by the preprocessor during compilation:
+
+```c
+#define NAME ...
+```
+
+For types that don't have literals that describe their constants, we can use **compound literals** on the replacement side of the macro:
+
+```c
+#define NAME (T){ INIT }
+```
+
+%%ANKI
+Basic
+What preprocessor directive is used to define macros?
+Back: `#define`
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419429-->
+END%%
+
+%%ANKI
+Basic
+How are compound literals specified in a macro definition, say `MACRO`?
+Back:
+```c
+#define MACRO (T){ INIT }
+```
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419447-->
+END%%
+
+%%ANKI
+Basic
+What term is used to refer to the replacement side of the following macro?
+```c
+#define MACRO (T){ INIT }
+```
+Back: A compound literal.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419481-->
+END%%
+
+%%ANKI
+Basic
+What is the difference between the following two lines?
+```c
+#define MACRO (T){ INIT }
+# define MACRO (T){ INIT }
+```
+Back: N/A. They are equivalent.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419485-->
+END%%
+
+%%ANKI
+Basic
+What is the difference between the following two lines?
+```c
+#define MACRO (T){ INIT }
+#define MACRO(T){ INIT }
+```
+Back: The first defines a compound literal. The latter defines a function-like macro.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419489-->
+END%%
+
+%%ANKI
+Basic
+What is `T` a reference to in the following compound literal?
+```c
+#define MACRO (T){ INIT }
+```
+Back: A type-specifier.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419492-->
+END%%
+
+%%ANKI
+Basic
+What is `INIT` a reference to in the following compound literal?
+```c
+#define MACRO (T){ INIT }
+```
+Back: An initializer.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419495-->
+END%%
+
+%%ANKI
+Basic
+Why aren't compound literals suitable for ICE?
+Back: They are objects, not constants.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419498-->
+END%%
+
+%%ANKI
+Basic
+How can the following be rewritten so that `MACRO` is an object?
+```c
+#define MACRO 5
+```
+Back:
+```c
+#define MACRO (int){5}
+```
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419500-->
+END%%
+
+%%ANKI
+Basic
+What is the difference between the following two lines?
+```c
+#define MACRO 5
+#define MACRO (int){5}
+```
+Back: The former is a literal whereas the latter is a compound literal.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419503-->
+END%%
+
+%%ANKI
+Basic
+Why should compound literals be, generally speaking, `const`-qualified?
+Back: Doing so gives the optimizer more room to generate good binary code.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419506-->
+END%%
+
+%%ANKI
+Basic
+How do we write macro definitions that span more than one line?
+Back: Ending all but the last line with a `\` character.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419508-->
+END%%
+
+%%ANKI
+Basic
+Generally speaking, what character should *not* be specified at the end of a macro definition?
+Back: `;`
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727432419511-->
+END%%
+
+## Bibliography
+
+* Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).

notes/formal-system/proof-system/natural-deduction.md

@@ -46,7 +46,7 @@ END%%
 
 %%ANKI
 Basic
-How are natural deduction's inference rules categorized into two?
+With respect to propositional logic, how are natural deduction's inference rules divided into two categories?
 Back: As introduction and elimination rules.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1721655978499-->
@@ -54,7 +54,7 @@ END%%
 
 %%ANKI
 Basic
-With respect to propositional logic, how are natural deduction's inference rules categorized into five?
+With respect to propositional logic, how are natural deduction's inference rules divided into five categories?
 Back: As an introduction and elimination rule per propositional logic operator.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1721655978506-->

notes/hashing/addressing.md

@@ -611,7 +611,7 @@ END%%
 
 %%ANKI
 Basic
-Consider open addressed hash table with $m$ slots. What condition must every probe sequence satisfy?
+Consider an open addressed hash table with $m$ slots. What condition must every probe sequence satisfy?
 Back: Each sequence must be a permutation of $\langle 0, 1, \ldots, m - 1 \rangle$.
 Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
 <!--ID: 1722080563937-->

notes/posix/signals.md

@@ -245,7 +245,16 @@ Tags: c17
 <!--ID: 1709131892349-->
 END%%
 
+%%ANKI
+Basic
+What does it mean for a program to (perform a) trap?
+Back: It is terminated abruptly before its usual end.
+Reference: Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).
+<!--ID: 1727433781278-->
+END%%
+
 ## Bibliography
 
 * Cooper, Mendel. “Advanced Bash-Scripting Guide,” n.d., 916.
 * Dowling, “A List of Signals and What They Mean.”
+* Jens Gustedt, _Modern C_ (Shelter Island, NY: Manning Publications Co, 2020).

notes/set/natural-numbers.md

@@ -838,6 +838,123 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1727019806554-->
 END%%
 
+## Recursion Theorem
+
+The recursion theorem guarantees recursively defined functions exist. More formally, let $A$ be a set, $a \in A$, and $F \colon A \rightarrow A$. Then there exists a unique function $h \colon \omega \rightarrow A$ such that, for every $n \in \omega$, $$\begin{align*} h(0) & = a \\ h(n^+) & = F(h(n)) \end{align*}$$
+
+%%ANKI
+Basic
+*Why* is the recursion theorem important?
+Back: It guarantees recursively defined functions exist.
+Reference: “Recursion,” in _Wikipedia_, September 23, 2024, [https://en.wikipedia.org/w/index.php?title=Recursion#The_recursion_theorem](https://en.wikipedia.org/w/index.php?title=Recursion&oldid=1247328220#The_recursion_theorem).
+<!--ID: 1727492422625-->
+END%%
+
+%%ANKI
+Basic
+What entities does the recursion theorem presume the existence of?
+Back: A set $A$, an element $a \in A$, and a function $F \colon A \rightarrow A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727492422632-->
+END%%
+
+%%ANKI
+Basic
+Let $a \in A$ and $F \colon A \rightarrow A$. The recursion theorem implies existence of what?
+Back: A unique function $h \colon \omega \rightarrow A$ such that $h(0) = a$ and $h(n^+) = F(h(n))$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727492422636-->
+END%%
+
+%%ANKI
+Basic
+What function "signature" is considered in the consequent of the recursion theorem?
+Back: $h \colon \omega \rightarrow A$ for some set $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727492422666-->
+END%%
+
+%%ANKI
+Basic
+What function "signature" is considered in the antecedent of the recursion theorem?
+Back: $F \colon A \rightarrow A$ for some set $A$ and function $F$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727492422673-->
+END%%
+
+%%ANKI
+Basic
+Suppose the recursion theorem proves $h \colon \omega \rightarrow A$ exists. What does $h(0)$ equal?
+Back: A fixed member of $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727492422679-->
+END%%
+
+%%ANKI
+Basic
+The recursion theorem proves $h \colon \omega \rightarrow A$ exists. What does $h(n^+)$ equal?
+Back: $F(h(n))$ for a fixed $F \colon A \rightarrow A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727492422685-->
+END%%
+
+%%ANKI
+Basic
+*Why* is the recursion theorem named the way it is?
+Back: It guarantees recursively defined functions exist.
+Reference: “Recursion,” in _Wikipedia_, September 23, 2024, [https://en.wikipedia.org/w/index.php?title=Recursion#The_recursion_theorem](https://en.wikipedia.org/w/index.php?title=Recursion&oldid=1247328220#The_recursion_theorem).
+<!--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 $h$?
+Back: $\omega$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727492422707-->
+END%%
+
+%%ANKI
+Basic
+The recursion theorem proves function $h$ exists. What is the codomain of $h$?
+Back: A fixed set.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727492422711-->
+END%%
+
+%%ANKI
+Basic
+The recursion theorem proves $h \colon \omega \rightarrow A$ exists. How do we compute $h(n)$?
+Back: By applying $F$ to a fixed initial element $n$ times.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727492422716-->
+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?
+Back: The recursion theorem implies existence of $h \colon \omega \rightarrow A$ satisfying $h(4) = F(F(F(F(a))))$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727492422721-->
+END%%
+
+%%ANKI
+Basic
+Which theorem in set theory implies existence of recursively defined functions?
+Back: The recursion theorem (on $\omega$).
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1727492422724-->
+END%%
+
 ## Bibliography
 
-* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+* “Recursion,” in _Wikipedia_, September 23, 2024, [https://en.wikipedia.org/w/index.php?title=Recursion#The_recursion_theorem](https://en.wikipedia.org/w/index.php?title=Recursion&oldid=1247328220#The_recursion_theorem).