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Macros and the recursion theorem.

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Joshua Potter 2024-09-27 21:02:33 -06:00
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---
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 $\omega$).

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

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@ -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).

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---
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).

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@ -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-->

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@ -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-->

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@ -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).

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@ -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).
* “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).