r
/
notebook
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Predicate transformers, set axioms, `leaq`.

c-declarations
Joshua Potter 2024-05-15 07:59:08 -06:00
parent b865693d9d
commit 5eb338be7e
14 changed files with 785 additions and 231 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

21
notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
View File

@ -326,7 +326,7 @@
"_journal/2024-03-22.md": "8da8cda07d3de74f7130981a05dce254",
"_journal/2024-03/2024-03-21.md": "cd465f71800b080afa5c6bdc75bf9cd3",
"x86-64/declarations.md": "75bc7857cf2207a40cd7f0ee056af2f2",
"x86-64/instructions.md": "eef92b487920c7a15f38955c97a48a2d",
"x86-64/instructions.md": "c447f09c6b0ad504726c3232ff5871b4",
"git/refs.md": "e20c2c9b14ba6c2bd235416017c5c474",
"set/trees.md": "0d21b947917498f107da140cc9fb93a7",
"_journal/2024-03-24.md": "1974cdb9fc42c3a8bebb8ac76d4b1fd6",
@ -390,12 +390,12 @@
"_journal/2024-04-28.md": "46726bf76a594b987c63ba8b9b6d13d3",
"_journal/2024-04/2024-04-27.md": "b0f3753821c232bf819b00fb49415bd0",
"_journal/2024-04/2024-04-26.md": "3ce37236a9e09e74b547a4f7231df5f0",
"algorithms/sorting/heapsort.md": "c5641af1431021751cba9bef96ee30bb",
"algorithms/sorting/heapsort.md": "d4f5a4d023b7a2e55035a8812acc71b6",
"_journal/2024-04-29.md": "7888f4e9497c9d8bd6f4aa759d9abc4d",
"_journal/2024-04/2024-04-28.md": "b34a9fe3bccb1f224b96ca00e78ad061",
"programming/assertions.md": "d663ab31fc7e7296b6636edbffe9d8c7",
"programming/text-sub.md": "4bf5287353fc6571dd15be8f9f509bee",
"programming/equiv-trans.md": "fbadd593688e364ea92f74f23e54bcfc",
"programming/text-sub.md": "4ffcaf134858b478ffc3087b58026ee8",
"programming/equiv-trans.md": "77b02845208902d55d3048bc05584d6e",
"programming/index.md": "bb082325e269a95236aa6aff9307fe59",
"_journal/2024-04-30.md": "369f98b9d91de89cc1f4f581bc530c0d",
"_journal/2024-04/2024-04-29.md": "b4fa2fd62e1b4fe34c1f71dc1e9f5b0b",
@ -404,7 +404,7 @@
"_journal/2024-05-01.md": "959ff67fe3db585ba6a7b121d853bbac",
"_journal/2024-05-02.md": "d7d6ba7e065d807986f0bd77281c0bb1",
"data-structures/priority-queues.md": "8c5c6bf62b1a39d8f1f72b800fcb17ff",
"data-structures/heaps.md": "0317372dcae8d81e4345c3ffa5c0cf90",
"data-structures/heaps.md": "a0289574930328d422b9b53047936274",
"data-structures/index.md": "2605977fad54956b5dc2d8dda9be2b10",
"abstract-data-types/priority-queues.md": "d3dad736cb05c47bdc93c52a3a4af083",
"abstract-data-types/index.md": "6b110b20393c561497629ca4c3e09472",
@ -427,9 +427,16 @@
"_journal/2024-05/2024-05-08.md": "0f1b1b9e2abcf3203b511b9e034e86f4",
"_journal/2024-05/2024-05-07.md": "4b1dde039a251f9a6dc7e606de98616d",
"_journal/2024-05/2024-05-06.md": "bc9306348b7063b87741768391d9d8a7",
"_journal/2024-05-13.md": "2dfa53ff061ba94fcbe45261aad6bb30",
"_journal/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b",
"_journal/2024-05/2024-05-12.md": "ca9f3996272152ef89924bb328efd365",
"git/remotes.md": "2208e34b3195b6f1ec041024a66fb38b"
"git/remotes.md": "2208e34b3195b6f1ec041024a66fb38b",
"programming/pred-trans.md": "241dc727a7a1cb5c899e5ddc7abb92fb",
"set/axioms.md": "8459540f07443284f80e8fb112c69d0a",
"_journal/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5",
"_journal/2024-05/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b",
"x86-64/registers.md": "be5e22ceb14084c34f446dd91702477b",
"_journal/2024-05-15.md": "17e60d1515be44ac4121eb86060ec4dd",
"_journal/2024-05/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5"
},
"fields_dict": {
"Basic": [

12
notes/_journal/2024-05-13.md
View File

@ -1,12 +0,0 @@
---
title: "2024-05-13"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
* Notes on [[graphs#Subgraphs|subgraphs]] and induced subgraphs.
* Notes on [[remotes]].

11
notes/_journal/2024-05-15.md Normal file
View File

@ -0,0 +1,11 @@
---
title: "2024-05-15"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
* Finished MOV instruction class practice problems in "Computer Systems: A Programmer's Perspective". Also notes on `leaq`.

14
notes/_journal/2024-05/2024-05-13.md Normal file
View File

@ -0,0 +1,14 @@
---
title: "2024-05-13"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
* Notes on [[graphs#Subgraphs|subgraphs]] and induced subgraphs.
* Notes on [[remotes]].
* Read through chapter 7 of "The Science of Programming", touching on the $wp$ predicate transformer.
* Read chapter 1 of "Elements of Set Theory". Made some progress on chapter 2 which touches on the basic axiomatic foundations.

9
notes/_journal/2024-05/2024-05-14.md Normal file
View File

@ -0,0 +1,9 @@
---
title: "2024-05-14"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)

4
notes/algorithms/sorting/heapsort.md
View File

@ -184,7 +184,7 @@ END%%
%%ANKI
Basic
What is initialization of `HEAPSORT`'s loop invariant?
What is initialization of `HEAPSORT`'s extraction-based loop invariant?
Back: The input array is a max-heap.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714410566845-->
@ -192,7 +192,7 @@ END%%
%%ANKI
Basic
What is maintenance of `HEAPSORT`'s loop invariant?
What is maintenance of `HEAPSORT`'s extraction-based loop invariant?
Back: Swap the root with the last position of the heap. Heapify the new root.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714410566846-->

4
notes/data-structures/heaps.md
View File

@ -289,7 +289,7 @@ END%%
%%ANKI
Basic
How many internal nodes precede the first external node of a heap of size $n$?
How many internal nodes precede the first external node of a binary heap of size $n$?
Back: $\lfloor n / 2 \rfloor$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714403425296-->
@ -297,7 +297,7 @@ END%%
%%ANKI
Basic
What is the height of a binary heap defined?
How is the height of a binary heap defined?
Back: The height of the heap's root when viewed as a complete binary tree.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714403425300-->

130
notes/programming/assertions.md
View File

@ -1,130 +0,0 @@
---
title: Assertions
TARGET DECK: Obsidian::STEM
FILE TAGS: programming::assertions
tags:
- assertions
- programming
---
## Overview
Define $\{Q\}\; S\; \{R\}$ as the predicate:
> If execution of $S$ is begun in a state satisfying $Q$, then it is guaranteed to terminate in a finite amount of time in a state satisfying $R$.
%%ANKI
Basic
*What* is $Q$ in predicate $\{Q\}\; S\; \{R\}$?
Back: A predicate.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640219-->
END%%
%%ANKI
Basic
What name is given to $Q$ in $\{Q\}\; S\; \{R\}$?
Back: The precondition of $S$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640222-->
END%%
%%ANKI
Basic
*What* is $R$ in predicate $\{Q\}\; S\; \{R\}$?
Back: A predicate.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640224-->
END%%
%%ANKI
Basic
What name is given to $R$ in $\{Q\}\; S\; \{R\}$?
Back: The postcondition of $S$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640226-->
END%%
%%ANKI
Basic
*What* is $S$ in predicate $\{Q\}\; S\; \{R\}$?
Back: A program (a sequence of statements).
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640227-->
END%%
%%ANKI
Basic
What is the antecedent of $\{Q\}\; S\; \{R\}$ in English?
Back: $S$ is executed in a state satisfying $Q$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640229-->
END%%
%%ANKI
Basic
What is the consequent of $\{Q\}\; S\; \{R\}$ in English?
Back: $S$ terminates in a finite amount of time in a state satisfying $R$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640231-->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ defined?
Back: If $S$ is executed in a state satisfying $Q$, it terminates in a finite amount of time in a state satisfying $R$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640232-->
END%%
%%ANKI
Basic
How is $\{x = X \land y = Y\}\; swap\; \{x = Y \land y = X\}$ rewritten without free identifiers?
Back: $\forall x, y, X, Y, \{x = X \land y = Y\}\; swap\; \{x = Y \land y = X\}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640234-->
END%%
%%ANKI
Basic
What name is given to $X$ in e.g. $\{x = X\}\; S\; \{y = Y\}$?
Back: The initial value of $x$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640235-->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ augmented so that $x$ has initial value $X$?
Back: $\{Q \land x = X\}\; S\; \{R\}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640237-->
END%%
%%ANKI
Basic
What name is given to $Y$ in e.g. $\{x = X\}\; S\; \{y = Y\}$?
Back: The final value of $y$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640238-->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ augmented so that $y$ has final value $X$?
Back: $\{Q\}\; S\; \{R \land y = X\}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640240-->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ augmented so that $y$ has initial value $X$?
Back: $\{Q \land y = X\}\; S\; \{R\}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640241-->
END%%
## Bibliography
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

16
notes/programming/equiv-trans.md
View File

@ -153,6 +153,14 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
<!--ID: 1706994861343-->
END%%
%%ANKI
Basic
If $p \Rightarrow q$, which of $p$ or $q$ is considered stronger?
Back: $p$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869202-->
END%%
%%ANKI
Basic
When is $p$ weaker than $q$?
@ -161,6 +169,14 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
<!--ID: 1706994861346-->
END%%
%%ANKI
Basic
If $p \Rightarrow q$, which of $p$ or $q$ is considered weaker?
Back: $q$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869207-->
END%%
%%ANKI
Basic
A proposition is well-defined with respect to what?

258
notes/programming/pred-trans.md Normal file
View File

@ -0,0 +1,258 @@
---
title: Predicate Transformers
TARGET DECK: Obsidian::STEM
FILE TAGS: programming::pred-trans
tags:
- pred_trans
- programming
---
## Overview
Define $\{Q\}\; S\; \{R\}$ as the predicate:
> If execution of $S$ is begun in a state satisfying $Q$, then it is guaranteed to terminate in a finite amount of time in a state satisfying $R$.
%%ANKI
Basic
*What* is $Q$ in predicate $\{Q\}\; S\; \{R\}$?
Back: A predicate.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640219-->
END%%
%%ANKI
Basic
What name is given to $Q$ in $\{Q\}\; S\; \{R\}$?
Back: The precondition of $S$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640222-->
END%%
%%ANKI
Basic
*What* is $R$ in predicate $\{Q\}\; S\; \{R\}$?
Back: A predicate.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640224-->
END%%
%%ANKI
Basic
What name is given to $R$ in $\{Q\}\; S\; \{R\}$?
Back: The postcondition of $S$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640226-->
END%%
%%ANKI
Basic
*What* is $S$ in predicate $\{Q\}\; S\; \{R\}$?
Back: A program (a sequence of statements).
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640227-->
END%%
%%ANKI
Basic
What is the antecedent of $\{Q\}\; S\; \{R\}$ in English?
Back: $S$ is executed in a state satisfying $Q$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640229-->
END%%
%%ANKI
Basic
What is the consequent of $\{Q\}\; S\; \{R\}$ in English?
Back: $S$ terminates in a finite amount of time in a state satisfying $R$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640231-->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ defined?
Back: If $S$ is executed in a state satisfying $Q$, it eventually terminates in a state satisfying $R$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640232-->
END%%
%%ANKI
Basic
How is $\{x = X \land y = Y\}\; swap\; \{x = Y \land y = X\}$ rewritten without free identifiers?
Back: $\forall x, y, X, Y, \{x = X \land y = Y\}\; swap\; \{x = Y \land y = X\}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640234-->
END%%
%%ANKI
Basic
What name is given to $X$ in e.g. $\{x = X\}\; S\; \{y = Y\}$?
Back: The initial value of $x$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640235-->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ augmented so that $x$ has initial value $X$?
Back: $\{Q \land x = X\}\; S\; \{R\}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640237-->
END%%
%%ANKI
Basic
What name is given to $Y$ in e.g. $\{x = X\}\; S\; \{y = Y\}$?
Back: The final value of $y$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640238-->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ augmented so that $y$ has final value $X$?
Back: $\{Q\}\; S\; \{R \land y = X\}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640240-->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ augmented so that $y$ has initial value $X$?
Back: $\{Q \land y = X\}\; S\; \{R\}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640241-->
END%%
%%ANKI
Basic
*Why* is $\{T\}\; \text{while }T\text{ do skip}\; \{T\}$ everywhere false?
Back: Because $\text{while }T\text{ do skip}$ never terminates.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869132-->
END%%
## Weakest Precondition
For any command $S$ and predicate $R$, we define the **weakest precondition** of $S$ with respect to $R$, denoted $wp(S, R)$, as
> the set of *all* states such that execution of $S$ begun in any one of them is guaranteed to terminate in a finite amount of time in a state satisfying $R$.
Expression $\{Q\}\; S\; \{R\}$ is equivalent to $Q \Rightarrow wp(S, R)$.
%%ANKI
Basic
What is the predicate transformer $wp$ an acronym for?
Back: The **w**eakest **p**recondition.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869137-->
END%%
%%ANKI
Basic
Given command $S$ and predicate $R$, how is $wp(S, R)$ defined?
Back: As the set of *all* states such that execution of $S$ in any one of them eventually terminates in a state satisfying $R$.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869141-->
END%%
%%ANKI
Basic
In terms of implications, how does a precondition compare to the weakest precondition?
Back: A precondition implies the weakest precondition but not the other way around.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869144-->
END%%
%%ANKI
Basic
In terms of sets of states, how does a precondition compare to the weakest precondition?
Back: A precondition represents a subset of the states the weakest precondition represents.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869148-->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ equivalently written as a predicate involving $wp$?
Back: $Q \Rightarrow wp(S, R)$
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869153-->
END%%
%%ANKI
Basic
How is $Q \Rightarrow wp(S, R)$ equivalently written as a predicate using assertions?
Back: $\{Q\}\; S\; \{R\}$
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869157-->
END%%
%%ANKI
Basic
What kind of mathematical object is the $wp$ transformer?
Back: A function.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869161-->
END%%
%%ANKI
Basic
Given command $S$ and predicate $R$, what kind of mathematical object is $wp(S, R)$?
Back: A set (of states).
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869165-->
END%%
%%ANKI
Basic
What does the term "predicate transformer" refer to?
Back: A function that transforms one predicate into another.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869170-->
END%%
%%ANKI
Basic
What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = y)$$
Back: $y \geq x$
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869174-->
END%%
%%ANKI
Basic
What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = y - 1)$$
Back: $F$
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869179-->
END%%
%%ANKI
Basic
What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = y + 1)$$
Back: $x = y + 1$
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869184-->
END%%
%%ANKI
Basic
What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = max(x, y))$$
Back: $T$
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869188-->
END%%
%%ANKI
Basic
Given command $S$, how is $wp(S, T)$ interpreted?
Back: As the set of all states such that execution of $S$ in any of them terminates in a finite amount of time.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869196-->
END%%
## Bibliography
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

6
notes/programming/text-sub.md
View File

@ -445,15 +445,15 @@ END%%
%%ANKI
Basic
Given valid expression $(b; [i]{\circ}s{:}e)$, what is the type of $b$?
Back: A function.
Back: A function (an array).
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714395640896-->
END%%
%%ANKI
Basic
Given valid expression $(b; \epsilon{\circ}s{:}e)$, what is the type of $b$?
Back: A scalar or function.
Given valid expression $(b; \epsilon{:}e)$, what is the type of $b$?
Back: A function or scalar.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714395640898-->
END%%

260
notes/set/axioms.md Normal file
View File

@ -0,0 +1,260 @@
---
title: Axioms
TARGET DECK: Obsidian::STEM
FILE TAGS: set
tags:
- set
---
## Overview
Enderton describes ten different axioms in total which serve as the foundation of our set theory.
## Extensionality
If two sets have exactly the same members, then they are equal: $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
%%ANKI
Basic
What does the extensionality axiom state?
Back: If two sets have exactly the same members, then they are equal.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649069247-->
END%%
%%ANKI
Basic
How is the extensionality axiom expressed using first-order logic?
Back: $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649734312-->
END%%
%%ANKI
Basic
The following encodes which set theory axiom? $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
Back: The extensionality axiom.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649069254-->
END%%
%%ANKI
Basic
How many sets exist with no members?
Back: Exactly one.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649069256-->
END%%
%%ANKI
Basic
Which set theory axiom proves uniqueness of $\varnothing$?
Back: The extensionality axiom.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649069259-->
END%%
## Empty Set Axiom
There exists a set having no members: $$\exists B, \forall x, x \not\in B$$
%%ANKI
Basic
What does the empty set axiom state?
Back: There exists a set having no members.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649734322-->
END%%
%%ANKI
Basic
How is the empty set axiom expressed using first-order logic?
Back: $$\exists B, \forall x, x \not\in B$$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649734327-->
END%%
%%ANKI
Basic
The following encodes which set theory axiom? $$\exists B, \forall x, x \not\in B$$
Back: The empty set axiom.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649734332-->
END%%
%%ANKI
Basic
Which set theory axiom proves existence of $\varnothing$?
Back: The empty set axiom.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649069259-->
END%%
%%ANKI
Basic
What two properties ensures definition $\varnothing$ is well-defined?
Back: The empty set exists and is unique.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034312-->
END%%
## Pairing Axiom
For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$: $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
%%ANKI
Basic
What does the pairing axiom state?
Back: For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649734337-->
END%%
%%ANKI
Basic
How is the pairing axiom expressed using first-order logic?
Back: $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649734341-->
END%%
%%ANKI
Basic
The following encodes which set theory axiom? $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
Back: The pairing axiom.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649734346-->
END%%
%%ANKI
Basic
Which set theory axiom proves existence of set $\{x, y\}$ where $x \neq y$?
Back: The pairing axiom.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649734351-->
END%%
%%ANKI
Basic
Which set theory axiom proves existence of set $\{x\}$?
Back: The pairing axiom.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715649734357-->
END%%
%%ANKI
Basic
For sets $u$ and $v$, what name is given to set $\{u, v\}$?
Back: The pair set of $u$ and $v$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034322-->
END%%
%%ANKI
Basic
In set theory, what does a singleton refer to?
Back: A set with exactly one member.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034325-->
END%%
%%ANKI
Basic
What set theory axiom is used to prove existence of singletons?
Back: The pairing axiom.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034329-->
END%%
## Union Axiom
### Preliminary Form
For any sets $a$ and $b$, there exists a set whose members are those sets belonging either to $a$ or to $b$ (or both): $$\forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)$$
%%ANKI
Basic
What does the union axiom (preliminary form) state?
Back: For any sets $a$ and $b$, there exists a set whose members are all in either $a$ or $b$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034333-->
END%%
%%ANKI
Basic
How is the union axiom (preliminary form) expressed using first-order logic?
Back: $$\forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)$$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034337-->
END%%
%%ANKI
Basic
The following encodes which set theory axiom? $$\forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)$$
Back: The union axiom (preliminary form).
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034341-->
END%%
%%ANKI
Basic
How is the union of sets $a$ and $b$ denoted?
Back: $a \cup b$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034346-->
END%%
%%ANKI
Basic
What two set theory axioms proves existence of e.g. $\{x_1, x_2, x_3\}$?
Back: The pairing axiom and union axiom.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034351-->
END%%
## Power Set Axiom
For any set $a$, there is a set whose members are exactly the subsets of $a$: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$
%%ANKI
Basic
What does the power set axiom state?
Back: For any set $a$, there exists a set whose members are exactly the subsets of $a$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034356-->
END%%
%%ANKI
Basic
How is the power set axiom expressed using first-order logic?
Back: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034361-->
END%%
%%ANKI
Basic
The following encodes which set theory axiom? $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$
Back: The power set axiom.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034368-->
END%%
%%ANKI
Basic
How is $x \subseteq a$ rewritten using first-order logic and $\in$?
Back: $\forall t, t \in x \Rightarrow t \in a$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034375-->
END%%
%%ANKI
Basic
How is the power set of set $a$ denoted?
Back: $\mathscr{P}{a}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1715688034381-->
END%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

183
notes/x86-64/instructions.md
View File

@ -62,81 +62,6 @@ END%%
## Instruction Classes
An x86-64 CPU contains a set of 16 general-purpose registers storing 64-bit values. They are used to store integers and pointers.
1 Byte | 2 Bytes | 4 Bytes | 8 Bytes | Purpose
------- | ------- | ------- | ------- | -------
`%al` | `%ax` | `%eax` | `%rax` | Return value
`%bl` | `%bx` | `%ebx` | `%rbx` | Callee saved
`%cl` | `%cx` | `%ecx` | `%rcx` | 4th argument
`%dl` | `%dx` | `%edx` | `%rdx` | 3rd argument
`%sil` | `%si` | `%esi` | `%rsi` | 2nd argument
`%dil` | `%di` | `%edi` | `%rdi` | 1st argument
`%bpl` | `%bp` | `%ebp` | `%rbp` | Callee saved
`%spl` | `%sp` | `%esp` | `%rsp` | Stack pointer
`%r8b` | `%r8w` | `%r8d` | `%r8` | 5th argument
`%r9b` | `%r9w` | `%r9d` | `%r9` | 6th argument
`%r10b` | `%r10w` | `%r10d` | `%r10` | Caller saved
`%r11b` | `%r11w` | `%r11d` | `%r11` | Caller saved
`%r12b` | `%r12w` | `%r12d` | `%r12` | Callee saved
`%r13b` | `%r13w` | `%r13d` | `%r13` | Callee saved
`%r14b` | `%r14w` | `%r14d` | `%r14` | Callee saved
`%r15b` | `%r15w` | `%r15d` | `%r15` | Callee saved
%%ANKI
Basic
How many general-purpose registers are available to x86-64 instructions?
Back: 16
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889856-->
END%%
%%ANKI
Cloze
The x86 64-bit registers all start with prefix {`%r`}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889861-->
END%%
%%ANKI
Cloze
The x86 32-bit registers all start with prefix {`%e`}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889864-->
END%%
%%ANKI
Basic
Instructions that generate 1-byte quantities do what to the remaining bytes of a register?
Back: Leaves them alone.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889866-->
END%%
%%ANKI
Basic
Instructions that generate 2-byte quantities do what to the remaining bytes of a register?
Back: Leaves them alone.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889869-->
END%%
%%ANKI
Basic
Instructions that generate 4-byte quantities do what to the remaining bytes of a register?
Back: Zeroes them out.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889872-->
END%%
%%ANKI
Basic
Instructions that generate 8-byte quantities do what to the remaining bytes of a register?
Back: N/A
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889874-->
END%%
There are three types of operands:
* **Immediates**. These denote constant values. In ATT assembly, they are written with a `$` followed by an integer using standard C notation.
@ -774,6 +699,114 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1715377284985-->
END%%
### Load Effective Address
| Instruction | Operands | Effect | Description |
| ----------- | -------- | ------- | ---------------------- |
| `leaq` | S, D | D <- &S | Load effective address |
`leaq` is a variant of MOV. The first operand appears to be a memory address, but instead of reading from the designated location, the instruction copies the effective address to the designated location (a register).
%%ANKI
Basic
`leaq` is considered a variant of what other instruction class?
Back: `MOV`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1715780601450-->
END%%
%%ANKI
Basic
Why is the `leaq` instruction named the way it is?
Back: It stands for **l**oad **e**ffective **a**ddress.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1715780601455-->
END%%
%%ANKI
Cloze
The {`leaq`} instruction is to x86-64 as the {`&`} operator is to C.
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: 1715780601458-->
END%%
%%ANKI
Basic
Which x86-64 instruction is used to generate pointers?
Back: `leaq`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1715780601461-->
END%%
%%ANKI
Basic
Why doesn't `leaq` have any other size variants?
Back: x96-64 addresses are always 64-bit.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1715780601464-->
END%%
%%ANKI
Basic
Suppose `%rdx` contains $x$. Use `leaq` to set `%rax` to $5x + 7$.
Back: `leaq 7(%rdx, %rdx, 4), %rax`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1715780601467-->
END%%
%%ANKI
Basic
Besides effect memory computations, how else is `leaq` used?
Back: For certain arithmetic operations.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1715780601469-->
END%%
%%ANKI
Basic
Assume `%rbx` holds $p$ and `%rdx` holds $q$. What is the value of `%rax` in the following?
```asm
leaq 9(%rdx),%rax
```
Back: $9 + q$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1715781031929-->
END%%
%%ANKI
Basic
Assume `%rbx` holds $p$ and `%rdx` holds $q$. What is the value of `%rax` in the following?
```asm
leaq (%rdx, %rbx),%rax
```
Back: $q + q$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1715781031935-->
END%%
%%ANKI
Basic
Assume `%rbx` holds $p$ and `%rdx` holds $q$. What is the value of `%rax` in the following?
```asm
leaq 2(%rbx, %rbx, 7),%rax
```
Back: $2 + 8p$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1715781031938-->
END%%
%%ANKI
Basic
Assume `%rbx` holds $p$ and `%rdx` holds $q$. What is the value of `%rax` in the following?
```asm
leaq 0xE(, %rdx, 3),%rax
```
Back: $14 + 3q$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1715781031941-->
END%%
## Bibliography
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

88
notes/x86-64/registers.md Normal file
View File

@ -0,0 +1,88 @@
---
title: Registers
TARGET DECK: Obsidian::STEM
FILE TAGS: x86-64
tags:
- x86-64
---
## Overview
An x86-64 CPU contains a set of 16 general-purpose registers storing 64-bit values. They are used to store integers and pointers.
1 Byte | 2 Bytes | 4 Bytes | 8 Bytes | Purpose
------- | ------- | ------- | ------- | -------
`%al` | `%ax` | `%eax` | `%rax` | Return value
`%bl` | `%bx` | `%ebx` | `%rbx` | Callee saved
`%cl` | `%cx` | `%ecx` | `%rcx` | 4th argument
`%dl` | `%dx` | `%edx` | `%rdx` | 3rd argument
`%sil` | `%si` | `%esi` | `%rsi` | 2nd argument
`%dil` | `%di` | `%edi` | `%rdi` | 1st argument
`%bpl` | `%bp` | `%ebp` | `%rbp` | Callee saved
`%spl` | `%sp` | `%esp` | `%rsp` | Stack pointer
`%r8b` | `%r8w` | `%r8d` | `%r8` | 5th argument
`%r9b` | `%r9w` | `%r9d` | `%r9` | 6th argument
`%r10b` | `%r10w` | `%r10d` | `%r10` | Caller saved
`%r11b` | `%r11w` | `%r11d` | `%r11` | Caller saved
`%r12b` | `%r12w` | `%r12d` | `%r12` | Callee saved
`%r13b` | `%r13w` | `%r13d` | `%r13` | Callee saved
`%r14b` | `%r14w` | `%r14d` | `%r14` | Callee saved
`%r15b` | `%r15w` | `%r15d` | `%r15` | Callee saved
%%ANKI
Basic
How many general-purpose registers are available to x86-64 instructions?
Back: 16
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889856-->
END%%
%%ANKI
Cloze
The x86 64-bit registers all start with prefix {`%r`}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889861-->
END%%
%%ANKI
Cloze
The x86 32-bit registers all start with prefix {`%e`}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889864-->
END%%
%%ANKI
Basic
Instructions that generate 1-byte quantities do what to the remaining bytes of a register?
Back: Leaves them alone.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889866-->
END%%
%%ANKI
Basic
Instructions that generate 2-byte quantities do what to the remaining bytes of a register?
Back: Leaves them alone.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889869-->
END%%
%%ANKI
Basic
Instructions that generate 4-byte quantities do what to the remaining bytes of a register?
Back: Zeroes them out.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889872-->
END%%
%%ANKI
Basic
Instructions that generate 8-byte quantities do what to the remaining bytes of a register?
Back: N/A
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1713212889874-->
END%%
## Bibliography
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.