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Two's-complement addition and consolidate `shifts.md`.

c-declarations
Joshua Potter 2024-02-26 09:23:01 -07:00
parent eb3cc5743a
commit a1447e824c
6 changed files with 346 additions and 213 deletions
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notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
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---
title: "2024-02-25"
---
- [x] Anki Flashcards
- [ ] KoL
- [ ] Sheet Music (10 min.)
- [ ] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)

14
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---
title: "2024-02-26"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
* Added flashcards on two's-complement addition.
* Consolidated `integer-encoding.md` with `shifts.md`.

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---
title: "2024-02-25"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [x] OGS (1 Life & Death Problem)
- [x] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
* Read 흥부와 놀부 (Heungbuwa Nolbu).
* 101weiqi (serial numbers)
* Q-79448
* Q-5508
* Q-359007
* Q-12406
* Q-72569
* Q-259158
* Q-93867
* Q-84628
* Q-342740
* Q-54992
* Q-49888
* Q-10602
* Q-261719
* Q-178790

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@ -822,6 +822,190 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708701087982-->
END%%
## Shifting
Left shift operations (`<<`) drop the `k` most significant bits and fills the right end of the result with `k` zeros. Right shift operations (`>>`) are classified in two ways:
* **Logical**
* Drops the `k` least significant bits and fills the left end of the result with `k` zeros.
* This mode is always used when calling `>>` on unsigned data.
* Sometimes denoted as `>>>` to disambiguate from arithmetic right shifts.
* **Arithmetic**
* Drops the `k` least significant bits and fills the left end of the result with `k` copies of the most significant bit.
* This mode is usually used when calling `>>` on signed data.
%%ANKI
Basic
How is decimal value $2^n$ written in binary?
Back: As `1` followed by $n$ zeros.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707432641574-->
END%%
%%ANKI
Basic
What kinds of left shift operations are there?
Back: Just logical.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589773-->
END%%
%%ANKI
Basic
How many significant bits are dropped on a left shift by `k`?
Back: `k`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708784904518-->
END%%
%%ANKI
Basic
How many `0`s exist in the result of a left shift by `k`?
Back: At least `k`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708784904521-->
END%%
%%ANKI
Basic
What kinds of right shift operations are there?
Back: Logical and arithmetic
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589784-->
END%%
%%ANKI
Basic
What is a logical right shift operation?
Back: One that fills the left end of the result with zeros.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589786-->
END%%
%%ANKI
Basic
What is an arithmetic right shift operation?
Back: One that fills the left end of the result with copies of the most significant bit.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589789-->
END%%
%%ANKI
Basic
What kind of right shift operation is *usually* applied to signed numbers?
Back: Arithmetic.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589801-->
END%%
%%ANKI
Basic
What kind of right shift operation is applied to unsigned numbers?
Back: Logical.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589804-->
END%%
%%ANKI
Basic
What portability issue do shift operations introduce?
Back: There is no standard on whether right shifts of signed numbers are logical or arithmetic.
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: 1707854589808-->
END%%
%%ANKI
Cloze
{1:Arithmetic} right shifts are to {1:signed} numbers whereas {2:logical} right shifts are to {2:unsigned} numbers.
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: 1707854589813-->
END%%
In C, it is undefined behavior to shift by more than the width $w$ of an integral type.
%%ANKI
Basic
Assuming two's-complement, what is the result of shifting an `int32_t` value by `32`?
Back: It is undefined behavior.
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: 1708785613342-->
END%%
%%ANKI
Basic
Assuming two's-complement, what is the result of shifting an `int32_t` value by `31`?
Back: $-2^{31}$
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: 1708785613370-->
END%%
%%ANKI
Basic
What is the result of shifting an `int32_t` value by `-1`?
Back: It is undefined behavior.
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: 1708785613376-->
END%%
%%ANKI
Basic
What is the result of shifting an `uint32_t` value by `32`?
Back: It is undefined behavior.
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: 1708785613383-->
END%%
%%ANKI
Basic
What is the result of shifting an `uint32_t` value by `31`?
Back: $2^{31}$
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: 1708785613389-->
END%%
%%ANKI
Basic
What is the result of shifting an `uint32_t` value by `-1`?
Back: It is undefined behavior.
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: 1708785613393-->
END%%
%%ANKI
Basic
How is $2^n$ written using bitwise shift operators?
Back: `1 << n`
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: 1708784904524-->
END%%
%%ANKI
Basic
What decimal value does `1 << n` translate to?
Back: $2^n$
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: 1708784904526-->
END%%
%%ANKI
Basic
How is $x \bmod 2^k$ equivalently written as a bit mask?
Back: `x & ((1 << k) - 1)`
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: 1707873410780-->
END%%
## Arithmetic
### Addition
@ -871,8 +1055,8 @@ END%%
Unsigned addition of $w$-bit integral types, denoted $+_w^u$, behaves like so:
$$x +_w^u y = \begin{cases}
x + y & \text{if } x + y < 2^w \\
x + y - 2^w & \text{otherwise}
x + y - 2^w & \text{if } x + y \geq 2^w \\
x + y & \text{otherwise}
\end{cases}$$
This is more simply expressed as $x +_w^u y = (x + y) \bmod 2^w$.
@ -928,14 +1112,14 @@ END%%
%%ANKI
Cloze
Without using modulo arithmetic, $x +_w^u y =$ {$x + y$} if {$x + y < 2^w$}.
Without using modular arithmetic, $x +_w^u y =$ {$x + y$} if {$x + y < 2^w$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678758-->
END%%
%%ANKI
Cloze
Without using modulo arithmetic, $x +_w^u y =$ {$x + y - 2^w$} if {$x + y \geq 2^w$}.
Without using modular arithmetic, $x +_w^u y =$ {$x + y - 2^w$} if {$x + y \geq 2^w$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708799678761-->
END%%
@ -995,7 +1179,119 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708808252017-->
END%%
Two's-complement addition, denoted $+_w^t$ operates similarly:
$$x +_w^u y = \begin{cases}
x + y - 2^w & \text{if } x + y \geq 2^{w-1} \\
x + y + 2^w & \text{if } x + y < -2^{w-1} \\
x + y & \text{otherwise}
\end{cases}$$
Unlike with unsigned addition, there is no simpler modulus operation that can be applied.
%%ANKI
Basic
What kind of overflows does two's-complement addition potentially exhibit?
Back: Positive and negative overflow.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376220-->
END%%
%%ANKI
Basic
Why is two's-complement addition overflow UB?
Back: Because the C standard does not mandate any particular signed integer encoding.
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: 1708964376225-->
END%%
%%ANKI
Basic
What does $+_w^t$ denote?
Back: Two's-complement addition of $w$-bit integral types.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376228-->
END%%
%%ANKI
Basic
*Why* doesn't two's-complement addition perform modular arithmetic?
Back: Because negative values are representable.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376231-->
END%%
%%ANKI
Cloze
$x +_w^t y =$ {$x + y - 2^w$} if {$x + y \geq 2^{w-1}$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376235-->
END%%
%%ANKI
Cloze
$x +_w^t y =$ {$x + y + 2^w$} if {$x + y < -2^{w-1}$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376238-->
END%%
%%ANKI
Cloze
$x +_w^t y =$ {$x + y$} if {$-2^{w-1} \leq x + y < 2^{w-1}$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376242-->
END%%
%%ANKI
Basic
How do we detect $x +_w^t y$ positive overflowed?
Back: This happens iff $x > 0$, $y > 0$, and $x +_w^t y \leq 0$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376246-->
END%%
%%ANKI
Basic
How do we detect $x +_w^t y$ negative overflowed?
Back: This happens iff $x < 0$, $y < 0$, and $x +_w^t y \geq 0$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376250-->
END%%
%%ANKI
Basic
How can we write $x +_w^t y$ in terms of unsigned addition?
Back: $x +_w^t y = U2T_w(T2U_w(x) +_w^u T2U_w(y))$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376254-->
END%%
%%ANKI
*Why* are we able to characterize $+_w^t$ in terms of $+_w^u$?
Back: Because two's-complement addition has the same bit-level representation as unsigned addition.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
END%%
%%ANKI
Basic
How would you complete the body of this function?
```c
/* Determine whether arguments can be added without overflow */
int tadd_ok(int x, int y);
```
Back:
```c
int pos_over = x > 0 && y > 0 && (x + y) <= 0;
int neg_over = x < 0 && y < 0 && (x + y) >= 0;
return !pos_over && !neg_over;
```
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708964376259-->
END%%
## References
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
* Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
* “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).

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@ -1,196 +0,0 @@
---
title: Shifts
TARGET DECK: Obsidian::STEM
FILE TAGS: binary
tags:
- binary
---
## Overview
Left shift operations (`<<`) drop the `k` most significant bits and fills the right end of the result with `k` zeros. Right shift operations (`>>`) are classified in two ways:
* **Logical**
* Drops the `k` least significant bits and fills the left end of the result with `k` zeros.
* This mode is always used when calling `>>` on unsigned data.
* Sometimes denoted as `>>>` to disambiguate from arithmetic right shifts.
* **Arithmetic**
* Drops the `k` least significant bits and fills the left end of the result with `k` copies of the most significant bit.
* This mode is usually used when calling `>>` on signed data.
%%ANKI
Basic
How is decimal value $2^n$ written in binary?
Back: As `1` followed by $n$ zeros.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707432641574-->
END%%
%%ANKI
Basic
What kinds of left shift operations are there?
Back: Just logical.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589773-->
END%%
%%ANKI
Basic
How many significant bits are dropped on a left shift by `k`?
Back: `k`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708784904518-->
END%%
%%ANKI
Basic
How many `0`s exist in the result of a left shift by `k`?
Back: At least `k`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708784904521-->
END%%
%%ANKI
Basic
What kinds of right shift operations are there?
Back: Logical and arithmetic
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589784-->
END%%
%%ANKI
Basic
What is a logical right shift operation?
Back: One that fills the left end of the result with zeros.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589786-->
END%%
%%ANKI
Basic
What is an arithmetic right shift operation?
Back: One that fills the left end of the result with copies of the most significant bit.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589789-->
END%%
%%ANKI
Basic
What kind of right shift operation is *usually* applied to signed numbers?
Back: Arithmetic.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589801-->
END%%
%%ANKI
Basic
What kind of right shift operation is applied to unsigned numbers?
Back: Logical.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1707854589804-->
END%%
%%ANKI
Basic
What portability issue do shift operations introduce?
Back: There is no standard on whether right shifts of signed numbers are logical or arithmetic.
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: 1707854589808-->
END%%
%%ANKI
Cloze
{1:Arithmetic} right shifts are to {1:signed} numbers whereas {2:logical} right shifts are to {2:unsigned} numbers.
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: 1707854589813-->
END%%
In C, it is undefined behavior to shift by more than the width $w$ of an integral type.
%%ANKI
Basic
Assuming two's-complement, what is the result of shifting an `int32_t` value by `32`?
Back: It is undefined behavior.
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: 1708785613342-->
END%%
%%ANKI
Basic
Assuming two's-complement, what is the result of shifting an `int32_t` value by `31`?
Back: $-2^{31}$
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: 1708785613370-->
END%%
%%ANKI
Basic
What is the result of shifting an `int32_t` value by `-1`?
Back: It is undefined behavior.
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: 1708785613376-->
END%%
%%ANKI
Basic
What is the result of shifting an `uint32_t` value by `32`?
Back: It is undefined behavior.
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: 1708785613383-->
END%%
%%ANKI
Basic
What is the result of shifting an `uint32_t` value by `31`?
Back: $2^{31}$
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: 1708785613389-->
END%%
%%ANKI
Basic
What is the result of shifting an `uint32_t` value by `-1`?
Back: It is undefined behavior.
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: 1708785613393-->
END%%
%%ANKI
Basic
How is $2^n$ written using bitwise shift operators?
Back: `1 << n`
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: 1708784904524-->
END%%
%%ANKI
Basic
What decimal value does `1 << n` translate to?
Back: $2^n$
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: 1708784904526-->
END%%
%%ANKI
Basic
How is $x \bmod 2^k$ equivalently written as a bit mask?
Back: `x & ((1 << k) - 1)`
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: 1707873410780-->
END%%
## References
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
* Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).