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More integer encoding notes.

c-declarations
Joshua Potter 2024-02-22 14:21:08 -07:00
parent 2538533aee
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12
notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
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notes/_journal/2024-02-21.md
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---
title: "2024-02-21"
---
- [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)
* Read through "Stars and Bars" section in "Discrete Mathematics: An Open Introduction".
* Added notes on radices.
* Spent time thinking about how two's-complement works.

19
notes/_journal/2024-02-22.md Normal file
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@ -0,0 +1,19 @@
---
title: "2024-02-22"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [x] Interview Prep (1 Practice Problem)
- [x] Log Work Hours (Max 3 hours)
* Briefly discussed first chapter of "Designing Data-Intensive Applications" with Mike.
* Continued reading more on integer encodings.
* Focused on conversions between unsigned encoding and two's-complement, though still need to create flashcards for these.
* Did Leetcode problems
* [3Sum Closest](https://leetcode.com/problems/3sum-closest/description/)
* [Letter Combinations of a Phone Number](https://leetcode.com/problems/letter-combinations-of-a-phone-number/description/)
* [Remove Nth Node From End of List](https://leetcode.com/problems/remove-nth-node-from-end-of-list/description/)

20
notes/_journal/2024-02/2024-02-21.md Normal file
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@ -0,0 +1,20 @@
---
title: "2024-02-21"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [x] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
* Read through "Stars and Bars" section in "Discrete Mathematics: An Open Introduction".
* Added notes on radices.
* Spent time thinking about how two's-complement works.
* Read the first chapter on "Designing Data-Intensive Applications".
* Leetcode Problems
* [Integer to Roman](https://leetcode.com/problems/integer-to-roman/description/)
* [Roman to Integer](https://leetcode.com/problems/roman-to-integer/description/)
* [Container With Most Water](https://leetcode.com/problems/container-with-most-water/description/)

18
notes/algebra/radices.md
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@ -267,6 +267,24 @@ Tags: binary::hex
<!--ID: 1707432641596-->
END%%
%%ANKI
Basic
Which hexadecimal digits have a leading `1` bit?
Back: `8` through `F`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1708631918825-->
END%%
%%ANKI
Basic
Which hexadecimal digits have a leading `0` bit?
Back: `0` through `7`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1708631918829-->
END%%
## References
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

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notes/binary/integer-encoding.md
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@ -54,7 +54,7 @@ END%%
%%ANKI
Basic
An integral value of 0 has what encoding?
An integral value of $0_{10}$ likely has what encoding?
Back: Either unsigned or two's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246105-->
@ -62,7 +62,7 @@ END%%
%%ANKI
Basic
An integral value of 100 has what encoding?
An integral value of $100_{10}$ likely has what encoding?
Back: Either unsigned or two's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246109-->
@ -70,7 +70,7 @@ END%%
%%ANKI
Basic
An integral value of -100 has what encoding?
An integral value of $-100_{10}$ likely has what encoding?
Back: Two's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708177246114-->
@ -87,7 +87,7 @@ END%%
%%ANKI
Basic
What integral values share the same binary representation in unsigned encoding and two's-complement?
Back: Nonnegative values $\leq |TMax|$.
Back: Nonnegative values $\leq TMax$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708454709515-->
END%%
@ -108,9 +108,83 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708455064696-->
END%%
%%ANKI
Basic
According to the C standard, Is `unsigned` underflow/overflow safe?
Back: Yes
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708551236389-->
END%%
%%ANKI
Basic
According to the C standard, Is `signed` underflow/overflow safe?
Back: No
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708551236392-->
END%%
%%ANKI
Basic
Why is `signed` underflow/overflow considered UB?
Back: Because there is no requirement on how `signed` integers are encoded.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708551236395-->
END%%
%%ANKI
Basic
How does $UMax$ relate to $TMax$?
Back: $UMax = 2 \cdot TMax + 1$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398445-->
END%%
%%ANKI
Basic
*Why* is it $UMax = 2 \cdot TMax + 1$?
Back: All bit patterns denoting negative numbers in two's-complement are positive in unsigned encoding.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708613447880-->
END%%
%%ANKI
Basic
What are the binary encodings of $UMax_4$ and $TMax_4$?
Back: $1111_2$ and $0111_2$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398449-->
END%%
%%ANKI
Basic
Reinterpret $TMax$ in unsigned encoding. What arithmetic operations yield $UMax$?
Back: Multiply by two and add one.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398454-->
END%%
%%ANKI
Basic
Reinterpret $TMax$ in unsigned encoding. What bitwise operations yield $UMax$?
Back: One-bit left shift and add one.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398459-->
END%%
%%ANKI
Basic
Reinterpret $UMax$ in two's-complement. What decimal value do you have?
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: 1708453398469-->
END%%
### Unsigned Encoding
Always represents nonnegative numbers. Given an integral type $\vec{x}$ of $w$ bits, we convert binary to its unsigned encoding with: $$B2U_w(\vec{x}) = \sum_{i=0}^{w-1} 2^ix_i$$
Always represents nonnegative numbers. Given an integral type $\vec{x}$ of $w$ bits, we convert binary to its unsigned encoding with: $$B2U_w(\vec{x}) = 2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$$
Note we unfold the summation on the RHS by one term to make it's relationship to $T2U_w$ clearer.
%%ANKI
Basic
@ -171,7 +245,7 @@ END%%
%%ANKI
Basic
How does Bryant et al. define $B2U_w$?
Back: $B2U_w(\vec{x}) = \sum_{i=0}^{w-1} 2^ix_i$
Back: $B2U_w(\vec{x}) = 2^{w-1}x_{w-1} + \sum_{i=0}^{w-2} 2^ix_i$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147785-->
END%%
@ -179,11 +253,19 @@ END%%
%%ANKI
Basic
What is $B2U_w$ an acronym for?
Back: **B**inary "to" **u**nsigned.
Back: **B**inary to **u**nsigned, width $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147791-->
END%%
%%ANKI
Basic
What is $U2B_w$ an acronym for?
Back: **U**nsigned to **b**inary, width $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708613447885-->
END%%
%%ANKI
Basic
What does $w$ in $B2U_w$ represent?
@ -195,11 +277,19 @@ END%%
%%ANKI
Basic
What is the domain of $B2U_w$?
Back: Bit vectors of size $w$.
Back: Bit strings of size $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179147798-->
END%%
%%ANKI
Basic
What is the domain of $U2B_w$?
Back: $[0, 2^w)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708613447888-->
END%%
%%ANKI
Basic
What is the range of $B2U_w$?
@ -210,10 +300,10 @@ END%%
%%ANKI
Basic
What is the hexadecimal representation of $UMin_4$?
Back: `0x0000`
What is the range of $U2B_w$?
Back: Bit strings of length $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398386-->
<!--ID: 1708613447891-->
END%%
%%ANKI
@ -224,14 +314,6 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708453398392-->
END%%
%%ANKI
Basic
What is the hexadecimal representation of $UMax_4$?
Back: `0xFFFF`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398397-->
END%%
%%ANKI
Basic
How is the largest unsigned integer formatted in hexadecimal?
@ -250,7 +332,7 @@ END%%
%%ANKI
Basic
With unsigned encoding, *why* does `n + ~n = UMax`?
Using unsigned encoding, *why* does `n + ~n = UMax`?
Back: Because this always yields a bit string of all `1`s.
Reference: “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).
<!--ID: 1708545574154-->
@ -284,6 +366,13 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708545383255-->
END%%
%%ANKI
Basic
How do $TMin$ and $TMax$ relate to one another?
Back: $TMin = -TMax - 1$
<!--ID: 1708609869518-->
END%%
%%ANKI
Basic
What half-open interval represents the possible $w$-bit two's-complement decimal values?
@ -326,7 +415,6 @@ END%%
Basic
What is the weight of the sign bit in $w$-bit two's-complement?
Back: $-2^{w-1}$
The {sign bit} refers to the {most significant bit} in two's-complement.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649887-->
END%%
@ -350,11 +438,19 @@ END%%
%%ANKI
Basic
What is $B2T_w$ an acronym for?
Back: **B**inary "to" **t**wo's-complement.
Back: **B**inary to **t**wo's-complement, width $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649907-->
END%%
%%ANKI
Basic
What is $T2B_w$ an acronym for?
Back: **T**wo's-complement to **b**inary, width $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708613447895-->
END%%
%%ANKI
Basic
What does $w$ in $B2T_w$ represent?
@ -366,11 +462,19 @@ END%%
%%ANKI
Basic
What is the domain of $B2T_w$?
Back: Bit vectors of size $w$.
Back: Bit strings of size $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708179649921-->
END%%
%%ANKI
Basic
What is the domain of $T2B_w$?
Back: $[-2^{w-1}, 2^{w-1})$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708613447899-->
END%%
%%ANKI
Basic
What is the range of $B2T_w$?
@ -381,10 +485,10 @@ END%%
%%ANKI
Basic
What is the hexadecimal representation of $TMin_4$?
Back: `0x8000`
What is the range of $T2B_w$?
Back: Bit strings of length $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398408-->
<!--ID: 1708613447903-->
END%%
%%ANKI
@ -395,14 +499,6 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708453398413-->
END%%
%%ANKI
Basic
What is the hexadecimal representation of $TMax_4$?
Back: `0x7FFF`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398418-->
END%%
%%ANKI
Basic
How is the largest two's-complement integer formatted in hexadecimal?
@ -429,54 +525,12 @@ END%%
%%ANKI
Basic
Why is two's-complement encoding's range asymmetric?
Back: Leading `1`s correspond to negatives but leading `0`s corerspond to nonnegative numbers (which include $0$.)
Why is two's-complement's encoding range asymmetric?
Back: Leading `1`s correspond to negatives but leading `0`s corerspond to nonnegative numbers (which include $0$).
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398440-->
END%%
%%ANKI
Basic
How does $UMax$ relate to $TMax$?
Back: $|UMax| = 2|TMax| + 1$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398445-->
END%%
%%ANKI
Basic
What are the binary encodings of $UMax_4$ and $TMax_4$?
Back: $1111_2$ and $0111_2$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398449-->
END%%
%%ANKI
Basic
Reinterpret $TMax$ in unsigned encoding. What arithmetic operations yield $UMax$?
Back: Multiply by two and add one.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398454-->
END%%
%%ANKI
Basic
Reinterpret $TMax$ in unsigned encoding. What bitwise operations yield $UMax$?
Back: One-bit left shift and add one.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708453398459-->
END%%
%%ANKI
Basic
Reinterpret $UMax$ in two's-complement. What decimal value do you have?
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: 1708453398469-->
END%%
The "two's-complement" of an integer often refers to the binary representation of the integer's additive inverse. For example, $0110_2 = 6$ has two's-complement $1010_2 = -6$. In contrast, the "complement" of a nonnegative integer is the other nonnegative integer such that together add up to $2^{w-1}$. For example, $0110_2 = 6$ has complement $0010_2 = 2$ (with respect to $2^3 = 8$).
%%ANKI
Basic
What are the median values of two's-complement's encoding range?
@ -492,90 +546,122 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708545383265-->
END%%
## Casting
Most implementations of C cast an object of one type to another by simply re-interpreting their binary representation. This casting may happen implicitly if comparing or operating on e.g. `signed` and `unsigned` objects in the same expression. $T2U$ and $U2T$ reflect this method of casting:
$$T2U_w(x) = \begin{cases}
x + 2^w & x < 0 \\
x & x \geq 0
\end{cases}$$
$$U2T_w(x) = \begin{cases}
x & x \leq TMax_w \\
x - 2^w & x > TMax_w
\end{cases}$$
%%ANKI
Basic
How does `n` relate to `~n` in two's-complement?
Back: `~n = -n - 1`
Reference: “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).
<!--ID: 1708545383267-->
How do most implementations of C perform casting?
Back: As a reinterpretation of the same byte pattern of the object being casted.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1708615249879-->
END%%
%%ANKI
Basic
What value does not have a two's-complement?
Back: $TMin$
Reference: “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).
<!--ID: 1708545383268-->
What is $T2U_w$ an acronym for?
Back: **T**wo's-complement to **u**nsigned, width $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249883-->
END%%
%%ANKI
Basic
*Why* doesn't $TMin$ have a two's-complement?
Back: Because there is one less representable positive number than negative number.
Reference: “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).
<!--ID: 1708545383270-->
For what decimal values $x$ does $T2U_w$ and $U2T_w$ serve as the identity function?
Back: $0 \leq x \leq TMax_w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249887-->
END%%
%%ANKI
Basic
What values $x$ are unaffected when casting from `signed` to `unsigned`?
Back: $0 \leq x \leq TMax_w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1708615249891-->
END%%
%%ANKI
Basic
What values $x$ are unaffected when casting from `unsigned` to `signed`?
Back: $0 \leq x \leq TMax_w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1708615249897-->
END%%
%%ANKI
Basic
How are casts implicitly performed in operations containing `signed` and `unsigned` objects?
Back: `signed` objects are cast to `unsigned` objects.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: c
<!--ID: 1708615249903-->
END%%
%%ANKI
Cloze
$TMin$ is called the "{weird number}" because {it is it's own two's-complement}.
Reference: “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).
END%%
%%ANKI
Basic
What is the result of applying the two's-complement operation on $TMin$?
Back: $TMin$
Reference: “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).
<!--ID: 1708545383271-->
END%%
%%ANKI
Basic
What does the two's-complement of $n$ refer to?
Back: The binary representation of $-n$.
Reference: “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).
<!--ID: 1708545383273-->
END%%
%%ANKI
Basic
Using two's-complement, what is the *complement* of $0101_2$?
Back: $0011_2$
Reference: “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).
<!--ID: 1708545383274-->
END%%
%%ANKI
Basic
Using two's-complement, the *complement* of $0101_2$ is found with respect to what value?
Back: $2^3 = 8$
Reference: “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).
<!--ID: 1708545383276-->
END%%
%%ANKI
Basic
What bitwise operations yield the two's-complement of $n$?
Back: `~n + 1`
Reference: “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).
<!--ID: 1708545383277-->
END%%
%%ANKI
Basic
Why is two's-complement named the way it is?
Back: For $x \geq 0$, $-x$'s $w$-bit representation is found using $2^w - x$.
For {$x < 0$}, $T2U_w(x) =$ {$x + 2^w$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708545383279-->
<!--ID: 1708615249908-->
END%%
%%ANKI
Cloze
For {$x \geq 0$}, $T2U_w(x) =$ {$x$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249914-->
END%%
%%ANKI
Basic
What identity show $3$ and $5$ are complements with respect to $2^3$?
Back: $3 + 5 = 8$
Reference: “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).
<!--ID: 1708545383280-->
How is $T2U_w$ written as a function composition?
Back: $T2U_w = B2U_w \circ T2B_w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249920-->
END%%
%%ANKI
Basic
What is $U2T_w$ an acronym for?
Back: **U**nsigned to **t**wo's-complement, width $w$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249925-->
END%%
%%ANKI
Basic
How is $U2T_w$ written as a function composition?
Back: $U2T_w = B2T_w \circ U2B_w$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249930-->
END%%
%%ANKI
Cloze
For {$x > TMax_w$}, $U2T_w(x) =$ {$x - 2^w$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249935-->
END%%
%%ANKI
Cloze
For {$x \leq TMax_w$}, $U2T_w(x) =$ {$x$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249939-->
END%%
## References

4
notes/binary/shifts.md
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@ -45,7 +45,7 @@ END%%
%%ANKI
Basic
What is a logical right shift operation?
Back: One that fills the left end of the result with `k` zeros.
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%%
@ -53,7 +53,7 @@ END%%
%%ANKI
Basic
What is an arithmetic right shift operation?
Back: One that fills the left end of the result with `k` copies of the most significant bit.
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%%

149
notes/c/declarations.md
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@ -28,6 +28,14 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1707835869728-->
END%%
%%ANKI
Basic
Which header file contains `INT32_MAX`?
Back: `<stdint.h>`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249870-->
END%%
%%ANKI
Basic
What does the "width" of an integer type refer to?
@ -195,6 +203,147 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1707493017244-->
END%%
## Integer Literals
Negative integer literals are typed in a counterintuitive way. When the compiler sees a number of form `-X`, the type of `X` is first determined *before* then being negated. Promotion rules are as follows:
C90 (Decimal) | C90 (Other) | C99 (Decimal) | C99 (Other)
--------------- | --------------- | ------------- | ---------
`int` | `int` | `int` | `int`
`long` | `unsigned` | `long` | `unsigned`
`unsigned` | `long` | `long long` | `long`
`unsigned long` | `unsigned long` | `-` | `unsigned long`
`-` | `-` | `-` | `long long`
`-` | `-` | `-` | `unsigned long long`
%%ANKI
Basic
How does the compiler process integer literal `-X`?
Back: By first determining the type of `X` and then negating the value.
Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820805-->
END%%
%%ANKI
Basic
What simplification did C99 introduce to decimal integer literals?
Back: The integer constant is guaranteed a `signed` type.
Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820816-->
END%%
%%ANKI
Basic
Since what standard was it guaranteed decimal integer literals were `signed`?
Back: C99
Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820820-->
END%%
%%ANKI
Basic
In ISO C90, what integer literals are guaranteed `signed`?
Back: None.
Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820823-->
END%%
%%ANKI
Basic
In ISO C99, what integer literals are guaranteed `signed`?
Back: Decimal integer constants.
Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820826-->
END%%
%%ANKI
Basic
Why avoid negative octal integer literals?
Back: Depending on value, the resulting type may be `unsigned`.
Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820829-->
END%%
%%ANKI
Basic
Why avoid negative hexadecimal integer literals?
Back: Depending on value, the resulting type may be `unsigned`.
Reference: Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820833-->
END%%
%%ANKI
Basic
Which header file contains `INT_MAX`?
Back: `<limits.h>`
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249864-->
END%%
%%ANKI
Cloze
{`INT_MAX`} is to `signed` whereas {`UINT_MAX`} is to `unsigned`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820837-->
END%%
%%ANKI
Basic
How does `<limits.h>` define `INT_MIN`?
Back: As `(-INT_MAX - 1)`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820840-->
END%%
%%ANKI
Basic
*Why* is `INT_MIN` defined as `(-INT_MAX - 1)` instead of directly as e.g. `-2147483648`?
Back: Because `2147483648` (without `-`) would be sized as a non-`int` before being negated.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820843-->
END%%
%%ANKI
Cloze
`INT_MAX` is to {`<limits.h>`} whereas `INT32_MAX` is to {`<stdint.h>`}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249873-->
END%%
%%ANKI
Basic
What suffix can be used to denote an `unsigned` integer literal?
Back: Case-insensitive `U`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708615249876-->
END%%
%%ANKI
Basic
What suffix can be used to denote a `long` integer literal?
Back: Case-insensitive `L`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820847-->
END%%
%%ANKI
Basic
What suffix can be used to denote a `long long` integer literal?
Back: Case-insensitive `LL`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820850-->
END%%
%%ANKI
Basic
What suffix can be used to denote an `unsigned long long` integer literal?
Back: Case-insensitive `ULL`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1708631820856-->
END%%
## Pointers
Pointers have the same size as the machine's word size since it should be able to refer to any virtual address.
%%ANKI

2
notes/logic/equiv-trans.md
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@ -788,7 +788,7 @@ END%%
%%ANKI
Basic
How is $(s; x{:}v)$ written instead using set-theoretical notation?
Back: $(s - \{(x, s(x))\}) \cup \{(x, v)\}$
Back: $(s - \{\langle x, s(x) \rangle\}) \cup \{\langle x, v \rangle\}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707937867053-->
END%%