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More notes on two's-complement.

c-declarations
Joshua Potter 2024-02-27 12:12:25 -07:00
parent cf0daeab7d
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notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
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@ -81,7 +81,9 @@
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notes/_journal/2024-02-27.md
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@ -8,4 +8,7 @@ title: "2024-02-27"
- [x] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
- [x] Log Work Hours (Max 3 hours)
* Added many flashcards on $\Theta$-, $O$-, and $\Omega$-notation.
* Yet more notes on two's-complement.

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notes/algorithms/order-growth.md
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@ -155,6 +155,20 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
<!--ID: 1708974221797-->
END%%
%%ANKI
What does it mean for function $f(n)$ to be asymptotically nonnegative?
Back: $f(n) \geq 0$ whenever $n$ is sufficiently large.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
END%%
%%ANKI
Basic
What does it mean for function $f(n)$ to be asymptotically positive?
Back: $f(n) > 0$ whenever $n$ is sufficiently large.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708974221871-->
END%%
## $\Theta$-notation
![[theta-notation.png]]
@ -163,7 +177,7 @@ $\Theta$-notation refers to a strict lower- and upper-bound. It is defined as se
%%ANKI
Basic
What kind of mathematical object is $\Theta(n)$?
What kind of mathematical object is $\Theta(g(n))$?
Back: A set.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708974221801-->
@ -250,14 +264,14 @@ END%%
%%ANKI
Cloze
Given $f(n) = \Theta(g(n))$, we say {1:$g(n)$} is an asymptotically {2:tight} bound for {1:$f(n)$}.
Given $f(n) = \Theta(g(n))$, we say {1:$g(n)$} is an asymptotic {2:tight} bound for {1:$f(n)$}.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708974221851-->
END%%
%%ANKI
Basic
Which notation corresponds to asymptotically tight bounds?
Which notation corresponds to asymptotic tight bounds?
Back: $\Theta$-notation.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708974221857-->
@ -271,20 +285,6 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
<!--ID: 1708974221864-->
END%%
%%ANKI
What does it mean for function $f(n)$ to be asymptotically nonnegative?
Back: $f(n) \geq 0$ whenever $n$ is sufficiently large.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
END%%
%%ANKI
Basic
What does it mean for function $f(n)$ to be asymptotically positive?
Back: $f(n) > 0$ whenever $n$ is sufficiently large.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708974221871-->
END%%
%%ANKI
Basic
What condition must $g(n)$ satisfy such that otherwise $\Theta(g(n))$ is empty?
@ -336,11 +336,333 @@ END%%
%%ANKI
Basic
What values does the $y$-axis implicitly range over in the following?
For $n < n_0$, what values does the $y$-axis take on?
![[theta-notation.png]]
Back: Indeterminate.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708974221909-->
END%%
%%ANKI
Basic
For $n \geq n_0$, what values does the $y$-axis take on?
![[theta-notation.png]]
Back: Nonnegative values.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708974221909-->
<!--ID: 1709053894064-->
END%%
## $O$-notation
![[big-o-notation.png]]
$O$-notation refers to a strict upper-bound. It is defined as set $$O(g(n)) = \{ f(n) \mid \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq f(n) \leq cg(n) \}$$
%%ANKI
Basic
What kind of mathematical object is $O(g(n))$?
Back: A set.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894066-->
END%%
%%ANKI
Basic
Using typical identifiers found in $O(g(n))$, what values do $c$ and $n_0$ take on?
Back: Positive constants.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894068-->
END%%
%%ANKI
Basic
What names are usually given to the existentially quantified identifers in $O(g(n))$'s definition?
Back: $c$ and $n_0$.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894070-->
END%%
%%ANKI
Basic
What name is usually given to the universally quantified identifer in $O(g(n))$'s definition?
Back: $n$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894072-->
END%%
%%ANKI
Cloze
Using typical identifiers, $f(n) = O(g(n))$ satisfies {$0$} $\leq$ {$f(n)$} $\leq$ {$cg(n)$}.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894074-->
END%%
%%ANKI
Basic
Using typical identifiers, what is the lower bound of $cg(n)$ in $O(g(n))$?
Back: $f(n)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894076-->
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of $cg(n)$ in $O(g(n))$?
Back: N/A
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894078-->
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of $f(n)$ in $O(g(n))$?
Back: $0$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894080-->
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of $f(n)$ in $O(g(n))$?
Back: $cg(n)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
END%%
%%ANKI
Cloze
Given $f(n) = O(g(n))$, we say {1:$g(n)$} is an asymptotic {2:upper} bound for {1:$f(n)$}.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894084-->
END%%
%%ANKI
Basic
Which notation corresponds to asymptotic upper bounds?
Back: $O$-notation.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894088-->
END%%
%%ANKI
Basic
Every member of $O(g(n))$ is expected to be asymptotically what?
Back: Nonnegative.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894091-->
END%%
%%ANKI
Basic
What condition must $g(n)$ satisfy such that otherwise $O(g(n))$ is empty?
Back: $g(n)$ must be asymptotically nonnegative.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894093-->
END%%
%%ANKI
Basic
How is $O(g(n))$ defined?
Back: $\{ \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq f(n) \leq cg(n) \}$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894096-->
END%%
%%ANKI
Basic
Which asymptotic notation is this image demonstrating?
![[big-o-notation.png]]
Back: $O$-notation
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894098-->
END%%
%%ANKI
Basic
When is it guaranteed $y$-values are nonnegative in the following?
![[big-o-notation.png]]
Back: When $n \geq n_0$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894100-->
END%%
%%ANKI
Basic
In set-theoretic notation, what does it mean for $\Theta$-notation to be stronger than $O$-notation?
Back: $\Theta(g(n)) \subseteq O(g(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894101-->
END%%
%%ANKI
Basic
What notation corresponds to worst-case running times?
Back: $O$-notation
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894103-->
END%%
%%ANKI
Basic
Why does Cormen et al. say "insertion sort's running time is $O(n^2)$" is an abuse of notation?
Back: Because technically its running time depends on the particular input of size $n$.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894105-->
END%%
## $\Omega$-notation
![[big-omega-notation.png]]
$\Omega$-notation refers to a strict lower-bound. It is defined as set $$\Omega(g(n)) = \{ f(n) \mid \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) \leq f(n) \}$$
%%ANKI
Basic
What kind of mathematical object is $\Omega(g(n))$?
Back: A set.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157375-->
END%%
%%ANKI
Basic
Using typical identifiers found in $\Omega(g(n))$, what values do $c$ and $n_0$ take on?
Back: Positive constants.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157377-->
END%%
%%ANKI
Basic
What names are usually given to the existentially quantified identifers in $\Omega(g(n))$'s definition?
Back: $c$ and $n_0$.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157379-->
END%%
%%ANKI
Basic
What name is usually given to the universally quantified identifer in $\Omega(g(n))$'s definition?
Back: $n$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157381-->
END%%
%%ANKI
Cloze
Using typical identifiers, $f(n) = \Omega(g(n))$ satisfies {$0$} $\leq$ {$cg(n)$} $\leq$ {$f(n)$}.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157383-->
END%%
%%ANKI
Basic
Using typical identifiers, what is the lower bound of $cg(n)$ in $\Omega(g(n))$?
Back: $0$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157384-->
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of $cg(n)$ in $\Omega(g(n))$?
Back: $f(n)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157386-->
END%%
%%ANKI
Basic
Using typical identifiers, what is the lower bound of $f(n)$ in $\Omega(g(n))$?
Back: $cg(n)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157388-->
END%%
%%ANKI
Basic
Using typical identifiers, what is the upper bound of $f(n)$ in $\Omega(g(n))$?
Back: N/A
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157390-->
END%%
%%ANKI
Cloze
Given $f(n) = \Omega(g(n))$, we say {1:$g(n)$} is an asymptotic {2:lower} bound for {1:$f(n)$}.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157392-->
END%%
%%ANKI
Basic
Which notation corresponds to asymptotic lower bounds?
Back: $\Omega$-notation.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157393-->
END%%
%%ANKI
Basic
Every member of $\Omega(g(n))$ is expected to be asymptotically what?
Back: Nonnegative.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157394-->
END%%
%%ANKI
Basic
What condition must $g(n)$ satisfy such that otherwise $\Omega(g(n))$ is empty?
Back: $g(n)$ must be asymptotically nonnegative.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157396-->
END%%
%%ANKI
Basic
How is $\Omega(g(n))$ defined?
Back: $\{ \exists c, n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) \leq f(n) \}$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157397-->
END%%
%%ANKI
Basic
Which asymptotic notation is this image demonstrating?
![[big-omega-notation.png]]
Back: $\Omega$-notation
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157399-->
END%%
%%ANKI
Basic
In set-theoretic notation, what does it mean for $\Theta$-notation to be stronger than $\Omega$-notation?
Back: $\Theta(g(n)) \subseteq \Omega(g(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157401-->
END%%
%%ANKI
Basic
What notation corresponds to best-case running times?
Back: $\Omega$-notation
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157402-->
END%%
%%ANKI
Cloze
{1:$O$}-notation is to asymptotic {2:upper}-bounds whereas {2:$\Omega$}-notation is to asymptotic {1:lower}-bounds.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157404-->
END%%
%%ANKI
Basic
What theorem relates $\Theta(g(n))$, $O(g(n))$, and $\Omega(g(n))$?
Back: $f(n) = \Theta(g(n))$ if and only if $f(n) \in O(g(n))$ and $f(n) \in \Omega(g(n))$.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157406-->
END%%
## References

111
notes/binary/integer-encoding.md
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@ -530,6 +530,116 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
<!--ID: 1708545383265-->
END%%
%%ANKI
Basic
Why is two's-complement named the way it is?
Back: Because there is only one $2$ in $2^w - x$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709060837130-->
END%%
%%ANKI
Basic
Given two's-complement $x \geq 0$, what is the significance of $2^w - x$?
Back: The result is the binary representation of $-x$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1709060849456-->
END%%
%%ANKI
Basic
Let $x$ be a $w$-bit two's-complement number. What is it's complement?
Back: The number $y$ such that $x + y = 2^w$.
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: 1709060837141-->
END%%
%%ANKI
Basic
What is the precise definition of the two's-complement of a $w$-bit number?
Back: The complement of $x$ with respect to $2^w$.
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: 1709060837145-->
END%%
%%ANKI
Basic
With respect to two's-complement encoding, what is the "weird number"?
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: 1709060837149-->
END%%
%%ANKI
Basic
Why is $TMin$ called the "weird number"?
Back: It is the only number that is it's own 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).
<!--ID: 1709060837151-->
END%%
%%ANKI
Basic
How is $2^w - x$ written schematically, fixed to $w = 8$ bits?
Back:
```
00000000
- x
----------
...
```
Reference: Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html).
<!--ID: 1709060837154-->
END%%
%%ANKI
Basic
How is the following rewritten to emphasize why "two's-complement" is named the way it is?
```
00000000
- 01010101
----------
...
```
Back:
```
100000000
- 01010101
-----------
...
```
Reference: Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html).
<!--ID: 1709060837156-->
END%%
%%ANKI
Basic
How is the following rewritten to emphasize two's-complement's idea of "invert and add one"?
```
100000000
- 01010101
-----------
...
```
Back:
```
1
+ 11111111
- 01010101
----------
...
```
Reference: Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html).
<!--ID: 1709060837160-->
END%%
%%ANKI
Cloze
Subtracting $x$ from {a number of all `1` bits} is equivalent to {`~x`}.
Reference: Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html).
<!--ID: 1709061032778-->
END%%
## Casting
Most implementations of C cast an object of one type to another by simply re-interpreting the object's 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:
@ -1397,5 +1507,6 @@ END%%
## References
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
* Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html).
* 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).