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Heaps and science of programming notes.

c-declarations
Joshua Potter 2024-04-29 07:29:03 -06:00
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---
title: "2024-04-29"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [x] Log Work Hours (Max 3 hours)
* Notes on chapter 5.3 of "The Science of Programming". Covered nested arrays.
* Read chapter 6 of "The Science of Programming". Still need to convert into notes though.

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---
title: "2024-04-26"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [x] Log Work Hours (Max 3 hours)

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---
title: "2024-04-27"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
* Hide-and-Seek Application
* Finished most game logic. Began working on theming.

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---
title: "2024-04-28"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [x] Log Work Hours (Max 3 hours)
* Hide-and-Seek Application
* Added booting and teardown game server logic.
* Begin theming according to Fort Collins vendor guidelines.
* Notes on chapter 5.2 in "The Science of Programming".
* Notes on floor/ceiling identities associated with complete $k$-ary trees.
* Start adding notes/flashcards on heaps and heapsort.

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@ -157,7 +157,7 @@ END%%
%%ANKI
Basic
What C operator corresponds to ceiling division?
Back: None.
Back: N/A.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779716-->
END%%
@ -352,6 +352,136 @@ Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete M
<!--ID: 1708115683366-->
END%%
## Identities
For integers $x$ and $y > 0$, $$\begin{align*} \left\lfloor \frac{x}{y} \right\rfloor & = \left\lceil \frac{x}{y} - \frac{y - 1}{y} \right\rceil \\ \left\lceil \frac{x}{y} \right\rceil & = \left\lfloor \frac{x}{y} + \frac{y - 1}{y} \right\rfloor \end{align*}$$
%%ANKI
Basic
If $n$ is even, what integer value does $\lfloor n / 2 \rfloor$ evaluate to?
Back: $n / 2$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1714182124789-->
END%%
%%ANKI
Basic
If $n$ is odd, what integer value does $\lfloor n / 2 \rfloor$ evaluate to?
Back: $(n - 1) / 2$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1714182124796-->
END%%
%%ANKI
Basic
If $n$ is even, what integer value does $\lceil n / 2 \rceil$ evaluate to?
Back: $n / 2$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1714182124804-->
END%%
%%ANKI
Basic
If $n$ is odd, what integer value does $\lceil n / 2 \rceil$ evaluate to?
Back: $(n + 1) / 2$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1714182124809-->
END%%
%%ANKI
Basic
Given integers $x$ and $y > 0$, what value of $Bias$ satisfies the following identity? $$\left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x}{y} + Bias \right\rfloor$$
Back: $(y - 1) / y$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714182124840-->
END%%
%%ANKI
Basic
Given integers $x$ and $y > 0$, what value of $Bias$ satisfies the following identity? $$\left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x + Bias}{y} \right\rfloor$$
Back: $(y - 1)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714184300367-->
END%%
%%ANKI
Basic
Given integers $x$ and $y > 0$, what operator satisfies the following identity? $$\left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x}{y} \;\square\; \frac{y - 1}{y} \right\rfloor$$
Back: $+$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714182124853-->
END%%
%%ANKI
Basic
What intuition explains why the following identity holds for integers $x$ and $y > 0$? $$\left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x}{y} + \frac{y - 1}{y} \right\rfloor$$
Back: $(y - 1) / y$ only affects the RHS if and only if $x / y$ is not an integer.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714182124860-->
END%%
%%ANKI
Basic
Given integers $x$ and $y > 0$, what value of $Bias$ satisfies the following identity? $$\left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x}{y} - Bias \right\rceil$$
Back: $(y - 1) / y$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714182124874-->
END%%
%%ANKI
Basic
Given integers $x$ and $y > 0$, what value of $Bias$ satisfies the following identity? $$\left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x - Bias}{y} \right\rceil$$
Back: $(y - 1)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714184300372-->
END%%
%%ANKI
Basic
Given integers $x$ and $y > 0$, what operator satisfies the following identity? $$\left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x}{y} \;\square\; \frac{y - 1}{y} \right\rceil$$
Back: $-$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714182124867-->
END%%
%%ANKI
Basic
What intuition explains why the following identity holds for integers $x$ and $y > 0$? $$\left\lfloor \frac{x}{y} \right\rfloor = \left\lceil \frac{x}{y} - \frac{y - 1}{y} \right\rceil$$
Back: $(y - 1) / y$ only affects the RHS if and only if $x / y$ is not an integer.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714182124884-->
END%%
%%ANKI
Cloze
For any integer $n$, floor expression {$\lfloor n / 2 \rfloor$} is equal to ceiling expression {$\lceil (n - 1) / 2 \rceil$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714349367669-->
END%%
%%ANKI
Cloze
For any integer $n$, ceiling expression {$\lceil n / 2 \rceil$} is equal to floor expression {$\lfloor (n + 1) / 2 \rfloor$}.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714349367676-->
END%%
%%ANKI
Basic
What identity generalizes the following? $$\left\lfloor \frac{n}{2} \right\rfloor = \left\lceil \frac{n - 1}{2} \right\rceil$$
Back: $$\left\lfloor \frac{n}{d} \right\rfloor = \left\lceil \frac{n - (d - 1)}{d} \right\rceil$$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714349367682-->
END%%
%%ANKI
Basic
What identity generalizes the following? $$\left\lceil \frac{n}{2} \right\rceil = \left\lfloor \frac{n + 1}{2} \right\rfloor$$
Back: $$\left\lceil \frac{n}{d} \right\rceil = \left\lfloor \frac{n + (d - 1)}{d} \right\rfloor$$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714349367688-->
END%%
## Bibliography
* 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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## Overview
TODO
The **binary heap** data structure is an array object that can be viewed as a [[trees#Positional Trees|complete binary tree]].
%%ANKI
Cloze
A binary heap is an {array} that can be viewed as a {binary tree}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379014-->
END%%
%%ANKI
Basic
Is the following a valid binary heap?
![[perfect-tree.png]]
Back: Yes.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379021-->
END%%
%%ANKI
Basic
Is the following a valid binary heap?
![[complete-tree.png]]
Back: Yes.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379024-->
END%%
%%ANKI
Basic
Is the following a valid binary heap?
![[non-complete-tree.png]]
Back: No.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379030-->
END%%
%%ANKI
Basic
Why can't the following be a binary heap?
![[non-complete-tree.png]]
Back: A heap is equivalently viewed as a *complete* binary tree.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379034-->
END%%
%%ANKI
Basic
What two sizes are associated with binary heaps?
Back: The number of valid elements and the size of the underlying array.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379038-->
END%%
%%ANKI
Basic
What are the two kinds of binary heaps?
Back: Max-heaps and min-heaps.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379042-->
END%%
%%ANKI
Basic
What is the max-heap property?
Back: Every parent node is greater than or equal to its children in value.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379046-->
END%%
%%ANKI
Basic
Where is the largest element of a max-heap?
Back: At the root.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379052-->
END%%
%%ANKI
Basic
Where is the smallest element of a max-heap?
Back: At the leaves.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379059-->
END%%
%%ANKI
Basic
What is the min-heap property?
Back: Every parent node is less than or equal to its children in value.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379072-->
END%%
%%ANKI
Basic
Where is the smallest element of a min-heap?
Back: At the root.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379077-->
END%%
%%ANKI
Basic
Where is the largest element of a min-heap?
Back: At the leaves.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379083-->
END%%
%%ANKI
Basic
How is the following binary heap viewed as an array?
![[max-heap-tree.png]]
Back:
![[max-heap-array.png]]
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356379065-->
END%%
%%ANKI
Basic
How is the following binary heap instead viewed as a binary tree?
![[max-heap-array.png]]
Back:
![[max-heap-tree.png]]
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356442370-->
END%%
%%ANKI
Basic
What kind of binary heap is the following?
![[max-heap-array.png]]
Back: A max-heap.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714356546616-->
END%%
## Bibliography
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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---
title: Heapsort
TARGET DECK: Obsidian::STEM
FILE TAGS: algorithm::sorting
tags:
- algorithm
- sorting
---
## Overview
Property | Value
----------- | --------
Best Case | -
Worst Case | -
Avg. Case | -
Aux. Memory | -
Stable | -
Adaptive | -
![[heapsort.gif]]
```c
inline int left_child(int i) { return (i << 1) + 1; }
inline int right_child(int i) { return (i << 1) + 2; }
void max_heapify(int n, int H[static n], int i) {
while (true) {
int lc = left_child(i);
int rc = right_child(i);
int next = i;
if (lc < n && H[next] < H[lc]) {
next = lc;
}
if (rc < n && H[next] < H[rc]) {
next = rc;
}
if (next == i) {
return;
}
swap(H, i, next);
i = next;
}
}
void build_max_heap(int n, int H[static n]) {
for (int i = n / 2 - 1; i >= 0; --i) {
max_heapify(n, H, i);
}
}
void heapsort(int n, int H[static n]) {
build_max_heap(n, H);
while (n > 1) {
swap(A, 0, --n);
max_heapify(n, A, 0);
}
}
```
## Bibliography
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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@ -2114,6 +2114,67 @@ Tags: c17
<!--ID: 1709831032461-->
END%%
%%ANKI
Basic
Assuming no overflow, rewrite expression `x >> k` to instead yield $\lceil x / 2^k \rceil$.
Back: `(x + (1 << k) - 1) >> k`
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: 1714184300343-->
END%%
%%ANKI
Basic
Assuming no overflow, what is the result of `(x + (1 << k) - 1) >> k`?
Back: $\lceil x / 2^k \rceil$
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: 1714184300349-->
END%%
%%ANKI
Basic
What value of $Bias$ satisfies the following identity? $$\left\lceil \frac{x}{2^k} \right\rceil = \left\lfloor \frac{x}{2^k} + Bias \right\rfloor$$
Back: $(2^k - 1) / 2^k$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714184300352-->
END%%
%%ANKI
Basic
What value of $Bias$ satisfies the following identity? $$\left\lceil \frac{x}{2^k} \right\rceil = \left\lfloor \frac{x + Bias}{2^k} \right\rfloor$$
Back: $2^k - 1$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
<!--ID: 1714184300355-->
END%%
%%ANKI
Basic
What floor/ceiling identity does expression `(x + (1 << k) - 1) >> k` exploit?
Back: $$\left\lceil \frac{x}{y} \right\rceil = \left\lfloor \frac{x + y - 1}{y} \right\rfloor$$
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: 1714184300359-->
END%%
%%ANKI
Basic
In two's-complement, how do we use `>>` to perform integer division of `x > 0` by $2^k$?
Back: `x >> k`
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: 1714184300362-->
END%%
%%ANKI
Basic
In two's-complement, how do we use `>>` to perform integer division of `x < 0` by $2^k$?
Back: `(x + (1 << k) - 1) >> k`
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: 1714184300364-->
END%%
## Bibliography
* 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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---
title: Assertions
TARGET DECK: Obsidian::STEM
FILE TAGS: programming::assertions
tags:
- assertions
- programming
---
## Overview
TODO
## Bibliography
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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---
title: Equivalence Transformation
TARGET DECK: Obsidian::STEM
FILE TAGS: programming::equiv-trans
tags:
- equiv-trans
- logic
- programming
---
## Overview
**Equivalence-transformation** refers to a class of calculi for [[propositional|propositional logic]] derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and equality ($=$). Gries covers two in "The Science of Programming": a system of evaluation and a formal system. The system of evaluation mirrors how a computer processes instructions, at least in an abstract sense. The formal system serves as a theoretical framework for reasoning about propositions and their transformations without resorting to "lower-level" operations like substitution.
%%ANKI
Basic
Who is the author of "The Science of Programming"?
Back: David Gries
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861286-->
END%%
%%ANKI
Basic
What are constant propositions?
Back: Propositions that contain only constants as operands.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707422675517-->
END%%
%%ANKI
Cloze
Gries replaces logical operator {$\Leftrightarrow$} in favor of {$=$}.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861295-->
END%%
%%ANKI
Basic
How does Lean define propositional equality?
Back: Expressions `a` and `b` are propositionally equal iff `a = b` is true.
Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
Tags: lean
<!--ID: 1706994861298-->
END%%
%%ANKI
Basic
How does Lean define `propext`?
Back:
```lean
axiom propext {a b : Prop} : (a ↔ b) → (a = b)
```
Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
Tags: lean
<!--ID: 1706994861300-->
END%%
%%ANKI
Basic
What Lean theorem justifies Gries' choice of $=$ over $\Leftrightarrow$?
Back: `propext`
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
Tags: lean
<!--ID: 1706994861302-->
END%%
%%ANKI
Basic
Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?
Back: Yes.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861318-->
END%%
%%ANKI
Basic
Is $(b \lor d)$ well-defined in $\{(b, T), (c, F)\}$?
Back: No.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861320-->
END%%
%%ANKI
Basic
What proposition represents states $\{(b, T)\}$ and $\{(c, F)\}$?
Back: $b \lor \neg c$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861337-->
END%%
%%ANKI
Basic
What set of states does $a \land b$ represent?
Back: The set containing just state $\{(a, T), (b, T)\}$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861339-->
END%%
%%ANKI
Basic
What is sloppy about phrase "the states in $b \lor \neg c$"?
Back: $b \lor \neg c$ is not a set but a representation of a set (of states).
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861341-->
END%%
%%ANKI
Basic
What is the weakest proposition?
Back: $T$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861348-->
END%%
%%ANKI
Basic
What set of states does $T$ represent?
Back: The set of all states.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861350-->
END%%
%%ANKI
Basic
What is the strongest proposition?
Back: $F$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861352-->
END%%
%%ANKI
Basic
What set of states does $F$ represent?
Back: The set of no states.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861354-->
END%%
%%ANKI
Basic
What does a proposition *represent*?
Back: The set of states in which it is true.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861335-->
END%%
%%ANKI
Basic
When is $p$ stronger than $q$?
Back: When $p \Rightarrow q$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861343-->
END%%
%%ANKI
Basic
When is $p$ weaker than $q$?
Back: When $q \Rightarrow p$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861346-->
END%%
%%ANKI
Basic
A proposition is well-defined with respect to what?
Back: A state to evaluate against.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861316-->
END%%
%%ANKI
Basic
Why is $b \land c$ stronger than $b \lor c$?
Back: The former represents a subset of the states the latter represents.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861356-->
END%%
%%ANKI
Basic
What is a state?
Back: A function mapping identifiers to values.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861314-->
END%%
%%ANKI
Basic
What are the two calculi Gries describes equivalence-transformation with?
Back: A formal system and a system of evaluation.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673342-->
END%%
## Equivalence Schemas
A proposition is said to be a **tautology** if it evaluates to $T$ in every state it is well-defined in. We say propositions $E1$ and $E2$ are **equivalent** if $E1 = E2$ is a tautology. In this case, we say $E1 = E2$ is an **equivalence**.
%%ANKI
Basic
What does it mean for a proposition to be a tautology?
Back: That the proposition is true in every state it is well-defined in.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861323-->
END%%
%%ANKI
Basic
How is tautology $e$ written equivalently with a quantifier?
Back: For free identifiers $i_1, \ldots, i_n$ in $e$, as $\forall (i_1, \ldots, i_n), e$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707937867032-->
END%%
%%ANKI
Basic
The term "equivalent" refers to a comparison between what two objects?
Back: Expressions.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673345-->
END%%
%%ANKI
Basic
What does it mean for two propositions to be equivalent?
Back: Given propositions $E1$ and $E2$, it means $E1 = E2$ is a tautology.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673347-->
END%%
%%ANKI
Basic
What is an equivalence?
Back: Given propositions $E1$ and $E2$, tautology $E1 = E2$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673348-->
END%%
* Commutative Laws
* $(E1 \land E2) = (E2 \land E1)$
* $(E1 \lor E2) = (E2 \lor E1)$
* $(E1 = E2) = (E2 = E1)$
%%ANKI
Basic
Which of the basic logical operators do the commutative laws apply to?
Back: $\land$, $\lor$, and $=$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673350-->
END%%
%%ANKI
Basic
What do the commutative laws allow us to do?
Back: Reorder operands.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673351-->
END%%
%%ANKI
Basic
What is the commutative law of e.g. $\land$?
Back: $E1 \land E2 = E2 \land E1$
<!--ID: 1707251673353-->
END%%
* Associative Laws
* $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
* $E1 \lor (E2 \lor E3) = (E1 \lor E2) \lor E3$
%%ANKI
Basic
Which of the basic logical operators do the associative laws apply to?
Back: $\land$ and $\lor$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673354-->
END%%
%%ANKI
Basic
What do the associative laws allow us to do?
Back: Remove parentheses.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673355-->
END%%
%%ANKI
Basic
What is the associative law of e.g. $\land$?
Back: $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673357-->
END%%
* Distributive Laws
* $E1 \lor (E2 \land E3) = (E1 \lor E2) \land (E1 \lor E3)$
* $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
%%ANKI
Basic
Which of the basic logical operators do the distributive laws apply to?
Back: $\land$ and $\lor$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673358-->
END%%
%%ANKI
Basic
What do the distributive laws allow us to do?
Back: "Factor" propositions.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673360-->
END%%
%%ANKI
Basic
What is the distributive law of e.g. $\land$ over $\lor$?
Back: $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673361-->
END%%
* De Morgan's Laws
* $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
* $\neg (E1 \lor E2) = \neg E1 \land \neg E2$
%%ANKI
Basic
Which of the basic logical operators do De Morgan's Laws apply to?
Back: $\neg$, $\land$, and $\lor$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673363-->
END%%
%%ANKI
Basic
What is De Morgan's Law of e.g. $\land$?
Back: $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673364-->
END%%
* Law of Negation
* $\neg (\neg E1) = E1$
%%ANKI
Basic
What does the Law of Negation say?
Back: $\neg (\neg E1) = E1$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673365-->
END%%
* Law of the Excluded Middle
* $E1 \lor \neg E1 = T$
%%ANKI
Basic
Which of the basic logical operators does the Law of the Excluded Middle apply to?
Back: $\lor$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673367-->
END%%
%%ANKI
Basic
What does the Law of the Excluded Middle say?
Back: $E1 \lor \neg E1 = T$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673368-->
END%%
%%ANKI
Basic
Which equivalence schema is "refuted" by sentence, "This sentence is false."
Back: Law of the Excluded Middle
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251779153-->
END%%
* Law of Contradiction
* $E1 \land \neg E1 = F$
%%ANKI
Basic
Which of the basic logical operators does the Law of Contradiction apply to?
Back: $\land$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673370-->
END%%
%%ANKI
Basic
What does the Law of Contradiction say?
Back: $E1 \land \neg E1 = F$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673371-->
END%%
%%ANKI
Cloze
The Law of {1:the Excluded Middle} is to {2:$\lor$} whereas the Law of {2:Contradiction} is to {1:$\land$}.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707251673373-->
END%%
Gries lists other "Laws" but they don't seem as important to note here.
%%ANKI
Basic
How is $\Rightarrow$ written in terms of other logical operators?
Back: $p \Rightarrow q$ is equivalent to $\neg p \lor q$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861358-->
END%%
%%ANKI
Basic
How is $\Leftrightarrow$/$=$ written in terms of other logical operators?
Back: $p \Leftrightarrow q$ is equivalent to $(p \Rightarrow q) \land (q \Rightarrow p)$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1706994861360-->
END%%
%%ANKI
Basic
What distinguishes an equality from an equivalence?
Back: An equivalence is an equality that is also a tautology.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707316178709-->
END%%
## Equivalence Rules
* Rule of Substitution
* Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
* Rule of Transitivity
* Let $E1 = E2$ and $E2 = E3$ be equivalences. Then $E1 = E3$ is an equivalence.
%%ANKI
Basic
What two inference rules make up the equivalence-transformation formal system?
Back: Substitution and transitivity.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707253246450-->
END%%
%%ANKI
Basic
Which of the two inference rules that make up the equivalence-transformation formal system is redundant?
Back: Transitivity.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707432641598-->
END%%
%%ANKI
Basic
What does the rule of substitution say in the system of evaluation?
Back: Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707253246452-->
END%%
%%ANKI
Basic
How is the rule of substitution written as an inference rule (in standard form)?
Back:
$$
\begin{matrix}
E1 = E2 \\
\hline P(E1) = P(E2)
\end{matrix}
$$
<!--ID: 1707253246454-->
END%%
%%ANKI
Basic
What does the rule of transitivity state in the system of evaluation?
Back: Let $E1 = E2$ and $E2 = E3$. Then $E1 = E3$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707253246455-->
END%%
%%ANKI
Basic
How is the rule of transitivity written as an inference rule (in standard form)?
Back:
$$
\begin{matrix}
E1 = E2, E2 = E3 \\
\hline E1 = E3
\end{matrix}
$$
<!--ID: 1707253246457-->
END%%
%%ANKI
Cloze
The system of evaluation has {equivalences} whereas the formal system has {theorems}.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707253246458-->
END%%
%%ANKI
Basic
What is a "theorem" in the equivalence-transformation formal system?
Back: An equivalence derived from the axioms and inference rules.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707316178712-->
END%%
%%ANKI
Basic
How is e.g. the Law of Implication proven in the system of evaluation?
Back: With truth tables
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707316178714-->
END%%
%%ANKI
Basic
How is e.g. the Law of Implication proven in the formal system?
Back: It isn't. It is an axiom.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707316178715-->
END%%
%%ANKI
Cloze
The system of evaluation and formal system are connected by the following biconditional: {$e$ is a tautology} iff {$e = T$ is a theorem}.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707316178717-->
END%%
%%ANKI
Cloze
The {1:system of evaluation} is to {2:"$e$ is a tautology"} whereas the {2:formal system} is to {1:"$e = T$ is a theorem"}.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707316276203-->
END%%
## Bibliography
* Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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---
title: Programming
tags:
- programming
---

1071
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108
notes/set/trees.md
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@ -960,7 +960,7 @@ END%%
%%ANKI
Basic
What does it mean for a $k$-ary tree to be complete?
Back: The last level is not filled but all leaves have the same depth and are leftmost arranged.
Back: All levels, except maybe the last, are filled. All leaves have the same depth and are leftmost arranged.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714080353480-->
END%%
@ -989,6 +989,86 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
<!--ID: 1714082676022-->
END%%
%%ANKI
Basic
How many internal nodes are in a complete $k$-ary tree of $n$ nodes?
Back: $\lceil (n - 1) / k \rceil$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367630-->
END%%
%%ANKI
Basic
What value of $k$ is used in the following description of a complete $k$-ary tree?
$$\begin{array}{c|c|c}
n & \text{external} & \text{internal} \\
\hline
1 & 1 & 0 \\
2 & 1 & 1 \\
3 & 2 & 1 \\
4 & 3 & 1 \\
5 & 4 & 1 \\
6 & 4 & 2 \\
7 & 5 & 2 \\
8 & 6 & 2
\end{array}$$
Back: $4$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367637-->
END%%
%%ANKI
Basic
What value of $k$ is used in the following description of a complete $k$-ary tree?
$$\begin{array}{c|c|c}
n & \text{external} & \text{internal} \\
\hline
1 & 1 & 0 \\
2 & 1 & 1 \\
3 & 2 & 1 \\
4 & 2 & 2 \\
5 & 3 & 2 \\
6 & 3 & 3 \\
7 & 4 & 3 \\
8 & 4 & 4
\end{array}$$
Back: $2$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367640-->
END%%
%%ANKI
Basic
When does the number of external nodes increment in a growing $k$-ary tree?
Back: When the next node added already has a sibling.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367644-->
END%%
%%ANKI
Basic
When does the number of external nodes remain static in a growing $k$-ary tree?
Back: When the next node added has no sibling.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367647-->
END%%
%%ANKI
Basic
When does the number of internal nodes increment in a growing $k$-ary tree?
Back: When the next node added has no sibling.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367651-->
END%%
%%ANKI
Basic
When does the number of internal nodes remain static in a growing $k$-ary tree?
Back: When the next node added already has a sibling.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367655-->
END%%
#### Binary Trees
A **binary tree** $T$ is a structure defined on a finite set of nodes that either
@ -1165,7 +1245,7 @@ Is the following a perfect binary tree?
![[perfect-tree.png]]
Back: Yes.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714080353484-->
<!--ID: 1714180419777-->
END%%
%%ANKI
@ -1174,7 +1254,7 @@ Is the following a complete binary tree?
![[perfect-tree.png]]
Back: Yes.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714080353488-->
<!--ID: 1714180419781-->
END%%
%%ANKI
@ -1183,7 +1263,7 @@ Is the following a full binary tree?
![[perfect-tree.png]]
Back: Yes.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714088438768-->
<!--ID: 1714180419784-->
END%%
%%ANKI
@ -1192,7 +1272,7 @@ Is the following a perfect binary tree?
![[complete-tree.png]]
Back: No.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714080353491-->
<!--ID: 1714180419787-->
END%%
%%ANKI
@ -1201,7 +1281,7 @@ Is the following a complete binary tree?
![[complete-tree.png]]
Back: Yes.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714080353495-->
<!--ID: 1714180419789-->
END%%
%%ANKI
@ -1210,7 +1290,7 @@ Is the following a full binary tree?
![[complete-tree.png]]
Back: No.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714088438773-->
<!--ID: 1714180419793-->
END%%
%%ANKI
@ -1219,7 +1299,7 @@ Is the following a perfect binary tree?
![[non-complete-tree.png]]
Back: No.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714080353498-->
<!--ID: 1714180419802-->
END%%
%%ANKI
@ -1228,7 +1308,7 @@ Is the following a complete binary tree?
![[non-complete-tree.png]]
Back: No.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714080353502-->
<!--ID: 1714180419809-->
END%%
%%ANKI
@ -1237,7 +1317,7 @@ Is the following a full binary tree?
![[non-complete-tree.png]]
Back: No.
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714088438777-->
<!--ID: 1714180419813-->
END%%
%%ANKI
@ -1366,6 +1446,14 @@ Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition
<!--ID: 1712409466653-->
END%%
%%ANKI
Basic
How many internal nodes are in a complete binary tree of $n$ nodes?
Back: $\lceil (n - 1) / 2 \rceil = \lfloor n / 2 \rfloor$
Reference: Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1714349367662-->
END%%
## Bibliography
* “Binary Tree,” in _Wikipedia_, March 13, 2024, [https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees](https://en.wikipedia.org/w/index.php?title=Binary_tree&oldid=1213529508#Types_of_binary_trees).