Add notes on sorting algorithms and equiv-trans.

2024-02-08 07:02:59 -07:00 · 2024-02-08 07:02:59 -07:00 · 00f23d5f94
parent 9cabe16101
commit 00f23d5f94
12 changed files with 652 additions and 27 deletions
--- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
+++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
@ -72,11 +72,14 @@
      "**/*.excalidraw.md"
    ]
  },
-  "Added Media": [],
+  "Added Media": [
+    "insertion-sort.gif",
+    "selection-sort.gif"
+  ],
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-    "algorithms/sorting/insertion-sort.md": "6ad645860c9c281eb8eb06c93135746f",
+    "algorithms/sorting/index.md": "6fada1f3d5d3af64687719eb465a5b97",
+    "algorithms/sorting/insertion-sort.md": "a531d611f4e2908fc4153b1b92077661",
    "bash/index.md": "3b5296277f095acdf16655adcdf524af",
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+    "posix/regexp.md": "e41bf86b770958316df1e20578d6020f",
    "journal/2024-02-04.md": "e2b5678fc53d7284b71ed6820c02b954",
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+    "algorithms/index 1.md": "6fada1f3d5d3af64687719eb465a5b97"
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  "fields_dict": {
    "Basic": [
--- a/notes/_journal/2024-02-07.md
+++ b/notes/_journal/2024-02-07.md
@ -4,10 +4,13 @@ title: "2024-02-07"

 - [x] Anki Flashcards
 - [x] KoL
- [ ] Sheet Music (10 min.)
+- [x] Sheet Music (10 min.)
 - [ ] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
+- [x] Korean (Read 1 Story)
+- [x] Interview Prep (1 Practice Problem)
+- [x] Log Work Hours (Max 3 hours)

-* Read section 4.1 of "GAWK: Effective AWK Programming".
+* Read section 4.1 of "GAWK: Effective AWK Programming".
+* Began translating more of "The Science of Programming" into flashcards.
+* Begin re-reading order of growth concepts.
+* Solved [Palindrome Number](https://leetcode.com/problems/palindrome-number/) and [Regular Expression Matching](https://leetcode.com/problems/regular-expression-matching/description/) (though my solution in the latter is subpar, using recursion instead of dynamic programming).
--- a/notes/_journal/2024-02-08.md
+++ b/notes/_journal/2024-02-08.md
@ -0,0 +1,13 @@
+---
+title: "2024-02-08"
+---
+
+- [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)
+
+* Add notes on selection sort.
--- a/notes/algorithms/sorting/index.md
+++ b/notes/algorithms/sorting/index.md
@ -1,7 +1,7 @@
 ---
 title: Sorting
 TARGET DECK: Obsidian::STEM
-FILE TAGS: algorithm sorting
+FILE TAGS: algorithm::sorting
 tags:
  - algorithm
  - sorting
--- a/notes/algorithms/loop-invariant.md
+++ b/notes/algorithms/loop-invariant.md
@ -0,0 +1,90 @@
+---
+title: Loop Invariant
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algorithm
+tags:
+  - algorithm
+---
+
+## Overview
+
+A loop invariant $P$ is a condition that holds before, during, and after each iteration of a loop (e.g. `for` or `while`). These "timings" correspond to the three necessary properties of an invariant:
+
+* Initialization
+	* $P$ is true before the first iteration of the loop.
+* Maintenance
+	* If $P$ is true before an iteration, $P$ is also true before the next iteration.
+* Termination
+	* $P$ provides a condition used to prove an algorithm's correctness.
+
+%%ANKI
+Basic
+What are the three necessary properties of a loop invariant?
+Back: Initialization, maintenance, and termination.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707329732922-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for loop invariant $P$ to respect initialization?
+Back: $P$ is true before the first iteration of the loop.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707329732926-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for loop invariant $P$ to respect maintenance?
+Back: If $P$ is true before an iteration, $P$ is also true before the next iteration.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707329732928-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for loop invariant $P$ to respect termination?
+Back: $P$ provides a condition used to prove an algorithm's correctness.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707329732929-->
+END%%
+
+%%ANKI
+Basic
+At what point in the following should initialization of a loop invariant be checked?
+```c
+for (int i = 0; i < n; ++i) { ... }
+```
+Back: After `int i = 0` but before `i < n`.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+Tags: c
+<!--ID: 1707329732931-->
+END%%
+
+Notice loop invariants mirror mathematical induction. Initialization is analogous to an inductive base case while iteration is analogous to the inductive step.
+
+%%ANKI
+Cloze
+Loop invariants are to {initialization} whereas mathematical induction is to {a base case}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707329951146-->
+END%%
+
+%%ANKI
+Cloze
+Loop invariants are to {maintenance} whereas mathematical induction is to {the inductive step}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707329951148-->
+END%%
+
+%%ANKI
+Basic
+Which loop invariant property has no analogy to mathematical induction?
+Back: Termination
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707329951150-->
+END%%
+
+## References
+
+* Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
--- a/notes/algorithms/order-growth.md
+++ b/notes/algorithms/order-growth.md
@ -0,0 +1,96 @@
+---
+title: Order of Growth
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algorithm::complexity
+tags:
+  - algorithm
+  - complexity
+---
+
+## Overview
+
+The **running time** of an algorithm is usually considered as a function of its **input size**. How input size is measured depends on the problem at hand. For instance, [[index 1|sorting]] algorithms have an input size corresponding to the number of elements to sort.
+
+%%ANKI
+Basic
+How is the running time of a program traditionally measured?
+Back: As a function of its input size.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707334419352-->
+END%%
+
+%%ANKI
+Basic
+How do you determine the input size used to measure an algorithm's running time?
+Back: This depends entirely on the specific problem/algorithm.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707334419356-->
+END%%
+
+%%ANKI
+Basic
+What *concrete* measure is typically used to measure running time?
+Back: The number of primitive operations executed.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707334419359-->
+END%%
+
+%%ANKI
+Basic
+What *abstract* measure is typically used to measure running time?
+Back: It's order of growth.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707344177499-->
+END%%
+
+%%ANKI
+Basic
+Why does Cormen et al. state the scope of average-case analysis is limited?
+Back: What constitutes an "average" input isn't always clear.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707334419363-->
+END%%
+
+%%ANKI
+Basic
+What about running time are algorithm designers mostly interested in?
+Back: It's order of growth.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707344177503-->
+END%%
+
+%%ANKI
+Basic
+How does order of growth relate to running time?
+Back: Order of growth measures how quickly running time grows with respect to input size.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707344177506-->
+END%%
+
+%%ANKI
+Basic
+Why are lower-ordered terms ignored when determining order of growth?
+Back: They become less significant as input size grows.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707344177510-->
+END%%
+
+%%ANKI
+Basic
+Why are leading coefficients ignored when determining order of growth?
+Back: They become less significant as input size grows.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707344177513-->
+END%%
+
+%%ANKI
+Basic
+Polynomials describing order of growth usually have what two parts ignored?
+Back: Coefficients and lower-ordered terms.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707344177515-->
+END%%
+
+## Reference
+
+* Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
--- a/notes/algorithms/sorting/insertion-sort.gif
+++ b/notes/algorithms/sorting/insertion-sort.gif
--- a/notes/algorithms/sorting/insertion-sort.md
+++ b/notes/algorithms/sorting/insertion-sort.md
@ -1,7 +1,7 @@
 ---
 title: Insertion Sort
 TARGET DECK: Obsidian::STEM
-FILE TAGS: algorithm sorting
+FILE TAGS: algorithm::sorting
 tags:
  - algorithm
  - sorting
@ -9,16 +9,16 @@ tags:

 ## Overview

-| Property   | Value    |
-| ---------- | -------- |
-| Best Case  | $O(n)$   |
-| Worst Case | $O(n^2)$ |
-| Avg. Case  | $O(n^2)$ |
-| Memory     | $O(1)$   |
-| In place   | Yes      |
-| Stable     | Yes      |
+Property   | Value
+---------- | --------
+Best Case  | $O(n)$
+Worst Case | $O(n^2)$
+Avg. Case  | $O(n^2)$
+Memory     | $O(1)$
+In place   | Yes
+Stable     | Yes

-Insertion sort works by advancing an index `i` through an array `A[1..n]` such that `A[1..i]` is kept in sorted order.
+![[insertion-sort.gif]]

 %%ANKI
 Basic
@ -52,6 +52,14 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
 <!--ID: 1706926586951-->
 END%%

+%%ANKI
+Basic
+What is insertion sort's average case runtime?
+Back: $O(n^2)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707329732933-->
+END%%
+
 %%ANKI
 Basic
 Is insertion sort in place?
@ -83,10 +91,49 @@ void insertion_sort(const int n, int A[static n]) {

 %%ANKI
 Basic
-What loop invariant is maintained in insertion sort?
-Back: `A[1..i]` is in sorted order.
+What sorting algorithm does the following demonstrate?
+![[insertion-sort.gif]]
+Back: Insertion sort.
 Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
-<!--ID: 1706927594718-->
+<!--ID: 1707400559085-->
+END%%
+
+## Loop Invariant
+
+Consider [[loop-invariant|loop invariant]] $P$ given by
+
+> `A[0..i-1]` consists of the original `A[0..i-1]` elements but in sorted order.
+
+We prove $P$ maintains the requisite properties:
+
+* Initialization
+	* When `i = 1`, `A[0..0]` contains a single element. This trivially satisfies $P$.
+* Maintenance
+	* Suppose $P$ holds for some `1 ≤ i < n`. Then `A[0..i-1]` consists of the original `A[0..i-1]` elements but in sorted order. On iteration `i + 1`, the nested for loop puts `A[0..i]` in sorted order. At the end of the iteration, `i` is incremented meaning `A[0..i-1]` still satisfies $P$.
+* Termination
+	* The loop ends because `i < n` is no longer true. Then `i = n`. Since $P$ holds, this means `A[0..n-1]`, the entire array, is in sorted order.
+
+%%ANKI
+Basic
+Given array `A[0..n-1]`, what is insertion sort's loop invariant?
+Back: `A[0..i-1]` consists of the original `A[0..i-1]` elements but in sorted order.
+<!--ID: 1707332638371-->
+END%%
+
+%%ANKI
+Basic
+What is initialization of insertion sort's loop invariant?
+Back: Sorting starts with an singleton array which is trivially sorted.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707332638373-->
+END%%
+
+%%ANKI
+Basic
+What is maintenance of insertion sort's loop invariant?
+Back: Each iteration puts the current element into sorted order.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707332638375-->
 END%%

 ## Analogy
@ -111,6 +158,39 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
 <!--ID: 1706927594732-->
 END%%

+%%ANKI
+Basic
+How does insertion sort partition its input array?
+Back:
+```
+[ sorted | unsorted ]
+```
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707399790957-->
+END%%
+
+%%ANKI
+Basic
+How many comparisons does insertion sort typically perform with `x`?
+```
+[ sorted | x : unsorted ]
+```
+Back: One plus however many elements in `sorted` are greater than `x`.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707399790958-->
+END%%
+
+%%ANKI
+Basic
+Which element will insertion sort move to `sorted`?
+```
+[ sorted | unsorted ]
+```
+Back: The first element of `unsorted`.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707399790960-->
+END%%
+
 ## References

 * Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
--- a/notes/algorithms/sorting/selection-sort.gif
+++ b/notes/algorithms/sorting/selection-sort.gif
--- a/notes/algorithms/sorting/selection-sort.md
+++ b/notes/algorithms/sorting/selection-sort.md
@ -0,0 +1,155 @@
+---
+title: Selection Sort
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algorithm::sorting
+tags:
+  - algorithm
+  - sorting
+---
+
+## Overview
+
+Property   | Value
+---------- | --------
+Best Case  | $O(n^2)$
+Worst Case | $O(n^2)$
+Avg. Case  | $O(n^2)$
+Memory     | $O(1)$
+In Place   | Yes
+Stable     | Yes
+
+![[selection-sort.gif]]
+
+%%ANKI
+Basic
+What is selection sort's best case runtime?
+Back: $O(n^2)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707398773323-->
+END%%
+
+%%ANKI
+Basic
+What is selection sort's worst case runtime?
+Back: $O(n^2)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707398773326-->
+END%%
+
+%%ANKI
+Basic
+What is selection sort's average case runtime?
+Back: $O(n^2)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707398773327-->
+END%%
+
+%%ANKI
+Basic
+Is selection sort in place?
+Back: Yes
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707398773328-->
+END%%
+
+%%ANKI
+Basic
+Is selection sort stable?
+Back: Yes
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707398773330-->
+END%%
+
+```c
+void swap(int i, int j, int *A) {
+  int tmp = A[i];
+  A[i] = A[j];
+  A[j] = tmp;
+}
+
+void selection_sort(const int n, int A[static n]) {
+  for (int i = 0; i < n - 1; ++i) {
+	int mini = i;
+    for (int j = i + 1; j < n; ++j) {
+      if (A[j] < A[mini]) {
+	    mini = j;
+      }
+    }
+    swap(i, mini, A);
+  }
+}
+```
+
+%%ANKI
+Basic
+What sorting algorithm does the following demonstrate?
+![[selection-sort.gif]]
+Back: Selection sort.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707400943836-->
+END%%
+
+## Loop Invariant
+
+Consider [[loop-invariant|loop invariant]] $P$ given by
+
+> On each iteration, `A[0..i-1]` is a sorted array of the `i` least elements of `A`.
+
+We prove $P$ maintains the requisite properties:
+
+* Initialization
+	* When `i = 0`, `A[0..-1]` is an empty array. This trivially satisfies $P$.
+* Maintenance
+	* Suppose $P$ holds for some `0 ≤ i < n - 1`. Then `A[0..i-1]` is a sorted array of the `i` least elements of `A`. Our inner loop then finds the smallest element in `A[i..n]` and swaps it with `A[i]`. Therefore `A[0..i]` is not a sorted array of the `i + 1` least elements of `A`. At the end of the iteration, `i` is incremented meaning `A[0..i-1]` still satisfies $P$.
+* Termination
+	* On termination, `i = n - 1` and `A[0..n-2]` are the `n - 1` least elements of `A` in sorted order. But, by exhaustion, `A[n-1]` must be the largest element meaning `A[0..n-1]`, the entire array, is in sorted order.
+
+%%ANKI
+Basic
+Given array `A[0..n-1]`, what is selection sort's loop invariant?
+Back: `A[0..i-1]` is a sorted array of the `i` least elements of `A`.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707398773331-->
+END%%
+
+%%ANKI
+Basic
+What is initialization of selection sort's loop invariant?
+Back: Sorting starts with an empty array which is trivially sorted.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707398773333-->
+END%%
+
+%%ANKI
+Basic
+What is maintenance of selection sort's loop invariant?
+Back: Each iteration puts the next least element into the sorted subarray.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707398773334-->
+END%%
+
+%%ANKI
+Basic
+How does selection sort partition its input array?
+Back:
+```
+[ sorted | unsorted ]
+```
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707399790952-->
+END%%
+
+%%ANKI
+Basic
+Which element will selection sort move to `sorted`?
+```
+[ sorted | unsorted ]
+```
+Back: The least element in `unsorted`.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1707399790955-->
+END%%
+
+## References
+
+* Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
--- a/notes/logic/equiv-trans.md
+++ b/notes/logic/equiv-trans.md
@ -499,6 +499,14 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--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
@ -563,6 +571,44 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1707253246458-->
 END%%

+%%ANKI
+Basic
+What is a "theorem" in the equivalence-transformation formal system?
+Back: An equality 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%%
+
 ## Normal Forms

 Every proposition can be written in **disjunctive normal form** (DNF) and **conjunctive normal form** (CNF). This is evident with the use of truth tables. To write a proposition in DNF, write its corresponding truth table and $\lor$ each row that evaluates to $T$. To write the same proposition in CNF, apply $\lor$ to each row that evaluates to $F$ and negate it.
@ -644,6 +690,138 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1707311869003-->
 END%%

+## Short-Circuit Evaluation
+
+The $\textbf{cand}$ and $\textbf{cor}$ operator allows short-circuiting evaluation in the case of undefined ($U$) values.
+
+%%ANKI
+Basic
+What truth values do short-circuit evaluation operators act on?
+Back: $T$, $F$, and $U$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708622-->
+END%%
+
+%%ANKI
+Basic
+What C operator corresponds to $\textbf{cand}$?
+Back: `&&`
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Tags: c
+<!--ID: 1707316606004-->
+END%%
+
+%%ANKI
+Basic
+Why is $\textbf{cand}$ named the way it is?
+Back: It is short for **c**onditional **and**.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708625-->
+END%%
+
+%%ANKI
+Basic
+How is $p \textbf{ cand } q$ written as a conditional?
+Back: $\textbf{if } p \textbf{ then } q \textbf{ else } F$ 
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708627-->
+END%%
+
+%%ANKI
+Basic
+When can $\textbf{cand}$ evaluate to a non-$U$ value despite being given a $U$ operand?
+Back: $F \textbf{ cand } U = F$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708628-->
+END%%
+
+%%ANKI
+Basic
+What C operator corresponds to $\textbf{cor}$?
+Back: `||`
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Tags: c
+<!--ID: 1707316606007-->
+END%%
+
+%%ANKI
+Basic
+Why is $\textbf{cor}$ named the way it is?
+Back: It is short for **c**onditional **or**.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708630-->
+END%%
+
+%%ANKI
+Basic
+How is $p \textbf{ cor } q$ written as a conditional?
+Back: $\textbf{if } p \textbf{ then } T \textbf{ else } q$ 
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708632-->
+END%%
+
+%%ANKI
+Basic
+When can $\textbf{cor}$ evaluate to a non-$U$ value despite being given a $U$ operand?
+Back: $T \textbf{ cor } U = T$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708633-->
+END%%
+
+* Associative Laws
+	* $E1 \textbf{ cand } (E2 \textbf{ cand } E3) = (E1 \textbf{ cand } E2) \textbf{ cand } E3$
+	* $E1 \textbf{ cor } (E2 \textbf{ cor } E3) = (E1 \textbf{ cor } E2) \textbf{ cor } E3$
+
+%%ANKI
+Basic
+Which of the short-circuit logical operators do the commutative laws apply to?
+Back: Neither of them.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708635-->
+END%%
+
+%%ANKI
+Basic
+Which of the short-circuit logical operators do the associative laws apply to?
+Back: $\textbf{cand}$ and $\textbf{cor}$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708636-->
+END%%
+
+* Distributive Laws
+	* $E1 \textbf{ cand } (E2 \textbf{ cor } E3) = (E1 \textbf{ cand } E2) \textbf{ cor } (E1 \textbf{ cand } E3)$
+	* $E1 \textbf{ cor } (E2 \textbf{ cand } E3) = (E1 \textbf{ cor } E2) \textbf{ cand } (E1 \textbf{ cor } E3)$
+
+%%ANKI
+Basic
+What is the distributive law of e.g. $\textbf{cor}$ over $\textbf{cand}$?
+Back: $E1 \textbf{ cor } (E2 \textbf{ cand } E3) = (E1 \textbf{ cor } E2) \textbf{ cand } (E1 \textbf{ cor } E3)$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708638-->
+END%%
+
+* De Morgan's Laws
+	* $\neg (E1 \textbf{ cand } E2) = \neg E1 \textbf{ cor } \neg E2$
+	* $\neg (E1 \textbf{ cor } E2) = \neg E1 \textbf{ cand } \neg E2$
+
+%%ANKI
+Basic
+Which of the short-circuit logical operators do De Morgan's Laws apply to?
+Back: $\textbf{cand}$ and $\textbf{cor}$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708640-->
+END%%
+
+%%ANKI
+Basic
+What is De Morgan's Law of e.g. $\textbf{cor}$?
+Back: $\neg (E1 \textbf{ cor } E2) = \neg E1 \textbf{ cand } \neg E2$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707317708642-->
+END%%
+
+Gries lists other "Laws" but they don't seem as important to note here. What's worth noting is that the other [[#Equivalence Schemas]] listed above still apply if we can limit operands to just $T$ and $F$.
+
 ## References

 * Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
--- a/notes/posix/regexp.md
+++ b/notes/posix/regexp.md
@ -132,7 +132,7 @@ END%%

 %%ANKI
 Basic
-`^` and `$` belong to what operator category?
+`^` and `$$` belong to what operator category?
 Back: Anchors
 Reference: “POSIX Basic Regular Expressions,” accessed February 4, 2024, [https://en.wikibooks.org/wiki/Regular_Expressions/POSIX_Basic_Regular_Expressions](https://en.wikibooks.org/wiki/Regular_Expressions/POSIX_Basic_Regular_Expressions).
 <!--ID: 1707050923643-->
@ -243,7 +243,7 @@ Notation for describing a class of characters specific to a given locale/charact

 %%ANKI
 Basic
-What inconsistency do character classes introduce?
+What portability issue do character classes introduce?
 Back: Matching characters are dependent on locale/character set.
 Reference: Robbins, Arnold D. “GAWK: Effective AWK Programming,” October 2023. [https://www.gnu.org/software/gawk/manual/gawk.pdf](https://www.gnu.org/software/gawk/manual/gawk.pdf)
 <!--ID: 1707050923719-->