Additional notes on binary shifting.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

@@ -80,7 +80,7 @@
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-    "algorithms/sorting/insertion-sort.md": "d40da14992d8331a07cebe1c4cfa41d4",
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@@ -89,7 +89,7 @@
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@@ -114,14 +114,14 @@
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     "binary/index.md": "27bdd4423e323cd961ddb307ab28d977",
@@ -152,12 +152,14 @@
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     "logic/truth-tables.md": "7892ceaa416c9a65acc79ca1e6ff778f",
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-    "logic/boolean-algebra.md": "c5d699d407e03d280aa1f201bc3e8b02",
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   "fields_dict": {
     "Basic": [

notes/_journal/2024-02-15.md

@@ -0,0 +1,15 @@
+---
+title: "2024-02-15"
+---
+
+- [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)
+
+* Finished "Ingestion" chapter of "Fundamentals of Data Engineering".
+* Read first chapter of "Concrete Mathematics" and added additional notes on bit shifting.
+* Met with Gus on Soft Skills workshop.

notes/_journal/2024-02/2024-02-14.md

@@ -7,9 +7,10 @@ title: "2024-02-14"
 - [ ] Sheet Music (10 min.)
 - [ ] OGS (1 Life & Death Problem)
 - [x] Korean (Read 1 Story)
-- [ ] Interview Prep (1 Practice Problem)
+- [x] Interview Prep (1 Practice Problem)
 - [x] Log Work Hours (Max 3 hours)
 
 * More flashcards on `printf`. This time on flags.
 * Read 금덩이를 버린 형제 (Two brothers Who Threw Away Their Lumps of Gold).
 * Flashcards are growing much more rapidly with my Obsidian_to_Anki plugin approach. Increased new card limit to 40 and will evaluate how well that works.
+* Notes on textual substitution.

notes/algorithms/sorting/insertion-sort.md

@@ -153,10 +153,10 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
 END%%
 
 %%ANKI
-Basic
+Cloze
 insertion sort makes fewer {comparisons} than selection sort in the average case.
 Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
-<!--ID: 1707500283783-->
+<!--ID: 1708002185982-->
 END%%
 
 ## Analogy

notes/algorithms/sorting/selection-sort.md

@@ -171,7 +171,7 @@ END%%
 Cloze
 Selection sort makes fewer {swaps} than insertion sort in the average case.
 Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
-<!--ID: 1707500283779-->
+<!--ID: 1708002177782-->
 END%%
 
 ## References

notes/binary/shifts.md

@@ -150,7 +150,7 @@ END%%
 
 %%ANKI
 Cloze
-{Arithmetic} right shifts are to {signed} numbers whereas {logical} right shifts are to {unsigned} numbers.
+{1:Arithmetic} right shifts are to {1:signed} numbers whereas {2:logical} right shifts are to {2:unsigned} numbers.
 Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
 Tags: c
 <!--ID: 1707854589813-->
@@ -194,6 +194,102 @@ Tags: c
 <!--ID: 1707873410780-->
 END%%
 
+## Arithmetic
+
+Shifting left by a single bit is equivalent to multiplication by $2$. Likewise, logically shifting right is equivalent to floor division by $2$. This is most clearly seen by examining the decimal expansion of a binary value. For example, consider $$001101_2 = 2^3 + 2^2 + 2^0$$
+
+Multiplying by $2$ yields $$2 \cdot (2^3 + 2^2 + 2^0) = 2^4 + 2^3 + 2^1 = 011010_2$$ 
+
+Dividing by $2$ yields $$(2^3 + 2^2 + 2^0) / 2 = 2^2 + 2^1 = 000110_2$$
+
+%%ANKI
+Basic
+Multiplication by 2 is equivalent to what bitwise shift operation?
+Back: Left shifting by 1 bit.
+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: 1708009783552-->
+END%%
+
+%%ANKI
+Basic
+Floor division by 2 is equivalent to what bitwise shift operation?
+Back: Logical right shifting by 1 bit.
+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: 1708009783562-->
+END%%
+
+%%ANKI
+Basic
+Ceiling division by 2 is equivalent to what bitwise shift operation?
+Back: None.
+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: 1708009783565-->
+END%%
+
+%%ANKI
+Cloze
+{1:Multiplication} by 2 is to {2:left shifts} whereas {2:floor division} by 2 is to {1:logical right shifts}.
+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: 1708009783568-->
+END%%
+
+%%ANKI
+Basic
+What arithmetic operation is equivalent to shifting left by $n$ bits?
+Back: Multiplication by $2^n$.
+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: 1708009783571-->
+END%%
+
+%%ANKI
+Basic
+When is $2n + 1$ equivalent to a one-bit cyclic shift left?
+Back: When $n$'s leading bit is `1`.
+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: 1708012138055-->
+END%%
+
+%%ANKI
+Basic
+What arithmetic operation is equivalent to a one-bit cyclic shift left?
+Back: Given $n$ with no leading zeros, $2n + 1$.
+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: 1708012138062-->
+END%%
+
+%%ANKI
+Basic
+What arithmetic operation is equivalent to logically shifting right by $n$ bits?
+Back: Floor division by $2^n$.
+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: 1708009783574-->
+END%%
+
+%%ANKI
+Basic
+Given $n = 0110_2$, what is the binary value of $2n$?
+Back: $1100_2$
+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: 1708012138065-->
+END%%
+
+%%ANKI
+Basic
+Given $n = 0110_2$, what is the binary value of $2n + 1$?
+Back: $1101_2$
+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: 1708012138068-->
+END%%
+
+%%ANKI
+Basic
+Given $n = 0110_2$, what is the binary value of $\lfloor n / 2 \rfloor$?
+Back: $0011_2$
+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: 1708012138071-->
+END%%
+
 ## References
 
 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+* Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).

notes/gawk/variables.md

@@ -192,7 +192,7 @@ END%%
 
 %%ANKI
 Basic
-What value is `${NF + 1}` given when we run `${NF + 2} = "test"`?
+What value is `$${NF + 1}` given when we run `${NF + 2} = "test"`?
 Back: `""`
 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: 1707491299854-->

notes/logic/boolean-algebra.md

@@ -11,6 +11,67 @@ tags:
 
 **Boolean algebra** refers to an algebraic system characterised by a set of axioms. This is something I'll explore further, probably after reading more on abstract algebra. The basic operations of Boolean algebra are negation ($\neg$), conjunction ($\land$), and disjunction ($\lor$). 
 
+%%ANKI
+Basic
+What name is given to $\land$ operands?
+Back: Conjuncts
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861304-->
+END%%
+
+%%ANKI
+Basic
+What name is given to $\lor$ operands?
+Back: Disjuncts
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1706994861306-->
+END%%
+
+%%ANKI
+Basic
+What C logical operator corresponds to $\neg$?
+Back: `!`
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Tags: c
+<!--ID: 1706994861325-->
+END%%
+
+%%ANKI
+Basic
+What C logical operator corresponds to $\land$?
+Back: There isn't one.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Tags: c
+<!--ID: 1706994861327-->
+END%%
+
+%%ANKI
+Basic
+What C logical operator corresponds to $\lor$?
+Back: There isn't one.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Tags: c
+<!--ID: 1706994861329-->
+END%%
+
+%%ANKI
+Basic
+What C logical operator corresponds to $\Rightarrow$?
+Back: There isn't one.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Tags: c
+<!--ID: 1706994861331-->
+END%%
+
+%%ANKI
+Basic
+What C logical operator corresponds to $\Leftrightarrow$?
+Back: `==`
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+Tags: c
+<!--ID: 1706994861333-->
+END%%
+
 %%ANKI
 Basic
 What C bit-level operator corresponds to $\neg$?

notes/logic/equiv-trans.md

@@ -72,22 +72,6 @@ Tags: lean
 <!--ID: 1706994861302-->
 END%%
 
-%%ANKI
-Basic
-What name is given to $\land$ operands?
-Back: Conjuncts
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861304-->
-END%%
-
-%%ANKI
-Basic
-What name is given to $\lor$ operands?
-Back: Disjuncts
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-<!--ID: 1706994861306-->
-END%%
-
 %%ANKI
 Basic
 What name is given to operand $a$ in $a \Rightarrow b$?
@@ -120,51 +104,6 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1706994861320-->
 END%%
 
-%%ANKI
-Basic
-What C logical operator corresponds to $\neg$?
-Back: `!`
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-Tags: c
-<!--ID: 1706994861325-->
-END%%
-
-%%ANKI
-Basic
-What C logical operator corresponds to $\land$?
-Back: There isn't one.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-Tags: c
-<!--ID: 1706994861327-->
-END%%
-
-%%ANKI
-Basic
-What C logical operator corresponds to $\lor$?
-Back: There isn't one.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-Tags: c
-<!--ID: 1706994861329-->
-END%%
-
-%%ANKI
-Basic
-What C logical operator corresponds to $\Rightarrow$?
-Back: There isn't one.
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-Tags: c
-<!--ID: 1706994861331-->
-END%%
-
-%%ANKI
-Basic
-What C logical operator corresponds to $\Leftrightarrow$?
-Back: `==`
-Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
-Tags: c
-<!--ID: 1706994861333-->
-END%%
-
 %%ANKI
 Basic
 What proposition represents states $\{(b, T)\}$ and $\{(c, F)\}$?
@@ -294,6 +233,7 @@ 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
@@ -637,11 +577,36 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1707762304123-->
 END%%
 
+%%ANKI
+Basic
+What term refers to $x$ in textual substitution $E_e^x$?
+Back: The reference.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707939006275-->
+END%%
+
+%%ANKI
+Basic
+What term refers to $e$ in textual substitution $E_e^x$?
+Back: The expression.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707939006283-->
+END%%
+
+%%ANKI
+Basic
+What term refers to both $x$ and $e$ together in textual substitution $E_e^x$?
+Back: The reference-expression pair.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707939006288-->
+END%%
+
 %%ANKI
 Basic
 What identifier is guaranteed to not occur in $E_e^x$?
 Back: None.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867036-->
 END%%
 
 %%ANKI
@@ -649,6 +614,7 @@ Basic
 What identifier is guaranteed to not occur in $E_{s(e)}^x$?
 Back: $x$.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867039-->
 END%%
 
 %%ANKI
@@ -656,6 +622,7 @@ Basic
 *Why* does $x$ not occur in $E_{s(e)}^x$?
 Back: Because $s(e)$ evaluates to a constant proposition.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867042-->
 END%%
 
 %%ANKI
@@ -751,6 +718,7 @@ Basic
 In textual substitution, what does e.g. $\bar{x}$ denote?
 Back: A list of *distinct* identifiers.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867046-->
 END%%
 
 %%ANKI
@@ -838,6 +806,7 @@ Basic
 How does Gries denote state $s$ with identifer $x$ set to value $v$?
 Back: $(s; x{:}v)$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867049-->
 END%%
 
 %%ANKI
@@ -845,6 +814,7 @@ Basic
 How is $(s; x{:}v)$ written instead using set-theoretical notation?
 Back: $(s - \{(x, s(x))\}) \cup \{(x, v)\}$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867053-->
 END%%
 
 %%ANKI
@@ -852,6 +822,7 @@ Basic
 Execution of `x := e` in state $s$ terminates in what new state?
 Back: $(s; x{:}s(e))$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867058-->
 END%%
 
 %%ANKI
@@ -859,6 +830,7 @@ Basic
 Given state $s$, what statement does $(s; x{:}s(e))$ derive from?
 Back: `x := e`
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867062-->
 END%%
 
 %%ANKI
@@ -866,6 +838,7 @@ Basic
 What missing value guarantees state $s$ satisfies equivalence $s = (s; x{:}\_)$?
 Back: $s(x)$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867067-->
 END%%
 
 %%ANKI
@@ -873,35 +846,62 @@ Basic
 Given state $s$, why is it that $s = (s; x{:}s(x))$?
 Back: Evaluating $x$ in state $s$ yields $s(x)$.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867072-->
 END%%
 
 * $s(E_e^x) = s(E_{s(e)}^x)$
 	* Substituting $x$ with $e$ and then evaluating is the same as substituting $x$ with the evaluation of $e$.
-	* This follows directly from the recursive definition of state evaluation.
 
 %%ANKI
 Basic
 How can we simplify $s(E_{s(e)}^x)$?
 Back: As $s(E_e^x)$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867076-->
 END%%
 
 %%ANKI
 Basic
-What equivalence relates $E_e^x$ with $E_{s(e)}^x$?
+Given state $s$, what equivalence relates $E_e^x$ with $E_{s(e)}^x$?
 Back: $s(E_e^x) = s(E_{s(e)}^x)$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707937867080-->
 END%%
 
-TODO
-
 * Let $s$ be a state and $s' = (s; x{:}s(e))$. Then $s'(E) = s(E_e^x)$.
 
-TODO
+%%ANKI
+Cloze
+Let $s$ be a state and $s' = (${$s; x{:}s(e)$}$)$. Then $s'(${$E$}$) = s(${$E_e^x$}$)$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707938187973-->
+END%%
+
+%%ANKI
+Basic
+How can we write an expression equivalent to $s(E_e^x)$ that doesn't use textual substitution?
+Back: Write $s'(E)$ where $s' = (s; x{:}e)$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707939006292-->
+END%%
+
+%%ANKI
+Basic
+Given defined value $v \neq s(x)$, when is $s(E) = (s; x{:}v)(E)$?
+Back: When $x$ is not free in $E$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707939315519-->
+END%%
 
 * Given identifiers $\bar{x}$ and fresh identifiers $\bar{u}$, $(E_{\bar{u}}^{\bar{x}})_{\bar{x}}^{\bar{u}} = E$.
 
-TODO
+%%ANKI
+Basic
+When is $(E_{\bar{u}}^{\bar{x}})_{\bar{x}}^{\bar{u}} = E$ guaranteed to be an equivalence?
+Back: When $\bar{x}$ and $\bar{u}$ are all distinct identifiers.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707939006297-->
+END%%
 
 ## References