Notes on normal forms.

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

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   "fields_dict": {
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notes/_journal/2024-02-11.md

@@ -10,4 +10,5 @@ title: "2024-02-11"
 - [ ] Interview Prep (1 Practice Problem)
 - [ ] Log Work Hours (Max 3 hours)
 
-* Notes on endianness.
+* Notes on endianness.
+* Notes on prenex normal form and normal forms in general.

notes/algorithms/order-growth.md

@@ -91,6 +91,6 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
 <!--ID: 1707344177515-->
 END%%
 
-## Reference
+## References
 
 * Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).

notes/binary/endianness.md

@@ -114,6 +114,6 @@ int main() {
 
 The above snippet can be used to check endianness on the current machine. If big-endian, the output should be `01 23 45 67`. If little-endian, `67 45 23 01`.
 
-## Reference
+## References
 
 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

notes/binary/hexadecimal.md

@@ -222,6 +222,6 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1707432641596-->
 END%%
 
-## Reference
+## References
 
 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

notes/c/declarations.md

@@ -164,6 +164,6 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1707493017246-->
 END%%
 
-## Reference
+## References
 
 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

notes/logic/normal-form.md

@@ -0,0 +1,43 @@
+---
+title: Normal Form
+TARGET DECK: Obsidian::STEM
+FILE TAGS: logic
+tags:
+  - logic
+---
+
+## Overview
+
+An object is said to be in **normal form** if it cannot be reduced any further. Examples of normal form include:
+
+* [[equiv-trans#Normal Forms|Conjunctive Normal Form]]
+* [[equiv-trans#Normal Forms|Disjunctive Normal Form]]
+* [[quantification#Identifiers|Prenex Normal Form]]
+
+%%ANKI
+Basic
+What does it mean for an object to be in normal form?
+Back: It cannot be rewritten/reduced any further.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707675146194-->
+END%%
+
+%%ANKI
+Basic
+What zero-order logical normal form(s) have only $\land$ and $\lor$ operators?
+Back: CNF and DNF
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707675369145-->
+END%%
+
+%%ANKI
+Basic
+What first-order logical normal form(s) writes bound identifiers before free ones?
+Back: PNF
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707675369187-->
+END%%
+
+## References
+
+* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

notes/logic/quantification.md

@@ -9,7 +9,25 @@ tags:
 
 ## Overview
 
-* **Existential quantification** asserts the existence of a member in a set (denoted the **range**) satisfying a property. There may be multiple members that satisfy the property; so long as one does, the existential quantification is considered true.
+A **quantifier** refers to an operator that specifies how many members of a set satisfy some formula. The most common quantifiers are $\exists$ and $\forall$, though others (such as the counting quantifier) are also used.
+
+%%ANKI
+Basic
+What are the most common first-order logic quantifiers?
+Back: $\exists$ and $\forall$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707674796763-->
+END%%
+
+%%ANKI
+Basic
+What term refers to operators like $\exists$ and $\forall$?
+Back: Quantifiers.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707674796766-->
+END%%
+
+* **Existential quantification** ($\exists$) asserts the existence of at least one member in a set satisfying a property.
 
 %%ANKI
 Basic
@@ -43,7 +61,7 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1707494819970-->
 END%%
 
-* **Universal quantification** asserts that every member of a set satisfies a property.
+* **Universal quantification** ($\forall$) asserts that every member of a set satisfies a property.
 
 %%ANKI
 Basic
@@ -98,11 +116,11 @@ Back: $\neg \forall x : S, \neg P(x)$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 END%%
 
-* **Counting quantification** asserts that a number of members of a set satisfy a property.
+* **Counting quantification** ($\exists^{=k}$ or $\exists^{\geq k}$) asserts that (at least) $k$ (say) members of a set satisfy a property.
 
 %%ANKI
 Basic
-What symbol denotes counting quantification (of exactly $k$ members)?
+What symbol denotes counting quantification (of *exactly* $k$ members)?
 Back: $\exists^{=k}$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1707494819983-->
@@ -110,7 +128,7 @@ END%%
 
 %%ANKI
 Basic
-What symbol denotes counting quantification (of at least $k$ members)?
+What symbol denotes counting quantification (of *at least* $k$ members)?
 Back: $\exists^{\geq k}$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1707494819985-->
@@ -132,6 +150,81 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1707494832058-->
 END%%
 
-## Reference
+## Identifiers
+
+Identifiers are said to be **bound** if they are parameters to a quantifier. Identifiers that are not bound are said to be **free**. A first-order logic formula is said to be in **prenex normal form** (PNF) if written in two parts: the first consisting of quantifiers and bound variables (the **prefix**), and the second consisting of no quantifiers (the **matrix**).
+
+%%ANKI
+Basic
+When is an identifier said to be bound?
+Back: When it is specified as a parameter to a quantifier.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707674796768-->
+END%%
+
+%%ANKI
+Basic
+When is an identifier said to be free?
+Back: When it isn't specified as a parameter to a quantifier.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707674796770-->
+END%%
+
+%%ANKI
+Cloze
+An identifier that is not {bound} is instead {free}.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707674796772-->
+END%%
+
+%%ANKI
+Basic
+Prenex normal form consists of what two parts?
+Back: The prefix and the matrix.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707674796773-->
+END%%
+
+%%ANKI
+Basic
+How is the prefix of a formula in PNF formatted?
+Back: As only quantifiers and bound variables.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707674796775-->
+END%%
+
+%%ANKI
+Basic
+How is the matrix of a formula in PNF formatted?
+Back: Without quantifiers.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707674796776-->
+END%%
+
+%%ANKI
+Basic
+Which identifiers in the following are bound? $$\exists x, P(x) \land P(y)$$
+Back: Just $x$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707674796777-->
+END%%
+
+%%ANKI
+Basic
+Which identifiers in the following are free? $$\exists x, P(x) \land P(y)$$
+Back: Just $y$.
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707674796779-->
+END%%
+
+%%ANKI
+Basic
+How is the following rewritten in PNF? $$(\exists x, P(x)) \land (\exists y, P(y))$$
+Back: $\exists x \;y, P(x) \land P(y)$
+Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
+<!--ID: 1707675399517-->
+END%%
+
+## References
 
 * Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

notes/nix/callPackage.md

@@ -111,6 +111,6 @@ Reference: Yin, Ryan. “NixOS and Flakes Book.” Nix, February 1, 2024. [https
 <!--ID: 1706828225240-->
 END%%
 
-## Reference
+## References
 
 * Yin, Ryan. “NixOS and Flakes Book.” Nix, February 1, 2024. [https://github.com/ryan4yin/nixos-and-flakes-book](https://github.com/ryan4yin/nixos-and-flakes-book)