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Notes on normal forms.

c-declarations
Joshua Potter 2024-02-11 12:05:17 -07:00
parent 6b6d109da7
commit 99e807f807
9 changed files with 157 additions and 19 deletions
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notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
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notes/_journal/2024-02-11.md
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@ -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.

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notes/algorithms/order-growth.md
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@ -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).

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@ -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.

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@ -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.

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@ -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.

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---
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.

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@ -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.

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@ -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)