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Notes on textual substitution.

c-declarations
Joshua Potter 2024-02-12 11:27:16 -07:00
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notes/logic/equiv-trans.md
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@ -617,218 +617,158 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
<!--ID: 1707316276203-->
END%%
## Normal Forms
## Textual Substitution
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.
$$\neg (a \Rightarrow b) \Leftrightarrow c$$
It's truth table looks like
$$\begin{array}{c|c|c|c|c|c}
\neg & (a & \Rightarrow & b) & \Leftrightarrow & c \\
\hline
F & T & T & T & F & T \\
F & T & T & T & T & F \\
T & T & F & F & T & T \\
T & T & F & F & F & F \\
F & F & T & T & F & T \\
F & F & T & T & T & F \\
F & F & T & F & F & T \\
F & F & T & F & T & F
\end{array}$$
and it's DNF looks like
$$
(a \land b \land \neg c) \lor
(a \land \neg b \land c) \lor
(\neg a \land b \land \neg c) \lor
(\neg a \land \neg b \land \neg c)
$$
It's CNF results from applying De Morgan's Law to the truth table's "complement":
$$
\neg(
(a \land b \land c) \lor
(a \land \neg b \land \neg c) \lor
(\neg a \land b \land c) \lor
(\neg a \land \neg b \land c)
)
$$
**Textual substitution** refers to the simultaneous replacement of a free identifier with an expression, introducing parentheses as necessary. This concept is just the [[#Equivalence Rules|Substitution Rule]] with different notation. For example, let $E$ and $e$ be expressions and $x$ an identifer. Then $$E_e^x$$ denotes the simultaneous replacement of all free occurrences of $x$ with $e$.
%%ANKI
Basic
What construct is used to prove every proposition can be written in DNF or CNF?
Back: Truth tables
Textual substitution is derived from what equivalence rule?
Back: The substitution rule.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311868994-->
<!--ID: 1707762304123-->
END%%
%%ANKI
Basic
Where are $\land$ and $\lor$ found within a DNF proposition?
Back: $\lor$ separates disjuncts containing $\land$.
What is $E$'s role in textual substitution $E_e^x$?
Back: It is the expression that free occurrences of $x$ are replaced with $e$ in.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311868998-->
<!--ID: 1707762304126-->
END%%
%%ANKI
Basic
What is DNF an acronym for?
Back: **D**isjunctive **N**ormal **F**orm.
What is $e$'s role in textual substitution $E_e^x$?
Back: It is the expression that free occurrences of $x$ in $E$ are substituted with.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311869000-->
<!--ID: 1707762304127-->
END%%
%%ANKI
Basic
What is CNF an acronym for?
Back: **C**onjunctive **N**ormal **F**orm.
What is $x$'s role in textual substitution $E_e^x$?
Back: It is the identifier matching free occurrences in $E$ that are replaced with $e$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311869002-->
<!--ID: 1707762304129-->
END%%
%%ANKI
Basic
Where are $\land$ and $\lor$ found within a CNF proposition?
Back: $\land$ separates conjuncts containing $\lor$.
How is textual substitution $E_e^x$ interpreted as a function?
Back: As $E(e)$, where $E$ is a function of $x$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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-->
<!--ID: 1707762304130-->
END%%
%%ANKI
Basic
What C operator corresponds to $\textbf{cand}$?
Back: `&&`
Why does Gries prefer notation $E_e^x$ over e.g. $E(e)$?
Back: The former indicates the identifier to replace.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
Tags: c
<!--ID: 1707316606004-->
<!--ID: 1707762304132-->
END%%
%%ANKI
Basic
Why is $\textbf{cand}$ named the way it is?
Back: It is short for **c**onditional **and**.
What two scenarios ensure $E_e^x = E$ is an equivalence?
Back: $x = e$ or no free occurrences of $x$ exist in $E$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707317708625-->
<!--ID: 1707762304133-->
END%%
%%ANKI
Basic
How is $p \textbf{ cand } q$ written as a conditional?
Back: $\textbf{if } p \textbf{ then } q \textbf{ else } F$
Why might $E_e^x = E$ be an equivalence despite identifier $x$ existing in $E$?
Back: If the only occurrences of $x$ in $E$ are bound.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707317708627-->
<!--ID: 1707762304135-->
END%%
%%ANKI
Basic
When can $\textbf{cand}$ evaluate to a non-$U$ value despite being given a $U$ operand?
Back: $F \textbf{ cand } U = F$
What is required for $E_e^x$ to be valid?
Back: Substitution must result in a syntactically valid expression.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707317708628-->
<!--ID: 1707762304137-->
END%%
%%ANKI
Basic
What C operator corresponds to $\textbf{cor}$?
Back: `||`
What is the result of the following? $$(x < y \land (\forall i : 0 \leq i < n : b[i] < y))_z^x$$
Back: $$(z < y \land (\forall i : 0 \leq i < n : b[i] < y))$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
Tags: c
<!--ID: 1707316606007-->
<!--ID: 1707762304139-->
END%%
%%ANKI
Basic
Why is $\textbf{cor}$ named the way it is?
Back: It is short for **c**onditional **or**.
What is the result of the following? $$(x < y \land (\forall i : 0 \leq i < n : b[i] < y))_z^y$$
Back: $$(x < z \land (\forall i : 0 \leq i < n : b[i] < z))$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707317708630-->
<!--ID: 1707762304140-->
END%%
%%ANKI
Basic
How is $p \textbf{ cor } q$ written as a conditional?
Back: $\textbf{if } p \textbf{ then } T \textbf{ else } q$
What is the result of the following? $$(x < y \land (\forall i : 0 \leq i < n : b[i] < y))_z^i$$
Back: $$(x < y \land (\forall i : 0 \leq i < n : b[i] < y))$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707317708632-->
<!--ID: 1707762304141-->
END%%
* $(E_u^x)_v^x = E_{u_v^x}^x$
* The only possible free occurrences of $x$ that may appear after the first of the sequential substitutions occur in $u$.
* If $y$ is not free in $E$, then $(E_u^x)_v^y = E_{u_v^y}^x$.
* $y$ may not be free in $E$ but substituting $x$ with $u$ can introduce a free occurrence. It doesn't matter if we perform the substitution first or second though.
%%ANKI
Basic
How do we simplify $(E_u^x)_v^x$?
Back: As $E_{u_v^x}^x$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707762304143-->
END%%
%%ANKI
Basic
When can $\textbf{cor}$ evaluate to a non-$U$ value despite being given a $U$ operand?
Back: $T \textbf{ cor } U = T$
How is $E_{u_v^x}^x$ rewritten as sequential substitution?
Back: As $(E_u^x)_v^x$
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-->
<!--ID: 1707762304145-->
END%%
%%ANKI
Basic
Which of the short-circuit logical operators do the associative laws apply to?
Back: $\textbf{cand}$ and $\textbf{cor}$
Why is $(E_u^x)_v^x = E_{u_v^x}^x$ an equivalence?
Back: After the first substitution, the only possible free occurrences of $x$ are in $u$.
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-->
<!--ID: 1707762304146-->
END%%
%%ANKI
Basic
What is De Morgan's Law of e.g. $\textbf{cor}$?
Back: $\neg (E1 \textbf{ cor } E2) = \neg E1 \textbf{ cand } \neg E2$
In what two scenarios is $(E_u^x)_v^y = E_{u_v^y}^x$ always an equivalence?
Back: $x = y$ or $y$ is not free in $E$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707317708642-->
<!--ID: 1707762304148-->
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 $F$.
%%ANKI
Basic
If $x \neq y$, when is $(E_u^x)_v^y = E_{u_v^y}^x$?
Back: When $y$ is not free in $E$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707762304150-->
END%%
%%ANKI
Basic
Why should $y$ not be free in $E$ for $(E_u^x)_v^y = E_{u_v^y}^x$ to be an equivalence?
Back: If it were, a $v$ would exist in the LHS that doesn't in the RHS.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707762304152-->
END%%
## References

4
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@ -10,8 +10,8 @@ tags:
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]]
* [[truth-tables|Conjunctive Normal Form]]
* [[truth-tables|Disjunctive Normal Form]]
* [[quantification#Identifiers|Prenex Normal Form]]
%%ANKI

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@ -0,0 +1,143 @@
---
title: Short-Circuit
TARGET DECK: Obsidian::STEM
FILE TAGS: logic
tags:
- logic
---
## Overview
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 [[equiv-trans#Equivalence Schemas|equivalence schemas]] still apply if we can limit operands to just $T$ and $F$.
## References
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

92
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@ -0,0 +1,92 @@
---
title: Truth Tables
TARGET DECK: Obsidian::STEM
FILE TAGS: logic
tags:
- logic
---
## Overview
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.
$$\neg (a \Rightarrow b) \Leftrightarrow c$$
It's truth table looks like
$$\begin{array}{c|c|c|c|c|c}
\neg & (a & \Rightarrow & b) & \Leftrightarrow & c \\
\hline
F & T & T & T & F & T \\
F & T & T & T & T & F \\
T & T & F & F & T & T \\
T & T & F & F & F & F \\
F & F & T & T & F & T \\
F & F & T & T & T & F \\
F & F & T & F & F & T \\
F & F & T & F & T & F
\end{array}$$
and it's DNF looks like
$$
(a \land b \land \neg c) \lor
(a \land \neg b \land c) \lor
(\neg a \land b \land \neg c) \lor
(\neg a \land \neg b \land \neg c)
$$
It's CNF results from applying De Morgan's Law to the truth table's "complement":
$$
\neg(
(a \land b \land c) \lor
(a \land \neg b \land \neg c) \lor
(\neg a \land b \land c) \lor
(\neg a \land \neg b \land c)
)
$$
%%ANKI
Basic
What construct is used to prove every proposition can be written in DNF or CNF?
Back: Truth tables
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311868994-->
END%%
%%ANKI
Basic
Where are $\land$ and $\lor$ found within a DNF proposition?
Back: $\lor$ separates disjuncts containing $\land$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311868998-->
END%%
%%ANKI
Basic
What is DNF an acronym for?
Back: **D**isjunctive **N**ormal **F**orm.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311869000-->
END%%
%%ANKI
Basic
What is CNF an acronym for?
Back: **C**onjunctive **N**ormal **F**orm.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311869002-->
END%%
%%ANKI
Basic
Where are $\land$ and $\lor$ found within a CNF proposition?
Back: $\land$ separates conjuncts containing $\lor$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1707311869003-->
END%%
## References
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.