776 lines
26 KiB
Markdown
776 lines
26 KiB
Markdown
---
|
||
title: Equivalence Transformation
|
||
TARGET DECK: Obsidian::STEM
|
||
FILE TAGS: logic::equiv-trans
|
||
tags:
|
||
- logic
|
||
- equiv-trans
|
||
---
|
||
|
||
## Overview
|
||
|
||
**Equivalence-transformation** refers to a class of calculi for manipulating propositions derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and equality ($=$). Gries covers two in "The Science of Programming": a system of evaluation and a formal system. The system of evaluation mirrors how a computer processes instructions, at least in an abstract sense. The formal system serves as a theoretical framework for reasoning about propositions and their transformations without resorting to "lower-level" operations like substitution.
|
||
|
||
%%ANKI
|
||
Basic
|
||
Who is the author of "The Science of Programming"?
|
||
Back: David Gries
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861286-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What are constant propositions?
|
||
Back: Propositions that contain only constants as operands.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707422675517-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What are the basic propositional logical operators?
|
||
Back: $\neg$, $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$/$=$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861291-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Cloze
|
||
Gries replaces logical operator {$\Leftrightarrow$} in favor of {$=$}.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861295-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How does Lean define propositional equality?
|
||
Back: Expressions `a` and `b` are propositionally equal iff `a = b` is true.
|
||
Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
|
||
Tags: lean
|
||
<!--ID: 1706994861298-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How does Lean define `propext`?
|
||
Back:
|
||
```lean
|
||
axiom propext {a b : Prop} : (a ↔ b) → (a = b)
|
||
```
|
||
Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
|
||
Tags: lean
|
||
<!--ID: 1706994861300-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What Lean theorem justifies Gries' choice of $=$ over $\Leftrightarrow$?
|
||
Back: `propext`
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
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 $a \Rightarrow b$?
|
||
Back: The antecedent
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861308-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What name is given to operand $b$ in $a \Rightarrow b$?
|
||
Back: The consequent
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861310-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?
|
||
Back: Yes
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861318-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Is $(b \lor d)$ well-defined in $\{(b, T), (c, F)\}$?
|
||
Back: No
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--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 $\{(c, F)\}$?
|
||
Back: $b \lor \neg c$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861337-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What set of states does $a \land b$ represent?
|
||
Back: The set containing just state $\{(a, T), (b, T)\}$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861339-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is sloppy about phrase "the states in $b \lor \neg c$"?
|
||
Back: $b \lor \neg c$ is not a set but a representation of a set (of states).
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861341-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the weakest proposition?
|
||
Back: $T$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861348-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What set of states does $T$ represent?
|
||
Back: The set of all states.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861350-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the strongest proposition?
|
||
Back: $F$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861352-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What set of states does $F$ represent?
|
||
Back: The set of no states.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861354-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What does a proposition *represent*?
|
||
Back: The set of states in which it is true.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861335-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
When is $p$ stronger than $q$?
|
||
Back: When $p \Rightarrow q$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861343-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
When is $p$ weaker than $q$?
|
||
Back: When $q \Rightarrow p$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861346-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
A proposition is well-defined with respect to what?
|
||
Back: A state to evaluate against.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861316-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Why is $b \land c$ stronger than $b \lor c$?
|
||
Back: The former represents a subset of the states the latter represents.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861356-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is a state?
|
||
Back: A function mapping identifiers to values.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861314-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What are the two calculi Gries describes equivalence-transformation with?
|
||
Back: A formal system and a system of evaluation.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673342-->
|
||
END%%
|
||
|
||
## Equivalence Schemas
|
||
|
||
A proposition is said to be a **tautology** if it evaluates to $T$ in every state it is well-defined in. We say propositions $E1$ and $E2$ are **equivalent** if $E1 = E2$ is a tautology. In this case, we say $E1 = E2$ is an **equivalence**.
|
||
|
||
%%ANKI
|
||
Basic
|
||
What does it mean for a proposition to be a tautology?
|
||
Back: That the proposition is true in every state it is well-defined in.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861323-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
The term "equivalent" refers to a comparison between what two objects?
|
||
Back: Propositions.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673345-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What does it mean for two propositions to be equivalent?
|
||
Back: Given propositions $E1$ and $E2$, it means $E1 = E2$ is a tautology.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673347-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is an equivalence?
|
||
Back: Given propositions $E1$ and $E2$, tautology $E1 = E2$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673348-->
|
||
END%%
|
||
|
||
* Commutative Laws
|
||
* $(E1 \land E2) = (E2 \land E1)$
|
||
* $(E1 \lor E2) = (E2 \lor E1)$
|
||
* $(E1 = E2) = (E2 = E1)$
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which of the basic logical operators do the commutative laws apply to?
|
||
Back: $\land$, $\lor$, and $=$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673350-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What do the commutative laws allow us to do?
|
||
Back: Reorder operands.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673351-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the commutative law of e.g. $\land$?
|
||
Back: $E1 \land E2 = E2 \land E1$
|
||
<!--ID: 1707251673353-->
|
||
END%%
|
||
|
||
* Associative Laws
|
||
* $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
|
||
* $E1 \lor (E2 \lor E3) = (E1 \lor E2) \lor E3$
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which of the basic logical operators do the associative laws apply to?
|
||
Back: $\land$ and $\lor$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673354-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What do the associative laws allow us to do?
|
||
Back: Remove parentheses.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673355-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the associative law of e.g. $\land$?
|
||
Back: $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673357-->
|
||
END%%
|
||
|
||
* Distributive Laws
|
||
* $E1 \lor (E2 \land E3) = (E1 \lor E2) \land (E1 \lor E3)$
|
||
* $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which of the basic logical operators do the distributive laws apply to?
|
||
Back: $\land$ and $\lor$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673358-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What do the distributive laws allow us to do?
|
||
Back: "Factor" propositions.
|
||
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673360-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the distributive law of e.g. $\land$ over $\lor$?
|
||
Back: $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673361-->
|
||
END%%
|
||
|
||
* De Morgan's Laws
|
||
* $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
|
||
* $\neg (E1 \lor E2) = \neg E1 \land \neg E2$
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which of the basic logical operators do De Morgan's Laws apply to?
|
||
Back: $\neg$, $\land$, and $\lor$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673363-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is De Morgan's Law of e.g. $\land$?
|
||
Back: $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673364-->
|
||
END%%
|
||
|
||
* Law of Negation
|
||
* $\neg (\neg E1) = E1$
|
||
|
||
%%ANKI
|
||
Basic
|
||
What does the Law of Negation say?
|
||
Back: $\neg (\neg E1) = E1$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673365-->
|
||
END%%
|
||
|
||
* Law of the Excluded Middle
|
||
* $E1 \lor \neg E1 = T$
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which of the basic logical operators does the Law of the Excluded Middle apply to?
|
||
Back: $\lor$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673367-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What does the Law of the Excluded Middle say?
|
||
Back: $E1 \lor \neg E1 = T$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673368-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which equivalence schema is "refuted" by sentence, "This sentence is false."
|
||
Back: Law of the Excluded Middle
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251779153-->
|
||
END%%
|
||
|
||
* Law of Contradiction
|
||
* $E1 \land \neg E1 = F$
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which of the basic logical operators does the Law of Contradiction apply to?
|
||
Back: $\land$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673370-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What does the Law of Contradiction say?
|
||
Back: $E1 \land \neg E1 = F$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673371-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Cloze
|
||
The Law of {1:the Excluded Middle} is to {2:$\lor$} whereas the Law of {2:Contradiction} is to {1:$\land$}.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707251673373-->
|
||
END%%
|
||
|
||
Gries lists other "Laws" but they don't seem as important to note here.
|
||
|
||
%%ANKI
|
||
Basic
|
||
How is $\Rightarrow$ written in terms of other logical operators?
|
||
Back: $p \Rightarrow q$ is equivalent to $\neg p \lor q$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1706994861358-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How is $\Leftrightarrow$/$=$ written in terms of other logical operators?
|
||
Back: $p \Leftrightarrow q$ is equivalent to $(p \Rightarrow q) \land (q \Rightarrow p)$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--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
|
||
* Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
|
||
* Rule of Transitivity
|
||
* Let $E1 = E2$ and $E2 = E3$ be equivalences. Then $E1 = E3$ is an equivalence.
|
||
|
||
%%ANKI
|
||
Basic
|
||
What two inference rules make up the equivalence-transformation formal system?
|
||
Back: Substitution and transitivity.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707253246450-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Which of the two inference rules that make up the equivalence-transformation formal system is redundant?
|
||
Back: Transitivity.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707432641598-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What does the rule of substitution say in the system of evaluation?
|
||
Back: Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707253246452-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How is the rule of substitution written as an inference rule (in standard form)?
|
||
Back:
|
||
$$
|
||
\begin{matrix}
|
||
E1 = E2 \\
|
||
\hline P(E1) = P(E2)
|
||
\end{matrix}
|
||
$$
|
||
<!--ID: 1707253246454-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What does the rule of transitivity state in the system of evaluation?
|
||
Back: Let $E1 = E2$ and $E2 = E3$. Then $E1 = E3$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707253246455-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How is the rule of transitivity written as an inference rule (in standard form)?
|
||
Back:
|
||
$$
|
||
\begin{matrix}
|
||
E1 = E2, E2 = E3 \\
|
||
\hline E1 = E3
|
||
\end{matrix}
|
||
$$
|
||
<!--ID: 1707253246457-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Cloze
|
||
The system of evaluation has {equivalences} whereas the formal system has {theorems}.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707253246458-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is a "theorem" in the equivalence-transformation formal system?
|
||
Back: An equivalence 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%%
|
||
|
||
## Textual Substitution
|
||
|
||
**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
|
||
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: 1707762304123-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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: 1707762304126-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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: 1707762304127-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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: 1707762304129-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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: 1707762304130-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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.
|
||
<!--ID: 1707762304132-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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: 1707762304133-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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: 1707762304135-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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: 1707762304137-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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.
|
||
<!--ID: 1707762304139-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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: 1707762304140-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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: 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
|
||
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: 1707762304145-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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: 1707762304146-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
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: 1707762304148-->
|
||
END%%
|
||
|
||
%%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
|
||
|
||
* Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
|
||
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. |