893 lines
31 KiB
Markdown
893 lines
31 KiB
Markdown
---
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title: Equivalence Transformation
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TARGET DECK: Obsidian::STEM
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FILE TAGS: logic::equiv-trans
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tags:
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- logic
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- equiv-trans
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---
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## Overview
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**Equivalence-transformation** refers to a class of calculi for [[propositional|propositional logic]] 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.
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%%ANKI
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Basic
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Who is the author of "The Science of Programming"?
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Back: David Gries
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861286-->
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END%%
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%%ANKI
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Basic
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What are constant propositions?
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Back: Propositions that contain only constants as operands.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707422675517-->
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END%%
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%%ANKI
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Cloze
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Gries replaces logical operator {$\Leftrightarrow$} in favor of {$=$}.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861295-->
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END%%
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%%ANKI
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Basic
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How does Lean define propositional equality?
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Back: Expressions `a` and `b` are propositionally equal iff `a = b` is true.
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Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
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Tags: lean
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<!--ID: 1706994861298-->
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END%%
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%%ANKI
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Basic
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How does Lean define `propext`?
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Back:
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```lean
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axiom propext {a b : Prop} : (a ↔ b) → (a = b)
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```
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Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
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Tags: lean
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<!--ID: 1706994861300-->
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END%%
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%%ANKI
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Basic
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What Lean theorem justifies Gries' choice of $=$ over $\Leftrightarrow$?
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Back: `propext`
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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Tags: lean
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<!--ID: 1706994861302-->
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END%%
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%%ANKI
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Basic
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Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?
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Back: Yes
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861318-->
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END%%
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%%ANKI
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Basic
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Is $(b \lor d)$ well-defined in $\{(b, T), (c, F)\}$?
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Back: No
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861320-->
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END%%
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%%ANKI
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Basic
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What proposition represents states $\{(b, T)\}$ and $\{(c, F)\}$?
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Back: $b \lor \neg c$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861337-->
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END%%
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%%ANKI
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Basic
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What set of states does $a \land b$ represent?
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Back: The set containing just state $\{(a, T), (b, T)\}$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861339-->
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END%%
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%%ANKI
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Basic
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What is sloppy about phrase "the states in $b \lor \neg c$"?
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Back: $b \lor \neg c$ is not a set but a representation of a set (of states).
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861341-->
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END%%
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%%ANKI
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Basic
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What is the weakest proposition?
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Back: $T$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861348-->
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END%%
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%%ANKI
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Basic
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What set of states does $T$ represent?
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Back: The set of all states.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861350-->
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END%%
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%%ANKI
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Basic
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What is the strongest proposition?
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Back: $F$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861352-->
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END%%
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%%ANKI
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Basic
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What set of states does $F$ represent?
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Back: The set of no states.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861354-->
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END%%
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%%ANKI
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Basic
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What does a proposition *represent*?
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Back: The set of states in which it is true.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861335-->
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END%%
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%%ANKI
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Basic
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When is $p$ stronger than $q$?
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Back: When $p \Rightarrow q$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861343-->
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END%%
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%%ANKI
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Basic
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When is $p$ weaker than $q$?
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Back: When $q \Rightarrow p$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861346-->
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END%%
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%%ANKI
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Basic
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A proposition is well-defined with respect to what?
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Back: A state to evaluate against.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861316-->
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END%%
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%%ANKI
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Basic
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Why is $b \land c$ stronger than $b \lor c$?
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Back: The former represents a subset of the states the latter represents.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861356-->
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END%%
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%%ANKI
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Basic
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What is a state?
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Back: A function mapping identifiers to values.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861314-->
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END%%
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%%ANKI
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Basic
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What are the two calculi Gries describes equivalence-transformation with?
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Back: A formal system and a system of evaluation.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673342-->
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END%%
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## Equivalence Schemas
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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**.
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%%ANKI
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Basic
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What does it mean for a proposition to be a tautology?
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Back: That the proposition is true in every state it is well-defined in.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861323-->
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END%%
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%%ANKI
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Basic
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How is tautology $e$ written equivalently with a quantifier?
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Back: For free identifiers $i_1, \ldots, i_n$ in $e$, as $\forall (i_1, \ldots, i_n), e$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707937867032-->
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END%%
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%%ANKI
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Basic
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The term "equivalent" refers to a comparison between what two objects?
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Back: Expressions.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673345-->
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END%%
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%%ANKI
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Basic
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What does it mean for two propositions to be equivalent?
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Back: Given propositions $E1$ and $E2$, it means $E1 = E2$ is a tautology.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673347-->
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END%%
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%%ANKI
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Basic
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What is an equivalence?
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Back: Given propositions $E1$ and $E2$, tautology $E1 = E2$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673348-->
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END%%
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* Commutative Laws
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* $(E1 \land E2) = (E2 \land E1)$
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* $(E1 \lor E2) = (E2 \lor E1)$
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* $(E1 = E2) = (E2 = E1)$
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%%ANKI
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Basic
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Which of the basic logical operators do the commutative laws apply to?
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Back: $\land$, $\lor$, and $=$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673350-->
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END%%
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%%ANKI
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Basic
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What do the commutative laws allow us to do?
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Back: Reorder operands.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673351-->
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END%%
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%%ANKI
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Basic
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What is the commutative law of e.g. $\land$?
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Back: $E1 \land E2 = E2 \land E1$
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<!--ID: 1707251673353-->
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END%%
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* Associative Laws
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* $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
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* $E1 \lor (E2 \lor E3) = (E1 \lor E2) \lor E3$
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%%ANKI
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Basic
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Which of the basic logical operators do the associative laws apply to?
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Back: $\land$ and $\lor$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673354-->
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END%%
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%%ANKI
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Basic
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What do the associative laws allow us to do?
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Back: Remove parentheses.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673355-->
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END%%
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%%ANKI
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Basic
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What is the associative law of e.g. $\land$?
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Back: $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673357-->
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END%%
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* Distributive Laws
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* $E1 \lor (E2 \land E3) = (E1 \lor E2) \land (E1 \lor E3)$
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* $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
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%%ANKI
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Basic
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Which of the basic logical operators do the distributive laws apply to?
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Back: $\land$ and $\lor$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673358-->
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END%%
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%%ANKI
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Basic
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What do the distributive laws allow us to do?
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Back: "Factor" propositions.
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Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673360-->
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END%%
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%%ANKI
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Basic
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What is the distributive law of e.g. $\land$ over $\lor$?
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Back: $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673361-->
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END%%
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* De Morgan's Laws
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* $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
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* $\neg (E1 \lor E2) = \neg E1 \land \neg E2$
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%%ANKI
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Basic
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Which of the basic logical operators do De Morgan's Laws apply to?
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Back: $\neg$, $\land$, and $\lor$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673363-->
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END%%
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%%ANKI
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Basic
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What is De Morgan's Law of e.g. $\land$?
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Back: $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673364-->
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END%%
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* Law of Negation
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* $\neg (\neg E1) = E1$
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%%ANKI
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Basic
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What does the Law of Negation say?
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Back: $\neg (\neg E1) = E1$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673365-->
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END%%
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* Law of the Excluded Middle
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* $E1 \lor \neg E1 = T$
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%%ANKI
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Basic
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Which of the basic logical operators does the Law of the Excluded Middle apply to?
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Back: $\lor$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673367-->
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END%%
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%%ANKI
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Basic
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What does the Law of the Excluded Middle say?
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Back: $E1 \lor \neg E1 = T$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673368-->
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END%%
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%%ANKI
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Basic
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Which equivalence schema is "refuted" by sentence, "This sentence is false."
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Back: Law of the Excluded Middle
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251779153-->
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END%%
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* Law of Contradiction
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* $E1 \land \neg E1 = F$
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%%ANKI
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Basic
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Which of the basic logical operators does the Law of Contradiction apply to?
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Back: $\land$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673370-->
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END%%
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%%ANKI
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Basic
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What does the Law of Contradiction say?
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Back: $E1 \land \neg E1 = F$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673371-->
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END%%
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%%ANKI
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Cloze
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The Law of {1:the Excluded Middle} is to {2:$\lor$} whereas the Law of {2:Contradiction} is to {1:$\land$}.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707251673373-->
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END%%
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Gries lists other "Laws" but they don't seem as important to note here.
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%%ANKI
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Basic
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How is $\Rightarrow$ written in terms of other logical operators?
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Back: $p \Rightarrow q$ is equivalent to $\neg p \lor q$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861358-->
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END%%
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%%ANKI
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Basic
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How is $\Leftrightarrow$/$=$ written in terms of other logical operators?
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Back: $p \Leftrightarrow q$ is equivalent to $(p \Rightarrow q) \land (q \Rightarrow p)$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1706994861360-->
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END%%
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%%ANKI
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Basic
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What distinguishes an equality from an equivalence?
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Back: An equivalence is an equality that is also a tautology.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707316178709-->
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END%%
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## Equivalence Rules
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* Rule of Substitution
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* Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
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* Rule of Transitivity
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* Let $E1 = E2$ and $E2 = E3$ be equivalences. Then $E1 = E3$ is an equivalence.
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%%ANKI
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Basic
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What two inference rules make up the equivalence-transformation formal system?
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Back: Substitution and transitivity.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707253246450-->
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END%%
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%%ANKI
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Basic
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Which of the two inference rules that make up the equivalence-transformation formal system is redundant?
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Back: Transitivity.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707432641598-->
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END%%
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%%ANKI
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Basic
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What does the rule of substitution say in the system of evaluation?
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Back: Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707253246452-->
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END%%
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%%ANKI
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Basic
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How is the rule of substitution written as an inference rule (in standard form)?
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Back:
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$$
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\begin{matrix}
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E1 = E2 \\
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\hline P(E1) = P(E2)
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\end{matrix}
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$$
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<!--ID: 1707253246454-->
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END%%
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%%ANKI
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Basic
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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. Let $\bar{x}$ denote a list of distinct identifiers. If $\bar{e}$ is a list of expressions of the same length as $\bar{x}$, then simultaneous substitution of $\bar{x}$ by $\bar{e}$ in expression $E$ is denoted as $$E_{\bar{e}}^{\bar{x}}$$
|
||
Note that simultaneous substitution is different than sequential substitution.
|
||
|
||
%%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 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 freely in $E_e^x$?
|
||
Back: N/A.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707937867036-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What identifier is guaranteed to not occur freely 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
|
||
Basic
|
||
*Why* does $x$ not occur freely 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
|
||
Basic
|
||
What is the role of $E$ in textual substitution $E_e^x$?
|
||
Back: It is the expression in which free occurrences of $x$ are replaced.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1708347042194-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the role of $e$ in textual substitution $E_e^x$?
|
||
Back: It is the expression that is evaluated and substituted into $E$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1708347042199-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the role of $x$ in textual substitution $E_e^x$?
|
||
Back: It is the identifier matching free occurrences in $E$ that are replaced.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1708347042203-->
|
||
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
|
||
If $x \neq e$, why might $E_e^x = E$ be an equivalence despite $x$ 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%%
|
||
|
||
%%ANKI
|
||
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
|
||
Basic
|
||
What is the role of $E$ in textual substitution $E_{\bar{e}}^{\bar{x}}$?
|
||
Back: It is the expression in which free occurrences of $\bar{x}$ are replaced.
|
||
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 the role of $\bar{e}$ in textual substitution $E_{\bar{e}}^{\bar{x}}$?
|
||
Back: It is the expressions that are substituted into $E$.
|
||
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 the role of $\bar{x}$ in textual substitution $E_{\bar{e}}^{\bar{x}}$?
|
||
Back: It is the distinct identifiers matching free occurrences in $E$ that are replaced.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707762304129-->
|
||
END%%
|
||
|
||
### Theorems
|
||
|
||
* $(E_u^x)_v^x = E_{u_v^x}^x$
|
||
* The only possible free occurrences of $x$ that may appear after the first of the sequential substitutions occur in $u$.
|
||
|
||
%%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%%
|
||
|
||
* 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
|
||
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%%
|
||
|
||
%%ANKI
|
||
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
|
||
Basic
|
||
How is $(s; x{:}v)$ written instead using set-theoretical notation?
|
||
Back: $(s - \{\langle x, s(x) \rangle\}) \cup \{\langle x, v \rangle\}$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1707937867053-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
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
|
||
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
|
||
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
|
||
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 $e$.
|
||
|
||
%%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
|
||
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%%
|
||
|
||
* Let $s$ be a state and $s' = (s; x{:}s(e))$. Then $s'(E) = s(E_e^x)$.
|
||
|
||
%%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
|
||
If $s' = (s; x{:}s(e))$, then $s'(E) = s(E_e^x)$. Why do we not say $s' = (s; x{:}e)$ instead?
|
||
Back: The value of a state's identifier must always be a constant proposition.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1708693353856-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How do you define $s'$ such that $s(E_e^x) = s'(E)$?
|
||
Back: $s' = (s; x{:}s(e))$.
|
||
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 $\bar{u}$, $(E_{\bar{u}}^{\bar{x}})_{\bar{x}}^{\bar{u}} = E$.
|
||
|
||
%%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%%
|
||
|
||
## Bibliography
|
||
|
||
* 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. |