929 lines
34 KiB
Markdown
929 lines
34 KiB
Markdown
---
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title: Equivalence Transformation
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TARGET DECK: Obsidian::STEM
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FILE TAGS: programming::equiv-trans
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tags:
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- equiv-trans
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- logic
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- programming
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---
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## Overview
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**Equivalence-transformation** refers to a class of calculi for [[prop-logic|propositional logic]] derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and equality ($=$).
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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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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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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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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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%%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?
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Back: Let $E1 = E2$ and $E2 = E3$. Then $E1 = E3$.
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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: 1707253246455-->
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END%%
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%%ANKI
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Basic
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How is the rule of transitivity written as an inference rule (in standard form)?
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Back:
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$$
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\begin{matrix}
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E1 = E2, E2 = E3 \\
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\hline E1 = E3
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\end{matrix}
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$$
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<!--ID: 1707253246457-->
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END%%
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%%ANKI
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Cloze
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The system of evaluation has {equivalences} whereas the formal system has {theorems}.
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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: 1707253246458-->
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END%%
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%%ANKI
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Basic
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What is a "theorem" in the equivalence-transformation formal system?
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Back: An equivalence derived from the axioms and inference rules.
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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: 1707316178712-->
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END%%
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%%ANKI
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Basic
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How is e.g. the Law of Implication proven in the system of evaluation?
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Back: With truth tables.
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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: 1707316178714-->
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END%%
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%%ANKI
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Basic
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How is e.g. the Law of Implication proven in the formal system?
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Back: It isn't. It is an axiom.
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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: 1707316178715-->
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END%%
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%%ANKI
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Cloze
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The system of evaluation and formal system are connected by the following biconditional: {$e$ is a tautology} iff {$e = T$ is a theorem}.
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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: 1707316178717-->
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END%%
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%%ANKI
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Cloze
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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"}.
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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: 1707316276203-->
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END%%
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## Selectors
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A **selector** denotes a finite sequence of subscript expressions, each enclosed in brackets. $\epsilon$ denotes the empty selector. For example, variable $x$ is equivalently denoted as $x \circ \epsilon$ whereas for array $b$, $b[i]$ is equivalently denoted as $b \circ [i]$.
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**Selector update** syntax allows specifying a new value with previous subscripted values overridden. For instance, $(b; i{:}e)$ denotes $b$ with $b[i]$ now referring to $e$. More formally, for any $j \in \mathop{domain}(b)$, $$(b; i{:}e)[j] = \begin{cases} i = j \rightarrow e \\ i \neq j \rightarrow b[j] \end{cases}$$
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%%ANKI
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Basic
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Let $b$ be an array. What does $b.lower$ denote?
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Back: The lower subscript bound of the array.
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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: 1713793130015-->
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END%%
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%%ANKI
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Basic
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Let $b$ be an array. What does $b.upper$ denote?
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Back: The upper subscript bound of the array.
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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: 1713793130019-->
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END%%
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%%ANKI
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Basic
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Let $b$ be an array. How is $domain(b)$ defined in set-theoretic notation?
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Back: $\{i \mid b.lower \leq i \leq b.upper\}$
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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: 1713793130022-->
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END%%
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%%ANKI
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Basic
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Let $b[0{:}2]$ be an array. What is $b.lower$?
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Back: $0$
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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: 1713793130025-->
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END%%
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%%ANKI
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Basic
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Let $b[0{:}2]$ be an array. What is $b.upper$?
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Back: $2$
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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: 1713793130028-->
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END%%
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%%ANKI
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Basic
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Execution of `b[i] := e` of array $b$ in state $s$ yields what new value of $b$?
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Back: $b = (b; i{:}s(e))$
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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: 1713793130031-->
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END%%
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%%ANKI
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Basic
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Let $s$ be a state. What *is* $x$ in $(s; x{:}e)$?
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Back: An identifier found in $s$.
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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: 1713793130034-->
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END%%
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%%ANKI
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Basic
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Let $s$ be a state. What *is* $e$ in $(s; x{:}e)$?
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Back: An expression.
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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: 1713793130037-->
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END%%
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%%ANKI
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Basic
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Let $s$ be a state. What is $e$'s type in $(s; x{:}e)$?
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Back: A type matching $x$'s declaration.
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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: 1713793130041-->
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END%%
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%%ANKI
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Basic
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Let $b$ be an array. What *is* $x$ in $(b; x{:}e)$?
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Back: An expression that evaluates to a member of $domain(b)$.
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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: 1713793130045-->
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END%%
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%%ANKI
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Basic
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Let $b$ be an array. What is $e$'s type in $(b; x{:}e)$?
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Back: A type matching $b$'s member declaration.
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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: 1713793130050-->
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END%%
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%%ANKI
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Basic
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Let $b$ be an array. What case analysis does $(b; i{:}e)[j]$ evaluate to?
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Back: $$(b; i{:}e)[j] = \begin{cases} i = j \rightarrow e \\ i \neq j \rightarrow b[j] \end{cases}$$
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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: 1713793130056-->
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END%%
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%%ANKI
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Basic
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Let $b$ be an array. How is $(((b; i{:}e_1); j{:}e_2); k{:}e_3)$ rewritten without nesting?
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Back: As $(b; i{:}e_1; j{:}e_2; k{:}e_3)$
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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: 1713793130062-->
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END%%
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%%ANKI
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Basic
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Let $b$ be an array. How is $(b; (i{:}e_1; (j{:}e_2; (k{:}e_3))))$ rewritten without nesting?
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Back: N/A. This is invalid syntax.
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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: 1713793130067-->
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END%%
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%%ANKI
|
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Basic
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Let $b$ be an array. How is $(b; i{:}e_1; j{:}e_2; k{:}e_3)$ rewritten with nesting?
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Back: As $(((b; i{:}e_1); j{:}e_2); k{:}e_3)$.
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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: 1713793130072-->
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END%%
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%%ANKI
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Basic
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Let $b$ be an array. What does $(b; i{:}2; i{:}3; i{:}4)[i]$ evaluate to?
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Back: $4$
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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: 1713793130077-->
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END%%
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%%ANKI
|
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Basic
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Let $b$ be an array. How is $(b; 0{:}8; 2{:}9; 0{:}7)[1]$ simplified?
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Back: As $b[1]$.
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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: 1713793130081-->
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END%%
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||
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%%ANKI
|
||
Basic
|
||
According to Gries, what is the traditional interpretation of an array?
|
||
Back: As a collection of subscripted independent variables (with a common name).
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1713793130086-->
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END%%
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||
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||
%%ANKI
|
||
Basic
|
||
According to Gries, what is the new interpretation of an array?
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||
Back: As a function.
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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: 1713793130090-->
|
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END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What propositional expression results from eliminating $(b; \ldots)$ notation from $(b; i{:}5)[j] = 5$?
|
||
Back: $(i = j \land 5 = 5) \lor (i \neq j \land b[j] = 5)$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1713793130095-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What logical axiom is used when eliminating $(b; \ldots)$ notation from e.g. $(b; i{:}5)[j] = 5$?
|
||
Back: The Law of the Excluded Middle.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1713793130100-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Cloze
|
||
For state $s$ and array $b$, {$(s; x{:}s(x))$} is analagous to {$(b; i{:}b[i])$}.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1713793130104-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the simplification of $(b; i{:}b[i]; j{:}b[j]; k{:}b[j])$?
|
||
Back: $(b; k{:}b[j])$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1713793130108-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Given array $b$, what terminology does Gries use to describe $i{:}j$ in e.g. $b[i{:}j]$?
|
||
Back: A section.
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||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640885-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Given array $b$, how many elements are in section $b[i{:}j]$?
|
||
Back: $j - i + 1$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714336859994-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Given array $b$ and fixed $j$, what is the largest possible value of $i$ in $b[i{:}j]$?
|
||
Back: $j + 1$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714336859997-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Given array $b$, how many elements are in $b[j{+}1{:}j]$?
|
||
Back: $0$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714336860000-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Given array $b$, what is $b[1{:}5] = x$ an abbreviation for?
|
||
Back: $\forall i, 1 \leq i \leq 5 \Rightarrow b[i] = x$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714336860003-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Given array $b$, what is $b[1{:}k{-}1] < x < b[k{:}n{-}1]$ an abbreviation for?
|
||
Back: $(\forall i, 1 \leq i < k \Rightarrow b[i] < x) \land (\forall i, k \leq i < n \Rightarrow x < b[i])$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714336860005-->
|
||
END%%
|
||
|
||
Generalizing further to all variable types $x$, $$\begin{align*} (x; \epsilon{:}e) & = e \\ (x; [i] {\circ} s{:}e)[j] & = \begin{cases} i \neq j \rightarrow x[j] \\ i = j \rightarrow (x[j]; s{:}e) \end{cases} \end{align*}$$
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is a selector?
|
||
Back: A finite sequence of subscript expressions.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640890-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Given valid expression $(x; [i]{\circ}s{:}e)$, what can be said about $i$?
|
||
Back: $i$ is in the domain of $x$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640893-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What is the base case of selector update syntax?
|
||
Back: $(x; \epsilon{:}e) = e$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640901-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How is the null selector usually denoted?
|
||
Back: $\epsilon$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640904-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
The null selector is the identity element of what operation?
|
||
Back: Subscript sequence concatenation.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640907-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How is assignment $x := e$ rewritten with a selector?
|
||
Back: $x \circ \epsilon := e$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640910-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How is $x \circ \epsilon := e$ rewritten using selector update syntax?
|
||
Back: $x := (x; \epsilon{:}e)$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640913-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How is command $x := (x; \epsilon{:}e)$ more compactly rewritten?
|
||
Back: $x := e$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640917-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What two assignments (i.e. using `:=`) are used to prove $e = (x; \epsilon{:}e)$?
|
||
Back: $x := e$ and $x \circ \epsilon := e$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640921-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
How do assignments $x := e$ and $x \circ \epsilon := e$ prove $e = (x; \epsilon{:}e)$?
|
||
Back: The assignments have the same effect and the latter can be written as $x := (x; \epsilon{:}e)$.
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640926-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Let $b$ be an array. How is $b[i][j] := e$ rewritten using selector update syntax?
|
||
Back: $b := (b; [i][j]{:}e)$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640930-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Let $b$ be an array. What does $(b; \epsilon{:}g)$ evaluate to?
|
||
Back: $g$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640934-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Let $b$ be an array and $i = j$. What does $(b; [i]{\circ}s{:}e)[j]$ evaluate to?
|
||
Back: $(b[j]; s{:}e)$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640938-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Let $b$ be an array and $i \neq j$. What does $(b; [i]{\circ}s{:}e)[j]$ evaluate to?
|
||
Back: $b[j]$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640942-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Maintaining selector update syntax, how is $(c; 1{:}3)[1]$ more explicitly written with a selector?
|
||
Back: $(c; [1]{:}3)[1]$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640948-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
Maintaining selector update syntax, how is $(c; 1{:}3)[1]$ rewritten with $[1]$ commuted as leftward as possible?
|
||
Back: $(c[1]; \epsilon{:}3)$
|
||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||
<!--ID: 1714395640953-->
|
||
END%%
|
||
|
||
## Substitution
|
||
|
||
**Textual substitution** refers to the replacement of a [[pred-logic#Identifiers|free]] identifier with an expression, introducing parentheses as necessary. This concept amounts to the [[#Equivalence Rules|Substitution Rule]] with different notation.
|
||
|
||
%%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%%
|
||
|
||
### Simple
|
||
|
||
If $x$ denotes a variable and $e$ an expression, substitution of $x$ by $e$ is denoted as $$\large{E_e^x}$$
|
||
|
||
%%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: The only occurrences of $x$ in $E$ may be 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%%
|
||
|
||
### General
|
||
|
||
We can generalize textual substitution to operate on a vector of reference-expression pairs, where each reference corresponds to some identifier concatenated with a selector. Let $\bar{x} = \langle x_1, \ldots, x_n \rangle$ denote a vector of identifiers concatenated with selectors and $\bar{e} = \langle e_1, \ldots, e_n \rangle$ denote a vector of expressions. Then textual substitition of $\bar{x}$ with $\bar{e}$ in expression $E$ is denoted as $$\large{E_{\bar{e}}^{\bar{x}}}$$
|
||
|
||
Substitution is defined recursively as follows:
|
||
|
||
1. If each $x_i$ is a distinct identifier with a null selector, then $E_{\bar{e}}^{\bar{x}}$ is the simultaneous substitution of $\bar{x}$ with $\bar{e}$.
|
||
2. Adjacent reference-expression pairs may be permuted as long as they begin with different identifiers. That is, for all distinct $b$ and $c$, $$\Large{E_{\bar{e}, \,f, \,h, \,\bar{g}}^{\bar{x}, \,b, \,c, \,\bar{y}} = E_{\bar{x}, \,h, \,f, \,\bar{g}}^{\bar{x}, \,c, \,b, \,\bar{y}}}$$
|
||
3. Multiple assignments to subparts of an object $b$ can be viewed as a single assignment to $b$. That is, provided $b$ does not begin any of the $x_i$, $$\Large{E_{e_1, \,\ldots, \,e_m, \,\bar{g}}^{b \circ s_1, \,\ldots, \,b \circ s_m, \,\bar{x}} = E_{(b; \,s_1{:}e_1; \,\cdots; \,s_m{:}e_m), \,\bar{g}}^{b, \,\bar{x}}}$$
|
||
|
||
Note that simultaneous substitution is different from sequential substitution.
|
||
|
||
TODO
|
||
|
||
### 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 $\bar{u}$ refer to 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. |