353 lines
13 KiB
Markdown
353 lines
13 KiB
Markdown
---
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title: Predicate Transformers
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TARGET DECK: Obsidian::STEM
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FILE TAGS: programming::pred-trans
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tags:
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- pred_trans
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- programming
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---
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## Overview
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Define $\{Q\}\; S\; \{R\}$ as the predicate:
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> If execution of $S$ is begun in a state satisfying $Q$, then it is guaranteed to terminate in a finite amount of time in a state satisfying $R$.
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%%ANKI
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Basic
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*What* is $Q$ in predicate $\{Q\}\; S\; \{R\}$?
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Back: A predicate.
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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: 1714420640219-->
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END%%
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%%ANKI
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Basic
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What name is given to $Q$ in $\{Q\}\; S\; \{R\}$?
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Back: The precondition of $S$.
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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: 1714420640222-->
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END%%
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%%ANKI
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Basic
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*What* is $R$ in predicate $\{Q\}\; S\; \{R\}$?
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Back: A predicate.
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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: 1714420640224-->
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END%%
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%%ANKI
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Basic
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What name is given to $R$ in $\{Q\}\; S\; \{R\}$?
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Back: The postcondition of $S$.
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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: 1714420640226-->
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END%%
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%%ANKI
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Basic
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*What* is $S$ in predicate $\{Q\}\; S\; \{R\}$?
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Back: A program (a sequence of statements).
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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: 1714420640227-->
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END%%
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%%ANKI
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Basic
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What is the antecedent of $\{Q\}\; S\; \{R\}$ in English?
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Back: $S$ is executed in a state satisfying $Q$.
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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: 1714420640229-->
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END%%
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%%ANKI
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Basic
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What is the consequent of $\{Q\}\; S\; \{R\}$ in English?
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Back: $S$ terminates in a finite amount of time in a state satisfying $R$.
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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: 1714420640231-->
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END%%
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%%ANKI
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Basic
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How is $\{Q\}\; S\; \{R\}$ defined?
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Back: If $S$ is executed in a state satisfying $Q$, it eventually terminates in a state satisfying $R$.
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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: 1714420640232-->
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END%%
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%%ANKI
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Basic
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How is $\{x = X \land y = Y\}\; swap\; \{x = Y \land y = X\}$ rewritten without free identifiers?
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Back: $\forall x, y, X, Y, \{x = X \land y = Y\}\; swap\; \{x = Y \land y = X\}$
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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: 1714420640234-->
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END%%
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%%ANKI
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Basic
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What name is given to $X$ in e.g. $\{x = X\}\; S\; \{y = Y\}$?
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Back: The initial value of $x$.
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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: 1714420640235-->
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END%%
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%%ANKI
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Basic
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How is $\{Q\}\; S\; \{R\}$ augmented so that $x$ has initial value $X$?
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Back: $\{Q \land x = X\}\; S\; \{R\}$
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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: 1714420640237-->
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END%%
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%%ANKI
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Basic
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What name is given to $Y$ in e.g. $\{x = X\}\; S\; \{y = Y\}$?
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Back: The final value of $y$.
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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: 1714420640238-->
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END%%
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%%ANKI
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Basic
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How is $\{Q\}\; S\; \{R\}$ augmented so that $y$ has final value $X$?
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Back: $\{Q\}\; S\; \{R \land y = X\}$
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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: 1714420640240-->
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END%%
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%%ANKI
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Basic
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How is $\{Q\}\; S\; \{R\}$ augmented so that $y$ has initial value $X$?
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Back: $\{Q \land y = X\}\; S\; \{R\}$
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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: 1714420640241-->
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END%%
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%%ANKI
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Basic
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*Why* is $\{T\}\; \text{while }T\text{ do skip}\; \{T\}$ everywhere false?
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Back: Because $\text{while }T\text{ do skip}$ never terminates.
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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: 1715631869132-->
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END%%
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## Weakest Precondition
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For any command $S$ and predicate $R$, we define the **weakest precondition** of $S$ with respect to $R$, denoted $wp(S, R)$, as
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> the set of *all* states such that execution of $S$ begun in any one of them is guaranteed to terminate in a finite amount of time in a state satisfying $R$.
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Expression $\{Q\}\; S\; \{R\}$ is equivalent to $Q \Rightarrow wp(S, R)$.
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%%ANKI
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Basic
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What is the predicate transformer $wp$ an acronym for?
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Back: The **w**eakest **p**recondition.
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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: 1715631869137-->
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END%%
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%%ANKI
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Basic
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Given command $S$ and predicate $R$, how is $wp(S, R)$ defined?
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Back: As the set of *all* states such that execution of $S$ in any one of them eventually terminates in a state satisfying $R$.
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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: 1715631869141-->
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END%%
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%%ANKI
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Basic
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In terms of implications, how does a precondition compare to the weakest precondition?
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Back: A precondition implies the weakest precondition but not the other way around.
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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: 1715631869144-->
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END%%
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%%ANKI
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Basic
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In terms of sets of states, how does a precondition compare to the weakest precondition?
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Back: A precondition represents a subset of the states the weakest precondition represents.
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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: 1715631869148-->
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END%%
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%%ANKI
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Basic
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How is $\{Q\}\; S\; \{R\}$ equivalently written as a predicate involving $wp$?
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Back: $Q \Rightarrow wp(S, R)$
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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: 1715631869153-->
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END%%
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%%ANKI
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Basic
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How is $Q \Rightarrow wp(S, R)$ equivalently written as a predicate using assertions?
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Back: $\{Q\}\; S\; \{R\}$
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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: 1715631869157-->
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END%%
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%%ANKI
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Basic
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What kind of mathematical object is the $wp$ transformer?
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Back: A function.
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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: 1715631869161-->
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END%%
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%%ANKI
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Basic
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Given command $S$ and predicate $R$, what kind of mathematical object is $wp(S, R)$?
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Back: A predicate, i.e. a set of states.
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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: 1715631869165-->
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END%%
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%%ANKI
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Basic
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What does the term "predicate transformer" refer to?
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Back: A function that transforms one predicate into another.
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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: 1715631869170-->
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END%%
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%%ANKI
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Basic
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What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = y)$$
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Back: $y \geq x$
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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: 1715631869174-->
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END%%
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%%ANKI
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Basic
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What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = y - 1)$$
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Back: $F$
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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: 1715631869179-->
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END%%
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%%ANKI
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Basic
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What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = y + 1)$$
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Back: $x = y + 1$
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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: 1715631869184-->
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END%%
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%%ANKI
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Basic
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What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = max(x, y))$$
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Back: $T$
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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: 1715631869188-->
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END%%
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%%ANKI
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Basic
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Given command $S$, how is $wp(S, T)$ interpreted?
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Back: As the set of all states such that execution of $S$ in any of them terminates in a finite amount of time.
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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: 1715631869196-->
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END%%
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### Law of the Excluded Miracle
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Given any command $S$, $$wp(S, F) = F$$
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%%ANKI
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Basic
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Given command $S$, what does $wp(S, F)$ evaluate to?
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Back: The empty set.
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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: 1715806256907-->
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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 Miracle state?
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Back: For any command $S$, $wp(S, F) = F$.
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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: 1715806256912-->
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END%%
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%%ANKI
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Basic
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What name is given to identity $wp(S, F) = F$?
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Back: The Law of the Excluded Miracle.
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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: 1715806256915-->
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END%%
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%%ANKI
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Basic
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Explain why the Law of the Excluded Miracle holds true.
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Back: No state satisfies $F$ so no precondition can either.
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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: 1715806256918-->
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END%%
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%%ANKI
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Basic
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Why is the Law of the Excluded Miracle named the way it is?
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Back: It would indeed be a miracle if execution could terminate in no state.
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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: 1715806256921-->
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END%%
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### Distributivity of Conjunction
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Given command $S$ and predicates $Q$ and $R$, $$wp(S, Q \land R) = wp(S, Q) \land wp(S, R)$$
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%%ANKI
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Basic
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What does Distributivity of Conjunction state?
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Back: Given command $S$ and predicates $Q$ and $R$, $wp(S, Q \land R) = wp(S, Q) \land wp(S, R)$.
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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: 1715969047060-->
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END%%
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%%ANKI
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Cloze
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Distributivity of Conjunction states {$wp(S, Q \land R)$} $=$ {$wp(S, Q) \land wp(S, R)$}.
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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: 1715969047062-->
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END%%
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%%ANKI
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Basic
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In Gries's exposition, is Distributivity of Conjunction taken as an axiom or a theorem?
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Back: An axiom.
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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: 1715969047064-->
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END%%
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%%ANKI
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Basic
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Does $wp(S, Q) \land wp(S, R) \Rightarrow wp(S, Q \land R)$ hold when $S$ is nondeterministic?
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Back: Yes.
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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: 1715969047065-->
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END%%
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%%ANKI
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Basic
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Does $wp(S, Q \land R) \Rightarrow wp(S, Q) \land wp(S, R)$ hold when $S$ is nondeterministic?
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Back: Yes.
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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: 1715969047067-->
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END%%
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%%ANKI
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Basic
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What does it mean for command $S$ to be nondeterministic?
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Back: Execution may not be the same even if begun in the same state.
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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: 1715969047068-->
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END%%
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## Bibliography
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* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. |