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---
title: Predicate Transformers
TARGET DECK: Obsidian::STEM
FILE TAGS: programming::pred-trans
tags:
- pred_trans
- programming
---
## Overview
Define $\{Q\}\; S\; \{R\}$ as the predicate:
> 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 $R$.
%%ANKI
Basic
*What* is $Q$ in predicate $\{Q\}\; S\; \{R\}$?
Back: A predicate.
Reference: Gries, David.
<!-- ID: 1714420640219 -->
END%%
%%ANKI
Basic
What name is given to $Q$ in $\{Q\}\; S\; \{R\}$?
Back: The precondition of $S$.
Reference: Gries, David.
<!-- ID: 1714420640222 -->
END%%
%%ANKI
Basic
*What* is $R$ in predicate $\{Q\}\; S\; \{R\}$?
Back: A predicate.
Reference: Gries, David.
<!-- ID: 1714420640224 -->
END%%
%%ANKI
Basic
What name is given to $R$ in $\{Q\}\; S\; \{R\}$?
Back: The postcondition of $S$.
Reference: Gries, David.
<!-- ID: 1714420640226 -->
END%%
%%ANKI
Basic
*What* is $S$ in predicate $\{Q\}\; S\; \{R\}$?
Back: A program (a sequence of statements).
Reference: Gries, David.
<!-- ID: 1714420640227 -->
END%%
%%ANKI
Basic
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Interpret $\{Q\}\; S\; \{R\}$ in English. What is the antecedent of the implication?
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Back: $S$ is executed in a state satisfying $Q$.
Reference: Gries, David.
<!-- ID: 1714420640229 -->
END%%
%%ANKI
Basic
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Interpret $\{Q\}\; S\; \{R\}$ in English. What is the consequent of the implication?
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Back: $S$ terminates in a finite amount of time in a state satisfying $R$.
Reference: Gries, David.
<!-- ID: 1714420640231 -->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ defined?
Back: If $S$ is executed in a state satisfying $Q$, it eventually terminates in a state satisfying $R$.
Reference: Gries, David.
<!-- ID: 1714420640232 -->
END%%
%%ANKI
Basic
How is $\{x = X \land y = Y\}\; swap\; \{x = Y \land y = X\}$ rewritten without free identifiers?
Back: $\forall x, y, X, Y, \{x = X \land y = Y\}\; swap\; \{x = Y \land y = X\}$
Reference: Gries, David.
<!-- ID: 1714420640234 -->
END%%
%%ANKI
Basic
What name is given to $X$ in e.g. $\{x = X\}\; S\; \{y = Y\}$?
Back: The initial value of $x$.
Reference: Gries, David.
<!-- ID: 1714420640235 -->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ augmented so that $x$ has initial value $X$?
Back: $\{Q \land x = X\}\; S\; \{R\}$
Reference: Gries, David.
<!-- ID: 1714420640237 -->
END%%
%%ANKI
Basic
What name is given to $Y$ in e.g. $\{x = X\}\; S\; \{y = Y\}$?
Back: The final value of $y$.
Reference: Gries, David.
<!-- ID: 1714420640238 -->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ augmented so that $y$ has final value $X$?
Back: $\{Q\}\; S\; \{R \land y = X\}$
Reference: Gries, David.
<!-- ID: 1714420640240 -->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ augmented so that $y$ has initial value $X$?
Back: $\{Q \land y = X\}\; S\; \{R\}$
Reference: Gries, David.
<!-- ID: 1714420640241 -->
END%%
%%ANKI
Basic
*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.
<!-- ID: 1715631869132 -->
END%%
## Weakest Precondition
For any command $S$ and predicate $R$, we define the **weakest precondition** of $S$ with respect to $R$, denoted $wp(S, R)$, as
> 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 $R$.
Expression $\{Q\}\; S\; \{R\}$ is equivalent to $Q \Rightarrow wp(S, R)$.
%%ANKI
Basic
What is the predicate transformer $wp$ an acronym for?
Back: The **w**eakest **p**recondition.
Reference: Reference: Gries, David.
<!-- ID: 1715631869137 -->
END%%
%%ANKI
Basic
Given command $S$ and predicate $R$, how is $wp(S, R)$ defined?
Back: As the set of *all* states such that execution of $S$ in any one of them eventually terminates in a state satisfying $R$.
Reference: Reference: Gries, David.
<!-- ID: 1715631869141 -->
END%%
%%ANKI
Basic
In terms of implications, how does a precondition compare to the weakest precondition?
Back: A precondition implies the weakest precondition but not the other way around.
Reference: Reference: Gries, David.
<!-- ID: 1715631869144 -->
END%%
%%ANKI
Basic
In terms of sets of states, how does a precondition compare to the weakest precondition?
Back: A precondition represents a subset of the states the weakest precondition represents.
Reference: Reference: Gries, David.
<!-- ID: 1715631869148 -->
END%%
%%ANKI
Basic
How is $\{Q\}\; S\; \{R\}$ equivalently written as a predicate involving $wp$?
Back: $Q \Rightarrow wp(S, R)$
Reference: Reference: Gries, David.
<!-- ID: 1715631869153 -->
END%%
%%ANKI
Basic
How is $Q \Rightarrow wp(S, R)$ equivalently written as a predicate using assertions?
Back: $\{Q\}\; S\; \{R\}$
Reference: Reference: Gries, David.
<!-- ID: 1715631869157 -->
END%%
%%ANKI
Basic
What kind of mathematical object is the $wp$ transformer?
Back: A function.
Reference: Reference: Gries, David.
<!-- ID: 1715631869161 -->
END%%
%%ANKI
Basic
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.
<!-- ID: 1715631869165 -->
END%%
%%ANKI
Basic
What does the term "predicate transformer" refer to?
Back: A function that transforms one predicate into another.
Reference: Reference: Gries, David.
<!-- ID: 1715631869170 -->
END%%
%%ANKI
Basic
What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = y)$$
Back: $y \geq x$
Reference: Reference: Gries, David.
<!-- ID: 1715631869174 -->
END%%
%%ANKI
Basic
What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = y - 1)$$
Back: $F$
Reference: Reference: Gries, David.
<!-- ID: 1715631869179 -->
END%%
%%ANKI
Basic
What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = y + 1)$$
Back: $x = y + 1$
Reference: Reference: Gries, David.
<!-- ID: 1715631869184 -->
END%%
%%ANKI
Basic
What does the following evaluate to? $$wp(''\text{if } x \geq y \text{ then } z := x \text{ else } z := y'', z = max(x, y))$$
Back: $T$
Reference: Reference: Gries, David.
<!-- ID: 1715631869188 -->
END%%
%%ANKI
Basic
Given command $S$, how is $wp(S, T)$ interpreted?
Back: As the set of all states such that execution of $S$ in any of them terminates in a finite amount of time.
Reference: Reference: Gries, David.
<!-- ID: 1715631869196 -->
END%%
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### Law of the Excluded Miracle
Given any command $S$, $$wp(S, F) = F$$
%%ANKI
Basic
Given command $S$, what does $wp(S, F)$ evaluate to?
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Back: $F$.
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Reference: Reference: Gries, David.
<!-- ID: 1715806256907 -->
END%%
%%ANKI
Basic
What does the Law of the Excluded Miracle state?
Back: For any command $S$, $wp(S, F) = F$.
Reference: Reference: Gries, David.
<!-- ID: 1715806256912 -->
END%%
%%ANKI
Basic
What name is given to identity $wp(S, F) = F$?
Back: The Law of the Excluded Miracle.
Reference: Reference: Gries, David.
<!-- ID: 1715806256915 -->
END%%
%%ANKI
Basic
Explain why the Law of the Excluded Miracle holds true.
Back: No state satisfies $F$ so no precondition can either.
Reference: Reference: Gries, David.
<!-- ID: 1715806256918 -->
END%%
%%ANKI
Basic
Why is the Law of the Excluded Miracle named the way it is?
Back: It would indeed be a miracle if execution could terminate in no state.
Reference: Reference: Gries, David.
<!-- ID: 1715806256921 -->
END%%
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%%ANKI
Basic
In Gries's exposition, is the Law of the Excluded Miracle taken as an axiom or a theorem?
Back: An axiom.
Reference: Reference: Gries, David.
<!-- ID: 1716227332852 -->
END%%
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### Distributivity of Conjunction
Given command $S$ and predicates $Q$ and $R$, $$wp(S, Q \land R) = wp(S, Q) \land wp(S, R)$$
%%ANKI
Basic
What does Distributivity of Conjunction state?
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: Gries, David.
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<!-- ID: 1715969047060 -->
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END%%
%%ANKI
Cloze
Distributivity of Conjunction states {$wp(S, Q \land R)$} $=$ {$wp(S, Q) \land wp(S, R)$}.
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Reference: Gries, David.
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<!-- ID: 1715969047062 -->
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END%%
%%ANKI
Basic
In Gries's exposition, is Distributivity of Conjunction taken as an axiom or a theorem?
Back: An axiom.
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Reference: Gries, David.
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<!-- ID: 1715969047064 -->
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END%%
%%ANKI
Basic
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Is $wp(S, Q) \land wp(S, R) \Rightarrow wp(S, Q \land R)$ true if $S$ is nondeterministic?
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Back: Yes.
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Reference: Gries, David.
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<!-- ID: 1715969047065 -->
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END%%
%%ANKI
Basic
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Is $wp(S, Q \land R) \Rightarrow wp(S, Q) \land wp(S, R)$ true if $S$ is nondeterministic?
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Back: Yes.
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Reference: Gries, David.
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<!-- ID: 1715969047067 -->
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END%%
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### Law of Monotonicity
Given command $S$ and predicates $Q$ and $R$, if $Q \Rightarrow R$, then $wp(S, Q) \Rightarrow wp(S, R)$.
%%ANKI
What does the Law of Monotonicity state?
Back: Given command $S$ and predicates $Q$ and $R$, if $Q \Rightarrow R$, then $wp(S, Q) \Rightarrow wp(S, R)$.
Reference: Reference: Gries, David.
END%%
%%ANKI
Cloze
Given command $S$, the Law of Monotonicity states that if {1:$Q$} $\Rightarrow$ {2:$R$}, then {2:$wp(S, Q)$} $\Rightarrow$ {1:$wp(S, R)$}.
Reference: Reference: Gries, David.
<!-- ID: 1716227332862 -->
END%%
%%ANKI
Basic
In Gries's exposition, is the Law of Monotonicity taken as an axiom or a theorem?
Back: A theorem.
Reference: Reference: Gries, David.
<!-- ID: 1716227332866 -->
END%%
%%ANKI
Basic
Is the Law of Monotonicity true if the relevant command is nondeterministic?
Back: Yes.
Reference: Reference: Gries, David.
<!-- ID: 1716227332870 -->
END%%
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### Distributivity of Disjunction
Given command $S$ and predicates $Q$ and $R$, $$wp(S, Q) \lor wp(S, R) \Rightarrow wp(S, Q \lor R)$$
%%ANKI
Basic
What does Distributivity of Disjunction state?
Back: Given command $S$ and predicates $Q$ and $R$, $wp(S, Q) \lor wp(S, R) \Rightarrow wp(S, Q \lor R)$.
Reference: Gries, David.
<!-- ID: 1716310927694 -->
END%%
%%ANKI
Cloze
Distributivity of Disjunction states {1:$wp(S, Q) \lor wp(S, r)$} $\Rightarrow$ {1:$wp(S, Q \lor R)$}.
Reference: Gries, David.
<!-- ID: 1716310927697 -->
END%%
%%ANKI
Basic
In Gries's exposition, is Distributivity of Disjunction taken as an axiom or a theorem?
Back: A theorem.
Reference: Gries, David.
<!-- ID: 1716310927698 -->
END%%
%%ANKI
Basic
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Assuming $S$ is nondeterministic, is the following a tautology? $$wp(S, Q \lor R) \Rightarrow wp(S, Q) \lor wp(S, R)$$
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Back: No.
Reference: Gries, David.
<!-- ID: 1716310927700 -->
END%%
%%ANKI
Basic
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Assuming $S$ is nondeterministic, is the following a tautology? $$wp(S, Q) \lor wp(S, R) \Rightarrow wp(S, Q \lor R)$$
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Back: Yes.
Reference: Gries, David.
<!-- ID: 1716310927701 -->
END%%
%%ANKI
Basic
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Assuming $S$ is deterministic, is the following a tautology? $$wp(S, Q \lor R) \Rightarrow wp(S, Q) \lor wp(S, R)$$
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Back: Yes.
Reference: Gries, David.
<!-- ID: 1716310927703 -->
END%%
%%ANKI
Basic
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Assuming $S$ is deterministic, is the following a tautology? $$wp(S, Q) \lor wp(S, R) \Rightarrow wp(S, Q \lor R)$$
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Back: Yes.
Reference: Gries, David.
<!-- ID: 1716310927710 -->
END%%
%%ANKI
Basic
What command does Gries use to demonstrate nondeterminism?
Back: The flipping of a coin.
Reference: Gries, David.
<!-- ID: 1716310927712 -->
END%%
%%ANKI
Basic
What does it mean for command $S$ to be nondeterministic?
Back: Execution may not be the same even if begun in the same state.
Reference: Gries, David.
<!-- ID: 1715969047068 -->
END%%
%%ANKI
Basic
Let $S$ flip a coin and $Q$ be flipping heads. What is $wp(S, Q)$?
Back: $F$
Reference: Gries, David.
<!-- ID: 1716310927713 -->
END%%
%%ANKI
Basic
Let $S$ flip a coin and $Q$ be flipping tails. What is $wp(S, Q)$?
Back: $F$
Reference: Gries, David.
<!-- ID: 1716310927715 -->
END%%
%%ANKI
Basic
Let $S$ flip a coin, $Q$ be flipping heads, and $R$ be flipping tails. What is $wp(S, Q \lor R)$?
Back: $T$
Reference: Gries, David.
<!-- ID: 1716310927716 -->
END%%
%%ANKI
Basic
*Why* does Distributivity of Disjunction use an implication instead of equality?
Back: Because the underlying command may be nondeterministic.
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Reference: Gries, David.
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<!-- ID: 1716311034191 -->
END%%
%%ANKI
Basic
*When* does Distributivity of Disjunction hold under equality (instead of implication)?
Back: When the underlying command is deterministic.
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Reference: Gries, David.
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<!-- ID: 1716311034194 -->
END%%
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## Commands
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### Skip
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For any predicate $R$, $wp(skip, R) = R$.
%%ANKI
Basic
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How is the $skip$ command defined in terms of $wp$?
Back: For any predicate $R$, $wp(skip, R) = R$.
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Reference: Gries, David.
<!-- ID: 1716810300099 -->
END%%
%%ANKI
Basic
Which command does Gries call the "identity transformation"?
Back: $skip$
Reference: Gries, David.
<!-- ID: 1716810300109 -->
END%%
%%ANKI
Cloze
Provide the specific command: for any predicate $R$, $wp(${$skip$}$, R) = R$.
Reference: Gries, David.
<!-- ID: 1716810300113 -->
END%%
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### Abort
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For any predicate $R$, $wp(abort, R) = F$.
%%ANKI
Basic
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How is the $abort$ command defined in terms of $wp$?
Back: For any predicate $R$, $wp(abort, R) = F$.
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Reference: Gries, David.
<!-- ID: 1716810300116 -->
END%%
%%ANKI
Cloze
Provide the specific command: for any predicate $R$, $wp(${$abort$}$, R) = F$.
Reference: Gries, David.
<!-- ID: 1716810300119 -->
END%%
%%ANKI
Basic
How is the $abort$ command executed?
Back: It isn't.
Reference: Gries, David.
<!-- ID: 1716810300126 -->
END%%
%%ANKI
Basic
*Why* can't the $abort$ command be executed?
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Back: By definition it executes in state $F$ which is impossible.
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Reference: Gries, David.
<!-- ID: 1716810300129 -->
END%%
%%ANKI
Basic
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Which command does Gries introduce as the only "constant" predicate transformer?
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Back: $abort$
Reference: Gries, David.
<!-- ID: 1716810300133 -->
END%%
%%ANKI
Basic
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How do we prove that $abort$ is the only "constant" predicate transformer?
Back: For any command $S$, the Law of the Excluded Miracle proves $wp(S, F) = F$.
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Reference: Gries, David.
<!-- ID: 1716810300137 -->
END%%
%%ANKI
Basic
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Suppose $makeTrue$ is defined as $wp(makeTrue, R) = T$ for all predicates $R$. What's wrong?
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Back: If $R = F$, $makeTrue$ violates the Law of the Excluded Miracle.
Reference: Gries, David.
<!-- ID: 1716810300145 -->
END%%
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### Sequential Composition
**Sequential composition** is one way of composing larger program segments from smaller segments. Let $S1$ and $S2$ be two commands. Then $S1; S2$ is defined as $$wp(''S1; S2'', R) = wp(S1, wp(S2, R))$$
%%ANKI
Basic
Let $S1$ and $S2$ be two commands. How is their sequential composition denoted?
Back: $S1; S2$
Reference: Gries, David.
<!-- ID: 1719019485648 -->
END%%
%%ANKI
Basic
How is $S1; S2$ defined in terms of $wp$?
Back: For any predicate $R$, $wp(''S1; S2'', R) = wp(S1, wp(S2, R))$.
Reference: Gries, David.
<!-- ID: 1719019485654 -->
END%%
%%ANKI
Basic
Is sequential composition commutative?
Back: No.
Reference: Gries, David.
<!-- ID: 1719019485662 -->
END%%
%%ANKI
Basic
Is sequential composition associative?
Back: Yes.
Reference: Gries, David.
<!-- ID: 1719019485666 -->
END%%
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### Assignment
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#### Simple
The assignment command has form $x \coloneqq e$, provided the types of $x$ and $e$ are the same. This command is read as "$x$ becomes $e$" and is defined as $$wp(''x \coloneqq e'', R) = domain(e) \textbf{ cand } R_e^x$$
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where $domain(e)$ is a predicate that describes the set of all states in which $e$ may be evaluated.
%%ANKI
Basic
The assignment command has what form?
Back: $x \coloneqq e$
Reference: Gries, David.
<!-- ID: 1720447926777 -->
END%%
%%ANKI
Basic
How is assignment "$x \coloneqq e$" pronounced?
Back: As "$x$ becomes $e$".
Reference: Gries, David.
<!-- ID: 1720447926789 -->
END%%
%%ANKI
Basic
How is assignment "$x \coloneqq e$" defined in terms of $wp$?
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Back: $wp(''x \coloneqq e'', R) = domain(e) \textbf{ cand } R_e^x$
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Reference: Gries, David.
<!-- ID: 1720447926794 -->
END%%
%%ANKI
Basic
In the $wp$ definition of "$x \coloneqq e$", what does $domain(e)$ refer to?
Back: A predicate that holds if $e$ is well-defined.
Reference: Gries, David.
<!-- ID: 1720447926799 -->
END%%
%%ANKI
Basic
In the $wp$ definition of "$x \coloneqq e$", $domain(e)$ must exclude which states?
Back: Those in which evaluation of $e$ would be undefined.
Reference: Gries, David.
<!-- ID: 1720447926804 -->
END%%
%%ANKI
Basic
What assumption is made when defining assignment as "$wp(''x \coloneqq e'', R) = R_e^x$"?
Back: $domain(e)$, i.e. evaluation of $e$ is well-defined.
Reference: Gries, David.
<!-- ID: 1720447926808 -->
END%%
%%ANKI
Basic
How is definition "$wp(''x \coloneqq e'', R) = R_e^x$" more completely stated?
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Back: $wp(''x \coloneqq e'', R) = domain(e) \textbf{ cand } R_e^x$
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Reference: Gries, David.
<!-- ID: 1720447926813 -->
END%%
%%ANKI
Basic
In what way is the $wp$ definition of assignment usually simplified?
Back: It is assumed evaluation of expressions (i.e. the RHS of $\coloneqq$) are always well-defined.
Reference: Gries, David.
<!-- ID: 1720447926818 -->
END%%
%%ANKI
Basic
What does Gries state is "bewildering at first" about definition "$wp(''x \coloneqq e'', R) = R_e^x$"?
Back: Operational habits make us feel the precondition should be $R$ and postcondition $R_e^x$.
Reference: Gries, David.
<!-- ID: 1720447926824 -->
END%%
%%ANKI
Basic
How is definition $wp(''x \coloneqq e'', R) = R_e^x$ informally justified?
Back: Since $x$ becomes $e$, $R$ is true after execution iff $R_e^x$ is true before execution.
Reference: Gries, David.
<!-- ID: 1720447926828 -->
END%%
%%ANKI
Basic
What does $wp(''x \coloneqq 5'', x = 5)$ evaluate to?
Back: $T$
Reference: Gries, David.
<!-- ID: 1720447926833 -->
END%%
%%ANKI
Basic
What does $wp(''x \coloneqq 5'', x \neq 5)$ evaluate to?
Back: $F$
Reference: Gries, David.
<!-- ID: 1720447926837 -->
END%%
%%ANKI
Basic
What does $wp(''x \coloneqq x + 1'', x < 0 )$ evaluate to ?
Back: $x < -1 $
Reference: Gries, David.
<!-- ID: 1720447926842 -->
END%%
%%ANKI
Basic
Given array $b$ with subscript range $0{:}100$, what does $wp(''x \coloneqq b[i]'', x = b[i])$ evaluate to?
Back: $0 \leq i \leq 100$
Reference: Gries, David.
<!-- ID: 1720447926846 -->
END%%
%%ANKI
Basic
Assume $c$ is constant and $x, y$ are distinct. What does $wp(''x \coloneqq e'', y = c)$ evaluate to?
Back: $y = c$
Reference: Gries, David.
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END%%
%%ANKI
Basic
What does evaluation "$wp(''x \coloneqq e'', y = c) = (y = c)$" demonstrate?
Back: That assignment (and expression evaluation) should exhibit no side effects.
Reference: Gries, David.
<!-- ID: 1720447926858 -->
END%%
2024-07-20 15:41:15 +00:00
#### General
TODO
2024-05-15 13:59:08 +00:00
## Bibliography
* Gries, David.
