notebook/notes/programming/pred-trans.md

1020 lines
39 KiB
Markdown
Raw Blame History

This file contains invisible Unicode characters!

This file contains invisible Unicode characters that may be processed differently from what appears below. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to reveal hidden characters.

---
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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640219-->
END%%
%%ANKI
Basic
What name is given to $Q$ in $\{Q\}\; S\; \{R\}$?
Back: The precondition of $S$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640222-->
END%%
%%ANKI
Basic
*What* is $R$ in predicate $\{Q\}\; S\; \{R\}$?
Back: A predicate.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640224-->
END%%
%%ANKI
Basic
What name is given to $R$ in $\{Q\}\; S\; \{R\}$?
Back: The postcondition of $S$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640226-->
END%%
%%ANKI
Basic
*What* is $S$ in predicate $\{Q\}\; S\; \{R\}$?
Back: A program (a sequence of statements).
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640227-->
END%%
%%ANKI
Basic
Interpret $\{Q\}\; S\; \{R\}$ in English. What is the antecedent of the implication?
Back: $S$ is executed in a state satisfying $Q$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640229-->
END%%
%%ANKI
Basic
Interpret $\{Q\}\; S\; \{R\}$ in English. What is the consequent of the implication?
Back: $S$ terminates in a finite amount of time in a state satisfying $R$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1714420640241-->
END%%
%%ANKI
Basic
*Why* is $\{T\}\; \text{while }T\text{ do skip}\; \{T\}$ everywhere false?
Back: Because "$\text{while }T\text{ do skip}$" never terminates.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869157-->
END%%
%%ANKI
Basic
What kind of mathematical object is the $wp$ transformer?
Back: A function.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869161-->
END%%
%%ANKI
Basic
Given command $S$ and predicate $R$, what kind of mathematical object is $wp(S, R)$?
Back: A predicate, i.e. a set of states.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715631869196-->
END%%
### 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?
Back: $F$.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715806256921-->
END%%
%%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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716227332852-->
END%%
### 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 $R$, $wp(S, Q \land R) = wp(S, Q) \land wp(S, R)$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047060-->
END%%
%%ANKI
Cloze
Distributivity of Conjunction states {$wp(S, Q \land R)$} $=$ {$wp(S, Q) \land wp(S, R)$}.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047062-->
END%%
%%ANKI
Basic
In Gries's exposition, is Distributivity of Conjunction taken as an axiom or a theorem?
Back: An axiom.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047064-->
END%%
%%ANKI
Basic
Is $wp(S, Q) \land wp(S, R) \Rightarrow wp(S, Q \land R)$ true if $S$ is nondeterministic?
Back: Yes.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047065-->
END%%
%%ANKI
Basic
Is $wp(S, Q \land R) \Rightarrow wp(S, Q) \land wp(S, R)$ true if $S$ is nondeterministic?
Back: Yes.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047067-->
END%%
### 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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716227332866-->
END%%
%%ANKI
Basic
Is the Law of Monotonicity true if the relevant command is nondeterministic?
Back: Yes.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716227332870-->
END%%
### 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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927698-->
END%%
%%ANKI
Basic
Assuming $S$ is nondeterministic, is the following a tautology? $$wp(S, Q \lor R) \Rightarrow wp(S, Q) \lor wp(S, R)$$
Back: No.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927700-->
END%%
%%ANKI
Basic
Assuming $S$ is nondeterministic, is the following a tautology? $$wp(S, Q) \lor wp(S, R) \Rightarrow wp(S, Q \lor R)$$
Back: Yes.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927701-->
END%%
%%ANKI
Basic
Assuming $S$ is deterministic, is the following a tautology? $$wp(S, Q \lor R) \Rightarrow wp(S, Q) \lor wp(S, R)$$
Back: Yes.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927703-->
END%%
%%ANKI
Basic
Assuming $S$ is deterministic, is the following a tautology? $$wp(S, Q) \lor wp(S, R) \Rightarrow wp(S, Q \lor R)$$
Back: Yes.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927710-->
END%%
%%ANKI
Basic
What command does Gries use to demonstrate nondeterminism?
Back: The flipping of a coin.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927716-->
END%%
%%ANKI
Basic
*Why* does Distributivity of Disjunction use an implication instead of equality?
Back: Because the underlying command may be nondeterministic.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716311034191-->
END%%
%%ANKI
Basic
*When* does Distributivity of Disjunction hold under equality (instead of implication)?
Back: When the underlying command is deterministic.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716311034194-->
END%%
## Commands
### Skip
For any predicate $R$, $wp(skip, R) = R$.
%%ANKI
Basic
How is the $skip$ command defined in terms of $wp$?
Back: For any predicate $R$, $wp(skip, R) = R$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300099-->
END%%
%%ANKI
Basic
Which command does Gries call the "identity transformation"?
Back: $skip$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300109-->
END%%
%%ANKI
Cloze
Provide the specific command: for any predicate $R$, $wp(${$skip$}$, R) = R$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300113-->
END%%
### Abort
For any predicate $R$, $wp(abort, R) = F$.
%%ANKI
Basic
How is the $abort$ command defined in terms of $wp$?
Back: For any predicate $R$, $wp(abort, R) = F$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300116-->
END%%
%%ANKI
Cloze
Provide the specific command: for any predicate $R$, $wp(${$abort$}$, R) = F$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300119-->
END%%
%%ANKI
Basic
How is the $abort$ command executed?
Back: It isn't.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300126-->
END%%
%%ANKI
Basic
*Why* can't the $abort$ command be executed?
Back: By definition it executes in state $F$ which is impossible.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300129-->
END%%
%%ANKI
Basic
Which command does Gries introduce as the only "constant" predicate transformer?
Back: $abort$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300133-->
END%%
%%ANKI
Basic
How do we prove that $abort$ is the only "constant" predicate transformer?
Back: For any command $S$, the Law of the Excluded Miracle proves $wp(S, F) = F$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300137-->
END%%
%%ANKI
Basic
Suppose $makeTrue$ is defined as $wp(makeTrue, R) = T$ for all predicates $R$. What's wrong?
Back: If $R = F$, $makeTrue$ violates the Law of the Excluded Miracle.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716810300145-->
END%%
### 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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1719019485654-->
END%%
%%ANKI
Basic
Is sequential composition commutative?
Back: No.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1719019485662-->
END%%
%%ANKI
Basic
Is sequential composition associative?
Back: Yes.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1719019485666-->
END%%
### Assignment
%%ANKI
Basic
What equivalence transformation rule is assignment related to?
Back: Substitution.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014026-->
END%%
#### 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 $$wp(''x \coloneqq e'', R) = domain(e) \textbf{ cand } R_e^x$$
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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1720447926777-->
END%%
%%ANKI
Basic
How is assignment "$x \coloneqq e$" pronounced?
Back: As "$x$ becomes $e$".
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1720447926789-->
END%%
%%ANKI
Basic
How is assignment "$x \coloneqq e$" defined in terms of $wp$?
Back: $wp(''x \coloneqq e'', R) = domain(e) \textbf{ cand } R_e^x$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1720447926808-->
END%%
%%ANKI
Basic
How is definition "$wp(''x \coloneqq e'', R) = R_e^x$" more completely stated?
Back: $wp(''x \coloneqq e'', R) = domain(e) \textbf{ cand } R_e^x$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1720447926828-->
END%%
%%ANKI
Basic
What does $wp(''x \coloneqq 5'', x = 5)$ evaluate to?
Back: $T$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1720447926833-->
END%%
%%ANKI
Basic
What does $wp(''x \coloneqq 5'', x \neq 5)$ evaluate to?
Back: $F$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1720447926837-->
END%%
%%ANKI
Basic
What does $wp(''x \coloneqq x + 1'', x < 0)$ evaluate to?
Back: $x < -1$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1720447926852-->
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. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1720447926858-->
END%%
#### General
The multiple assignment command has form $$x_1 \circ s_1, \cdots, x_n \circ s_n \coloneqq e_1, \cdots, e_n$$
where each $x_i$ is an identifier, each $s_i$ is a [[equiv-trans#Selectors|selector]], and each expression $e_i$ has the same type as $x_i \circ s_i$. We denote this assignment more compactly as $\bar{x} \coloneqq \bar{e}$. We define multiple assignment as $$wp(''\bar{x} \coloneqq \bar{e}'', R) = domain(\bar{e}) \textbf{ cand } R_{\bar{e}}^\bar{x}$$
%%ANKI
Basic
How is simple assignment $x \coloneqq e$ expressed in the following, more general form? $$x_1 \circ s_1, \ldots, x_n \circ s_n \coloneqq e_1, \ldots, e_n$$
Back: As $x \circ \epsilon \coloneqq e$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014033-->
END%%
%%ANKI
Basic
How is simple assignment $b[i] \coloneqq e$ expressed in the following, more general form? $$x_1 \circ s_1, \ldots, x_n \circ s_n \coloneqq e_1, \ldots, e_n$$
Back: As $b \circ [i] \coloneqq e$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014038-->
END%%
%%ANKI
Basic
Consider assignment command $\bar{x} \coloneqq \bar{e}$. In what order must $\bar{e}$ be evaluated?
Back: In any order.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014043-->
END%%
%%ANKI
Basic
Consider assignment command $\bar{x} \coloneqq \bar{e}$. In what order must assignment be performed?
Back: Left-to-right.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014047-->
END%%
%%ANKI
Basic
*Why* must assignment in $\bar{x} \coloneqq \bar{e}$ happen left-to-right?
Back: Because update selector syntax has right-to-left precedence.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014051-->
END%%
%%ANKI
Basic
Consider assignment command $\bar{x} \coloneqq \bar{e}$. When can assignment be performed in any order?
Back: When the identifiers in $\bar{x}$ are distinct.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014055-->
END%%
%%ANKI
Basic
The general assignment command has what form?
Back: $\bar{x} \coloneqq \bar{e}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014059-->
END%%
%%ANKI
Basic
How is assignment "$\bar{x} \coloneqq \bar{e}$" defined in terms of $wp$?
Back: $wp(''\bar{x} \coloneqq \bar{e}'', R) = domain(\bar{e}) \textbf{ cand } R_{\bar{e}}^{\bar{x}}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014064-->
END%%
%%ANKI
Basic
In the $wp$ definition of "$\bar{x} \coloneqq \bar{e}$", what does $domain(\bar{e})$ refer to?
Back: A predicate that holds if each member of $\bar{e}$ is well-defined.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014068-->
END%%
%%ANKI
Basic
In the $wp$ definition of "$\bar{x} \coloneqq \bar{e}$", $domain(\bar{e})$ must exclude which states?
Back: Those in which evaluation of any member of $\bar{e}$ would be undefined.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014072-->
END%%
%%ANKI
Basic
What assumption is made when defining assignment as the following? $$wp(''\bar{x} \coloneqq \bar{e}'', R) = R_{\bar{e}}^{\bar{x}}$$
Back: $domain(\bar{e})$, i.e. evaluation of each member of $\bar{e}$ is well-defined.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014076-->
END%%
%%ANKI
Basic
How is definition "$wp(''\bar{x} \coloneqq \bar{e}'', R) = R_{\bar{e}}^{\bar{x}}$" more completely stated?
Back: $wp(''\bar{x} \coloneqq \bar{e}'', R) = domain(\bar{e}) \textbf{ cand } R_{\bar{e}}^{\bar{x}}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014081-->
END%%
%%ANKI
Basic
Given $\bar{e} = \langle e_1, \ldots, e_n \rangle$, how is $\mathop{domain}(\bar{e})$ defined in predicate logic?
Back: $\forall i, 1 \leq i \leq n \Rightarrow \mathop{domain}(e_i)$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1721497014085-->
END%%
### Alternative
The general form of the **alternative command** is: $$\begin{align*} \textbf{if } & B_1 \rightarrow S_1 \\ \textbf{ | } & B_2 \rightarrow S_2 \\ & \quad\cdots \\ \textbf{ | } & B_n \rightarrow S_n \\ \textbf{fi } & \end{align*}$$
Each $B_i \rightarrow S_i$ is called a **guarded command**. To execute the alternative command, find one true guard and execute the corresponding command. Notice this is nondeterministic. We denote the alternative command as $\text{IF}$ and define $\text{IF}$ in terms of $wp$ as: $$\begin{align*} wp(\text{IF}, R) = \;& (\forall i, 1 \leq i \leq n \Rightarrow domain(B_i)) \;\land \\ & (\exists i, 1 \leq i \leq n \land B_i) \;\land \\ & (\forall i, 1 \leq i \leq n \Rightarrow (B_i \Rightarrow wp(S_i, R))) \end{align*}$$
%%ANKI
Basic
How is the alternative command compactly denoted?
Back: As $\text{IF}$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722256345848-->
END%%
%%ANKI
Basic
What kind of command is $\text{IF}$ a representation of?
Back: An alternative command.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722256906202-->
END%%
%%ANKI
Basic
What is the general form of the alternative command?
Back: $$\begin{align*} \textbf{if } & B_1 \rightarrow S_1 \\ \textbf{ | } & B_2 \rightarrow S_2 \\ & \quad\cdots \\ \textbf{ | } & B_n \rightarrow S_n \\ \textbf{fi } & \end{align*}$$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722256345855-->
END%%
%%ANKI
Basic
What do "guarded commands" refer to?
Back: Each $B_i \rightarrow S_i$ found in the alternative command.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722256345859-->
END%%
%%ANKI
Basic
*Why* are guarded commands named the way they are?
Back: The execution of the command is "guarded" by the truthiness of the condition.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722256345863-->
END%%
%%ANKI
Basic
How are alternative commands executed?
Back: By finding any true guard and executing the corresponding command.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722256345868-->
END%%
%%ANKI
Cloze
Consider $\text{IF}$ containing $B_i \rightarrow S_i$ for $1 \leq i \leq n$. Then $wp(\text{IF}, R)$ is the conjunction of:
* {$\forall i, 1 \leq i \leq n \Rightarrow domain(B_i)$}
* {$\exists i, 1 \leq i \leq n \land B_i$}
* {$\forall i, 1 \leq i \leq n \Rightarrow (B_i \Rightarrow wp(S_i, R))$}
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722256345873-->
END%%
%%ANKI
Basic
What assumption is made when defining $\text{IF}$ as follows? $$\begin{align*} wp(\text{IF}, R) = \;& (\exists i, 1 \leq i \leq n \land B_i) \;\land \\ & (\forall i, 1 \leq i \leq n \Rightarrow (B_i \Rightarrow wp(S_i, R))) \end{align*}$$
Back: $domain(B_i)$ for $1 \leq i \leq n$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722256906206-->
END%%
%%ANKI
Basic
Under what two conditions does the alternative command abort?
Back: If a condition isn't well-defined or no condition is satisfied.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722256906210-->
END%%
%%ANKI
Basic
In what way is the alternative command's execution different from traditional case statements?
Back: It is nondeterministic.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722256906214-->
END%%
%%ANKI
Basic
When *might* the following alternative command abort? $$\begin{align*} \textbf{if } & x > 0 \rightarrow z \coloneqq x \\ \textbf{ | } & x < 0 \rightarrow z \coloneqq -x \\ \textbf{fi } & \end{align*}$$
Back: When $x = 0$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722257348944-->
END%%
%%ANKI
Basic
When is the first guarded command of the following executed? $$\begin{align*} \textbf{if } & x \geq 0 \rightarrow z \coloneqq x \\ \textbf{ | } & x \leq 0 \rightarrow z \coloneqq -x \\ \textbf{fi } & \end{align*}$$
Back: When $x > 0$ or (possibly) when $x = 0$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722257348955-->
END%%
%%ANKI
Basic
When are both of the following guarded commands executed? $$\begin{align*} \textbf{if } & x \geq 0 \rightarrow z \coloneqq x \\ \textbf{ | } & x \leq 0 \rightarrow z \coloneqq -x \\ \textbf{fi } & \end{align*}$$
Back: N/A.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722257348960-->
END%%
%%ANKI
Basic
When are either of the following guarded commands executed? $$\begin{align*} \textbf{if } & x \geq 0 \rightarrow z \coloneqq x \\ \textbf{ | } & x \leq 0 \rightarrow z \coloneqq -x \\ \textbf{fi } & \end{align*}$$
Back: When $x = 0$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722257348966-->
END%%
%%ANKI
Cloze
Alternative command {$\textbf{if fi}$} is equivalent to command {$abort$}.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722259243605-->
END%%
%%ANKI
Basic
*Why* does command $\textbf{if fi}$ abort?
Back: Because no guarded command is true (vacuously) by time of execution.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722259243633-->
END%%
%%ANKI
Basic
How is command $skip$ wrapped in a no-op alternative command?
Back: As $\textbf{if } T \rightarrow skip \textbf{ fi}$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1722259243640-->
END%%
## Bibliography
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.