22 KiB
| title | TARGET DECK | FILE TAGS | tags | ||
|---|---|---|---|---|---|
| Predicate Transformers | Obsidian::STEM | programming::pred-trans |
|
Overview
Define \{Q\}\; S\; \{R\} as the predicate:
If execution of
Sis begun in a state satisfyingQ, then it is guaranteed to terminate in a finite amount of time in a state satisfyingR.
%%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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
Sbegun in any one of them is guaranteed to terminate in a finite amount of time in a state satisfyingR.
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 weakest precondition.
Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
END%%
Bibliography
- Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.