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title TARGET DECK FILE TAGS tags
Predicate Transformers Obsidian::STEM programming::pred-trans
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.

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 What is the antecedent of \{Q\}\; S\; \{R\} in English? 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 What is the consequent of \{Q\}\; S\; \{R\} in English? 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 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 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 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%%

Bibliography

  • Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.