34 KiB
| title | TARGET DECK | FILE TAGS | tags | |||
|---|---|---|---|---|---|---|
| Equivalence Transformation | Obsidian::STEM | programming::equiv-trans |
|
Overview
Equivalence-transformation refers to a class of calculi for prop-logic derived from negation (\neg), conjunction (\land), disjunction (\lor), implication (\Rightarrow), and equality (=).
Gries covers two in "The Science of Programming": a system of evaluation and a formal system. The system of evaluation mirrors how a computer processes instructions, at least in an abstract sense. The formal system serves as a theoretical framework for reasoning about propositions and their transformations without resorting to "lower-level" operations like substitution.
%%ANKI Basic Who is the author of "The Science of Programming"? Back: David Gries Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Cloze
Gries replaces logical operator {\Leftrightarrow} in favor of {=}.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What Lean theorem justifies Gries' choice of = over \Leftrightarrow?
Back: propext
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
Tags: lean
END%%
%%ANKI Basic What are the two calculi Gries describes equivalence-transformation with? Back: A formal system and a system of evaluation. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
Equivalence Schemas
A proposition is said to be a tautology if it evaluates to T in every state it is well-defined in. We say propositions E1 and E2 are equivalent if E1 = E2 is a tautology. In this case, we say E1 = E2 is an equivalence.
%%ANKI Basic What does it mean for a proposition to be a tautology? Back: That the proposition is true in every state it is well-defined in. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is tautology e written equivalently with a quantifier?
Back: For free identifiers i_1, \ldots, i_n in e, as \forall (i_1, \ldots, i_n), e.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic The term "equivalent" refers to a comparison between what two objects? Back: Expressions. 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 two propositions to be equivalent?
Back: Given propositions E1 and E2, it means E1 = E2 is a tautology.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is an equivalence?
Back: Given propositions E1 and E2, tautology E1 = E2.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is \Rightarrow written in terms of other logical operators?
Back: p \Rightarrow q is equivalent to \neg p \lor q.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is \Leftrightarrow/= written in terms of other logical operators?
Back: p \Leftrightarrow q is equivalent to (p \Rightarrow q) \land (q \Rightarrow p).
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic What distinguishes an equality from an equivalence? Back: An equivalence is an equality that is also a tautology. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
Equivalence Rules
- Rule of Substitution
- Let
P(r)be a predicate andE1 = E2be an equivalence. ThenP(E1) = P(E2)is an equivalence.
- Let
- Rule of Transitivity
- Let
E1 = E2andE2 = E3be equivalences. ThenE1 = E3is an equivalence.
- Let
%%ANKI Basic What two inference rules make up the equivalence-transformation formal system? Back: Substitution and transitivity. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic Which of the two inference rules that make up the equivalence-transformation formal system is redundant? Back: Transitivity. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What does the rule of substitution say in the system of evaluation?
Back: Let P(r) be a predicate and E1 = E2 be an equivalence. Then P(E1) = P(E2) is an equivalence.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic How is the rule of substitution written as an inference rule (in standard form)? Back:
\begin{matrix}
E1 = E2 \\
\hline P(E1) = P(E2)
\end{matrix}
END%%
%%ANKI
Basic
What does the rule of transitivity state in the system of evaluation?
Back: Let E1 = E2 and E2 = E3. Then E1 = E3.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic How is the rule of transitivity written as an inference rule (in standard form)? Back:
\begin{matrix}
E1 = E2, E2 = E3 \\
\hline E1 = E3
\end{matrix}
END%%
%%ANKI Cloze The system of evaluation has {equivalences} whereas the formal system has {theorems}. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic What is a "theorem" in the equivalence-transformation formal system? Back: An equivalence derived from the axioms and inference rules. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic How is e.g. the Law of Implication proven in the system of evaluation? Back: With truth tables. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic How is e.g. the Law of Implication proven in the formal system? Back: It isn't. It is an axiom. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Cloze
The system of evaluation and formal system are connected by the following biconditional: {e is a tautology} iff {e = T is a theorem}.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Cloze
The {1:system of evaluation} is to {2:"e is a tautology"} whereas the {2:formal system} is to {1:"e = T is a theorem"}.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
Substitution
Textual substitution refers to the simultaneous replacement of a free identifier with an expression, introducing parentheses as necessary. This concept is just the #Equivalence Rules with different notation. Let \bar{x} denote a list of distinct identifiers. If \bar{e} is a list of expressions of the same length as \bar{x}, then simultaneous substitution of \bar{x} by \bar{e} in expression E is denoted as $E_{\bar{e}}^{\bar{x}}$
Note that simultaneous substitution is different than sequential substitution.
%%ANKI Basic Textual substitution is derived from what equivalence rule? Back: The substitution rule. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What term refers to x in textual substitution E_e^x?
Back: The reference.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What term refers to e in textual substitution E_e^x?
Back: The expression.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What term refers to both x and e together in textual substitution E_e^x?
Back: The reference-expression pair.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What identifier is guaranteed to not occur freely in E_e^x?
Back: N/A.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What identifier is guaranteed to not occur freely in E_{s(e)}^x?
Back: x
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Why does x not occur freely in E_{s(e)}^x?
Back: Because s(e) evaluates to a constant proposition.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the role of E in textual substitution E_e^x?
Back: It is the expression in which free occurrences of x are replaced.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the role of e in textual substitution E_e^x?
Back: It is the expression that is evaluated and substituted into E.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the role of x in textual substitution E_e^x?
Back: It is the identifier matching free occurrences in E that are replaced.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is textual substitution E_e^x interpreted as a function?
Back: As E(e), where E is a function of x.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Why does Gries prefer notation E_e^x over e.g. E(e)?
Back: The former indicates the identifier to replace.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What two scenarios ensure E_e^x = E is an equivalence?
Back: x = e or no free occurrences of x exist in E.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
If x \neq e, why might E_e^x = E be an equivalence despite x existing in E?
Back: The only occurrences of x in E may be bound.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is required for E_e^x to be valid?
Back: Substitution must result in a syntactically valid expression.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the result of the following? (x < y \land (\forall i : 0 \leq i < n : b[i] < y))_z^x$$
Back: (z < y \land (\forall i : 0 \leq i < n : b[i] < y))
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the result of the following? (x < y \land (\forall i : 0 \leq i < n : b[i] < y))_z^y$$
Back: (x < z \land (\forall i : 0 \leq i < n : b[i] < z))
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the result of the following? (x < y \land (\forall i : 0 \leq i < n : b[i] < y))_z^i$$
Back: (x < y \land (\forall i : 0 \leq i < n : b[i] < y))
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
In textual substitution, what does e.g. \bar{x} denote?
Back: A list of distinct identifiers.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the role of E in textual substitution E_{\bar{e}}^{\bar{x}}?
Back: It is the expression in which free occurrences of \bar{x} are replaced.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the role of \bar{e} in textual substitution E_{\bar{e}}^{\bar{x}}?
Back: It is the expressions that are substituted into E.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the role of \bar{x} in textual substitution E_{\bar{e}}^{\bar{x}}?
Back: It is the distinct identifiers matching free occurrences in E that are replaced.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
Arrays
An array can be seen as a function from the domain of the array to the subscripted values found in the array. We denote array subscript assignment similarly to state identifier assignment: (b; i{:}e)[j] = \begin{cases} i = j \rightarrow e \ i \neq j \rightarrow b[j] \end{cases}
%%ANKI
Basic
Let b be an array. What does b.lower denote?
Back: The lower subscript bound of the array.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. What does b.upper denote?
Back: The upper subscript bound of the array.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. How is domain(b) defined in set-theoretic notation?
Back: \{i \mid b.lower \leq i \leq b.upper\}
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b[0{:}2] be an array. What is b.lower?
Back: 0
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b[0{:}2] be an array. What is b.upper?
Back: 2
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Execution of b[i] := e of array b in state s yields what new value of b?
Back: b = (b; i{:}s(e))
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let s be a state. What is x in (s; x{:}e)?
Back: An identifier found in s.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let s be a state. What is e in (s; x{:}e)?
Back: An expression.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let s be a state. What is e's type in (s; x{:}e)?
Back: A type matching x's declaration.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. What is x in (b; x{:}e)?
Back: An expression that evaluates to a member of domain(b).
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. What is e's type in (b; x{:}e)?
Back: A type matching b's member declaration.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. What case analysis does (b; i{:}e)[j] evaluate to?
Back: (b; i{:}e)[j] = \begin{cases} i = j \rightarrow e \ i \neq j \rightarrow b[j] \end{cases}
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. How is (((b; i{:}e_1); j{:}e_2); k{:}e_3) rewritten without nesting?
Back: As (b; i{:}e_1; j{:}e_2; k{:}e_3)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. How is (b; (i{:}e_1; (j{:}e_2; (k{:}e_3)))) rewritten without nesting?
Back: N/A. This is invalid syntax.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. How is (b; i{:}e_1; j{:}e_2; k{:}e_3) rewritten with nesting?
Back: As (((b; i{:}e_1); j{:}e_2); k{:}e_3).
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. What does (b; i{:}2; i{:}3; i{:}4)[i] evaluate to?
Back: 4
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. How is (b; 0{:}8; 2{:}9; 0{:}7)[1] simplified?
Back: As b[1].
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic According to Gries, what is the traditional interpretation of an array? Back: As a collection of subscripted independent variables (with a common name). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic According to Gries, what is the new interpretation of an array? Back: As a function. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What propositional expression results from eliminating (b; \ldots) notation from (b; i{:}5)[j] = 5?
Back: (i = j \land 5 = 5) \lor (i \neq j \land b[j] = 5)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What logical axiom is used when eliminating (b; \ldots) notation from e.g. (b; i{:}5)[j] = 5?
Back: The Law of the Excluded Middle.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Cloze
For state s and array b, {(s; x{:}s(x))} is analagous to {(b; i{:}b[i])}.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the simplification of (b; i{:}b[i]; j{:}b[j]; k{:}b[j])?
Back: (b; k{:}b[j])
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Given array b, what terminology does Gries use to describe i{:}j in e.g. b[i{:}j]?
Back: A section.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Given array b, how many elements are in section b[i{:}j]?
Back: j - i + 1
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Given array b and fixed j, what is the largest possible value of i in b[i{:}j]?
Back: j + 1
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Given array b, how many elements are in b[j{+}1{:}j]?
Back: 0
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Given array b, what is b[1{:}5] = x an abbreviation for?
Back: \forall i, 1 \leq i \leq 5 \Rightarrow b[i] = x
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Given array b, what is b[1{:}k{-}1] < x < b[k{:}n{-}1] an abbreviation for?
Back: (\forall i, 1 \leq i < k \Rightarrow b[i] < x) \land (\forall i, k \leq i < n \Rightarrow x < b[i])
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
Selector Update Syntax
A selector denotes a finite sequence of subscript expressions, each enclosed in brackets. \epsilon denotes the empty selector. We can generalize the above to all variable types as follows: \begin{align*} (b; \epsilon{:}g) & = g \ (b; [i] \circ s{:}e)[j] & = \begin{cases} i \neq j \rightarrow b[j] \ i = j \rightarrow (b[j]; s{:}e) \end{cases} \end{align*}
%%ANKI Basic What is a selector? Back: A finite sequence of subscript expressions. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Given valid expression (b; [i]{\circ}s{:}e), what can be said about i?
Back: i is in the domain of b.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What is the base case of selector update syntax?
Back: (b; \epsilon{:}g) = g
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
The null selector is usually denoted as what?
Back: \epsilon
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI Basic The null selector is the identity element of what operation? Back: Concatenation of sequences of subscripts. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is assignment x := e rewritten with a selector?
Back: x \circ \epsilon := e
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is x \circ \epsilon := e rewritten using selector update syntax?
Back: x := (x; \epsilon{:}e)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What assignment expression (i.e. using :=) is simpler but equivalent to x := (x; \epsilon{:}e)?
Back: x := e
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What two assignments (i.e. using :=) are used to prove e = (x; \epsilon{:}e)?
Back: x := e and x \circ \epsilon := e.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How do assignments x := e and x \circ \epsilon := e prove e = (x; \epsilon{:}e)?
Back: The assignments have the same effect and the latter can be written as x := (x; \epsilon{:}e).
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. How is b[i][j] := e rewritten using selector update syntax?
Back: b := (b; [i][j]{:}e)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array. What does (b; \epsilon{:}g) evaluate to?
Back: g
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array and i = j. What does (b; [i]{\circ}s{:}e)[j] evaluate to?
Back: (b[j]; s{:}e)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Let b be an array and i \neq j. What does (b; [i]{\circ}s{:}e)[j] evaluate to?
Back: b[j]
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Maintaining selector update syntax, how is (c; 1{:}3)[1] more explicitly written with a selector?
Back: (c; [1]{:}3)[1]
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Maintaining selector update syntax, how is (c; 1{:}3)[1] rewritten with [1] commuted as leftward as possible?
Back: (c[1]; \epsilon{:}3)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
Theorems
(E_u^x)_v^x = E_{u_v^x}^x- The only possible free occurrences of
xthat may appear after the first of the sequential substitutions occur inu.
- The only possible free occurrences of
%%ANKI
Basic
How do we simplify (E_u^x)_v^x?
Back: As E_{u_v^x}^x
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is E_{u_v^x}^x rewritten as sequential substitution?
Back: As (E_u^x)_v^x
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Why is (E_u^x)_v^x = E_{u_v^x}^x an equivalence?
Back: After the first substitution, the only possible free occurrences of x are in u.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
- If
yis not free inE, then(E_u^x)_v^y = E_{u_v^y}^x.ymay not be free inEbut substitutingxwithucan introduce a free occurrence. It doesn't matter if we perform the substitution first or second though.
%%ANKI
Basic
In what two scenarios is (E_u^x)_v^y = E_{u_v^y}^x always an equivalence?
Back: x = y or y is not free in E.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
If x \neq y, when is (E_u^x)_v^y = E_{u_v^y}^x?
Back: When y is not free in E.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Why should y not be free in E for (E_u^x)_v^y = E_{u_v^y}^x to be an equivalence?
Back: If it were, a v would exist in the LHS that doesn't in the RHS.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How does Gries denote state s with identifer x set to value v?
Back: (s; x{:}v)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How is (s; x{:}v) written instead using set-theoretical notation?
Back: (s - \{\langle x, s(x) \rangle\}) \cup \{\langle x, v \rangle\}
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Execution of x := e in state s terminates in what new state?
Back: (s; x{:}s(e))
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Given state s, what statement does (s; x{:}s(e)) derive from?
Back: x := e
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
What missing value guarantees state s satisfies equivalence s = (s; x{:}\_)?
Back: s(x)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Given state s, why is it that s = (s; x{:}s(x))?
Back: Evaluating x in state s yields s(x).
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
s(E_e^x) = s(E_{s(e)}^x)- Substituting
xwitheand then evaluating is the same as substitutingxwith the evaluation ofe.
- Substituting
%%ANKI
Basic
How can we simplify s(E_{s(e)}^x)?
Back: As s(E_e^x)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Given state s, what equivalence relates E_e^x with E_{s(e)}^x?
Back: s(E_e^x) = s(E_{s(e)}^x)
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
- Let
sbe a state ands' = (s; x{:}s(e)). Thens'(E) = s(E_e^x).
%%ANKI
Cloze
Let s be a state and s' = ({s; x{:}s(e)}). Then s'({E}) = s({E_e^x}).
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
If s' = (s; x{:}s(e)), then s'(E) = s(E_e^x). Why do we not say s' = (s; x{:}e) instead?
Back: The value of a state's identifier must always be a constant proposition.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
How do you define s' such that s(E_e^x) = s'(E)?
Back: s' = (s; x{:}s(e)).
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
%%ANKI
Basic
Given defined value v \neq s(x), when is s(E) = (s; x{:}v)(E)?
Back: When x is not free in E.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
- Given identifiers
\bar{x}and fresh identifiers\bar{u},(E_{\bar{u}}^{\bar{x}})_{\bar{x}}^{\bar{u}} = E.
%%ANKI
Basic
When is (E_{\bar{u}}^{\bar{x}})_{\bar{x}}^{\bar{u}} = E guaranteed to be an equivalence?
Back: When \bar{x} and \bar{u} are all distinct identifiers.
Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
END%%
Bibliography
- Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
- Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.