r
/
notebook
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
notebook/notes/logic/equiv-trans.md

39 KiB
Raw Blame History

title TARGET DECK FILE TAGS tags
Equivalence Transformation Obsidian::STEM programming::equiv-trans
equiv-trans
logic
programming

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%%

  • Commutative Laws
    • (E1 \land E2) = (E2 \land E1)
    • (E1 \lor E2) = (E2 \lor E1)
    • (E1 = E2) = (E2 = E1)

%%ANKI Basic Which of the basic logical operators do the commutative laws apply to? Back: \land, \lor, and = Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What do the commutative laws allow us to do? Back: Reorder operands. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the commutative law of e.g. \land? Back: E1 \land E2 = E2 \land E1

END%%

  • Associative Laws
    • E1 \land (E2 \land E3) = (E1 \land E2) \land E3
    • E1 \lor (E2 \lor E3) = (E1 \lor E2) \lor E3

%%ANKI Basic Which of the basic logical operators do the associative laws apply to? Back: \land and \lor Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What do the associative laws allow us to do? Back: Remove parentheses. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the associative law of e.g. \land? Back: E1 \land (E2 \land E3) = (E1 \land E2) \land E3 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

  • Distributive Laws
    • E1 \lor (E2 \land E3) = (E1 \lor E2) \land (E1 \lor E3)
    • E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)

%%ANKI Basic Which of the basic logical operators do the distributive laws apply to? Back: \land and \lor Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What do the distributive laws allow us to do? Back: "Factor" propositions. Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the distributive law of e.g. \land over \lor? Back: E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

  • De Morgan's Laws
    • \neg (E1 \land E2) = \neg E1 \lor \neg E2
    • \neg (E1 \lor E2) = \neg E1 \land \neg E2

%%ANKI Basic Which of the basic logical operators do De Morgan's Laws apply to? Back: \neg, \land, and \lor Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is De Morgan's Law of e.g. \land? Back: \neg (E1 \land E2) = \neg E1 \lor \neg E2 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

  • Law of Negation
    • \neg (\neg E1) = E1

%%ANKI Basic What does the Law of Negation say? Back: \neg (\neg E1) = E1 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

  • Law of the Excluded Middle
    • E1 \lor \neg E1 = T

%%ANKI Basic Which of the basic logical operators does the Law of the Excluded Middle apply to? Back: \lor 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 Middle say? Back: E1 \lor \neg E1 = T Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Which equivalence schema is "refuted" by sentence, "This sentence is false." Back: Law of the Excluded Middle Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

  • Law of Contradiction
    • E1 \land \neg E1 = F

%%ANKI Basic Which of the basic logical operators does the Law of Contradiction apply to? Back: \land 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 Contradiction say? Back: E1 \land \neg E1 = F Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Cloze The Law of {1:the Excluded Middle} is to {2:\lor} whereas the Law of {2:Contradiction} is to {1:\land}. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Gries lists other "Laws" but they don't seem as important to note here.

%%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 and E1 = E2 be an equivalence. Then P(E1) = P(E2) is an equivalence.
  • Rule of Transitivity
    • Let E1 = E2 and E2 = E3 be equivalences. Then E1 = E3 is an equivalence.

%%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 x that may appear after the first of the sequential substitutions occur in u.

%%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 y is not free in E, then (E_u^x)_v^y = E_{u_v^y}^x.
    • y may not be free in E but substituting x with u can 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 x with e and then evaluating is the same as substituting x with the evaluation of e.

%%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 s be a state and s' = (s; x{:}s(e)). Then s'(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.