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Equivalence Transformation Obsidian::STEM logic::equiv-trans
logic
equiv-trans

Overview

Equivalence-transformation refers to a class of calculi for manipulating propositions 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 Basic What are constant propositions? Back: Propositions that contain only constants as operands. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What are the basic propositional logical operators? Back: \neg, \land, \lor, \Rightarrow, and \Leftrightarrow/= 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 How does Lean define propositional equality? Back: Expressions a and b are propositionally equal iff a = b is true. Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d. Tags: lean

END%%

%%ANKI Basic How does Lean define propext? Back:

axiom propext {a b : Prop} : (a ↔ b) → (a = b)

Reference: Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d. Tags: lean

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 name is given to \land operands? Back: Conjuncts 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 \lor operands? Back: Disjuncts 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 operand a in a \Rightarrow b? Back: The antecedent 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 operand b in a \Rightarrow b? Back: The consequent Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Is (b \land c) well-defined in \{(b, T), (c, F)\}? Back: Yes Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Is (b \lor d) well-defined in \{(b, T), (c, F)\}? Back: No Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What C logical operator corresponds to \neg? Back: ! Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. Tags: c

END%%

%%ANKI Basic What C logical operator corresponds to \land? Back: There isn't one. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. Tags: c

END%%

%%ANKI Basic What C logical operator corresponds to \lor? Back: There isn't one. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. Tags: c

END%%

%%ANKI Basic What C logical operator corresponds to \Rightarrow? Back: There isn't one. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. Tags: c

END%%

%%ANKI Basic What C logical operator corresponds to \Leftrightarrow? Back: == Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. Tags: c

END%%

%%ANKI Basic What proposition represents states \{(b, T)\} and \{(c, F)\}? Back: b \lor \neg c Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What set of states does a \land b represent? Back: The set containing just state \{(a, T), (b, T)\}. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is sloppy about phrase "the states in b \lor \neg c"? Back: b \lor \neg c is not a set but a representation of a set (of states). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the weakest proposition? Back: T Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What set of states does T represent? Back: The set of all states. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the strongest proposition? Back: F Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What set of states does F represent? Back: The set of no states. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What does a proposition represent? Back: The set of states in which it is true. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic When is p stronger than q? Back: When p \Rightarrow q. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic When is p weaker than q? Back: When q \Rightarrow p. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic A proposition is well-defined with respect to what? Back: A state to evaluate against. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Why is b \land c stronger than b \lor c? Back: The former represents a subset of the states the latter represents. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is a state? Back: A function mapping identifiers to values. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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 The term "equivalent" refers to a comparison between what two objects? Back: Propositions. 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%%

Textual 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. For example, let E and e be expressions and x an identifer. Then $E_e^x denotes the simultaneous replacement of all free occurrences of $x with e.

%%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 is E's role in textual substitution E_e^x? Back: It is the expression that free occurrences of x are replaced with e in. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is e's role in textual substitution E_e^x? Back: It is the expression that free occurrences of x in E are substituted with. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is x's role in textual substitution E_e^x? Back: It is the identifier matching free occurrences in E that are replaced with e. 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 Why might E_e^x = E be an equivalence despite identifier x existing in E? Back: If the only occurrences of x in E are 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%%

  • (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.
  • 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 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%%

%%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%%

References

  • 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.