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Copy file name to clipboardExpand all lines: typst/hw4.typ
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// Fancy block
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#letBlock(body, ..args) = {
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#Box(align: right)[
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#Block[
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The cosmos, as Stanisław Lem's heroes have discovered, is full of logical puzzles.
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In this assignment, you encounter problems that arise from their adventures: logical contradictions that plague Captain Pirx, truth-telling paradoxes that confound Ijon Tichy, and formal reasoning challenges that challenge even the ingenious constructors Trurl and Klapaucius.
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At Orbital Station Cygnus-7, Ijon Tichy visits the *Emporium of Synthetic Companions*, where the proprietor Minik enforces strict regulations on robotic pet purchases:
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#Block[
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*Regulation 1:* If you purchase exactly one of the Giraffoid or Elephandroid, you must also purchase a Simianoid. (Solitary mega-fauna develop existential melancholy.)
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*Regulation 2:* Neither the Elephandroid nor the Simianoid may be purchased unless the Giraffoid is also purchased. (The Giraffoid serves as social anchor.)
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At the Interstellar Navigation Academy, a scandal erupts: a *Quantum Cheat-Leaf* (a banned mnemonic device) is discovered near the examination hall.
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Tichy interrogates three cadets who took the same exam.
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#Block[
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*Ivanko:* "I didn't use it. Sidorik used it. Petryn used it too."
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*Petryn:* "Ivanko used it. I didn't use it. About Sidorik, I cannot say."
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Aboard the research vessel *Asymptote*, Tichy investigates a reactor incident by interviewing three crew members.
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- *Knyazev* gives an initial testimony.
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- *Faraonov* says: "Knyazev's testimony is false."
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- *Tsaryov* states: "Faraonov is lying."
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Dr. Arkonada discovers three vaults in the Solaric Archives: Gold, Silver, and Lead.
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One contains the legendary *Crystallized Theorem*, the others are empty.
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Each vault bears an inscription:
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#Block[
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*Gold vault*: "The Theorem is not here."
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- *Gold vault*: "The Theorem is not here."
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- *Silver vault*: "The Theorem is in the Gold vault."
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- *Lead vault*: "The Theorem is here."
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*Silver vault*: "The Theorem is in the Gold vault."
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*Lead vault*: "The Theorem is here."
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]
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Ancient texts warn: "You cannot trust all the inscriptions, nor can you dismiss them entirely."
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A vast archive records which travelers have visited which worlds.
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Tichy must decode the archive's formal notation.
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*Predicates over the domain "all planets and travelers":*
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Predicates over the domain "all planets and travelers":
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- $P(x)$: "$x$ is a planet"
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- $I(x)$: "$x$ is inhabited"
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- $V(x, y)$: "$x$ has visited $y$"
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The archive contains two interpretations of "Only inhabited planets are worth visiting."
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The Archive Keeper and her assistant disagree on which is correct:
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*Keeper:*$forall x . thin (I(x) imply W(x))$
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*Assistant:*$forall x . thin (W(x) imply I(x))$
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Two ancient inscriptions appear in the archive, each describing a different visitation pattern.
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Translate them to English, paying close attention to how quantifier order changes the meaning:
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*Inscription A:*$exists x forall y . thin V(y, x)$
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*Inscription B:*$forall x exists y . thin V(y, x)$
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=== Chamber 1: The Disjunction
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_Constraint:_ Exactly one inscription is true, and exactly one is false.
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*Door A:* "In this room there is a lady, and in the other room there is a tiger."
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=== Chamber 2: The Conjunction
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_Constraint:_ Both inscriptions have the same truth value (either both true or both false).
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*Door A:* "At least one of these rooms contains a lady."
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=== Chamber 3: The Implication
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_Constraint:_ Both inscriptions have the same truth value (either both true or both false).
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*Door A:* "If there is no tiger in this room, then there is a lady in the other room."
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Captain Pirx analyzes communications among four field agents: *Bilion*, *Stevok*, *Tomix*, *Johnon*.
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Cryptanalysts reconstructed these constraints:
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- If Bilion contacted Stevok, then Tomix did not contact Stevok.
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- Bilion contacted Stevok iff Johnon contacted at least one of them (Bilion or Stevok).
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- If Johnon contacted Stevok, then Bilion did not contact Stevok.
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