--- title: "Resolution: Marcus and Caesar" subject: "Artificial Intelligence" chapter: 4 key: "semester_6/artificial_intelligence:resolution-pompeian-caesar" source: https://www.bctengineer.com/q/semester_6/artificial_intelligence/resolution-pompeian-caesar site: BCT Engineering --- # Resolution: Marcus and Caesar Artificial Intelligence · Chapter 4: Knowledge Representation, Inference & Reasoning - Asked in: 2072 Magh, 7 marks - Video: [Was Marcus Loyalto Ceasar? Was Marcus Hates Ceasar? using Predicate Logic in AI by Mahesh Huddar](https://www.youtube.com/watch?v=Z0vTGofo2n8) — Mahesh Huddar - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/artificial_intelligence/resolution-pompeian-caesar - Whole chapter: https://www.bctengineer.com/md/semester_6/artificial_intelligence/chapter-4 ## Question Using resolution solve the following statements: All Pompeian are Romans. All Romans were either loyal to Caesar or hated him. Everyone is loyal to someone. People only try to assassinate rulers they are not loyal to. Marcus tried to assassinate Caesar. Marcus was Pompeian. Find, did Marcus hate Caesar? ## Short answer Convert statements to clauses, add negated goal ¬Hate(Marcus,Caesar). Resolution yields empty clause after deriving ¬TryAssassinate(Marcus,Caesar) from given facts, so Marcus hated Caesar. ## Answer #### Step 1: Statements in FOPL Predicates: Pompeian(x), Roman(x), Loyal(x,y), Hate(x,y), Ruler(x), Person(x), TryAssassinate(x,y). 1. ∀x Pompeian(x) → Roman(x) 2. ∀x Roman(x) → Loyal(x, Caesar) ∨ Hate(x, Caesar) 3. ∀x ∃y Loyal(x, y) 4. ∀x ∀y [Person(x) ∧ Ruler(y) ∧ TryAssassinate(x, y)] → ¬Loyal(x, y) 5. TryAssassinate(Marcus, Caesar) 6. Pompeian(Marcus) Added common knowledge: 7. Person(Marcus), 8. Ruler(Caesar). **Goal:** Hate(Marcus, Caesar) #### Step 2: Clauses (In statement 3, ∃y is Skolemised as a function f(x).) | No. | Clause | |---|---| | C1 | ¬Pompeian(x) ∨ Roman(x) | | C2 | ¬Roman(x) ∨ Loyal(x, Caesar) ∨ Hate(x, Caesar) | | C3 | Loyal(x, f(x)) | | C4 | ¬Person(x) ∨ ¬Ruler(y) ∨ ¬TryAssassinate(x, y) ∨ ¬Loyal(x, y) | | C5 | TryAssassinate(Marcus, Caesar) | | C6 | Pompeian(Marcus) | | C7 | Person(Marcus) | | C8 | Ruler(Caesar) | #### Step 3: Negated goal C9: **¬Hate(Marcus, Caesar)** #### Step 4: Resolution | Step | Resolve | Substitution | Resolvent | |---|---|---|---| | 1 | C9 and C2 | x/Marcus | R1: ¬Roman(Marcus) ∨ Loyal(Marcus, Caesar) | | 2 | R1 and C1 | x/Marcus | R2: ¬Pompeian(Marcus) ∨ Loyal(Marcus, Caesar) | | 3 | R2 and C6 | | R3: Loyal(Marcus, Caesar) | | 4 | R3 and C4 | x/Marcus, y/Caesar | R4: ¬Person(Marcus) ∨ ¬Ruler(Caesar) ∨ ¬TryAssassinate(Marcus, Caesar) | | 5 | R4 and C7 | | R5: ¬Ruler(Caesar) ∨ ¬TryAssassinate(Marcus, Caesar) | | 6 | R5 and C8 | | R6: ¬TryAssassinate(Marcus, Caesar) | | 7 | R6 and C5 | | **R7: empty clause (□)** | The empty clause is derived, so Hate(Marcus, Caesar) follows from the statements. **Answer: Yes, Marcus hated Caesar.** (Reason: as a Roman he was loyal to Caesar or hated him; since he tried to assassinate Caesar, a ruler, he could not be loyal, so he hated him. C3 "everyone is loyal to someone" is not needed.) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. To answer 'Did Marcus hate Caesar?' by refutation, which clause is added? A. Hate(Marcus, Caesar) B. ¬Hate(Marcus, Caesar) C. ¬Loyal(Marcus, Caesar) D. Loyal(Marcus, Caesar) Answer: B. The goal Hate(Marcus,Caesar) is negated and shown to give a contradiction. (Key point: Negated goal) 2. How is 'everyone is loyal to someone' (∀x ∃y Loyal(x,y)) made into a clause? A. The ∃y is replaced by the constant Caesar B. The ∀x is dropped and y is kept C. The ∃y is replaced by a Skolem function: Loyal(x, f(x)) D. It is negated and split into two clauses Answer: C. Existential variables depending on x become Skolem functions f(x). (Key point: Skolemisation) 3. Why can Marcus not be loyal to Caesar? A. He tried to assassinate Caesar, a ruler, and people only try to assassinate rulers they are not loyal to B. He is a Pompeian, and Pompeians are never loyal C. Everyone is loyal to someone else D. Caesar is a person, not a ruler Answer: A. Clause C4 with Person(Marcus), Ruler(Caesar) and TryAssassinate yields ¬Loyal(Marcus,Caesar). (Key point: Key derivation) 4. Which statement is not needed in the proof? A. All Pompeians are Romans B. Marcus was Pompeian C. Marcus tried to assassinate Caesar D. Everyone is loyal to someone (C3) Answer: D. Roman gives loyal or hate; assassination rules out loyal, so hate. C3 is never used. 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