- 2072 Magh · 7 marks
Resolution: Marcus and 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.
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?
Step 1: Statements in FOPL
Predicates: Pompeian(x), Roman(x), Loyal(x,y), Hate(x,y), Ruler(x), Person(x), TryAssassinate(x,y).
- ∀x Pompeian(x) → Roman(x)
- ∀x Roman(x) → Loyal(x, Caesar) ∨ Hate(x, Caesar)
- ∀x ∃y Loyal(x, y)
- ∀x ∀y [Person(x) ∧ Ruler(y) ∧ TryAssassinate(x, y)] → ¬Loyal(x, y)
- TryAssassinate(Marcus, Caesar)
- 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.)
✓ Answer checked by Claude Opus 5.5 (high effort)
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