---
title: "Resolution: cup supports book"
subject: "Artificial Intelligence"
chapter: 4
key: "semester_6/artificial_intelligence:resolution-cup-supports-book"
source: https://www.bctengineer.com/q/semester_6/artificial_intelligence/resolution-cup-supports-book
site: BCT Engineering
---

# Resolution: cup supports book

Artificial Intelligence · Chapter 4: Knowledge Representation, Inference & Reasoning

- Asked in: 2071 Bhadra, 8 marks
- Video: [Resolution to Prove Predicate Facts to First Order Logic FOL Artificial Intelligence Mahesh Huddar](https://www.youtube.com/watch?v=CbI-Q2a5rUU) — 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-cup-supports-book
- Whole chapter: https://www.bctengineer.com/md/semester_6/artificial_intelligence/chapter-4

## Question

If X is on the top of Y, Y supports X. If X is above Y and they are touching each other, X is on the top of Y. A cup is above a book. A cup is touching a book. Show that supports (book, cup) is true.

## Short answer

Convert to FOPL clauses, negate goal ¬Supports(Book,Cup). Resolve with C1: ¬OnTop(Cup,Book); with C2: ¬Above(Cup,Book) ∨ ¬Touching(Cup,Book); with C3: ¬Touching(Cup,Book); with C4: empty clause. Contradiction proves Supports(Book,Cup) true.

## Answer

#### Step 1: Statements in FOPL
Predicates: OnTop(x,y), Supports(x,y) = x supports y, Above(x,y), Touching(x,y).

1. ∀x ∀y OnTop(x, y) → Supports(y, x)
2. ∀x ∀y [Above(x, y) ∧ Touching(x, y)] → OnTop(x, y)
3. Above(Cup, Book)
4. Touching(Cup, Book)

**Goal:** Supports(Book, Cup)

#### Step 2: Clauses
| No. | Clause |
|---|---|
| C1 | ¬OnTop(x, y) ∨ Supports(y, x) |
| C2 | ¬Above(x, y) ∨ ¬Touching(x, y) ∨ OnTop(x, y) |
| C3 | Above(Cup, Book) |
| C4 | Touching(Cup, Book) |

#### Step 3: Negated goal
C5: **¬Supports(Book, Cup)**

#### Step 4: Resolution
| Step | Resolve | Substitution | Resolvent |
|---|---|---|---|
| 1 | C5 and C1 | y/Book, x/Cup | R1: ¬OnTop(Cup, Book) |
| 2 | R1 and C2 | x/Cup, y/Book | R2: ¬Above(Cup, Book) ∨ ¬Touching(Cup, Book) |
| 3 | R2 and C3 | | R3: ¬Touching(Cup, Book) |
| 4 | R3 and C4 | | **R4: empty clause (□)** |

```text
¬Supports(Book,Cup) + C1 -> ¬OnTop(Cup,Book)
   + C2 -> ¬Above(Cup,Book) ∨ ¬Touching(Cup,Book)
   + C3 -> ¬Touching(Cup,Book)
   + C4 -> □
```
The empty clause is obtained, so the assumption is false. **Hence Supports(Book, Cup) is true.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In the resolution refutation, which clause is added to the clause set?
   A. Supports(Book, Cup)
   B. ¬Supports(Cup, Book)
   C. OnTop(Cup, Book)
   D. ¬Supports(Book, Cup)
   Answer: D. Refutation adds the negation of the goal and shows a contradiction. (Key point: Negated goal)

2. Which is the clause form of 'if X is above Y and touching Y then X is on top of Y'?
   A. ¬Above(x,y) ∨ ¬Touching(x,y) ∨ OnTop(x,y)
   B. Above(x,y) ∨ Touching(x,y) ∨ OnTop(x,y)
   C. ¬Above(x,y) ∧ ¬Touching(x,y) ∧ OnTop(x,y)
   D. ¬OnTop(x,y) ∨ Above(x,y) ∨ Touching(x,y)
   Answer: A. A ∧ B → C becomes ¬A ∨ ¬B ∨ C. (Key point: Rule 2 as a clause)

3. Resolving ¬Supports(Book,Cup) with ¬OnTop(x,y) ∨ Supports(y,x) gives which resolvent, and with what substitution?
   A. ¬OnTop(Book, Cup) with x/Book, y/Cup
   B. ¬OnTop(Cup, Book) with y/Book, x/Cup
   C. OnTop(Cup, Book) with y/Book, x/Cup
   D. ¬Above(Cup, Book) with x/Cup
   Answer: B. Supports(y,x) unifies with Supports(Book,Cup), so y = Book and x = Cup, leaving ¬OnTop(Cup,Book). (Key point: First resolution step)

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