Consider the following popular puzzle. A boy and a girl are talking. “I am aboy” said the child with black hair. “I am a girl” said the child with white hair.At least one of them is lying. Write down a knowledge base that describes thisriddle. Show with resolution that both of them are lying.
Question
Consider the following popular puzzle. A boy and a girl are talking. “I am aboy” said the child with black hair. “I am a girl” said the child with white hair.At least one of them is lying. Write down a knowledge base that describes thisriddle. Show with resolution that both of them are lying.
Solution
To solve this puzzle, we can use propositional logic and resolution. Let's denote:
B: The child with black hair is a boy. W: The child with white hair is a girl.
The statements given in the puzzle can be written as:
- B (The child with black hair said "I am a boy")
- W (The child with white hair said "I am a girl")
We are also given that at least one of them is lying. This can be represented as:
- ¬B ∨ ¬W (Either the child with black hair is not a boy or the child with white hair is not a girl)
We want to show that both of them are lying, which can be represented as:
- ¬B ∧ ¬W (The child with black hair is not a boy and the child with white hair is not a girl)
To prove this using resolution, we need to convert all our statements into Conjunctive Normal Form (CNF). Statements 1 and 2 are already in CNF. Statement 3 is also in CNF. Statement 4 can be converted into CNF by using De Morgan's laws, resulting in:
- ¬B ∨ ¬W
Now we can use resolution:
Resolving 1 and 4 gives ¬W (The child with white hair is not a girl) Resolving 2 and 4 gives ¬B (The child with black hair is not a boy)
So, we have shown that both children are lying, which is what we wanted to prove.
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