7.5 Reasoning Patterns in Propositional Logic

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Conjunctive Normal Form: CNF

1. Eliminate ⇔. Example: a ⇔ b.
2. Eliminate ⇒. Example: a ⇒ b.
3. Move negation inwards. Example: ¬(a ^ b).
4. Use distributive laws. Example: (a ^ b) ∨ c.

Resolution

• Unit resolution. (a ∨ b) and ¬b yields a.
• Full resolution. (a ∨ b ∨ c ∨ d) and (¬d ∨ e ∨ f) yields (a ∨ b ∨ c ∨ e ∨ f).
• Resolution is sound.

Resolution Theorem Proving

• The knowledge base must be converted to CNF.
• Assume the negation of what you are trying to prove.
• If a contradiction can be reached, what you are trying to prove is known to be true.
• Resolution theorem proving is refutation complete.

Exercise 7.9

If the unicorn is mythical, then it is immortal, but if it is not mythical, then it is a mortal mammal. If the unicorn is either immortal or a mammal, then it is horned. The unicorn is magical if it is horned.

Using resolution, prove that the unicorn is magical.

Horn Clauses

• A Horn clause is a disjunction of literals where at most one is positive.
• Horn clauses are the basis for Prolog!
• Here is a Prolog rule: a ^ b ^ c => d
• Here is a Prolog fact: e
• Deciding entailment with Horn clauses can be done in time that is linear in the size of the knowledge base!

Example

Consider the following Prolog rules and facts:

• a ^ b ⇒ c
• a ⇒ g
• a ⇒ h
• c ⇒ d
• d ^ e ⇒ f
• i ⇒ f
• a
• b
• e

Prove f first using forward chaining and then using backward chaining.