Predicate Logic and Resolution Methods for Knowledge Representation

1. Predicate Logic in Knowledge Representation

Predicate logic, also known as first-order logic (FOL), extends propositional logic by allowing representation of objects, their properties, and the relationships between them. It is widely used in knowledge representation and reasoning (KRR) in AI because it can express complex facts, rules, and structures beyond simple true/false propositions.

Key Components

• Predicates: Functions that express properties or relationships. Example: Father(John, Mary) means John is the father of Mary.
• Variables: Symbols that stand for arbitrary objects. Example: x in IsHuman(x).
• Constants: Specific objects or entities like John and Mary.
• Quantifiers:
  • Universal quantifier: "For all", written ∀x.
  • Existential quantifier: "There exists", written ∃x.

Why Predicate Logic Is Important in KRR

• Allows detailed and structured representation of knowledge involving multiple entities and their relationships.
• Supports general rules and inferences using quantifiers, such as "All humans are mortal."
• Enables AI systems to perform reasoning to derive new facts or answer queries based on stored knowledge.

Example

To represent "John is the father of Mary, and all fathers are male," predicate logic statements would be:

Father(John, Mary)
∀x, y ( Father(x, y) → Male(x) )

This allows an AI system to understand and infer properties and relationships systematically within its knowledge base.

In summary, predicate logic in knowledge representation provides a more expressive and formal framework than propositional logic for encoding and reasoning about complex real-world information in AI systems.

2. Resolution Method in Logic

Concept of Resolution.

Resolution is a rule of inference primarily used in automated theorem proving for propositional and first-order logic. It is a refutation-based method where the goal is to prove the unsatisfiability of the negation of a statement. In simpler terms, it works by assuming the negation of the theorem to be proved and deriving contradictions by combining clauses until an empty clause is found, indicating the original statement is true.

How Resolution Works

1. Express all logical statements in Conjunctive Normal Form (CNF), i.e., a conjunction of disjunctions of literals.
2. Identify two clauses that contain complementary literals (a literal and its negation).
3. Resolve these clauses by eliminating the complementary literals and merging the remaining parts into a new clause.
4. Repeat the process with the new and existing clauses.
5. If an empty clause is derived, it indicates the original set is contradictory, thereby proving the statement.