Lecture 9: Inference in First-Order Logic

Lecture 9: Inference in First-Order Logic¶

AIMA Chapter 9 — 1 hour¶

Learning Objectives¶

• Reduce FOL inference to propositional inference

• Apply unification for variable substitution

• Implement forward and backward chaining in FOL

• Use resolution for FOL theorem proving

Propositional vs. FOL Inference¶

• Propositionalization: Ground (instantiate) all variables

• Problem: Infinite Herbrand universe

• Solution: Lifted inference — keep variables, use unification

Unification¶

• Substitution: θ = {x/A, y/B}

• Unify(Knows(John,x), Knows(John,Jane)): θ = {x/Jane}

• Unify(Knows(John,x), Knows(y,Mother(y))): θ = {x/Mother(John), y/John}

• MGU: Most general unifier

Unification Algorithm¶

• Occurs check: x must not occur in term we substitute for x

• Recursive: Unify arguments, combine substitutions

• Failure: Incompatible (e.g., P(A) vs P(B))

Forward Chaining (FOL)¶

• Definite clauses: L₁ ∧ L₂ ∧ ... ⇒ L

• Ground: Substitute to get ground facts

• Efficient: Indexing, incremental

First-Order Definite Clauses¶

• Storage: Index by predicate

• Matching: Unify rule premises with facts

• Incremental: Only check rules affected by new fact

Backward Chaining (FOL)¶

• Goal: Prove query

• Find: Rules whose head unifies with goal

• Recurse: Prove body (subgoals)

• Prolog: Depth-first, left-to-right

Logic Programming (Prolog)¶

• Horn clauses: Definite clauses

• Resolution: Backward chaining

• Database semantics: Closed-world assumption

• Negation as failure: ¬P if P not provable

Redundant Inference¶

• Redundant: Derive same fact multiple ways

• Memoization: Cache proved goals

• Infinite loops: p(X) :- p(X) — need occurs check

Resolution for FOL¶

• Convert to CNF: Eliminate quantifiers, Skolemize

• Skolemization: ∃x P(x) → P(SkolemConstant)

• Resolution: Unify literals, resolve

Resolution Rule (FOL)¶

• Binary resolution: (L₁ ∨ ... ∨ Lₙ) and (¬M₁ ∨ ... ∨ Mₘ)

• Unify: Lᵢ and Mⱼ

• Resolvent: Combine clauses, remove Lᵢ and Mⱼ

Completeness of Resolution¶

• Resolution: Refutation-complete for FOL

• Lifting: FOL resolution lifts propositional resolution

• Herbrand: If unsatisfiable, finite ground subset

Resolution Strategies¶

• Unit preference: Prefer unit clauses

• Subsumption: Remove subsumed clauses

• Set of support: One clause from goal

Summary¶

• Unification: Variable substitution, MGU

• Forward chaining: Data-driven, definite clauses

• Backward chaining: Goal-driven, Prolog

• Resolution: Skolemize, resolve, complete

References¶

• Russell & Norvig, AIMA 4e, Ch. 9

• Chapter PDF: chapters/chapter-09.pdf

Questions?¶

Next lecture: Knowledge Representation (Chapter 10)