AGM Belief Revision

Origin. Alchourrón, Gärdenfors, Makinson (1985). Rational belief change. Contraction and revision. Postulates. Foundation of belief dynamics.

Models. Belief sets change rationally. Three operations: expansion, contraction, revision. AGM postulates characterize. Epistemic entrenchment. Rational belief management.

Formalism.

Belief set: K = deductively closed set of sentences. Cn(K) = K. Theory of beliefs. Consistent (usually).

Three operations: K + φ: expansion (add φ, close). K - φ: contraction (remove φ, minimal change). K * φ: revision (add φ, maintain consistency).

Levi identity: K * φ = (K - ¬φ) + φ. Revision = contract negation, then expand. Reduces revision to contraction.

Harper identity: K - φ = K ∩ (K * ¬φ). Contraction from revision. Interdefinability.

AGM postulates for revision: K1: K * φ is belief set. K2: φ ∈ K * φ. K3: K * φ ⊆ K + φ. K4: If ¬φ ∉ K, then K + φ ⊆ K * φ. K5: K * φ = K_⊥ only if ⊢ ¬φ. K6: If ⊢ φ ↔ ψ, then K * φ = K * ψ. K7, K8: supplementary postulates.

Epistemic entrenchment: φ ≤ ψ: ψ at least as entrenched. More entrenched = harder to give up. Ordering determines contraction. φ ∈ K - ψ iff φ ∈ K and (ψ < φ ∨ ψ or ⊢ ψ).

Grove spheres: Possible worlds semantics. Concentric spheres around K-worlds. Revision: closest φ-worlds. Lewis-style.

Symbols.

Metatheory. Rationality postulates. Entrenchment. Sphere semantics. Interdefinability.

Applies to. Epistemology. AI. Database update. Theory change.

Limitations. Idealization. Iterated revision. Computational cost. Single-agent focus.

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