Lecture 7: Logical Agents

Lecture 7: Logical Agents¶

AIMA Chapter 7 — 1 hour¶

Why Logic Matters¶

• Explicit knowledge: tell the agent facts and rules

• Inference: derive new facts without enumerating worlds

• Soundness and completeness: we can prove correctness

• Foundation for planning and diagnosis

Learning Objectives¶

• Design knowledge-based agents

• Understand the Wumpus world

• Represent knowledge in propositional logic

• Apply inference: resolution, forward/backward chaining

• Build agents for planning and state estimation

Knowledge-Based Agents¶

• Knowledge base (KB): Set of sentences

• Tell: Add sentence to KB

• Ask: Query KB (inference)

• Agent: Tell(percepts), Ask(what action?)

The Wumpus World¶

• Grid: 4×4, rooms

• Pit: Bottomless, breeze in adjacent

• Wumpus: Smell in adjacent

• Gold: Goal

• Agent: Perceive stench, breeze, glitter, bump, scream

Logic¶

• Syntax: Formal structure of sentences

• Semantics: Meaning (truth in a world)

• Entailment: α ⊨ β means α entails β (every model of α is model of β)

• Inference: Derive new sentences from KB

Propositional Logic¶

• Atomic sentences: True, False, or proposition symbols (P, Q, ...)

• Connectives: ¬, ∧, ∨, ⇒, ⇔

• Complex sentences: Built from atoms and connectives

Propositional Logic: Semantics¶

• Model: Assignment of truth values to symbols

• m ⊨ P: P is true in model m

• KB ⊨ α: α is true in every model of KB

Inference¶

• Sound: Only derives entailed sentences

• Complete: Derives all entailed sentences

• Model checking: Enumerate models (exponential)

• Theorem proving: Apply inference rules

Resolution¶

• Rule: (A ∨ B) ∧ (¬A ∨ C) ⊨ (B ∨ C)

• Resolvent: (B ∨ C)

• CNF: Conjunctive Normal Form — conjunction of clauses

• Resolution: Add resolvents until empty clause (contradiction)

Resolution Algorithm¶

1. Convert KB ∧ ¬α to CNF

2. Repeatedly apply resolution

3. If empty clause derived → KB ⊨ α

4. Complete for propositional logic

Horn Clauses¶

• Horn clause: At most one positive literal

• Definite clause: Exactly one positive (P₁ ∧ ... ∧ Pₙ ⇒ Q)

• Fact: Single positive literal

• Efficient inference for Horn KBs

Forward Chaining¶

• Start with facts

• Apply rules whose premises are satisfied

• Add conclusions to KB

• Repeat until no new facts

• Data-driven, O(n) for Horn KBs

Backward Chaining¶

• Start with query

• Find rules whose conclusion matches query

• Recursively prove premises

• Goal-driven

• Logic programming (Prolog)

SAT and Model Checking¶

• SAT: Is formula satisfiable?

• DPLL: Backtracking search for model

• Local search: WalkSAT, GSAT

• Random SAT: Phase transition

Logical State Estimation¶

• Belief state: Set of possible worlds

• Axiom: “Fluent holds at t iff it held at t-1 or was caused”

• Frame problem: What stays the same?

Planning by Propositional Inference¶

• Planning as SAT: Encode planning problem as propositional formula

• Variables: At(Location, t), Have(Arrow, t), etc.

• SAT solvers: Very efficient

Summary¶

• KB agents: Tell, Ask

• Propositional logic: Syntax, semantics

• Resolution: Complete for inference

• Forward/backward chaining: For Horn clauses

• Planning: Encode as SAT

References¶

• Russell & Norvig, AIMA 4e, Ch. 7

• Chapter PDF: chapters/chapter-07.pdf

• aima-python: logic.ipynb

Questions?¶

Next lecture: First-Order Logic (Chapter 8)