Resolution Theorem Proving - Artificial Intelligence - Lecture Slides, Slides of Artificial Intelligence

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Lecture 14 of 41

Resolution Theorem Proving

Conjunctive Normal ( aka Clausal) Form:

Conversion (Nilsson) and Mnemonic

• Implications Out
• Negations Out
• Standardize Variables Apart
• Existentials Out (Skolemize)
• Universals Made Implicit
• Distribute And Over Or (i.e., Disjunctions In)
• Operators Out
• Rename Variables
• A Memonic for Star Trek: The Next Generation Fans
  • Captain Picard:
• I’ll Notify Spock’s Eminent Underground Dissidents On Romulus
• I’ll Notify Sarek’s Eminent Underground Descendant On Romulus

Resolution Theorem Proving

Example:

Resolution Proof

Adapted from slides by S. Russell, UC Berkeley

Logic Programming vs. Imperative Programming

Adapted from slides by S. Russell, UC Berkeley

A Look Ahead:

Logic Programming as Horn Clause Resolution

Summary Points

• Previously: Logical Agents and Calculi, FOL in Practice
• Today: Resolution Theorem Proving
  • Conjunctive Normal Form (clausal form)
  • Inference rule
    • Single-resolvent form
    • General form
  • Proof procedure: refutation
  • Decidability properties
    • FOL-SAT
    • FOL-NOT-SAT (language of unsatisfiable sentences; complement of FOL-SAT)
    • FOL-VALID
    • FOL-NOT-VALID
• Thursday
  • More Prolog
  • Implementing unification

Terminology

• Properties of Knowledge Bases (KBs)
  • Satisfiability and validity
  • Entailment and provability
• Properties of Proof Systems
  • Soundness and completeness
  • Decidability, semi-decidability, undecidability
• Normal Forms: CNF, DNF, Horn; Clauses vs. Terms
• Resolution
• Refutation
• Satisfiability, Validity
• Unification