Automated Theorem Proving

What is a "Proof"?

• Flip a coin: Heads I win, Tails you lose.
  • Heads => IWin
  • Tails => YouLose
  • YouLose => IWin
  • Heads or Tails
  • not Heads or not Tails
• Inference Rule (Modus Ponens)
  • Condition: A and (A => B)
  • Conclusion: B
• Proof by Modus Ponens
• Inference Rule (Resolution)
  • Condition 1: A₁ or ... or A_i or not C or ... or A_m
  • Condition 2: B₁ or ... or B_j or C or ... or B_n
  • Conclusion: A₁ or ... A_m or B₁ or ... or B_n
• Proof by contradiction
  • Tautology: A or not A
  • Contradiction: not (A or not A) => A and not A

Satisfiability Problem

• CNF (Conjunctive Normal Form)
  • NP-Completeness
• Horn Clause
  • Each clause has at most one positive literal
  • (Definite Horn Clause): Each clause has exactly one positive literal
  • Leads to polynomial-time resolution inference

Expert Systems

• Build knowledge base using Horn Clauses
• Prolog
  • Backward-chaining (resolution)
• CLIPS
  • Forward-chaining (modus ponens)

Predicate Calculus

• "Abbreviated" predicate calculus
  • Allows parameterized predicates
  • Abbreviates propositional logic
• "Full" predicate calculus
  • Allows "function symbols"
  • Example: Peano Axioms:
    • NatNum(Zero)
    • NatNum(?N) => NatNum(S(?N))