Lecture 17: Resolution Techniques in First-Order Logic, Slides of Artificial Intelligence

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Lecture 17 of 41

Clausal (Conjunctive Normal) Form

and Resolution Techniques

Lecture Outline

• Today’s Reading
• Thursday’s Reading: Chapter 9, R&N
• Previously: Propositional and First-Order Logic
  • Two weeks ago
    • Logical agents: KR, inference, problem solving
    • Propositional logic: normal forms, sequent rules
    • Predicates and terms
    • First-order logic (FOL): quantifiers
  • Last week
    • FOL agents; frame problem; situation calculus, successor-state axioms
    • FOL KBs and forward search using sequent rules (sound but incomplete set)
• Today: Backward Inference
  • Resolution refutation (sound and complete proof procedure)
  • Computability (decidability) issues

Conjunctive Normal ( aka Clausal) Form [2]:

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

Offline Exercise:

Read-and-Explain Pairs

• For Class Participation (PS3, MP4)
• With Your Term Project Partner or Assigned Partner(s)
• Read: Chapter 9 (esp. 9.2, 9.5), Chapter 10 R&N 2e
• By Fri 08 Oct 2004, Fri 15 Oct 2004

Review:

Logic Programming (Prolog) Examples

Completeness of Resolution

• Any Set of Sentences S Is Representable in Clausal Form (Last Class)
• Assume S Is Unsatisfiable, and in Clasual Form
• (By Herbrand’s Theorem) Some Set S’ of Ground Instances is Unsatisfiable
• (By Ground Resolution Theorem) Resolution Derives ⊥ From S’
• (By Lifting Lemma) ∃ A Resolution Proof S  n ⊥

Figure 9.13 p. 301 R&N 2e

Unification Procedure:

General Idea

• Most General Unifier (Least-Commitment Substitution)
• See: Examples (p. 271 R&N, Nilsson and Genesereth)

Logic Programming – Tricks of The Trade [1]:

Dealing with Equality

• Problem
  • How to find appropriate inference rules for sentences with =?
  • Unification OK without it, but…
  • A = B doesn’t force P ( A ) and P ( B ) to unify
• Solutions
  • Demodulation
    • Generate substitution from equality term
    • Additional sequent rule: p. 284 R&N
  • Paramodulation
    • More powerful
    • Generate substitution from WFF containing equality constraint
    • e.g., ( x = y ) ∨ P(x)
    • Sequent rule sketch: p. 284 R&N

Logic Programming – Tricks of The Trade [3]:

Indexing Strategies

• Store and Fetch
  • Idea: store knowledge base in list of conjuncts
  • STORE: constant, i.e., O(1) worst-case running time
  • FETCH: linear, i.e., O( n ) time
• Table Based
  • Idea: store KB in hash table (key: ground literals)
  • STORE: O(1)
  • FETCH: O(1) expected case
  • Problems
    • Complex WFFs (other than negated atoms)
    • Variables
  • Solution: implicative normal form matching (Figure 10.1, p. 301 R&N)
• Tree-Based
  • What if there are many clauses for a predicate? (e.g., Brother (012-34-5678, x ))
  • Type of combined indexing: joint primary key – predicate and argument symbols
  • May need background knowledge for semantic query optimization (SQO)

Logic Programming – Tricks of The Trade [4]:

Compilation

• Intermediate Languages
  • Abstract machines
    • Warren Abstract Machine (WAM)
    • Java Virtual Machine (JVM)
  • Imperative intermediate representations (IRs)
    • C/C++
    • LISP / Scheme / SML – functional languages with imperative features
• Use in Genetic Programming (GLP): Later
• Beyond Scope of CIS 730: Compiling with Continuations (Appel)

Terminology

• Properties of Knowledge Bases (KBs)
  • Satisfiability and validity
  • Entailment and provability
• Properties of Proof Systems
  • Soundness and completeness
  • Decidability, semi-decidability, undecidability
• Resolution
• Refutation
• Satisfiability, Validity
• Unification
  • Occurs check
  • Most General Unifer
• Prolog: Tricks of The Trade
  • Demodulation, paramodulation
  • Unit resolution, set of support, input / linear resolution, subsumption
  • Indexing (table-based, tree-based)