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

A lecture outline from docsity.com covering clausal (conjunctive normal) form and resolution techniques in first-order logic. The lecture includes topics such as conjunctive normal form conversion, resolution refutation, computability issues, and automated deduction by forward chaining. Students are encouraged to read and explain pairs of sections for class participation.

Typology: Slides

2012/2013

Uploaded on 04/29/2013

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Lecture 17 of 41
Clausal (Conjunctive Normal) Form
and Resolution Techniques
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Download Lecture 17: Resolution Techniques in First-Order Logic and more Slides Artificial Intelligence in PDF only on Docsity!

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 DerivesFrom S’
  • (By Lifting Lemma)A Resolution Proof Sn

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)