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Unification and First-Order Logic Review - KSU Lecture

This lecture focuses on the concepts of unification and first-order logic (FOL) as part of the course at Kansas State University. Topics covered include resolution theorem proving, logic programming with Prolog, and the implementation details of unification. Students engage in offline exercises to enhance understanding, with a focus on group discussions about assigned reading sections from Russell and Norvig. The lecture aims to clarify complex topics such as the completeness of resolution and strategies for effective logic programming.

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Unification and First-Order Logic Review - KSU Lecture

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  1. Lecture 23 Unification and FOL Review Friday, 17 October 2003 William H. Hsu Department of Computing and Information Sciences, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading: Chapter 10, Russell and Norvig (next 1.5 weeks) Handout, Nilsson and Genesereth

  2. Lecture Outline • Today’s Reading • Chapter 9, Russell and Norvig (before midterm, Wed 20 Oct 2003) • Recommended references: Nilsson and Genesereth (excerpt of Chapter 5 online) • Next Week’s Reading: Chapter 11, Russell and Norvig • Previously: First-Order Logic • Theorem proving: forward and backward chaining • Resolution refutation (sound and complete proof procedure) • Today: Logic Programming by Resolution and Unification • Resolution theorem proving • Specific implementation: Prolog • Implementing unification: some details • Occurs check • Complexity • Other “industrial-strength” KR and inference methods

  3. Offline Exercise:Read-and-Explain Pairs • For Class Participation (MP3, PS4) • With Your Term Project Partner or Assigned Partner(s) • Read your assigned sections (take notes if needed) • Group A: R&N 2e Sections 9.1-9.3, p. 272-287; 9.6-9.7; 10.1-10.2; 10.5-6 • Group B: R&N 2e Sections 9.4-9.5. p. 287-309; 9.6-9.7; 10.3-10.4; 10.7 • Skim your partner’s sections • Meet with your partner (by e-mail, ICQ, IRC, or in person) • Explain your section • Key ideas – what’s important? • Important technical points • Discuss unclear points and write them down! • Optional: By Mon 20 Oct 2003 • Post • Confirmation to ksu-cis730-fall_2003 • Muddiest point: what is least clear in your understanding of your section? • Re-read your partner’s section as needed

  4. Review:Logic Programming (Prolog) Examples Adapted from slides by S. Russell, UC Berkeley

  5. Review:Resolution Inference Rule Adapted from slides by S. Russell, UC Berkeley

  6. 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 9.8 p. 287 R&N

  7. Decidability Revisited • See: Section 9.7 Sidebar, p. 288 R&N • Duals (Why?) • Complexity Classes • Understand: Reduction to Ld, LH

  8. Unification Procedure [1]:General Idea • Most General Unifier (Least-Commitment Substitution) • See: Examples (p. 271 R&N, Nilsson and Genesereth) Adapted from slides by S. Russell, UC Berkeley

  9. Unification Procedure [2]:Algorithm Figure 10.3 p. 303 R&N

  10. Example [1]:Sentences in FOL

  11. Example [2]:Clausal Form (CNF)

  12. Example [3]:Applying Resolution and Unification

  13. 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

  14. Logic Programming – Tricks of The Trade [2]:Resolution Strategies • Unit Preference • Idea: Prefer inferences that produce shorter sentences (compare: Occam’s Razor) • How? Prefer unit clause (single-literal) resolvents • Reason: trying to produce a short sentence (  True  False) • Set of Support • Idea: try to eliminate some potential resolutions (prevention as opposed to cure) • How? Maintain set SoS of resolution results and always take one resolvent from it • Caveat: need right choice for SoS to ensure completeness • Input Resolution and Linear Resolution • Idea: “diagonal” proof (proof “list” instead of proof tree) • How? Every resolution combines some input sentence with some other sentence • Input sentence: in original KB or query • Generalize to linear resolution: include any ancestor in proof tree to be used • Subsumption • Idea: eliminate sentences that sentences that are more specific than others • E.g., P(x) subsumesP(A)

  15. Summary Points • Previously: FOL, Forward and Backward Chaining, Resolution • Today: More Resolution Theorem Proving, Prolog, and Unification • Review: resolution inference rule • Single-resolvent form • General form • Application to logic programming • Review: decidability properties • FOL-SAT • FOL-NOT-SAT (language of unsatisfiable sentences; complement of FOL-SAT) • FOL-VALID • FOL-NOT-VALID • Unification • Next Week • Intro to classical planning • Inference as basis of planning

  16. 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)

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