Lecture 15 of 41 FOL Resolution Strategies Input

Lecture 15 of 41 FOL: Resolution Strategies Input, Linear, Unit, & Set of Support Resolution Handling Equality William H. Hsu Department of Computer Science, KSU Canvas course redirector: http: //bit. ly/kstate-ai-class Course web site: http: //kdd. cs. ksu. edu/Courses/CIS 530/ Instructor home page: http: //www. cs. ksu. edu/~bhsu Reading for Today: § 9. 5, p. 345 – 356, Russell & Norvig 3 rd edition Handout, Nilsson & Genesereth, Logical Foundations of Artificial Intelligence Reading for Next Class: Review Chapter 9 (Read & Explain Pairs), Russell & Norvig 3 rd edition CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University

Where we Are Today CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University

Clausal Form Mnemonic – INSEUDOR: Review l Implications Out (Replace with Disjunctive Clauses) l Negations Inward (De. Morgan’s Theorem) l Standardize Variables Apart (Eliminate Duplicate Names) l Existentials Out (Skolemize) l Universals Made Implicit l Distribute And Over Or (i. e. , Disjunctions Inward) l Operators Made Implicit (Convert to List of Lists of Literals) l Rename Variables (Independent Clauses) l 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 Adapted from: Nilsson and Genesereth (1987). Logical Foundations of Artificial Intelligence. http: //bit. ly/45 Cmqq CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University

Skolemization: Eliminating Existential Quantifiers Based on slide © 2004 - 2010 S. Russell & P. Norvig. Reused with permission. CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University

Resolution: Review © 2004 - 2010 S. Russell & P. Norvig. Reused with permission. CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University

Refutation-Based Resolution Theorem Proving Based on slide © 2004 - 2010 S. Russell & P. Norvig. Reused with permission. CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University

Resolution Proof Example [1] Based on slide © 2004 - 2010 S. Russell & P. Norvig. Reused with permission. CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University

Resolution Proof Example [2] Based on slide © 2004 - 2010 S. Russell & P. Norvig. Reused with permission. CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University

Dealing with Equality in FOL: Demodulation & Paramodulation l Problem: How To Find Inference Rules for Sentences With = Unification OK without it, but… A = B doesn’t force P(A) and P(B) to unify l Solutions Demodulation ð Generate substitution from equality term ð Additional sequent rule, Section 9. 5 (p. 304) R&N 2 e Paramodulation ð More powerful: e. g. , (x = y) P(x) © 2003 S. Russell & P. Norvig ð Generate substitution from WFF containing equality constraint ð Sequent rule sketch, Section 9. 5 (p. 304) R&N 2 e ð Full discussion in Nilsson & Genesereth CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University

Resolution Strategies [1]: Unit and Input Resolution l Unit Preference Idea: Prefer inferences that produce shorter sentences Compare: Occam’s Razor How? Prefer unit clause (single-literal) resolvents (α β with β α) Reason: trying to produce a short sentence ( True False) l Input Resolution Idea: “diagonal” proof (proof “list” instead of proof tree) Every resolution combines some input sentence with some other sentence Input sentence: in original KB or query Unit resolution Input resolutions CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University

Resolution Strategies [2]: Linear Resolution and Set-of-Support l Linear Resolution Generalization of input resolution Include any ancestor in proof tree to be used Linear resolutions l Set of Support (So. S) Idea: try to eliminate some potential resolutions Prevention as opposed to cure How? ð Maintain set So. S of resolution results ð Always take one resolvent from it Caveat: need right choice for So. S to ensure completeness CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University

Resolution Strategies [3]: Subsumption l Subsumption Idea: eliminate sentences that are more specific than others e. g. , P(x) subsumes P(A) l Putting It All Together CIS 530 / 730 Artificial Intelligence Lecture 15 of 41 Part A (1 of 2) Department of Computer Science Kansas State University