CS 621 Artificial Intelligence Lecture 4 – 05/08/05 Prof. Pushpak Bhattacharyya KNOWLEDGE REPRESENTATION & INFERENCING USING PREDICATE CALCULUS 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Prof. Pushpak Bhattacharyya, IIT Bombay Example Example: John, Jack & Jill are members of Alpine club. Every member of the club is either a mountain climber or a skier. All skiers like snow. No mountain climber likes rain. Jack dislikes whatever John likes, and likes whatever John dislikes. John likes rain and snow. Is there a member who is a mountain climber but not a skier. 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Knowledge Representation member (John, Alpine) member (Jack, Alpine) member (Jill, Alpine) x [member(x, Alpine) → mc(x) sk(x)] x [sk(x) → like(x, snow)] x [mc(x) → ~like(x, rain)] x [like(John, x) → ~like(John, x)] x [~like(John, x) → like(Jack, x)] like(John, rain) like(John, snow) Ques: x [member(x, Alpine) mc(x) ~sk(x)] 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Inference Strategy - RESOLUTION Basic Idea: REFUTATION of the goal Proof by contradiction Suppose the goal is false. Then show contradiction in the knowledge base. 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Prof. Pushpak Bhattacharyya, IIT Bombay Steps in Inferencing Convert all expressions, including the falsified goal, into clauses. member (John, Alpine) member (Jack, Alpine) member (Jill, Alpine) ~ member(x1, Alpine) ν mc(x1) ν sk(x1) ~ sk(x2) ν like(x2,snow) ~ mc(x3) ν ~ like(x3,rain) ~ like(John, x4) ν ~ like(Jack, x4) like(John, x5) ν like(Jack, x5) like(John, rain) like(John, snow) ~ member(x6, Alpine) ν ~ mc(x6) ν sk(x6) 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Prof. Pushpak Bhattacharyya, IIT Bombay Run Resolution By unification find value for x6 Theory of resolution: given P & ~P ν Q Resolvents we can obtain Q (Resolute) 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Prof. Pushpak Bhattacharyya, IIT Bombay Inverted Tree Diagram P ~P ν Q Q Aim C1 C2 Indicates contradiction null clause 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Prof. Pushpak Bhattacharyya, IIT Bombay Goal with Negation ~[ x{(member (x, Alpine) Λ mc(x) Λ ~ sk(x))}] x[~ member (x, Alpine) ν ~ mc(x) ν sk(x)] 11. ~ member(x6, Alpine) ν ~ mc(x6) ν sk(x6) 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Prof. Pushpak Bhattacharyya, IIT Bombay Monotonic Logic Every step of resolution increases the KB monotonically. Non-monotonic logic which used default reasoning. NEGATION BY FAILURE 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Which Pair of Clauses for Resolvents? 10 7 unification of x4 with snow snow/x4 snow/x4 12. ~ like(Jack, snow) 5 Jack/x2 13. ~ sk(Jack) 4 Jack/x1 14. ~ member(Jack, Alpine) ν mc(Jack) 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Prof. Pushpak Bhattacharyya, IIT Bombay Resolvents (Contd) 2 14 15. mc(Jack) 11 16. ~ member(Jack) ν sk(Jack) 13 ~ member(Jack) 2 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Prof. Pushpak Bhattacharyya, IIT Bombay Resolution Strategy Start from negated goal Use the derived clause as one of the pairs always - set of support strategy If the  is not reached, then The knowledge base is not complete. The inference rules are not adequate (modus ponens) Wrong inference path 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Modus Ponens & Modus Tolens P & P →Q gives Q Modus Tolens: ~Q and P →Q gives ~P 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay

Prof. Pushpak Bhattacharyya, IIT Bombay Prolog Predicate calculus (HORN Clause) + Resolution HORN Clauses: All the implications have single literal as consequent. A(antecedent) → B(consequent) B is a single literal, never contain any operator. Moreover B has to be a positive literal. 05-08-05 Prof. Pushpak Bhattacharyya, IIT Bombay