IF614 – Inteligensia Semu 09 / 01 - 13 Chapter 7 Symbolic Reasoning Under Uncertainty 7.1 Introduction to Nonmonotonic Reasoning Problems with Our Current Logic Knowledge must be complete. Knowledge must be consistent. Knowledge base must grow monotonically. BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu 09 / 02 - 13 7.2 Logic for Nonmonotonic Reasoning Models and Interpretations An interpretation of a set of wff ’s consists of : - A domain (D) - A function that assigns to each predicate a relation to each n-ary function an operator that maps from Dn into D to each constant an element of D A model of a set of wff ’s is an interpretation that satisfies them. BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu 09 / 03 - 13 Models, Wff’s, and Nonmonotonic Reasoning A B C BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu 09 / 04 - 13 Nonmonotomic Logic Example 1 : x,y : Related(x,y) M GetAlong(x,y) WillDefend(x,y) Example 2 : x : Republican(x) M Pacifist(x) Pacifist(x) x : Quaker(x) M Pacifist(x) Pacifist(x) Republican(Dick) Quaker(Dick) Rules are wff ’s AMBB A M B B We can derive the expression MBB BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu 09 / 05 - 13 Default Logic From : A:B C Use rules to extensions. compute one or more Rules are not wff ‘s. For example, given the two rules A : B B A:B B no conclusion about B can be drawn. BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu 09 / 06 - 13 Abduction Example : Given x : measles(x) Spots(x) Spots(Jill) conclude Measles(Jill) Definition : Given two wff’s (A B) and (B), for any expresions A and B, if it is consistent to assume A, do so. BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu 09 / 07 - 13 Inheritance in Default Logic Given : Baseball-Player(x) : height(x,6-1) height(x,6-1) x,y,z : height(x,y) height(x,z) y = z Pitcher(Three-Finger-Brown) Conclude : height(Three-Finger-Brown, 6-1) But this is blocked by height(Three-Finger-Brown, 5-11) Now we add : Adult - Male(x) : height(x,5-10) height(x,5-10) But now there are two extensions. BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu 09 / 08 - 13 Revised axiom : Adult-Male(x) : Baseball-Player(x) height(x,5-10) height(x,5-10) But this approach becomes unwiedly : Adult-Male(x) : Baseball-Player(x) Midget(x) Jockey(x) height(x,5-10) height(x,5-10) So we introduce AB predicates : x : Adult-male(x) AB(x,aspect1) height(x,510) x : Baseball-Player(x) AB(x, aspect1) x : Midget(x) AB(x, aspect1) x : Jockey(x) AB(x, aspect1) and single default rule : :AB(x,y) AB(x,y) BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu 09 / 09 - 13 The Closed World Assumption The only objects that satisfy any predicate P are those that must. Very useful for databases and AB predicates Problem : Some worlds are not closed The CWA is a purely syntactic reasoning process Example 1 From : : A(Joe) B(Joe) we derive : A(Joe) B(Joe) A (Joe) B (Joe) BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu Example 2 From : 09 / 010 - 13 : Single(John) Single(Mary) We derive : Single(Jane) From : Married(John) Married(Mary) We derive : Married(Jane) BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu 09 / 011 - 13 Circumscription Two advantages over CWA : Operates on whole formulas, not individual predicates. Allows some predicates to be marked as closed and others as open. Accomplished by adding axioms that force a minimal interpretation on a selected portion of the KB. Example 1 : x : Adult(x) AB(x,aspect1) Literate(x) Example 2 : A(Joe) B(Joe) BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu 09 / 012 - 13 7.3 Implementation Issues 7.4 Augmenting Problem Solver 7.5 Implementation: Depth First Search Justification - Based TMSs Used in conjunction problem solver. Connect nodes dependencies. via with a a separate network of Provide an algorithm for labeling nodes with their belief status. Search depth-first. BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999 IF614 – Inteligensia Semu 09 / 013 - 13 Justifications Suspect Abbott supported belief + - Beneficiary Abbott IN -list justification Alibi Abbott OUT - list A justification is valid if every assertion in the IN-list is believed and none of those in the OUT-list is. A justification is nonmonotonic if its OUTlist is not empty, or, recursively, if any assertion in its IN-list has a nonmonotonic justification. 7.6 Implemetation : Breadth-First Search BINA NUSANTARA Edisi : 1 Revisi : 3 Sept - 1999