1. Learning Three-Valued Logical Programs Evelina Lamma 1, Fabrizio Riguzzi 1, Luis Moniz Pereira 2 1 DEIS, Università di Bologna 2 Centro de Inteligencia Artificial (CENTRIA), Lisbon

2. 2 Extended Logic Programs A finite set of rules of the form:A finite set of rules of the form: L 0  L 1,...,L n,,not L n+1,...,not L m where each L i can be either A or ¬A ¬A is the explicit negation of A,¬A is the explicit negation of A, ¬fliles(X)  animal(X). fliles(X)  bird(X). ¬ fliles(X)  penguin(X). Three-valued semantics (WFSX)Three-valued semantics (WFSX)

3. 3 Learning Three-Valued Logical Programs Autonomous agent: acquisition of information by means of experimentsAutonomous agent: acquisition of information by means of experiments Experiment:Experiment: –execution of an action –evaluation of the results with respect to the achievement of a goal –positive and negative results Learning general rules on actions:Learning general rules on actions: –distinction among actions with a positive, negative or unknown outcome

4. 4 New Learning Framework Conditions on examples:Conditions on examples: P    E +,  E - (completeness) P   E -,  E + (consistency) LIVE (Learning In a three-Valued Environment)LIVE (Learning In a three-Valued Environment) Standard ILP techniques to learn p and  pStandard ILP techniques to learn p and  p –top-down or bottom-up Learning of:Learning of: –p using E +, E - as training set –  p using E -, E + as training set

5. 5 Intersection of definitions E+E+ E-E-  p p Exceptions to the positive concept: negative examples Exceptions to the negative concept: positive examples Unseen atoms

6. 6 Atoms in the intersection Unseen atoms both true and false are classified as unknown:Unseen atoms both true and false are classified as unknown: p(X)  p + (X), not  p(X).  p(X)  p - (X), not p(X).  p(X)  p - (X), not p(X). Training set atoms must be classified according to the training set:Training set atoms must be classified according to the training set: p(X)  p + (X), not ab p (X), not  p(X).  p(X)  p - (X), not ab  p (X), not p(X).

7. 7 Generality of Solutions Bottom-up methods:Bottom-up methods: –search from specific to general: General Solution (GS) –GOLEM (RLGG), CIGOL (Inverse Resolution) Top-down methods:Top-down methods: –search from general to specific: Specific Solution (SS) –FOIL, Progol

8. 8 Criteria for chosing the generality Risk that can derive from a classification errorRisk that can derive from a classification error –high risk GS –low riskSS Confidence in the set of negative examplesConfidence in the set of negative examples –high confidence GS –low confidenceSS

9. 9 Example B:bird(a).has_wings(a). jet(b).has_wings(b). angel(c).has_wings(c).has_limbs(c). penguin(d).has_wings(d).has_limbs(d). dog(e).has_limbs(e). cat(f).has_limbs(f). E + ={flies(a)}E - ={flies(d), flies(e)} flies + E+E+ abcf de flies - E-E-

10. 10 The theory can be revised differently depending on the type of literal that is found contradictory in the intersection. Strategies For Theory Refinement