Learning Three-Valued Logical ProgramsLearning Three-Valued Logical Programs

Evelina LammaEvelina Lamma11, Fabrizio Riguzzi, Fabrizio Riguzzi11, Luis Moniz Pereira, Luis Moniz Pereira22

11 DEIS, Università di Bologna DEIS, Università di Bologna 22 Centro de Inteligencia Artificial (CENTRIA), Lisbon Centro de Inteligencia Artificial (CENTRIA), Lisbon

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Extended Logic ProgramsExtended Logic Programs

• A finite set of rules of the form: A finite set of rules of the form:

LL00 L L11,...,L,...,Ln,n,,not L,not Ln+1n+1,...,not L,...,not Lmm

where each Lwhere each Lii can be either A or ¬A can be either A or ¬A

• ¬A is the explicit negation of A, ¬A is the explicit negation of A,

¬fliles(X) ¬fliles(X) animal(X). animal(X).

fliles(X) fliles(X) bird(X). bird(X).

¬ fliles(X) ¬ fliles(X) penguin(X). penguin(X).• Three-valued semantics (WFSX)Three-valued semantics (WFSX)

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Learning Three-Valued Logical ProgramsLearning Three-Valued Logical Programs

• Autonomous agent: acquisition of information Autonomous agent: acquisition of information by means of experimentsby means of experiments

• Experiment:Experiment:– execution of an actionexecution of an action– evaluation of the results with respect to the evaluation of the results with respect to the

achievement of a goalachievement of a goal– positive and negative resultspositive and negative results

• Learning general rules on actions:Learning general rules on actions:– distinction among actions with a positive, negative distinction among actions with a positive, negative

or unknown outcomeor unknown outcome

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New Learning FrameworkNew Learning Framework

• Conditions on examples:Conditions on examples:PPEE++, , EE-- (completeness)(completeness)

PP E E--, , EE++ (consistency)(consistency)

• LIVE (Learning In a three-Valued LIVE (Learning In a three-Valued Environment)Environment)

• Standard ILP techniques to learn p and Standard ILP techniques to learn p and p p – top-down or bottom-uptop-down or bottom-up

• Learning of:Learning of:– p using Ep using E++, E, E- - as training set as training set – p using Ep using E--, E, E+ + as training set as training set

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Intersection of definitionsIntersection of definitions

E+ E-

pp

Exceptions to the positive concept: negative examples

Exceptions to the negative concept: positive examples

Unseen atoms

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Atoms in the intersectionAtoms in the intersection

• Unseen atoms both true and false are Unseen atoms both true and false are classified as unknown:classified as unknown:

p(p(XX))pp++((XX), ), notnot p(p(XX).).

p(p(XX))pp--((XX), ), notnot p( p(XX).).• Training set atoms must be classified Training set atoms must be classified

according to the training set:according to the training set:

p(p(XX) ) pp++((XX), ), notnot ab abpp ( (XX), ), notnot p(p(XX).).

p(p(XX) ) pp--((XX), ), notnot ab abpp ( (XX), ), notnot p( p(XX).).

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Generality of SolutionsGenerality of Solutions

• Bottom-up methods: Bottom-up methods: – search from specific to general: General search from specific to general: General

Solution (GS)Solution (GS)– GOLEM (RLGG), CIGOL (Inverse Resolution) GOLEM (RLGG), CIGOL (Inverse Resolution)

• Top-down methods:Top-down methods:– search from general to specific: Specific search from general to specific: Specific

Solution (SS)Solution (SS)– FOIL, ProgolFOIL, Progol

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Criteria for chosing the generalityCriteria for chosing the generality

• Risk that can derive from a classification Risk that can derive from a classification errorerror– high risk high risk GSGS– low risklow risk SSSS

• Confidence in the set of negative examplesConfidence in the set of negative examples– high confidence high confidence GSGS– low confidencelow confidence SSSS

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ExampleExample

B:B: bird(a).bird(a). has_wings(a).has_wings(a).

jet(b).jet(b). has_wings(b).has_wings(b).

angel(c).angel(c). has_wings(c).has_wings(c). has_limbs(c).has_limbs(c).

penguin(d).penguin(d). has_wings(d).has_wings(d). has_limbs(d).has_limbs(d).

dog(e).dog(e). has_limbs(e).has_limbs(e).

cat(f).cat(f). has_limbs(f).has_limbs(f).

EE++={flies(a)}={flies(a)} EE--={flies(d), flies(e)}={flies(d), flies(e)}

flies+ E+

a b c fd e

flies-E-

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The theory can be revised differently The theory can be revised differently depending on the type of literal that is depending on the type of literal that is found contradictory in the intersection.found contradictory in the intersection.

Strategies For Theory RefinementStrategies For Theory Refinement