Chapter X: On the Relation between AA, ABA and LP Martin Caminada - PowerPoint PPT Presentation

Chapter X: On the Relation between AA, ABA and LP Martin Caminada Department of Computing Science University of Aberdeen

Overview ABA LP AA

Overview ABA LP AA

Overview ABA LP stable preferred grounded complete ideal AA

Overview ABA LP stable preferred grounded complete ideal semi-stable AA

Overview ABA LP stable preferred grounded complete ideal semi-stable AA

Overview ABA LP stable stable preferred preferred grounded grounded complete complete ideal ideal semi-stable semi-stable AA

Overview ABA LP stable stable preferred preferred grounded grounded complete complete ideal ideal semi-stable semi-stable AA

Overview stable preferred grounded complete ideal semi-stable ABA LP stable stable preferred preferred grounded grounded complete complete ideal ideal semi-stable semi-stable AA

Overview stable preferred grounded DeLP complete ideal semi-stable ABA LP stable stable preferred preferred grounded grounded complete complete ideal ideal semi-stable semi-stable AA

Overview stable preferred grounded DeLP complete ideal semi-stable ABA LP stable stable preferred preferred grounded grounded complete complete ideal ideal semi-stable semi-stable AA

Translating ABA to LP

Translating ABA to LP starting point: an ABA framework that is flat , unique-contrary and non-assumption contrary

Translating ABA to LP starting point: an ABA framework that is flat , unique-contrary and non-assumption contrary idea: - each ABA rule becomes an LP rule - replace each assumption α by not α

Translating ABA to LP starting point: an ABA framework that is flat , unique-contrary and non-assumption contrary idea: - each ABA rule becomes an LP rule - replace each assumption α by not α Example:

Translating ABA to LP starting point: an ABA framework that is flat , unique-contrary and non-assumption contrary idea: - each ABA rule becomes an LP rule - replace each assumption α by not α Example: a ← b, c, δ , ε

Translating ABA to LP starting point: an ABA framework that is flat , unique-contrary and non-assumption contrary idea: - each ABA rule becomes an LP rule - replace each assumption α by not α Example: a ← b, c, δ , ε a ← b, c, not d, not e (assuming δ = d and ε = e)

From LP model to ABA labelling

From LP model to ABA labelling starting point: (3-valued) model <T,F> of P F

From LP model to ABA labelling starting point: (3-valued) model <T,F> of P F α with α ∈ F becomes in α with α ∈ T becomes out α with α ∈ HB P F \ (T U F) becomes undec

From LP model to ABA labelling starting point: (3-valued) model <T,F> of P F α with α ∈ F becomes in (just like α with α Ï HB P F ) α with α ∈ T becomes out α with α ∈ HB P F \ (T U F) becomes undec

From LP model to ABA labelling starting point: (3-valued) model <T,F> of P F α with α ∈ F becomes in (just like α with α Ï HB P F ) α with α ∈ T becomes out α with α ∈ HB P F \ (T U F) becomes undec LP model assumption labelling 3-valued stable complete 2-valued stable stable regular preferred well-founded grounded ideal ideal

From ABA labelling to LP model

From ABA labelling to LP model starting point: assumption labelling of ABA framework F

From ABA labelling to LP model starting point: assumption labelling of ABA framework F T': each α ∈ HB P F where α is out F': each α ∈ HB P F where α is in

From ABA labelling to LP model starting point: assumption labelling of ABA framework F T': each α ∈ HB P F where α is out F': each α ∈ HB P F where α is in <T, F> = Φ (P F <T',F'> )

From ABA labelling to LP model starting point: assumption labelling of ABA framework F T': each α ∈ HB P F where α is out F': each α ∈ HB P F where α is in <T, F> = Φ (P F <T',F'> ) assumption labelling LP model complete 3-valued stable stable 2-valued stable preferred regular grounded well-founded ideal ideal

Translating LP to ABA P → F P → P F P

Translating LP to ABA P → F P → P F P = P

Translating LP to ABA P → F P → P F P = P LP model assumption labelling 3-valued stable complete 2-valued stable stable regular preferred well-founded grounded ideal ideal

Translating LP to ABA P → F P → P F P = P assumption labelling LP model complete 3-valued stable stable 2-valued stable preferred regular grounded well-founded ideal ideal

Take Home Message ABA = LP and LP = ABA ABA writes α , β , γ instead of not a , not b , not c ABA semantics are defined only on the weakly negated part of the program; apply GL-reduct to get entire model