Preemption Operators Ph. Besnard 1 . Grgoire 2 S. Ramon 2 1 IRIT CNRS - PowerPoint PPT Presentation

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption Operators Ph. Besnard 1 É. Grégoire 2 S. Ramon 2 1 IRIT CNRS (Toulouse, France) 2 Université d’Artois, CRIL CNRS (Lens, France) 14th International Workshop on N on- M onotonic R easoning 8th June 2012 - Roma 1 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Guideline Preemption vs. Revision 1 Postulates 2 Characterization 3 Hansson’s Replacement Operators 4 2 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Guideline Preemption vs. Revision 1 Postulates 2 Characterization 3 Hansson’s Replacement Operators 4 3 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g ➠ 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g ➠ Preemption is quite different from revision 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K inconsistent with g Aim : Giving up any belief in K contradicting g ➠ Preemption is quite different from revision 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g ➠ 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g Example 1 : Let K be a consistent belief base 1 paul _ at _ office ∨ paul _ at _ home ➠ 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g Example 1 : Let K be a consistent belief base 1 paul _ at _ office ∨ paul _ at _ home g paul _ at _ office ∨ paul _ at _ home ∨ paul _ at _ club ➠ 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g Example 1 : Let K be a consistent belief base 1 paul _ at _ office ∨ paul _ at _ home g paul _ at _ office ∨ paul _ at _ home ∨ paul _ at _ club ➠ g conveys some uncertainty that Paul’s office or home are where he is now 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g Example 1 : Let K be a consistent belief base 1 paul _ at _ office ∨ paul _ at _ home g paul _ at _ office ∨ paul _ at _ home ∨ paul _ at _ club ➠ The belief in K subsuming g should no longer be deduced from K 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g ➠ 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g Example 2 : Let K be a consistent belief base 1 dana _ agrees ⇒ we _ begin _ tomorrow ➠ 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g Example 2 : Let K be a consistent belief base 1 dana _ agrees ⇒ we _ begin _ tomorrow g ( dana _ agrees ∧ alexander _ agrees ) ⇒ we _ begin _ tomorrow ➠ 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g Example 2 : Let K be a consistent belief base 1 dana _ agrees ⇒ we _ begin _ tomorrow g ( dana _ agrees ∧ alexander _ agrees ) ⇒ we _ begin _ tomorrow ➠ g conveys some uncertainty that Dana’s agreement is now sufficient 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g Example 2 : Let K be a consistent belief base 1 dana _ agrees ⇒ we _ begin _ tomorrow g ( dana _ agrees ∧ alexander _ agrees ) ⇒ we _ begin _ tomorrow ➠ The belief in K subsuming g should no longer be deduced from K 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g ➠ 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g Solution : Contract f , for all strict implicants f of g , then add g ➠ 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g Solution : Contract f , for all strict implicants f of g , then add g ➠ Not enough : introducing g might enable strict implicants f of g 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption [BGR11b, BGR11a] Context : Insertion of a new piece of information g into a belief base K consistent with g Aim : Giving up any belief in K subsuming g Solution : Contract g ⇒ f , for all strict implicants f of g , then add g ➠ 4 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption vs. Revision [AGM85] K ∗ g : revision of K (a consistent belief base) by g (a formula) ( K ∗ 1 ) (closure) K ∗ g is a theory. ( K ∗ 2 ) (success) g ∈ K ∗ g . ( K ∗ 3 ) (inclusion) K ∗ g ⊆ K + g . ( K ∗ 4 ) (vacuity) If ¬ g / ∈ K then K + g ⊆ K ∗ g . ( K ∗ 5 ) (consistent) K ∗ g = K ⊥ iff ⊢ ¬ g . ( K ∗ 6 ) (extensionality) If g ≡ h then K ∗ g = K ∗ h . ➠ Existing operators for revision are inadequate for preemption 5 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Preemption vs. Revision [AGM85] K ∗ g : revision of K (a consistent belief base) by g (a formula) ( K ∗ 1 ) (closure) K ∗ g is a theory. ( K ∗ 2 ) (success) g ∈ K ∗ g . ( K ∗ 3 ) (inclusion) K ∗ g ⊆ K + g . ( K ∗ 4 ) (vacuity) If ¬ g / ∈ K then K + g ⊆ K ∗ g . ( K ∗ 5 ) (consistent) K ∗ g = K ⊥ iff ⊢ ¬ g . ( K ∗ 6 ) (extensionality) If g ≡ h then K ∗ g = K ∗ h . ➠ ( K ∗ 4 ) expresses that no information must be expelled when K and g are not contradictory 5 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Guideline Preemption vs. Revision 1 Postulates 2 Characterization 3 Hansson’s Replacement Operators 4 6 / 14

Preemption vs. Revision Postulates Characterization Hansson’s Replacement Operators Postulates K ⊛ g : preemption of g (a clause) over K (a consistent belief base) ( K ⊛ 1 ) (closure) K ⊛ g is a theory. ( K ⊛ 2 ) (success of insertion) g ∈ K ⊛ g . ( K ⊛ 3 ) ∈ K ⊛ g for all clausal strict implicants f of g . f / (success of preemption) ( K ⊛ 4 ) (inclusion) K ⊛ g ⊆ K + g . ( K ⊛ 5 ) (vacuity) If ( g ⇒ f ) / ∈ K for all clausal strict impli- cants f of g then K + g ⊆ K ⊛ g . ( K ⊛ 6 ) (extensionality) If g ≡ h then K ⊛ g = K ⊛ h . 7 / 14