C LASSIFICATION OF S TRUCTURED O BJECTS I O H H C H H Methanol - PowerPoint PPT Presentation

C OMPUTING S TABLE M ODELS FOR N ONMONOTONIC E XISTENTIAL R ULES Despoina Magka, Markus Krötzsch, Ian Horrocks Department of Computer Science, University of Oxford IJCAI, 2013

T HE OWL S ARE NOT WHAT THEY SEEM OWL widely used for authoring biomedical ontologies 1

T HE OWL S ARE NOT WHAT THEY SEEM OWL widely used for authoring biomedical ontologies 1

T HE OWL S ARE NOT WHAT THEY SEEM OWL widely used for authoring biomedical ontologies Not marked for its ability to model cyclic structures 1

T HE OWL S ARE NOT WHAT THEY SEEM OWL widely used for authoring biomedical ontologies Not marked for its ability to model cyclic structures Such structures abound in life science (and other) domains 1

T HE OWL S ARE NOT WHAT THEY SEEM OWL widely used for authoring biomedical ontologies Not marked for its ability to model cyclic structures Such structures abound in life science (and other) domains hasParticipant locatedIn reactant ✷✲❦❡t♦❣❧✉t❛r❛t❡ ♠❛❧❛t❡ product ♠✐t♦❝❤♦♥❞r✐♦♥ tr❛♥s♣♦rt❘❡❛❝t✐♦♥ ✷✲❦❡t♦❣❧✉t❛r❛t❡✲♠❛❧❛t❡✲❛♥t✐♣♦rt ❝②t♦s♦❧ ✷✲❦❡t♦❣❧✉t❛r❛t❡ ♠❛❧❛t❡ 1

N ONMONOTONIC E XISTENTIAL R ULES Rules with nonmonotonic negation in the body and existentials in the head B 1 ∧ . . . ∧ B n ∧ not B n + 1 ∧ . . . ∧ not B m → ∃ y . H 1 ∧ . . . ∧ H k 2

N ONMONOTONIC E XISTENTIAL R ULES Rules with nonmonotonic negation in the body and existentials in the head B 1 ∧ . . . ∧ B n ∧ not B n + 1 ∧ . . . ∧ not B m → ∃ y . H 1 ∧ . . . ∧ H k Interpreted under stable model semantics 2

N ONMONOTONIC E XISTENTIAL R ULES Rules with nonmonotonic negation in the body and existentials in the head B 1 ∧ . . . ∧ B n ∧ not B n + 1 ∧ . . . ∧ not B m → ∃ y . H 1 ∧ . . . ∧ H k Interpreted under stable model semantics Good for representing non-tree-shaped structures 2

N ONMONOTONIC E XISTENTIAL R ULES Rules with nonmonotonic negation in the body and existentials in the head B 1 ∧ . . . ∧ B n ∧ not B n + 1 ∧ . . . ∧ not B m → ∃ y . H 1 ∧ . . . ∧ H k Interpreted under stable model semantics Good for representing non-tree-shaped structures Existentials allow us to infer new structures 2

N ONMONOTONIC E XISTENTIAL R ULES Rules with nonmonotonic negation in the body and existentials in the head B 1 ∧ . . . ∧ B n ∧ not B n + 1 ∧ . . . ∧ not B m → ∃ y . H 1 ∧ . . . ∧ H k Interpreted under stable model semantics Good for representing non-tree-shaped structures Existentials allow us to infer new structures Nonmonotonicity adds extra expressivity in modelling 2

N ONMONOTONIC E XISTENTIAL R ULES Rules with nonmonotonic negation in the body and existentials in the head B 1 ∧ . . . ∧ B n ∧ not B n + 1 ∧ . . . ∧ not B m → ∃ y . H 1 ∧ . . . ∧ H k Interpreted under stable model semantics Good for representing non-tree-shaped structures Existentials allow us to infer new structures Nonmonotonicity adds extra expressivity in modelling Stable model semantics supported by many tools: DLV, clasp, . . . 2

C LASSIFICATION OF S TRUCTURED O BJECTS I O H H C H H Methanol molecule 3

C LASSIFICATION OF S TRUCTURED O BJECTS I O H H C H H Methanol molecule methanol ( x ) → ∃ 6 i = 1 y i . ∧ 6 i = 1 hasAtom ( x , y i ) ∧ c ( y 1 ) ∧ o ( y 2 ) ∧ ∧ 6 i = 3 h ( y i ) ∧ ∧ 5 i = 2 bond ( y 1 , y i ) ∧ bond ( y 2 , y 6 ) 3

C LASSIFICATION OF S TRUCTURED O BJECTS I c o f 2 ( m ) f 3 ( m ) O H h m f 6 ( m ) f 1 ( m ) f 4 ( m ) H C H hasAtom methanol bond H f 5 ( m ) Methanol molecule methanol ( x ) → ∃ 6 i = 1 y i . ∧ 6 i = 1 hasAtom ( x , y i ) ∧ c ( y 1 ) ∧ o ( y 2 ) ∧ ∧ 6 i = 3 h ( y i ) ∧ ∧ 5 i = 2 bond ( y 1 , y i ) ∧ bond ( y 2 , y 6 ) 3

C LASSIFICATION OF S TRUCTURED O BJECTS I c o f 2 ( m ) f 3 ( m ) O H h m f 6 ( m ) f 1 ( m ) f 4 ( m ) H C H hasAtom methanol bond H f 5 ( m ) Methanol molecule methanol ( x ) → ∃ 6 i = 1 y i . ∧ 6 i = 1 hasAtom ( x , y i ) ∧ c ( y 1 ) ∧ o ( y 2 ) ∧ ∧ 6 i = 3 h ( y i ) ∧ ∧ 5 i = 2 bond ( y 1 , y i ) ∧ bond ( y 2 , y 6 ) ∧ 3 i = 1 hasAtom ( x , z i ) ∧ c ( z 1 ) ∧ o ( z 2 ) ∧ h ( z 3 ) ∧ bond ( z 1 , z 2 ) ∧ bond ( z 2 , z 3 ) → organicHydroxy ( x ) 3

C LASSIFICATION OF S TRUCTURED O BJECTS I c o f 2 ( m ) f 3 ( m ) O H h m f 6 ( m ) f 1 ( m ) f 4 ( m ) H C H hasAtom methanol bond H organicHydroxy f 5 ( m ) Methanol molecule methanol ( x ) → ∃ 6 i = 1 y i . ∧ 6 i = 1 hasAtom ( x , y i ) ∧ c ( y 1 ) ∧ o ( y 2 ) ∧ ∧ 6 i = 3 h ( y i ) ∧ ∧ 5 i = 2 bond ( y 1 , y i ) ∧ bond ( y 2 , y 6 ) ∧ 3 i = 1 hasAtom ( x , z i ) ∧ c ( z 1 ) ∧ o ( z 2 ) ∧ h ( z 3 ) ∧ bond ( z 1 , z 2 ) ∧ bond ( z 2 , z 3 ) → organicHydroxy ( x ) methanol ⊑ organicHydroxy ✓ 3

C LASSIFICATION OF S TRUCTURED O BJECTS I c o f 2 ( m ) f 3 ( m ) O H h m f 6 ( m ) f 1 ( m ) f 4 ( m ) H C H hasAtom methanol bond H organicHydroxy f 5 ( m ) Methanol molecule methanol ( x ) → ∃ 6 i = 1 y i . ∧ 6 i = 1 hasAtom ( x , y i ) ∧ c ( y 1 ) ∧ o ( y 2 ) ∧ ∧ 6 i = 3 h ( y i ) ∧ ∧ 5 i = 2 bond ( y 1 , y i ) ∧ bond ( y 2 , y 6 ) ∧ 3 i = 1 hasAtom ( x , z i ) ∧ c ( z 1 ) ∧ o ( z 2 ) ∧ h ( z 3 ) ∧ bond ( z 1 , z 2 ) ∧ bond ( z 2 , z 3 ) → organicHydroxy ( x ) hasAtom ( x , z ) ∧ o ( z ) → hasOxygen ( x ) methanol ⊑ organicHydroxy ✓ 3

C LASSIFICATION OF S TRUCTURED O BJECTS I c o f 2 ( m ) f 3 ( m ) O H h m f 6 ( m ) f 1 ( m ) f 4 ( m ) H C H hasAtom methanol bond H organicHydroxy f 5 ( m ) hasOxygen Methanol molecule methanol ( x ) → ∃ 6 i = 1 y i . ∧ 6 i = 1 hasAtom ( x , y i ) ∧ c ( y 1 ) ∧ o ( y 2 ) ∧ ∧ 6 i = 3 h ( y i ) ∧ ∧ 5 i = 2 bond ( y 1 , y i ) ∧ bond ( y 2 , y 6 ) ∧ 3 i = 1 hasAtom ( x , z i ) ∧ c ( z 1 ) ∧ o ( z 2 ) ∧ h ( z 3 ) ∧ bond ( z 1 , z 2 ) ∧ bond ( z 2 , z 3 ) → organicHydroxy ( x ) hasAtom ( x , z ) ∧ o ( z ) → hasOxygen ( x ) methanol ⊑ organicHydroxy ✓ methanol ⊑ hasOxygen ✓ 3

C LASSIFICATION OF S TRUCTURED O BJECTS II O H C Organic hydroxy group 4

C LASSIFICATION OF S TRUCTURED O BJECTS II O H C Organic hydroxy group organicHydroxy ( x ) → ∃ 3 i = 1 y i . ∧ 3 i = 1 hasAtom ( x , y i ) ∧ c ( y 1 ) ∧ o ( y 2 ) ∧ h ( y 3 ) ∧ bond ( y 1 , y 2 ) ∧ bond ( y 2 , y 3 ) 4

C LASSIFICATION OF S TRUCTURED O BJECTS II O H c o g 2 ( h ) g 3 ( h ) h C h g 1 ( h ) hasAtom organicHydroxy bond Organic hydroxy group organicHydroxy ( x ) → ∃ 3 i = 1 y i . ∧ 3 i = 1 hasAtom ( x , y i ) ∧ c ( y 1 ) ∧ o ( y 2 ) ∧ h ( y 3 ) ∧ bond ( y 1 , y 2 ) ∧ bond ( y 2 , y 3 ) 4

C LASSIFICATION OF S TRUCTURED O BJECTS II O H c o g 2 ( h ) g 3 ( h ) h C h g 1 ( h ) hasAtom organicHydroxy bond Organic hydroxy group organicHydroxy ( x ) → ∃ 3 i = 1 y i . ∧ 3 i = 1 hasAtom ( x , y i ) ∧ c ( y 1 ) ∧ o ( y 2 ) ∧ h ( y 3 ) ∧ bond ( y 1 , y 2 ) ∧ bond ( y 2 , y 3 ) hasAtom ( x , z ) ∧ o ( z ) → hasOxygen ( x ) 4

C LASSIFICATION OF S TRUCTURED O BJECTS II O H c o g 2 ( h ) g 3 ( h ) h C h g 1 ( h ) hasAtom organicHydroxy bond hasOxygen Organic hydroxy group organicHydroxy ( x ) → ∃ 3 i = 1 y i . ∧ 3 i = 1 hasAtom ( x , y i ) ∧ c ( y 1 ) ∧ o ( y 2 ) ∧ h ( y 3 ) ∧ bond ( y 1 , y 2 ) ∧ bond ( y 2 , y 3 ) hasAtom ( x , z ) ∧ o ( z ) → hasOxygen ( x ) organicHydroxy ⊑ hasOxygen ✓ 4

I NCORRECT M ODELLING methanol ( x ) → ∃ 6 i = 1 y i . ∧ 6 i = 1 hasAtom ( x , y i ) ∧ . . . ∧ bond ( y 2 , y 6 ) ∧ 3 i = 1 hasAtom ( x , z i ) ∧ . . . ∧ bond ( z 2 , z 3 ) → organicHydroxy ( x ) organicHydroxy ( x ) → ∃ 3 i = 1 y i . ∧ 3 i = 1 hasAtom ( x , y i ) ∧ . . . ∧ bond ( y 2 , y 3 ) hasAtom ( x , z ) ∧ o ( z ) → hasOxygen ( x ) 5

I NCORRECT M ODELLING f 2 ( m ) f 3 ( m ) m f 6 ( m ) f 1 ( m ) f 4 ( m ) methanol f 5 ( m ) methanol ( x ) → ∃ 6 i = 1 y i . ∧ 6 i = 1 hasAtom ( x , y i ) ∧ . . . ∧ bond ( y 2 , y 6 ) ∧ 3 i = 1 hasAtom ( x , z i ) ∧ . . . ∧ bond ( z 2 , z 3 ) → organicHydroxy ( x ) organicHydroxy ( x ) → ∃ 3 i = 1 y i . ∧ 3 i = 1 hasAtom ( x , y i ) ∧ . . . ∧ bond ( y 2 , y 3 ) hasAtom ( x , z ) ∧ o ( z ) → hasOxygen ( x ) 5

I NCORRECT M ODELLING f 2 ( m ) f 3 ( m ) m f 6 ( m ) f 1 ( m ) f 4 ( m ) methanol organicHydroxy f 5 ( m ) methanol ⊑ organicHydroxy ✓ methanol ( x ) → ∃ 6 i = 1 y i . ∧ 6 i = 1 hasAtom ( x , y i ) ∧ . . . ∧ bond ( y 2 , y 6 ) ∧ 3 i = 1 hasAtom ( x , z i ) ∧ . . . ∧ bond ( z 2 , z 3 ) → organicHydroxy ( x ) organicHydroxy ( x ) → ∃ 3 i = 1 y i . ∧ 3 i = 1 hasAtom ( x , y i ) ∧ . . . ∧ bond ( y 2 , y 3 ) hasAtom ( x , z ) ∧ o ( z ) → hasOxygen ( x ) 5

I NCORRECT M ODELLING f 2 ( m ) f 3 ( m ) m f 6 ( m ) f 1 ( m ) f 4 ( m ) methanol organicHydroxy f 5 ( m ) hasOxygen methanol ⊑ hasOxygen ✓ methanol ( x ) → ∃ 6 i = 1 y i . ∧ 6 i = 1 hasAtom ( x , y i ) ∧ . . . ∧ bond ( y 2 , y 6 ) ∧ 3 i = 1 hasAtom ( x , z i ) ∧ . . . ∧ bond ( z 2 , z 3 ) → organicHydroxy ( x ) organicHydroxy ( x ) → ∃ 3 i = 1 y i . ∧ 3 i = 1 hasAtom ( x , y i ) ∧ . . . ∧ bond ( y 2 , y 3 ) hasAtom ( x , z ) ∧ o ( z ) → hasOxygen ( x ) 5