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

COMPUTING STABLE MODELS FOR NONMONOTONIC EXISTENTIAL RULES

Despoina Magka, Markus Krötzsch, Ian Horrocks

Department of Computer Science, University of Oxford

IJCAI, 2013

THE OWLS ARE NOT WHAT THEY SEEM

OWL widely used for authoring biomedical ontologies

1

THE OWLS ARE NOT WHAT THEY SEEM

OWL widely used for authoring biomedical ontologies

1

THE OWLS ARE NOT WHAT THEY SEEM

OWL widely used for authoring biomedical ontologies Not marked for its ability to model cyclic structures

1

THE OWLS 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

THE OWLS 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 product tr❛♥s♣♦rt❘❡❛❝t✐♦♥ ✷✲❦❡t♦❣❧✉t❛r❛t❡✲♠❛❧❛t❡✲❛♥t✐♣♦rt ✷✲❦❡t♦❣❧✉t❛r❛t❡ ♠❛❧❛t❡ ✷✲❦❡t♦❣❧✉t❛r❛t❡ ♠❛❧❛t❡ ♠✐t♦❝❤♦♥❞r✐♦♥ ❝②t♦s♦❧

1

NONMONOTONIC EXISTENTIAL RULES

Rules with nonmonotonic negation in the body and existentials in the head B1 ∧ . . . ∧ Bn ∧ not Bn+1 ∧ . . . ∧ not Bm → ∃y.H1 ∧ . . . ∧ Hk

2

NONMONOTONIC EXISTENTIAL RULES

Rules with nonmonotonic negation in the body and existentials in the head B1 ∧ . . . ∧ Bn ∧ not Bn+1 ∧ . . . ∧ not Bm → ∃y.H1 ∧ . . . ∧ Hk Interpreted under stable model semantics

2

NONMONOTONIC EXISTENTIAL RULES

Rules with nonmonotonic negation in the body and existentials in the head B1 ∧ . . . ∧ Bn ∧ not Bn+1 ∧ . . . ∧ not Bm → ∃y.H1 ∧ . . . ∧ Hk Interpreted under stable model semantics Good for representing non-tree-shaped structures

2

NONMONOTONIC EXISTENTIAL RULES

Rules with nonmonotonic negation in the body and existentials in the head B1 ∧ . . . ∧ Bn ∧ not Bn+1 ∧ . . . ∧ not Bm → ∃y.H1 ∧ . . . ∧ Hk Interpreted under stable model semantics Good for representing non-tree-shaped structures

Existentials allow us to infer new structures

2

NONMONOTONIC EXISTENTIAL RULES

Rules with nonmonotonic negation in the body and existentials in the head B1 ∧ . . . ∧ Bn ∧ not Bn+1 ∧ . . . ∧ not Bm → ∃y.H1 ∧ . . . ∧ Hk 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

NONMONOTONIC EXISTENTIAL RULES

Rules with nonmonotonic negation in the body and existentials in the head B1 ∧ . . . ∧ Bn ∧ not Bn+1 ∧ . . . ∧ not Bm → ∃y.H1 ∧ . . . ∧ Hk 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

CLASSIFICATION OF STRUCTURED OBJECTS I

C O H H H H

Methanol molecule

3

CLASSIFICATION OF STRUCTURED OBJECTS I

C O H H H H

Methanol molecule

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ c(y1) ∧ o(y2) ∧

∧6

i=3h(yi)∧ ∧5 i=2 bond(y1, yi) ∧ bond(y2, y6)

3

CLASSIFICATION OF STRUCTURED OBJECTS I

C O H H H H

Methanol molecule

hasAtom bond

c

• h

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ c(y1) ∧ o(y2) ∧

∧6

i=3h(yi)∧ ∧5 i=2 bond(y1, yi) ∧ bond(y2, y6)

3

CLASSIFICATION OF STRUCTURED OBJECTS I

C O H H H H

Methanol molecule

hasAtom bond

c

• h

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ c(y1) ∧ o(y2) ∧

∧6

i=3h(yi)∧ ∧5 i=2 bond(y1, yi) ∧ bond(y2, y6)

∧3

i=1 hasAtom(x, zi) ∧ c(z1)∧o(z2) ∧

h(z3) ∧ bond(z1, z2) ∧ bond(z2, z3)→ organicHydroxy(x)

3

CLASSIFICATION OF STRUCTURED OBJECTS I

C O H H H H

Methanol molecule

hasAtom bond

c

• h

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ c(y1) ∧ o(y2) ∧

∧6

i=3h(yi)∧ ∧5 i=2 bond(y1, yi) ∧ bond(y2, y6)

∧3

i=1 hasAtom(x, zi) ∧ c(z1)∧o(z2) ∧

h(z3) ∧ bond(z1, z2) ∧ bond(z2, z3)→ organicHydroxy(x) methanol ⊑ organicHydroxy ✓

3

CLASSIFICATION OF STRUCTURED OBJECTS I

C O H H H H

Methanol molecule

hasAtom bond

c

• h

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ c(y1) ∧ o(y2) ∧

∧6

i=3h(yi)∧ ∧5 i=2 bond(y1, yi) ∧ bond(y2, y6)

∧3

i=1 hasAtom(x, zi) ∧ c(z1)∧o(z2) ∧

h(z3) ∧ bond(z1, z2) ∧ bond(z2, z3)→ organicHydroxy(x) hasAtom(x, z) ∧ o(z) → hasOxygen(x) methanol ⊑ organicHydroxy ✓

3

CLASSIFICATION OF STRUCTURED OBJECTS I

C O H H H H

Methanol molecule

hasAtom bond

c

• h

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

hasOxygen

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ c(y1) ∧ o(y2) ∧

∧6

i=3h(yi)∧ ∧5 i=2 bond(y1, yi) ∧ bond(y2, y6)

∧3

i=1 hasAtom(x, zi) ∧ c(z1)∧o(z2) ∧

h(z3) ∧ bond(z1, z2) ∧ bond(z2, z3)→ organicHydroxy(x) hasAtom(x, z) ∧ o(z) → hasOxygen(x) methanol ⊑ organicHydroxy ✓ methanol ⊑ hasOxygen ✓

3

CLASSIFICATION OF STRUCTURED OBJECTS II

C O H

Organic hydroxy group

4

CLASSIFICATION OF STRUCTURED OBJECTS II

C O H

Organic hydroxy group

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ c(y1)

∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3)

4

CLASSIFICATION OF STRUCTURED OBJECTS II

C O H

Organic hydroxy group

hasAtom bond

c

• h

h

• rganicHydroxy

g1(h) g2(h) g3(h)

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ c(y1)

∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3)

4

CLASSIFICATION OF STRUCTURED OBJECTS II

C O H

Organic hydroxy group

hasAtom bond

c

• h

h

• rganicHydroxy

g1(h) g2(h) g3(h)

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ c(y1)

∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3) hasAtom(x, z) ∧ o(z) → hasOxygen(x)

4

CLASSIFICATION OF STRUCTURED OBJECTS II

C O H

Organic hydroxy group

hasAtom bond

c

• h

h

• rganicHydroxy

g1(h) g2(h) g3(h)

hasOxygen

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ c(y1)

∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3) hasAtom(x, z) ∧ o(z) → hasOxygen(x)

• rganicHydroxy ⊑ hasOxygen ✓

4

INCORRECT MODELLING

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) → organicHydroxy(x)

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) hasAtom(x, z) ∧ o(z) → hasOxygen(x)

5

INCORRECT MODELLING

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) → organicHydroxy(x)

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) hasAtom(x, z) ∧ o(z) → hasOxygen(x)

5

INCORRECT MODELLING

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

methanol ⊑ organicHydroxy ✓ methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) → organicHydroxy(x)

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) hasAtom(x, z) ∧ o(z) → hasOxygen(x)

5

INCORRECT MODELLING

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

hasOxygen

methanol ⊑ hasOxygen ✓ methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) → organicHydroxy(x)

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) hasAtom(x, z) ∧ o(z) → hasOxygen(x)

5

INCORRECT MODELLING

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

hasOxygen

h

• rganicHydroxy

g1(h) g2(h) g3(h)

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) → organicHydroxy(x)

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) hasAtom(x, z) ∧ o(z) → hasOxygen(x)

5

INCORRECT MODELLING

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

hasOxygen

h

• rganicHydroxy

g1(h) g2(h) g3(h)

hasOxygen

• rganicHydroxy ⊑ hasOxygen ✓

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) → organicHydroxy(x)

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) hasAtom(x, z) ∧ o(z) → hasOxygen(x)

5

INCORRECT MODELLING

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m) g1(m) g2(m) g3(m)

• rganicHydroxy

hasOxygen

h

• rganicHydroxy

g1(h) g2(h) g3(h)

hasOxygen

methanol ⊑ hasOneCarbon ✘ methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) → organicHydroxy(x)

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) hasAtom(x, z) ∧ o(z) → hasOxygen(x)

5

REPAIR WITH AUXILIARY PREDICATES

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) ∧ not gh(z1) ∧ not gh(z2) ∧ not gh(z3) → organicHydroxy(x) ∧ rh(x)

• rganicHydroxy(x)∧ not rh(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) ∧ ∧3

i=1gh(yi)

hasAtom(x, z) ∧ o(z) → hasOxygen(x)

6

REPAIR WITH AUXILIARY PREDICATES

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

hasOxygen

h

• rganicHydroxy

g1(h) g2(h) g3(h)

hasOxygen

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) ∧ not gh(z1) ∧ not gh(z2) ∧ not gh(z3) → organicHydroxy(x) ∧ rh(x)

• rganicHydroxy(x)∧ not rh(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) ∧ ∧3

i=1gh(yi)

hasAtom(x, z) ∧ o(z) → hasOxygen(x)

6

REPAIR WITH AUXILIARY PREDICATES

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

hasOxygen

h

• rganicHydroxy

g1(h) g2(h) g3(h)

hasOxygen

methanol ⊑ hasOneCarbon ✓ methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) ∧ not gh(z1) ∧ not gh(z2) ∧ not gh(z3) → organicHydroxy(x) ∧ rh(x)

• rganicHydroxy(x)∧ not rh(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) ∧ ∧3

i=1gh(yi)

hasAtom(x, z) ∧ o(z) → hasOxygen(x)

6

WHAT’S THE PROBLEM?

Reasoning is undecidable (even fact entailment, even without not)

7

WHAT’S THE PROBLEM?

Reasoning is undecidable (even fact entailment, even without not)

many known conditions for regaining decidability acyclicity conditions ensure finite models: (super)-weak acyclicity, joint acyclicity, aGRD, MSA, MFA, . . .

7

WHAT’S THE PROBLEM?

Reasoning is undecidable (even fact entailment, even without not)

many known conditions for regaining decidability acyclicity conditions ensure finite models: (super)-weak acyclicity, joint acyclicity, aGRD, MSA, MFA, . . .

Reasoning is hard (even for finite models)

7

WHAT’S THE PROBLEM?

Reasoning is undecidable (even fact entailment, even without not)

many known conditions for regaining decidability acyclicity conditions ensure finite models: (super)-weak acyclicity, joint acyclicity, aGRD, MSA, MFA, . . .

Reasoning is hard (even for finite models)

stable models lead to non-determinism stratification conditions ensure determinism

7

WHAT’S THE PROBLEM?

Reasoning is undecidable (even fact entailment, even without not)

many known conditions for regaining decidability acyclicity conditions ensure finite models: (super)-weak acyclicity, joint acyclicity, aGRD, MSA, MFA, . . .

Reasoning is hard (even for finite models)

stable models lead to non-determinism stratification conditions ensure determinism

Stratification Stable model uniqueness Deterministic reasoning

7

WHAT’S OUR PROBLEM?

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

hasOxygen

h

• rganicHydroxy

g1(h) g2(h) g3(h)

hasOxygen

Repaired program not stratified methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) ∧ not gh(z1) ∧ not gh(z2) ∧ not gh(z3) → organicHydroxy(x) ∧ rh(x)

• rganicHydroxy(x)∧ not rh(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) ∧ ∧3

i=1gh(yi)

hasAtom(x, z) ∧ o(z) → hasOxygen(x)

8

WHAT’S OUR PROBLEM?

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

hasOxygen

h

• rganicHydroxy

g1(h) g2(h) g3(h)

hasOxygen

Repaired program not stratified methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) ∧ not gh(z1) ∧ not gh(z2) ∧ not gh(z3) → organicHydroxy(x) ∧ rh(x)

• rganicHydroxy(x)∧ not rh(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) ∧ ∧3

i=1gh(yi)

hasAtom(x, z) ∧ o(z) → hasOxygen(x)

8

WHAT’S OUR PROBLEM?

m

methanol

f1(m) f2(m) f3(m) f4(m) f5(m) f6(m)

• rganicHydroxy

hasOxygen

h

• rganicHydroxy

g1(h) g2(h) g3(h)

hasOxygen

Repaired program not stratified methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y6) ∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) ∧ not gh(z1) ∧ not gh(z2) ∧ not gh(z3) → organicHydroxy(x) ∧ rh(x)

• rganicHydroxy(x)∧ not rh(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) ∧ ∧3

i=1gh(yi)

hasAtom(x, z) ∧ o(z) → hasOxygen(x)

8

RESULTS OVERVIEW

1 R-acyclicity and R-stratification conditions

R-stratification ensures stable model uniqueness Both coNP-complete to check

9

RESULTS OVERVIEW

1 R-acyclicity and R-stratification conditions

R-stratification ensures stable model uniqueness Both coNP-complete to check

2 Complexity of reasoning

Fact entailment Program comp. Data comp. R-acyclic coN2ExpTime-complete coNP-complete R-acyclic+R-stratified 2ExpTime-complete PTime-complete

9

RESULTS OVERVIEW

1 R-acyclicity and R-stratification conditions

R-stratification ensures stable model uniqueness Both coNP-complete to check

2 Complexity of reasoning

Fact entailment Program comp. Data comp. R-acyclic coN2ExpTime-complete coNP-complete R-acyclic+R-stratified 2ExpTime-complete PTime-complete

3 Generalise R-acyclicity and R-stratification with constraints

new conditions ΠP

2-complete to check

9

RESULTS OVERVIEW

1 R-acyclicity and R-stratification conditions

R-stratification ensures stable model uniqueness Both coNP-complete to check

2 Complexity of reasoning

Fact entailment Program comp. Data comp. R-acyclic coN2ExpTime-complete coNP-complete R-acyclic+R-stratified 2ExpTime-complete PTime-complete

3 Generalise R-acyclicity and R-stratification with constraints

new conditions ΠP

2-complete to check 4 Experiments over ChEBI with DLV

Performance gains in DLV using R-stratification Missing subsumptions from ChEBI ontology

9

POSITIVE RELIANCES

Rule r2 positively relies on r1 (written r1 + − → r2): there is a situation when r1 can trigger r2 to derive something new

10

POSITIVE RELIANCES

Rule r2 positively relies on r1 (written r1 + − → r2): there is a situation when r1 can trigger r2 to derive something new

EXAMPLE

r1 : ∧3

i=1 hasAtom(x, zi) ∧

c(z1) ∧ o(z2) ∧ h(z3) ∧ bond(z1, z2) ∧ bond(z2, z3) → organicHydroxy(x) r2 :

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧

c(y1) ∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3)

10

POSITIVE RELIANCES

Rule r2 positively relies on r1 (written r1 + − → r2): there is a situation when r1 can trigger r2 to derive something new

EXAMPLE

r1 : ∧3

i=1 hasAtom(x, zi) ∧

c(z1) ∧ o(z2) ∧ h(z3) ∧ bond(z1, z2) ∧ bond(z2, z3) → organicHydroxy(x) r2 :

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧

c(y1) ∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3) r1 + − → r2

10

POSITIVE RELIANCES

Rule r2 positively relies on r1 (written r1 + − → r2): there is a situation when r1 can trigger r2 to derive something new

EXAMPLE

r1 : ∧3

i=1 hasAtom(x, zi) ∧

c(z1) ∧ o(z2) ∧ h(z3) ∧ bond(z1, z2) ∧ bond(z2, z3) → organicHydroxy(x) r2 :

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧

c(y1) ∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3) r1 + − → r2 but r2

+

− → r1

10

POSITIVE RELIANCES

Rule r2 positively relies on r1 (written r1 + − → r2): there is a situation when r1 can trigger r2 to derive something new

EXAMPLE

r1 : ∧3

i=1 hasAtom(x, zi) ∧

c(z1) ∧ o(z2) ∧ h(z3) ∧ bond(z1, z2) ∧ bond(z2, z3) → organicHydroxy(x) r2 :

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧

c(y1) ∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3) r1 + − → r2 but r2

+

− → r1 NP-complete to check

(but only w.r.t. the size of the rules)

10

POSITIVE RELIANCES

Rule r2 positively relies on r1 (written r1 + − → r2): there is a situation when r1 can trigger r2 to derive something new

EXAMPLE

r1 : ∧3

i=1 hasAtom(x, zi) ∧

c(z1) ∧ o(z2) ∧ h(z3) ∧ bond(z1, z2) ∧ bond(z2, z3) → organicHydroxy(x) r2 :

• rganicHydroxy(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧

c(y1) ∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3) r1 + − → r2 but r2

+

− → r1 NP-complete to check

(but only w.r.t. the size of the rules)

10

R-ACYCLICITY

A program is R-acyclic: there is no cycle of positive reliances that involves a rule with an existential

Checking R-acyclicity is coNP-complete Similar to ≺-stratification [Deutsch et al., PODS, 2008]; extension of aGRD [Baget et al., RR, 2011]

11

R-ACYCLICITY

A program is R-acyclic: there is no cycle of positive reliances that involves a rule with an existential

Checking R-acyclicity is coNP-complete Similar to ≺-stratification [Deutsch et al., PODS, 2008]; extension of aGRD [Baget et al., RR, 2011]

Fact entailment for R-acyclic programs

Stable models bounded in size (double exp), but many models possible

11

R-ACYCLICITY

A program is R-acyclic: there is no cycle of positive reliances that involves a rule with an existential

Checking R-acyclicity is coNP-complete Similar to ≺-stratification [Deutsch et al., PODS, 2008]; extension of aGRD [Baget et al., RR, 2011]

Fact entailment for R-acyclic programs

Stable models bounded in size (double exp), but many models possible coN2ExpTime-complete w.r.t. program complexity

11

R-ACYCLICITY

A program is R-acyclic: there is no cycle of positive reliances that involves a rule with an existential

Checking R-acyclicity is coNP-complete Similar to ≺-stratification [Deutsch et al., PODS, 2008]; extension of aGRD [Baget et al., RR, 2011]

Fact entailment for R-acyclic programs

Stable models bounded in size (double exp), but many models possible coN2ExpTime-complete w.r.t. program complexity

11

R-ACYCLICITY

A program is R-acyclic: there is no cycle of positive reliances that involves a rule with an existential

Checking R-acyclicity is coNP-complete Similar to ≺-stratification [Deutsch et al., PODS, 2008]; extension of aGRD [Baget et al., RR, 2011]

Fact entailment for R-acyclic programs

Stable models bounded in size (double exp), but many models possible coN2ExpTime-complete w.r.t. program complexity coNP-complete w.r.t. data complexity

11

NEGATIVE RELIANCES

Rule r2 negatively relies on r1 (written r1 − − → r2): there is a situation when r1 can inhibit the application of r2

12

NEGATIVE RELIANCES

Rule r2 negatively relies on r1 (written r1 − − → r2): there is a situation when r1 can inhibit the application of r2

EXAMPLE

r1 : ∧3

i=1 hasAtom(x, zi) ∧ c(z1) ∧

• (z2) ∧ h(z3) ∧ bond(z1, z2) ∧

bond(z2, z3) ∧ not gh(z1) ∧ not gh(z2) ∧ not gh(z3) → organicHydroxy(x) ∧ rh(x) r2 :

• rganicHydroxy(x) ∧ not rh(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi)

∧ c(y1) ∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3) ∧ gh(y1) ∧ gh(y2) ∧ gh(y3)

12

NEGATIVE RELIANCES

Rule r2 negatively relies on r1 (written r1 − − → r2): there is a situation when r1 can inhibit the application of r2

EXAMPLE

r1 : ∧3

i=1 hasAtom(x, zi) ∧ c(z1) ∧

• (z2) ∧ h(z3) ∧ bond(z1, z2) ∧

bond(z2, z3) ∧ not gh(z1) ∧ not gh(z2) ∧ not gh(z3) → organicHydroxy(x) ∧ rh(x) r2 :

• rganicHydroxy(x) ∧ not rh(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi)

∧ c(y1) ∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3) ∧ gh(y1) ∧ gh(y2) ∧ gh(y3) r1 − − → r2

12

NEGATIVE RELIANCES

Rule r2 negatively relies on r1 (written r1 − − → r2): there is a situation when r1 can inhibit the application of r2

EXAMPLE

r1 : ∧3

i=1 hasAtom(x, zi) ∧ c(z1) ∧

• (z2) ∧ h(z3) ∧ bond(z1, z2) ∧

bond(z2, z3) ∧ not gh(z1) ∧ not gh(z2) ∧ not gh(z3) → organicHydroxy(x) ∧ rh(x) r2 :

• rganicHydroxy(x) ∧ not rh(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi)

∧ c(y1) ∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3) ∧ gh(y1) ∧ gh(y2) ∧ gh(y3) r1 − − → r2 but r2

−

− → r1

12

NEGATIVE RELIANCES

Rule r2 negatively relies on r1 (written r1 − − → r2): there is a situation when r1 can inhibit the application of r2

EXAMPLE

r1 : ∧3

i=1 hasAtom(x, zi) ∧ c(z1) ∧

• (z2) ∧ h(z3) ∧ bond(z1, z2) ∧

bond(z2, z3) ∧ not gh(z1) ∧ not gh(z2) ∧ not gh(z3) → organicHydroxy(x) ∧ rh(x) r2 :

• rganicHydroxy(x) ∧ not rh(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi)

∧ c(y1) ∧ o(y2) ∧ h(y3) ∧ bond(y1, y2) ∧ bond(y2, y3) ∧ gh(y1) ∧ gh(y2) ∧ gh(y3) r1 − − → r2 but r2

−

− → r1 Polynomial time to check

12

R-STRATIFICATION

A program P is R-stratified if there is a partition P1, . . . , Pn of P such that for Pi, Pj and rules r1 ∈ Pi and r2 ∈ Pj, we have: if r1 + − → r2 then i ≤ j and if r1 − − → r2 then i < j.

13

R-STRATIFICATION

A program P is R-stratified if there is a partition P1, . . . , Pn of P such that for Pi, Pj and rules r1 ∈ Pi and r2 ∈ Pj, we have: if r1 + − → r2 then i ≤ j and if r1 − − → r2 then i < j.

EXAMPLE

r2 r1 +

+

r3 r4 r5 r6 P1 P2 P3

− + + + + + −

13

R-STRATIFICATION

A program P is R-stratified if there is a partition P1, . . . , Pn of P such that for Pi, Pj and rules r1 ∈ Pi and r2 ∈ Pj, we have: if r1 + − → r2 then i ≤ j and if r1 − − → r2 then i < j.

EXAMPLE

r2 r1 +

+

r3 r4 r5 r6 P1 P2 P3 S1

• P = TP1(F)

− + + + + + −

13

R-STRATIFICATION

A program P is R-stratified if there is a partition P1, . . . , Pn of P such that for Pi, Pj and rules r1 ∈ Pi and r2 ∈ Pj, we have: if r1 + − → r2 then i ≤ j and if r1 − − → r2 then i < j.

EXAMPLE

r2 r1 +

+

r3 r4 r5 r6 P1 P2 P3 S1

• P = TP1(F)

S2

• P = TP2(S1
• P)

− + + + + + −

13

R-STRATIFICATION

A program P is R-stratified if there is a partition P1, . . . , Pn of P such that for Pi, Pj and rules r1 ∈ Pi and r2 ∈ Pj, we have: if r1 + − → r2 then i ≤ j and if r1 − − → r2 then i < j.

EXAMPLE

r2 r1 +

+

r3 r4 r5 r6 P1 P2 P3 S1

• P = TP1(F)

S2

• P = TP2(S1
• P)

S3

• P = TP3(S2
• P)

− + + + + + −

13

R-STRATIFICATION

A program P is R-stratified if there is a partition P1, . . . , Pn of P such that for Pi, Pj and rules r1 ∈ Pi and r2 ∈ Pj, we have: if r1 + − → r2 then i ≤ j and if r1 − − → r2 then i < j.

Strictly extends ‘classical’ stratification ensures stable model uniqueness coNP-complete to check

13

R-STRATIFICATION

A program P is R-stratified if there is a partition P1, . . . , Pn of P such that for Pi, Pj and rules r1 ∈ Pi and r2 ∈ Pj, we have: if r1 + − → r2 then i ≤ j and if r1 − − → r2 then i < j.

Strictly extends ‘classical’ stratification ensures stable model uniqueness coNP-complete to check

Fact entailment for R-acyclic, R-stratified programs

Stable models bounded in size (double exp), and at most one stable model

13

R-STRATIFICATION

A program P is R-stratified if there is a partition P1, . . . , Pn of P such that for Pi, Pj and rules r1 ∈ Pi and r2 ∈ Pj, we have: if r1 + − → r2 then i ≤ j and if r1 − − → r2 then i < j.

Strictly extends ‘classical’ stratification ensures stable model uniqueness coNP-complete to check

Fact entailment for R-acyclic, R-stratified programs

Stable models bounded in size (double exp), and at most one stable model 2ExpTime-complete w.r.t. program complexity

13

R-STRATIFICATION

A program P is R-stratified if there is a partition P1, . . . , Pn of P such that for Pi, Pj and rules r1 ∈ Pi and r2 ∈ Pj, we have: if r1 + − → r2 then i ≤ j and if r1 − − → r2 then i < j.

Strictly extends ‘classical’ stratification ensures stable model uniqueness coNP-complete to check

Fact entailment for R-acyclic, R-stratified programs

Stable models bounded in size (double exp), and at most one stable model 2ExpTime-complete w.r.t. program complexity PTime-complete w.r.t. data complexity

13

RELIANCES UNDER CONSTRAINTS

Restrict input sets of facts to relax R-acyclicity and R-stratification using constraints

14

RELIANCES UNDER CONSTRAINTS

Restrict input sets of facts to relax R-acyclicity and R-stratification using constraints

EXAMPLE

r1 : mol(x) ∧ hasAtom(x, z) ∧ c(z) → organic(x) r2 : mol(x) ∧ not organic(x) → inorganic(x) r3 : inorganic(x) → mol(x) ∧ geoOrigin(x)

14

RELIANCES UNDER CONSTRAINTS

Restrict input sets of facts to relax R-acyclicity and R-stratification using constraints

EXAMPLE

r1 : mol(x) ∧ hasAtom(x, z) ∧ c(z) → organic(x) r2 : mol(x) ∧ not organic(x) → inorganic(x) r3 : inorganic(x) → mol(x) ∧ geoOrigin(x) r1 − − → r2 + − → r3 + − → r1

14

RELIANCES UNDER CONSTRAINTS

Restrict input sets of facts to relax R-acyclicity and R-stratification using constraints

EXAMPLE

r1 : mol(x) ∧ hasAtom(x, z) ∧ c(z) → organic(x) r2 : mol(x) ∧ not organic(x) → inorganic(x) r3 : inorganic(x) → mol(x) ∧ geoOrigin(x) C = {inorganic(x) ∧ hasAtom(x, z) ∧ c(z) → ⊥} r1 − − → r2 + − → r3 + − → r1

14

RELIANCES UNDER CONSTRAINTS

Restrict input sets of facts to relax R-acyclicity and R-stratification using constraints

EXAMPLE

r1 : mol(x) ∧ hasAtom(x, z) ∧ c(z) → organic(x) r2 : mol(x) ∧ not organic(x) → inorganic(x) r3 : inorganic(x) → mol(x) ∧ geoOrigin(x) C = {inorganic(x) ∧ hasAtom(x, z) ∧ c(z) → ⊥} r1 − − → r2 + − → r3 + − → r1 but r3

+

− →C r1

14

RELIANCES UNDER CONSTRAINTS

Restrict input sets of facts to relax R-acyclicity and R-stratification using constraints

EXAMPLE

r1 : mol(x) ∧ hasAtom(x, z) ∧ c(z) → organic(x) r2 : mol(x) ∧ not organic(x) → inorganic(x) r3 : inorganic(x) → mol(x) ∧ geoOrigin(x) C = {inorganic(x) ∧ hasAtom(x, z) ∧ c(z) → ⊥} r1 − − → r2 + − → r3 + − → r1 but r3

+

− →C r1 Slightly more complex to check: Positive reliance Negative reliance R-acyclicity/R-stratification ΣP

2-complete

in ∆P

2

ΠP

2-complete

ΣP

2-hardness follows from satisfiability of a QBF ∃

p.∀ q.ϕ

14

EXPERIMENTAL SETUP

Chemical Entities of Biological Interest

Reference terminology adopted for chemical annotation by major bio-ontologies ~20,000 molecule and ~8,000 chemical class descriptions ChEBI taxonomy manually curated

15

EXPERIMENTAL SETUP

Chemical Entities of Biological Interest

Reference terminology adopted for chemical annotation by major bio-ontologies ~20,000 molecule and ~8,000 chemical class descriptions ChEBI taxonomy manually curated

Our knowledge base consisted of rules derived from ChEBI that represented

15

EXPERIMENTAL SETUP

Chemical Entities of Biological Interest

Reference terminology adopted for chemical annotation by major bio-ontologies ~20,000 molecule and ~8,000 chemical class descriptions ChEBI taxonomy manually curated

Our knowledge base consisted of rules derived from ChEBI that represented

500 molecules

EXAMPLE

methanol(x) → ∃6

i=1yi. ∧6 i=1 hasAtom(x, yi) ∧ . . . ∧ bond(y2, y6)

15

EXPERIMENTAL SETUP

Chemical Entities of Biological Interest

Reference terminology adopted for chemical annotation by major bio-ontologies ~20,000 molecule and ~8,000 chemical class descriptions ChEBI taxonomy manually curated

Our knowledge base consisted of rules derived from ChEBI that represented

500 molecules 30 molecular part descriptions

EXAMPLE

∧3

i=1 hasAtom(x, zi) ∧ . . . ∧

bond(z2, z3) ∧ not gh(z1) ∧ not gh(z2) ∧ not gh(z3) → organicHydroxy(x) ∧ rh(x)

• rganicHydroxy(x)∧ not rh(x) → ∃3

i=1yi. ∧3 i=1 hasAtom(x, yi) ∧ . . .

∧ bond(y2, y3) ∧ ∧3

i=1gh(yi)

15

EXPERIMENTAL SETUP

Chemical Entities of Biological Interest

Reference terminology adopted for chemical annotation by major bio-ontologies ~20,000 molecule and ~8,000 chemical class descriptions ChEBI taxonomy manually curated

Our knowledge base consisted of rules derived from ChEBI that represented

500 molecules 30 molecular part descriptions 50 chemical class descriptions

EXAMPLE

hasAtom(x, z) ∧ o(z) → hasOxygen(x)

15

EXPERIMENTAL SETUP

Chemical Entities of Biological Interest

Reference terminology adopted for chemical annotation by major bio-ontologies ~20,000 molecule and ~8,000 chemical class descriptions ChEBI taxonomy manually curated

Our knowledge base consisted of rules derived from ChEBI that represented

500 molecules 30 molecular part descriptions 50 chemical class descriptions

78,957 rules in total (R-stratified and R-acyclic)

15

EXPERIMENTAL SETUP

Chemical Entities of Biological Interest

Reference terminology adopted for chemical annotation by major bio-ontologies ~20,000 molecule and ~8,000 chemical class descriptions ChEBI taxonomy manually curated

Our knowledge base consisted of rules derived from ChEBI that represented

500 molecules 30 molecular part descriptions 50 chemical class descriptions

78,957 rules in total (R-stratified and R-acyclic) Used DLV for stable model computation

15

EMPIRICAL RESULTS

First attempt to compute the stable model of the overall program P failed (no result after 600 secs)

16

EMPIRICAL RESULTS

First attempt to compute the stable model of the overall program P failed (no result after 600 secs) Second attempt exploited partition of the program into two rule sets according to R-stratification

16

EMPIRICAL RESULTS

First attempt to compute the stable model of the overall program P failed (no result after 600 secs) Second attempt exploited partition of the program into two rule sets according to R-stratification Split into lowest R-stratum P1 and remaining four upper R-strata P5

2

16

EMPIRICAL RESULTS

First attempt to compute the stable model of the overall program P failed (no result after 600 secs) Second attempt exploited partition of the program into two rule sets according to R-stratification Split into lowest R-stratum P1 and remaining four upper R-strata P5

2

Computed stable model S1

• P of P1 ∪ F

16

EMPIRICAL RESULTS

First attempt to compute the stable model of the overall program P failed (no result after 600 secs) Second attempt exploited partition of the program into two rule sets according to R-stratification Split into lowest R-stratum P1 and remaining four upper R-strata P5

2

Computed stable model S1

• P of P1 ∪ F

Computed stable model S5

• P of P5

2 ∪ S1

• P

16

EMPIRICAL RESULTS

First attempt to compute the stable model of the overall program P failed (no result after 600 secs) Second attempt exploited partition of the program into two rule sets according to R-stratification Computed 8,639 subclass relations in 13.5 secs

16

EMPIRICAL RESULTS

First attempt to compute the stable model of the overall program P failed (no result after 600 secs) Second attempt exploited partition of the program into two rule sets according to R-stratification Computed 8,639 subclass relations in 13.5 secs Revealed missing subsumptions from the ChEBI ontology

16

EMPIRICAL RESULTS

First attempt to compute the stable model of the overall program P failed (no result after 600 secs) Second attempt exploited partition of the program into two rule sets according to R-stratification Computed 8,639 subclass relations in 13.5 secs Revealed missing subsumptions from the ChEBI ontology E.g. organicHydroxy ⊑ organoOxygenCompound ✓ ✘

16

CONCLUSIONS

R-acyclicity and R-stratification conditions (coNP-complete to check)

17

CONCLUSIONS

R-acyclicity and R-stratification conditions (coNP-complete to check) Fact entailment Program comp. Data comp. R-acyclic coN2ExpTime-complete coNP-complete R-acyclic+R-stratified 2ExpTime-complete PTime-complete

17

CONCLUSIONS

R-acyclicity and R-stratification conditions (coNP-complete to check) Fact entailment Program comp. Data comp. R-acyclic coN2ExpTime-complete coNP-complete R-acyclic+R-stratified 2ExpTime-complete PTime-complete Generalise with constraints (ΠP

2-complete to check)

17

CONCLUSIONS

R-acyclicity and R-stratification conditions (coNP-complete to check) Fact entailment Program comp. Data comp. R-acyclic coN2ExpTime-complete coNP-complete R-acyclic+R-stratified 2ExpTime-complete PTime-complete Generalise with constraints (ΠP

2-complete to check)

Performance gains in DLV & new subsumptions in ChEBI

17

CONCLUSIONS

R-acyclicity and R-stratification conditions (coNP-complete to check) Fact entailment Program comp. Data comp. R-acyclic coN2ExpTime-complete coNP-complete R-acyclic+R-stratified 2ExpTime-complete PTime-complete Generalise with constraints (ΠP

2-complete to check)

Performance gains in DLV & new subsumptions in ChEBI Future directions:

More general notions of ‘rule’ + equality in rule heads [LPNMR’13]

17

CONCLUSIONS

R-acyclicity and R-stratification conditions (coNP-complete to check) Fact entailment Program comp. Data comp. R-acyclic coN2ExpTime-complete coNP-complete R-acyclic+R-stratified 2ExpTime-complete PTime-complete Generalise with constraints (ΠP

2-complete to check)

Performance gains in DLV & new subsumptions in ChEBI Future directions:

More general notions of ‘rule’ + equality in rule heads [LPNMR’13] Compare performance with other ASP solvers [chemical classification problem, ASPCOMP’13]

17

CONCLUSIONS

R-acyclicity and R-stratification conditions (coNP-complete to check) Fact entailment Program comp. Data comp. R-acyclic coN2ExpTime-complete coNP-complete R-acyclic+R-stratified 2ExpTime-complete PTime-complete Generalise with constraints (ΠP

2-complete to check)

Performance gains in DLV & new subsumptions in ChEBI Future directions:

More general notions of ‘rule’ + equality in rule heads [LPNMR’13] Compare performance with other ASP solvers [chemical classification problem, ASPCOMP’13]

Thank you! Questions?!?

17