Modelling Structured Domains with Description Graphs and Logic Programming

MODELLING STRUCTURED DOMAINS USING

DESCRIPTION GRAPHS AND LOGIC

PROGRAMMING

Despoina Magka, Boris Motik and Ian Horrocks

Department of Computer Science, University of Oxford

May 29, 2012

OUTLINE

1 MOTIVATION

2 DGLPS, IMPLEMENTATION AND OVERVIEW

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MODELLING STRUCTURED DOMAINS WITH OWL

OWL used for the representation of complex structures:

Aerospace

Cellular biology

Human anatomy

Molecules

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Aerospace

Cellular biology

Human anatomy

Molecules

2



Aerospace

Cellular biology

Human anatomy

Molecules

2



Aerospace

Cellular biology

Human anatomy

Molecules

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Aerospace

Cellular biology

Human anatomy

Molecules

2

THE CHEBI ONTOLOGY

OWL ontology Chemical Entities of Biological Interest

Freely accessible dictionary of ‘small’ molecular entities

High quality annotation and taxonomy of chemicals

Interoperability between researchers

Drug discovery and elucidation of metabolic pathways

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THE CHEBI ONTOLOGY






3

THE CHEBI ONTOLOGY






3

THE CHEBI ONTOLOGY






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THE CHEBI ONTOLOGY






3

AUTOMATE CHEMICAL CLASSIFICATION

ChEBI is manually incremented

Currently contains approx. 28,000 fully annotated entities

Grows at a rate of ~1,500 entities per curator per year

Biologically interesting entities possibly > 1,000,000

Each new molecule is subsumed by several chemicalclasses

Is dinitrogen inorganic?Does cyclobutane contain a four-membered ring?Is acetylene a hydrocarbon?Does benzaldehyde contain a benzene ring?

Speed up curating tasks with automated reasoning tools

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Is dinitrogen inorganic?

Does cyclobutane contain a four-membered ring?Is acetylene a hydrocarbon?Does benzaldehyde contain a benzene ring?


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Is dinitrogen inorganic?Does cyclobutane contain a four-membered ring?

Is acetylene a hydrocarbon?Does benzaldehyde contain a benzene ring?


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Is dinitrogen inorganic?Does cyclobutane contain a four-membered ring?Is acetylene a hydrocarbon?

Does benzaldehyde contain a benzene ring?


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Is dinitrogen inorganic? YesDoes cyclobutane contain a four-membered ring? YesIs acetylene a hydrocarbon? YesDoes benzaldehyde contain a benzene ring? Yes


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(MIS)REPRESENTING RINGS WITH OWL

Chemical compounds with rings are highly frequent

Fundamental inability of OWL to represent cyclesAt least one tree-shaped model for each consistent OWLknowledge baseOWL-based reasoning support

Does cyclobutane contain a four-membered ring?Does benzaldehyde contain a benzene ring?

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Chemical compounds with rings are highly frequentFundamental inability of OWL to represent cycles

At least one tree-shaped model for each consistent OWLknowledge baseOWL-based reasoning support


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Chemical compounds with rings are highly frequentFundamental inability of OWL to represent cyclesAt least one tree-shaped model for each consistent OWLknowledge base

OWL-based reasoning support


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EXAMPLE

Cyclobutane v ∃(= 4)hasAtom.(Carbon u ∃(= 2)hasBond.Carbon)

C C

CC



5



EXAMPLE


C C

CC



5



EXAMPLE


C C

CC



5



EXAMPLE


C C

CC



5



EXAMPLE


C C

CC


Does cyclobutane contain a four-membered ring? 8

Does benzaldehyde contain a benzene ring? 8

5

OWL EXTENSIONS

Limitation of OWL to represent cycles (partially) remediedby extension of OWL with Description Graphs and rules[Motik et al., 2009]

A Description Graph represents structures by means of adirected labeled graphIs cyclobutadiene a hydrocarbon?

6

OWL EXTENSIONS

Limitation of OWL to represent cycles (partially) remediedby extension of OWL with Description Graphs and rules[Motik et al., 2009]A Description Graph represents structures by means of adirected labeled graph

Is cyclobutadiene a hydrocarbon?

6

OWL EXTENSIONS


EXAMPLE

C C

CC

1Cyclobutadiene

2Carbon 3 Carbon

4 Carbon5Carbon

∀hasAtom.(Carbon t Hydrogen) v Hydrocarbon


6

OWL EXTENSIONS


EXAMPLE

C C

CC

1Cyclobutadiene

2Carbon 3 Carbon

4 Carbon5Carbon



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OWL EXTENSIONS


EXAMPLE

C C

CC

1Cyclobutadiene

2Carbon 3 Carbon

4 Carbon5Carbon


Does cyclobutadiene have a conjugated four-memberedring?


6

OWL EXTENSIONS


EXAMPLE

C C

CC

1Cyclobutadiene

2Carbon 3 Carbon

4 Carbon5Carbon


Does cyclobutadiene have a conjugated four-memberedring? 3


6

OWL EXTENSIONS


EXAMPLE

C C

CC

1Cyclobutadiene

2Carbon 3 Carbon

4 Carbon5Carbon

Oxygen



6

OWL EXTENSIONS


EXAMPLE

C C

CC

1Cyclobutadiene

2Carbon 3 Carbon

4 Carbon5Carbon

Oxygen



6

OWL EXTENSIONS


EXAMPLE

C C

CC

1Cyclobutadiene

2Carbon 3 Carbon

4 Carbon5Carbon

Oxygen



6

OWL EXTENSIONS


EXAMPLE

C C

CC

1Cyclobutadiene

2Carbon 3 Carbon

4 Carbon5Carbon

Oxygen


Is cyclobutadiene a hydrocarbon? 8

6

RESULTS OVERVIEW

Key idea:

Switch from first-order logic to logic programming semantics

Use negation-as-failure to derive non-monotonic inferences

Expressive decidable logic-based formalism for modellingstructured entities: Description Graph Logic Programs(DGLPs)

DGLPS all cycles CWAOWL+DGS+RULES some cycles OWAOWL no cycles OWA

Negation-as-failure↔ Closed-world assumption↔ Missinginformation treated as falseClassical negation ↔ Open-world assumption↔ Missinginformation treated as not known

Prototypical implementation of DGLPs with application instructure-based chemical classification

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RESULTS OVERVIEW

Key idea:Switch from first-order logic to logic programming semantics






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RESULTS OVERVIEW







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RESULTS OVERVIEW







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RESULTS OVERVIEW







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RESULTS OVERVIEW





Negation-as-failure↔ Closed-world assumption↔ Missinginformation treated as false

Classical negation ↔ Open-world assumption↔ Missinginformation treated as not known


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RESULTS OVERVIEW







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RESULTS OVERVIEW







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RESULTS OVERVIEW







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OUTLINE

1 MOTIVATION

2 DGLPS, IMPLEMENTATION AND OVERVIEW

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WHAT IS A DGLP ONTOLOGY?The syntactic objects of a DGLP ontology:

Description graphsFunction-free FOL Horn rulesRules with negation-as-failureFacts

9


Description graphs

Function-free FOL Horn rulesRules with negation-as-failureFacts

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Description graphs

EXAMPLE

C C

CC

1Cyclobutane

2Carbon 3 Carbon

4 Carbon5Carbon

O O

1Dioxygen

2Oxygen 3 Oxygen

Function-free FOL Horn rulesRules with negation-as-failureFacts

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Description graphsFunction-free FOL Horn rules

Rules with negation-as-failureFacts

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EXAMPLEBond(x, y) → Bond(y, x)SingleBond(x, y) → Bond(x, y)

Rules with negation-as-failureFacts

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Rules with negation-as-failure

Facts

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EXAMPLEHasAtom(x, y) ∧ Carbon(y) → HasCarbon(x)Molecule(x)∧ not HasCarbon(x) → Inorganic(x)

Facts

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Facts

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Facts

EXAMPLE

Cyclobutane(c1), Dinitrogen(c2), . . .

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ENCODING DESCRIPTION GRAPHS

Translate DGs into logic programs with function symbols

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EXAMPLE

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EXAMPLE

Cyclobutane(x) →Gcb(x, f1(x), f2(x), f3(x), f4(x))Gcb(x, y1, y2, y3, y4)→Cyclobutane(x) ∧

Carbon(y1) ∧ Carbon(y2) ∧Carbon(y3) ∧ Carbon(y4) ∧HasAtom(x, y1) ∧ Bond(y1, y2) ∧HasAtom(x, y2) ∧ Bond(y2, y3) ∧HasAtom(x, y3) ∧ Bond(y3, y4) ∧HasAtom(x, y4) ∧ Bond(y4, y1)

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CLASSIFYING OBJECTS

EXAMPLE

Molecule(x) ∧ HasAtom(x, y) ∧ not Carbon(y) ∧ not Hydrogen(y)→ NotHydroCarbon(x)Molecule(x) ∧ not NotHydroCarbon(x)→ HydroCarbon(x)

C C

CC

Is cyclobutane ahydrocarbon? 3

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CLASSIFYING OBJECTS

EXAMPLE


C C

CC


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CLASSIFYING OBJECTS

EXAMPLE


C C

CC


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CLASSIFYING OBJECTS

EXAMPLE

Molecule(x) ∧∧

1≤i≤4

HasAtom(x, yi) ∧∧

1≤i≤3

Bond(yi, yi+1) ∧

Bond(y4, y1)∧

1≤i<j≤4

not yi = yj

→ MoleculeWith4MemberedRing(x)

C C

CC

Does cyclobutane contain afour-membered ring? 3

12

CLASSIFYING OBJECTS

EXAMPLE

Molecule(x) ∧∧

1≤i≤4


1≤i≤3

Bond(yi, yi+1) ∧

Bond(y4, y1)∧

1≤i<j≤4

not yi = yj


C C

CC


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CLASSIFYING OBJECTS

EXAMPLE

Molecule(x) ∧∧

1≤i≤4


1≤i≤3

Bond(yi, yi+1) ∧

Bond(y4, y1)∧

1≤i<j≤4

not yi = yj


C C

CC


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UNDECIDABILITY

Logic programs with function symbols can axiomatiseinfinitely large structures

Reasoning with DGLP ontologies is trivially undecidableWe are only interested in finite structures

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UNDECIDABILITY

Logic programs with function symbols can axiomatiseinfinitely large structuresReasoning with DGLP ontologies is trivially undecidable

We are only interested in finite structures

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UNDECIDABILITY

Logic programs with function symbols can axiomatiseinfinitely large structuresReasoning with DGLP ontologies is trivially undecidableWe are only interested in finite structures

13

UNDECIDABILITY

Logic programs with function symbols can axiomatiseinfinitely large structuresReasoning with DGLP ontologies is trivially undecidableWe are only interested in finite structures

EXAMPLE

Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

1AceticAcid

2Methyl

3Carboxyl

1Carboxyl

2Carbonyl

3Hydroxyl

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SYNTACTIC ACYCLICITY CONDITIONS

Chase [Maier et al., 1979] termination is undecidable

Problem extensively studied in theory of databasesVarious syntax-based acyclicity conditions

weak acyclicity [Fagin et al., ICDT, 2002]super-weak acyclicity [Marnette, PODS, 2009]joint acyclicity [Krötzsch and Rudolph, IJCAI, 2011]rule out naturally-arising nested structures

EXAMPLE

Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

1AceticAcid

2Methyl

3Carboxyl

1Carboxyl

2Carbonyl

3Hydroxyl

14


Chase [Maier et al., 1979] termination is undecidableProblem extensively studied in theory of databases

Various syntax-based acyclicity conditionsweak acyclicity [Fagin et al., ICDT, 2002]super-weak acyclicity [Marnette, PODS, 2009]joint acyclicity [Krötzsch and Rudolph, IJCAI, 2011]rule out naturally-arising nested structures

EXAMPLE

Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

1AceticAcid

2Methyl

3Carboxyl

1Carboxyl

2Carbonyl

3Hydroxyl

14


Chase [Maier et al., 1979] termination is undecidableProblem extensively studied in theory of databasesVarious syntax-based acyclicity conditions


EXAMPLE

Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

1AceticAcid

2Methyl

3Carboxyl

1Carboxyl

2Carbonyl

3Hydroxyl

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weak acyclicity [Fagin et al., ICDT, 2002]super-weak acyclicity [Marnette, PODS, 2009]joint acyclicity [Krötzsch and Rudolph, IJCAI, 2011]

rule out naturally-arising nested structures

EXAMPLE

Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

1AceticAcid

2Methyl

3Carboxyl

1Carboxyl

2Carbonyl

3Hydroxyl

14




EXAMPLE

Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

1AceticAcid

2Methyl

3Carboxyl

1Carboxyl

2Carbonyl

3Hydroxyl

14




EXAMPLE

Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

1AceticAcid

2Methyl

3Carboxyl

1Carboxyl

2Carbonyl

3Hydroxyl

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SEMANTIC ACYCLICITY

1 Transitive and irreflexive graph ordering which specifieswhich graph instances may imply the existence of othergraph instances

2 Extend the logic program with rules that detect violation ofthe graph ordering

3 Repetitive construction of graph instances during reasoningtriggers derivation of Cycle

A DGLP ontology O is semantically acyclic if O 6|= CycleDGLP ontology with acetic acid is semantically acyclic 3

Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

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SEMANTIC ACYCLICITY


EXAMPLE

1AceticAcid

2Methyl

3Carboxyl

1Carboxyl

2Carbonyl

3Hydroxyl

AceticAcid ≺ Carboxyl




Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

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SEMANTIC ACYCLICITY





Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

15

SEMANTIC ACYCLICITY





Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

15

SEMANTIC ACYCLICITY




A DGLP ontology O is semantically acyclic if O 6|= Cycle

DGLP ontology with acetic acid is semantically acyclic 3

Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

15

SEMANTIC ACYCLICITY





Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

15

SEMANTIC ACYCLICITY





Methyl

Carboxyl

Carbonyl

Hydroxyl

O

C

CH3 OH

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TECHNICAL RESULTS

1 Termination guarantee for semantically acyclic ontologies

2 Decidability of semantic acyclicity for negation-free DGLPontologies

3 Decidability of semantic acyclicity for DGLP ontologies withstratified negation

Semantically acyclic DGLP ontologies with stratifiednegation capture a wide range of chemical classes:


Does cyclobutane contain a four-membered ring?

Is acetylene a hydrocarbon?


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TECHNICAL RESULTS

1 Termination guarantee for semantically acyclic ontologies2 Decidability of semantic acyclicity for negation-free DGLP

ontologies

3 Decidability of semantic acyclicity for DGLP ontologies withstratified negation






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TECHNICAL RESULTS


ontologies3 Decidability of semantic acyclicity for DGLP ontologies with

stratified negation






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TECHNICAL RESULTS



stratified negation






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TECHNICAL RESULTS



stratified negation






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TECHNICAL RESULTS



stratified negation


Is dinitrogen inorganic? 3

Does cyclobutane contain a four-membered ring? 3

Is acetylene a hydrocarbon? 3

Does benzaldehyde contain a benzene ring? 3

16

EMPIRICAL EVALUATION

Data extracted from ChEBI in Molfile format

XSB logic programming engineChemical classes:

HydrocarbonsInorganic moleculesMolecules with exactly two carbonsMolecules with a four-membered ringMolecules with a benzene

Preliminary evaluation ranging from 10 to 70 moleculesResults:

All DGLP ontologies were found acyclicMolecules classified as expectedSuite of subsumption tests for largest ontology performed infew minutes

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Data extracted from ChEBI in Molfile formatXSB logic programming engine

Chemical classes:




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Data extracted from ChEBI in Molfile formatXSB logic programming engineChemical classes:




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Preliminary evaluation ranging from 10 to 70 molecules

Results:


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All DGLP ontologies were found acyclic

Molecules classified as expectedSuite of subsumption tests for largest ontology performed infew minutes

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All DGLP ontologies were found acyclicMolecules classified as expected

Suite of subsumption tests for largest ontology performed infew minutes

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OVERVIEW AND FUTURE DIRECTIONS

1 Expressive and decidable formalism for representation ofstructured objects

2 Novel acyclicity condition for logic programs with restricteduse of function symbols

3 Prototype for the structure-based classification of complexobjects

Future directions:

Generalise acyclicity condition for datalog rules withexistentials in the headRelax stratifiability criteria for negationUser-friendly surface syntaxFully-fledged classification system for graph-shaped objects

Thank you for listening. Questions?

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Future directions:



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Future directions:



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Future directions:Generalise acyclicity condition for datalog rules withexistentials in the head

Relax stratifiability criteria for negationUser-friendly surface syntaxFully-fledged classification system for graph-shaped objects


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Future directions:Generalise acyclicity condition for datalog rules withexistentials in the headRelax stratifiability criteria for negation

User-friendly surface syntaxFully-fledged classification system for graph-shaped objects


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Future directions:Generalise acyclicity condition for datalog rules withexistentials in the headRelax stratifiability criteria for negationUser-friendly surface syntax

Fully-fledged classification system for graph-shaped objects


18





Future directions:Generalise acyclicity condition for datalog rules withexistentials in the headRelax stratifiability criteria for negationUser-friendly surface syntaxFully-fledged classification system for graph-shaped objects


18





Future directions:Generalise acyclicity condition for datalog rules withexistentials in the headRelax stratifiability criteria for negationUser-friendly surface syntaxFully-fledged classification system for graph-shaped objects


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Technology

Modelling Structured Domains with Description Graphs and Logic Programming