Pay-as-you-go Query Answering with PAGOdA

BERNARDO CUENCA GRAU

Ontology-mediated Query Answering

Q

TA

B D

C

RDF Data

ab

•  (Meta)-data published in RDF

•  RDF resources reference an OWL 2 ontology

•  The ontology describes the meaning of data

RDF and OWL 2 well-established

•  Thousands of available OWL 2 ontologies

•  RDF ubiquitous on the Web

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Ontology-mediated Query Answering

Ontology languages offer a wide range modeling constructs

High expressive power à high worst-case complexity of reasoning How can we provide scalable query answering?

•  Restrict our ontology to a lightweight fragment of OWL EL, QL or RL profiles

•  Tolerate incompleteness

•  Rely on highly optimised pay-as-you-go systems •  Worst case optimal for lightweight fragments •  Rapidly computes easy answers •  Performance gracefully degrades with harder instances

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Datalog and the OWL 2 Profiles

Datalog is the quintessential rule-based KR language

•  Reasoning typically implemented via materialisation •  Our in-house system RDFox shows excellent performance

Query answering within the OWL 2 profiles

•  RL ontologies equivalent to Datalog programs •  EL and QL ontologies can be strengthened using Datalog

Query answering requires an additional filtration step

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Incomplete Reasoning

§  RL / EL reasoning w.r.t. arbitrary OWL ontology O dataset D and query q gives (in general) an incomplete answer L

P  Profile-specific reasoning via Datalog (relatively) scalable O Answers may be incomplete O Degree of incompleteness unknown O  Incompleteness may be pathological (empty answers)

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L = cert(q, hO`,Di) ✓ cert(q, hO,Di) with O |= O`

The idea behind PAGOdA

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Redistribute reasoning workload Datalog reasoner Fully-fledged OWL 2 reasoner

Resort to expensive OW2 reasoning as little as possible (if at all) Ensure sound and complete answers Do not restrict ontology language

Datalog reasoner OWL 2 reasoner

Step 1: Lower and Upper Bounds

Data

Lower

ELHO Lower

Data

Upper

Ontology

Query

Datalog

Eng

ine

Datalog Engine

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Profile-specific reasoning via Datalog gives a lower bound

L gives a subset of We transform O into strictly stronger Datalog ontology Ou

•  Normalise ontology into Datalog±,v rules

•  Eliminate ∨ by transforming to ∧

•  Replace existential variables with Skolem constants

Datalog reasoning w.r.t. Ou gives upper bound answer U

cert(q, hO,Di)

cert(q, hO,Di) ✓ cert(q, hOu,Di) = U

Step 2: Module extraction

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Checking possible answers in U \ L is expensive Compute a fragment of ontology + data sufficient to check each answer in U \ L. Fragment computation involves proof tracing in Ou

Achieved also using Datalog materialisation Relevant fragments are typically much smaller Size of the problem substantially reduced

DUDatalog Engine

Fragment

Step 3: Summarisation

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Summary

Summarisation

Full Reasoner Q

FragmentFurther reduce problem size by summarising the fragment

•  Technique introduced by the SHER team at IBM •  “Merge” constants that are instances of same concepts •  Check answers against summary using OWL 2 reasoner •  The summary of the fragment is typically very small

This is an orthogonal over-approximation to previous ones We further reduce the size of U \ L Sometimes we even make it empty !

Step 4: Dependency analysis

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Dependency AnalysisF

Full Reasoner QF

Output

Group remaining candidate answers •  If a and b are in the same group then a is an answer iff b is •  We can also establish dependencies between groups

Check group representatives against fragment using the fully-fledged reasoner.

Features of PAGOdA

PAGOdA provides PAYG query answering for OWL 2:

§  Uses Datalog reasoner “out of the box” §  Efficiently computes sound partial answers §  In “easy” cases, efficiently computes complete answers §  In “harder” cases, applies increasingly powerful but less scalable

reasoning techniques as needed to completely answer query §  The last step involving full reasoner is rarely needed in practice

§  Recent improvements §  Better and better upper bounds §  Smaller and smaller modules

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Queries answered by each technique

Scalability for lower and upper bound computation

LUBM UOBM FLY DBPedia NPD Total 24 15 6 441 329

Bounds 22 12 5 439 326

Sum 22 14 5 440 329

Full 24 15 6 441 329

Importing Lower Mat Upper Mat Ave QA LUBM1000 313s 190s 269s 12s UOBM500 356s 346s 734s 4s

Queries that require full reasoning

Lower Upper Gap Sum Groups LUBM100_q20 0 26 26 26 1 LUBM100_q22 0 14 14 14 1 UOBM1_q14 6271 6535 264 264 1 FLY_q5 0 344 344 344 1 DBPedia_q404 0 2 2 2 1

Lower Upper Frag Size (%) Sum Full

LUBM100_q20 0.2s 0.3s 14.5s .005/.04 1.2s 190.1s

LUBM100_q22 0.3s 0.2s 10.0s .005/.04 0.8s 46.1s

UOBM1_q14 0.1s 0.1s 0.7s .17/.076 0.5s 5.4s

FLY_q5 0.0s 0.0s 16.0s .34/.01 0.1s 0.2s

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Time distribution and fragment size

PAGOdA Team

§  Yujiao Zhou

§  Yavor Nenov

§  Bernardo Cuenca Grau

§  Ian Horrocks

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