Pay-as-you-go OWL Query Answering Using a Triple Store

Yujiao Zhou, Yavor Nenov, Bernardo Cuenca Grau and Ian Horrocks

Pay-as-you-go Approach

Intuition‣ to delegate the bulk of the

computational workload to a highly scalable datalog reasoner

!‣ to minimise the use of a fully-

fledged reasoner

Evaluation‣ Evaluated on LUBM(100,1000), UOBM(1, 60, 500), FLY, DBPedia+travel

and NPD FactPages.

Average time without OWL 2 reasoning

Average timeAcknowledgements This work was supported by the Royal Society, the EPSRC projects Score!, ExODA, and MaSI3, and the FP7 project OPTIQUE.

Data

Lower

ELHO Lower

Data

Upper

Ontology

DU

Query

Summary

Datalog

Eng

ine

Datalog Engine

Datalog Engine

Summarisation

Full Reasoner Q

Dependency Analysis

Fragment

F

Full Reasoner QF

Output

Tracking by datalog encoding

triple store OWL 2 reasoner

L=LRL ∪ LEL ∪ … U

L = U

σ(cert(q, F)) ⊆ cert(q, σ(F))

Incomplete endomorphisms

Arrange calls to the reasoner according to the dependencies heuristically

Rule out non-answers

Done

Diagram Over-approx to datalog

!!!!

‣ upper bound U answer of q w.r.t the resulting set of rules U(Σ) and D.

Lower bounds ‣ basic lower bound LRL

answer of q w.r.t. the datalog fragment of Σ and D; ‣ EL lower bound LEL

answer of q w.r.t. the ELHO fragment of Σ and D.

Tracking encoding in datalog Intuition: to compute all the rules and facts that participate in a proof of q(a) in Σ∪D. This goal can be archived using datalog encoding. ‣ Example:

‣ If B1(x1),…,Bm(xm) → H(x) is a rule in U(Σ), Ht(x), B1 (x1), . . . , Bm (xm) → S(cr)∧B1

t (x1 )∧ . . . ∧Bm

t(xm ) is added to the tracking rule.

‣ Involved rules: {r | S(cr) is derived} Involved facts: {P(a) ∈ D | Pt(a) is derived}

Summarisation & dependency between answers ‣ Let σ be the summary function, σ(cert(q, F)) ⊆ cert(q, σ(F)) ‣ If there is an endomorphism from a to b in F, then

a ∈ cert(q, F) implies b ∈ cert(q, F)

‣ Existential knowledge

{. . . , A uB, . . .}

{C} {C}

{A, . . .}x1

{A, . . .}x2

R

R

{A, . . .}

{C}c

x1{A, . . .}

x2

R

R

{. . . , A tB, . . .}

‣ Disjunctive knowledge

DL Ontology Dataset QueriesLUBM(n) SHI 93 ~100,000n 14 (std)+10

!UOBM(n) SHIN 314 ~200,000n 15FLY SRI 144,407 6,308

88 5

DBPedia SHOIN 1,757 12,119,662 441 (atomic)NPD SHIF 819 3,817,079 329 (atomic)

LUBM(1000) UOBM(100) FLY DBPedia NPDQueries 22/24 12/15 5/5 439/441 294/329 Time(s) 18.4 0.7 0.2 0.3 0.1

LUBM(100) UOBM(1) FLY DBPedia NPD

Time(s) 29.6 1.8 0.2 3 3

Problem Setting‣Ontology Σ — a set of rules of the form φ(x) → Vi ∃yi ψ(x, yi) ‣ Data D — a set of ground atoms of the form P(a) ‣ Conjunctive queries — FO formula of the form q(x) ← ∃y ψ(x, y)

where ψ and φ are conjunctions of atoms.