The Rise of Approximate Ontology Reasoning: Is It Mainstream Yet? --- Revisiting the two Towers of the Semantic Web

The Rise of Approximate Ontology
Reasoning:
- Revisiting the two Towers of the Semantic Web
Keynote at the 6th Joint International Conference on Semantic
Technologies (JIST2016)
Singapore, 4th Nov, 2016
Jeff Z. Pan
Department of Computing Science
University of Aberdeen, UKorg

Revisit the two Towers of the Semantic Web
[Slide credit: J. Hendler, ISWC2015]

4
[Slide credit: J. Hendler, ESWC2016]

5
[Slide credit: I. Horrocks, ISWC2015]
Powered by OWL

6
Powered by OWL

7
[Slide credit: C. Bizer, ISWC2016]

8
[Slide credit: J. Hendler, ESWC2016]
Practically, this would suggest
supporting undecidable logics
Approximate Reasoning is a natural choice

Outline
1.  Does approximate reasoning work?
•  If so, why it might work?
2.  What is the cost of constructing approximations?
3.  Are there any approximate reasoning approaches that
construct no approximations at all?
4.  What does it mean to semantic applications?
9

1. Does Approximate Reasoning Work?
10
[Mining Big Data with RDF Graph Technology – Oracle]
OWL-DBC

Does Approximate Reasoning Work?
Ontology Reasoner Evaluation (ORE 2014):
•  A competition for sound and complete reasoners
•  TrOWL (3rd place in OWL 2 DL ABox materialisation)
is an approximate reasoner from University of
Aberdeen
•  Outperforming many sound and complete
reasoners
•  Certainly the competition is very challenging for
approximate reasoners
•  since even getting 99.99% of the reasoning
results is not enough
•  only getting 100% is counted as a valid answer

Why Approximation Might Work?
13
3.1415926
1.414
[Credit: theoriginof.come; deviantart.com]

theory is an approximation to reality

General class axioms are widely used
•  such as the domain constraint
•  ObjectProperty(hasF domain (Person))
• 
Expansion rules for general class axioms
•  If a new node y is added, then add a general class
description D t :C in L(y) for each general class axiom
For example: if an ontology consists of 100 general
class axioms
•  then theoretically there are 2100 possible expansions
•  for each node

Classification is a complex reasoning service
•  Compute the class hierarchy
Classification can be reduced to subsumption
checking
•  between every pair of classes
•  which can be further reduced to class unsatisfiability
checking
For example, if an ontology consists of 100,000
classes
•  then a naïve implementation needs
•  100,000 * 100,000 class unsatisfiability checking

Decidability in Query
answering in OWL 2 DL is
still an open problem
[Garey & Johnson. Computers and Intractability: A Guide to the
Theory of NP-Completeness. Freeman, 1979.]
I can’t find any algorithms, but neither can all these famous people.
17

Quality
Expressive
Power
Efficiency
Trade-off
Tractable/Horn-DLs
•  Efficient
•  Quality guaranteed
•  Limited expressiveness
Approximate Reasoning
•  (designed to be) Efficient
•  Support expressive DLs
•  (potentially) Incomplete and/or unsound
Tableau-based algorithms
•  Quality guaranteed
•  Support expressive DLs
•  High worst case complexity
18

OWL 2 DL
OWL 1 DL
OWL 2 QL
OWL 2 RL OWL 2 EL
SROIQ
SHOIN
DL-Lite
EL++
OWL 2 Full
In AC0
PTime-
Complete
NExpTime-
Complete
2NExpTime-
Complete
Undecidable
19

Optimal Approximations
Idea: to compile a source ontology O (in more
expressive LS) into its upper/lower bound (in less
expressive LT)
Entailment set ES(O, LT) of O in LT
•  The set of all LT axioms that are entailed by O under NC,
NP and NI
MstMMwk
20

Semantic Approximation [Pan and Thomas, AAAI2007]
Strongest weaker approximation for QL ES(O, DL-
Litecore) of an OWL2 DL O is finite and unique.
Theorem 1: Given an ontology O, a conjunctive
query q(X) and an evaluation [X→S], if ES(O, DL-
Litecore) |= q[X→S], then O |= q[X→S].
Theorem 2: Given an ontology OS, a database-
style conjunctive query q(X) without non-
distinguished variables and an evaluation [X→S],
ES(OS, DL-Litecore) |= q[X→S] iff OS |= q[X→S].
21
Q
ES(O, LT)O
è

Theorem 2 suggests:
An approximate DL-Lite ontology can be better than
the original OWL 2 DL ontology, in terms of query
answering!

2. What is the Cost of Constructing
Approximations?

Semantic approximation
•  Pro: can achieve optimal approximation [Pan and
Thomas, AAAI2007]
•  Con: can be expensive to compute
Syntactic approximation
•  Pro: compute with low cost
•  Cons:
1.  Might not reduce the complexity [Groot et al. ESWC2005]
(still the same logic)
approximate ∀r.(∀s.C) as ∀r.(∀s.T)
approximate ∀r.(∀s.T) as ∀r.T
2.  Not necessarily guarantee reasoning quality and
performance
24
What is the Cost of Constructing Approximations?

Approximation used in Studying Motions of Planets
[Credit: spaceplex.com]

One idea: k-approximations [Santarelli et al., ISWC2014]
•  When k=|OS|, k-approximation is called GSA
(global semantic approximation)
•  When k=1, k-approximation is called LSA (local
semantic approximation)
26

•  Upper bound is needed when there existential variables
in the queries
•  Different approaches for computing upper bound
•  Summarisation [Dolby et al., AAAI2007; Fokoue et
al., AAAI2012]
•  Ontology strengthening [Tserendorj et al., RR2008]
•  Query generalisation [Pan et al., DL2009]
•  Testing the idea of computing Lower bound
Approximation (LA) and Upper bound Approximation (UA)
[Pan et al., DL2009; Zhao et al., AAAI2014]
•  In case LA = UA, we have the exact answers
28
Upper Bound Approximations

29
Prof1
source 2
C – Course
H – Head of Dept
B – Business Prof
CS – CS Prof
U – University
taughBy
in
Prof1 is Head of D: src3
Prof1 is CS Prof: src2
U1
C3
U3
CS2
source 3
B2
C2
U2
Prof1
B1
C1
U1
Summary 3
Course
University
CS
ProfHofD
§ Independent Local
Summaries:
•  Sources
independently build and
maintain their local
summary of their ABox
•  A local summary
node represents
individuals with same
explicit concepts and
same hash value
•  Property of summary:
completeness
preserving
approximation of ABox
Distributed Summarisation [Fokoue et al. AAAI2012]
hash(c) = hash(h)
source 1
Busi
ness
Prof
Course
University
Summary 1
taughtBy
taughtBy
taughtBy
CS
Prof
Summary 2
Busi
ness
Prof
Course
University
“Find courses taught
by a CS professor who
is a Head of Dept.”
C4
29

3. Approximate Reasoning without Approximations?

Why SROIQ (OWL2 DL) to EL++ (OWL2 EL)?
•  Minor syntactic gap results in major complexity
difference
• From most expressive/complex to most efficient for
classification/materialisation
DL SROIQ DL EL++
N2EXPTIME-complete PTIME-complete
31
Approximate Deduction [Pan et al., AIJ, 2016]

Example
Step 1: Represent non-OWL2-EL concepts with fresh named concepts
•  A ∀r.B
•  is replaced by
•  A X1, X1 ∀r.B
Step 2: Maintain semantic relations for these named concepts
•  complementary relations
•  cardinality relations
Step 3: Additional tractable completion Rules (on top of the EL ones), e.g.
•  Handling complement
•  E.g. B C => ¬C ¬B
ALL
r B
A
C
ALL
D Some
r nB
A
nC
Some D
B C
X1 X2
32

TBox Classification with ORE2013
Benchmark (186/203)
33

TBox Classification on ORE2013 Benchmark
Syntactic approximation improves recall
over naïve EL++ approximation
0%
20%
40%
60%
80%
100%
RELRecall
0%
20%
40%
60%
80%
100%
ELKRecall
34

4. What does it mean for Semantic Applications
35
•  Allow users to build
complex ontologies,
•  ... to explore more
expressive logics
•  Users don’t have to
understand the
differences among
different OWL profiles

What does it mean for Semantic Applications
36
•  being able to handle
more dynamic data and
knowledge
•  more demanding
reasoning services
•  as well as more
expressive query
languages
[Credit: steve-wheeler.co.uk]

The two many Towers of the Semantic Web

Take Home Message(s)
38
•  Approximation helps
identify / exploit important
information
•  Many good results on
approximate reasoning in
SW, still a lot to investigate
[Credit: theoriginof.come; deviantart.com; steve-wheeler.co.uk]

Further Reading on Knowledge
Graphs
Jeff Z. Pan, Guido Vetere, Jose Manuel
Gomez Perez and Honghan Wu (Eds.).
Exploiting Linked Data and Knowledge
Graphs for Large Organisations. Springer.
2016.
Jeff Z. Pan, Diego Calvanese, Thomas Eiter,
Ian Horrocks, Michael Kifer, Fangzhen Lin
and Yuting Zhao (Eds.). Logical Foundation
of Knowledge Graph Construction and
Querying Answering. Springer. 2016
39

Acknowledgements
•  Funding bodies: EPSRC, EC
•  Yuan Ren, Yuting Zhao, Edward Thomas, Zhe (Alan) Wu
and Achille Fokoue
•  My colleagues and students in KT@ABDN

References
•  [Pan and Thomas, AAAI2007] Jeff Z. Pan and Edward Thomas. Approximating OWL-DL
Ontologies. In Proc. of the 22nd AAAI Conference on Artificial Intelligence (AAAI-07).
1434-1439. 2007.
•  [Groot et al. ESWC2005] Perry Groot, Heiner Stuckenschmidt, Holger Wache. Approximating
Description Logic Classification for Semantic Web Reasoning. In Proc. of ESWC2005.
•  [Santarelli et al., ISWC2014] Marco Console, Jose Mora, Riccardo Rosati, Valerio Santarelli,
Domenico Fabio Savo. Effective Computation of Maximal Sound Approximations of
Description Logic Ontologies. In Proc. of ISWC2014.
•  [Tserendorj et al., RR2008] Tserendorj, T.; Rudolph, S.; Krötzsch, M.; and Hitzler, P.
Approximate OWL-reasoning with Screech. In Proc. of RR2008.
•  [Dolby et al., AAAI2007] Dolby, J.; Fokoue, A.; Kalyanpur, A.; Kershenbaum, A.; Schonberg, E.;
Srinivas, K.; and Ma, L. Scalable semantic retrieval through summarization and refinement. In
Proc. of AAAI2007.
•  [Fokoue et al., AAAI2012] Achille Fokoue, Felipe Meneguzzi, Murat Sensoy and Jeff Z. Pan.
Querying Linked Ontological Data through Distributed Summarization. In Proc. of AAAI2012.
•  [Pan et al., DL2009] Jeff Z. Pan, Edward Thomas and Yuting Zhao. Completeness Guaranteed
Approximation for OWL DL Query Answering. In Proc. of DL2009.
•  [Zhao et al., AAAI2014] Yujiao Zhou and Yavor Nenov and Bernardo Cuenca Grau and Ian
Horrocks. Pay-as-you-go OWL Query Answering Using a Triple Store. In Proc. of AAAI2014.
•  [Pan et al., AIJ, 2016] Jeff Z. Pan, Yuan Ren and Yuting Zhao. Tractable approximate
deduction for OWL. In Arteficial Intelligence, volumn 235: p95-155, 2016.

The Rise of Approximate Ontology
Reasoning:
- Revisiting the two Towers of the
Semantic Web
JIST2016
Thank you
… questions?

The Rise of Approximate Ontology Reasoning: Is It Mainstream Yet? --- Revisiting the two Towers of the Semantic Web

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