Integrating DLs with Logic Programming

1
Integrating DLs withLogic Programming

• Boris Motik, University of Manchester
• Joint work with
• Riccardo Rosati, University of Rome

2
Contents

• Description Logics and OWL
• What is Missing in DLs?
• Hybrid MKNF Knowledge Bases
• Reasoning Algorithm
• Conclusion

3
Description Logics and OWL

• OWL (Web Ontology Langage)
• language for ontology modeling in the Semantic
  Web
• standard of the W3C (http//www.w3.org/2004/OWL/)
• OWL is based on Description Logics (DLs)
• inspired by semantic networks
• DLs have a precise semantics based on first-order
  logics
• well-understood computational properties
• What can we say in OWL?

4
DLs vs. Logic Programming

• OWL
• has been very successful
• but many features are still needed in practice
• Logic programming seems to address many needs
• Political dimension The battle for The Semantic
  Web Language
• This work provides an integration framework for
  DLs and LP
• fully compatible with both systems

5
Contents

• Description Logics and OWL
• What is Missing in DLs?
• Hybrid MKNF Knowledge Bases
• Reasoning Algorithm
• Conclusion

6
Missing Features (I)

• Relational expressivity
• OWL can express only tree-like axioms
• Polyadic predicates
• e.g., Flight(From, To, Airline)
• Exceptions
• the heart is on the left, but in some cases it is
  on the right
• Human v HeartOnLeft, Dextrocardiac v
  Human,Dextrocardiac v HeartOnLeft
• the class Dextrocardiac is unsatisfiable
• we want to say with no contrary evidence, the
  heart is on the left

9 S.(9 R.C u 9 R.D) v Q , 8 x9 y S(x,y) Æ (9
x R(y,x) Æ C(x)) Æ (9 x R(y,x) Æ D(x)) !
Q(x) , 8 x,x1,x2,x3 S(x,x1) Æ R(x1,x2) Æ C(x2)
Æ R(x1,x3) Æ D(x3) ! Q(x)
7
Missing Features (II) Closed Worlds
Question is there a flight from MAN to MUC?
flight(MAN,STR) flight(MAN,LHR) flight(MAN,FRA) fl
ight(FRA,ZAG)
Open worlds (OWL) Dont know! We did not
specify thatwe know information aboutall
possible flights.
Closed worlds (LP) No. If we cannot prove
something, it must be false.

• Partial solution close off the predicate flight
• 8 x,y flight(x,y) (x ¼ MAN Æ y ¼ STR) Ç (x ¼
  MAN Æ y ¼ LHR) Ç
• cannot express many things (e.g., transitive
  closure)
• Closed-world is orthogonal to closed-domain
  reasoning
• Person v 9 father.Person Person(Peter) gt v
  Peter,Paul
• Peter and Paul are now the only objects (the
  domain is closed)
• we do not have CWA (e.g, we cannot derive
  father(Peter,Paul)

8
Missing Features (III) Constraints

• Each person must have an SSN
• naïve attempt
• Person u (9 hasSSN.SSN) v ?
• in FOL, this is equivalent to
• Person v 9 hasSSN.SSN
• assume that only Person(Peter) is given
• we expect the constraint to be violated (no SSN)
• but KB is satisfiable Peter has some unknown SSN
• FOL formulae
• speak about the general properties of worlds
• cannot reason about their own knowledge

9
Contents

• Description Logics and OWL
• What is Missing in DLs?
• Hybrid MKNF Knowledge Bases
• Reasoning Algorithm
• Conclusion

10
Main Idea

• OWA vs. CWA
• CWA requires introspection reasoning about own
  beliefs
• Modal logics allow reasoning about consequences
• KB ² A iff KB ² K A
• KB ² A iff KB ² K A ( looks like CWA

(Researcher t Programmer)(Boris) Researcher v
Employed Programmer v Employed
² Employed(Boris) ² Researcher(Boris) ²
Programmer(Boris)
² K Employed(Boris) ² K Researcher(Boris) ²
K Programmer(Boris)

• K is nonmonotonic
• if we assert Researcher(Boris), then
• K Researcher(Boris) holds
• K Researcher(Boris) does not hold any more

11
Minimal Knowledge and Negation as Failure

• Lifschitz IJCAI 91, Artificial Intelligence
  95
• Syntax FOL with modal operators K and not
• Semantics
• an FO interpretation I and two sets of FO
  interpretations M and N
• M is a model of ? if
• (I,M,M) ² ? and
• for each M ¾ M, there is some I 2 M such that
  (I,M,M) ² ?

Gelfond-Lifschitz reduct!
12
Hybrid MKNF Knowledge Bases

• MKNF Rule

• H1 Ç Ç Hn à B1, , Bm
• Hi are first-order or K-atoms
• Bi are first-order, K-, or not-atoms

P(t1, , tn) - first-order atom K P(t1, , tn) -
K-atom not P(t1, , tn) - not-atom

• Hybrid MKNF Knowledge Base K (O,P)
• O a FOL KB in some language DL
• P a finite set of MKNF rules
• Semantics by translation into MKNF

?(K) K ?(O) Æ Ær 2 P 8 x1,,xn H1 Ç Ç Hn ½
B1 Æ Æ Bm

• DL-safety
• each variable in each rule must occur in a body
  non-DL-K-atom
• makes rules applicable only to named objects
• necessary for decidability

13
Example (I)
default rule

• We derive seasideCity(Barcelona)
• assuming it does not lead to contradiction
• deriving seasideCity(Hamburg) would cause a
  contraction

• We derive Suggest(Barcelona)
• this involves standard DL reasoning
• we do not know the name of the beach in Barcelona

14
Example (II)
constraint

• We treat ¼ in a special way
• we minimize equality along with other predicates
• this yields intuitive consequences
• The constraint is satisfied
• HolyFamily is a church,
• the architect of SagradaFamilia has been
  specified, and
• HolyFamily and SagradaFamilia are synonyms

15
Compatibility

• Our formalism is fully compatible with DLs
• (O,) ² ? iff O ² ? for any FOL formula ?
• to achieve this, we modified MKNF slightly
• we must treat equality in a special way
• Our formalism is fully compatible with LP
• (,P) ² A iff P ² A for A a ground atom
• already shown by Lifschitz
• The combination seems quite intuitive
• as long as we do not mix modal and nonmodal atoms

16
Contents

• Description Logics and OWL
• What is Missing in DLs?
• Hybrid MKNF Knowledge Bases
• Reasoning Algorithm
• Conclusion

17
How to Represent Models

• An MKNF model M is a set of interpretations
• typically infinite!
• we need a finite representation
• We represent M by a FOL formula ? such that
• M I I ² ?
• We can consider only K-atoms from P
• (P,N) a partition of all K-atoms into positive
  and negative
• objective knowledge (?)
• obK,P O A K A 2 P
• our main task is to find a partition (P,N) that
  defines a model

18
The General Case
Grounding
Guess a partition that defines an MKNF model
Check whether the rules are satisfied in this
model.
Check whether this model is consistent with the
DL KB.
Check whether this is the model of minimal
knowledge.
Check whether the query does not hold in the
model.
These are the extensions to the standard
algorithm for disjunctive datalog.
19
Data Complexity

• If rules have special form, we can
• find (P,N) in an easier way (e.g.
  deterministically) and/or
• check the minimality condition easier
• Data complexity of answering ground atomic
  queries
• schema is fixed
• data is variable

• The notion of stratification is rather complex
• it must take into account recursion through the
  DL KB
• difficult to check ( undecidable) and relatively
  weak

20
Contents

• Description Logics and OWL
• What is Missing in DLs?
• Hybrid MKNF Knowledge Bases
• Reasoning Algorithm
• Conclusion

21
Conclusion

• Hybrid MKNF rules
• generalize most known combinations of DLs and
  rules
• they generalize Rosatis DL-log
• they do not generalize Eiters approach
• are fully compatible with both DLs and LP
• are intuitive
• have nice complexity
• Future work
• well-founded semantics
• not trivial, because MKNF is a two-valued logic
• implementation