Semantics and Reasoning Algorithmsfor a Faithful Integration of
Description Logics and Rules
Boris Motik, University of Oxford
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Contents
• Why Combine DLs with LP?
• Main Challenge: OWA vs. CVA
• Existing Approaches
• Minimal Knowledge and Negation as Failure
• MKNF Knowledge Bases
• Reasoning and Complexity
• Conclusion
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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 DLs?
UK cities are in UK regions.UK cities are in UK regions. UKCity v 9 isIn.UKRgnUKCity v 9 isIn.UKRgn 8 x : UKCity(x) ! 9 y : isIn(x,y) Æ UKRgn(y)8 x : UKCity(x) ! 9 y : isIn(x,y) Æ UKRgn(y)
UK regions are EU regions.UK regions are EU regions. UKRgn v EURgnUKRgn v EURgn 8 x : UKRgn(x) ! EURgn(x)8 x : UKRgn(x) ! EURgn(x)
Things in EU are parts of EU.Things in EU are parts of EU. 9 isIn.EURgn v EUPart9 isIn.EURgn v EUPart 8 x : [9 y : isIn(x,y) Æ EURgn(y)] ! EUPart(x)8 x : [9 y : isIn(x,y) Æ EURgn(y)] ! EUPart(x)
UK cities are parts of EU.UK cities are parts of EU. UKCity v EUPartUKCity v EUPart 8 x : UKCity(x) ! EUPart(x)8 x : UKCity(x) ! EUPart(x)
We can conclude:
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• Relational expressivity OWL can express only tree-like axioms
• Polyadic predicates e.g., Flight(From, To, Airline)
• Can be addressed by rules (LP or ASP)
Missing Features (I)
9 S.(9 R.C u 9 R.D) v Q ,8 x:{[9 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) }
x
x1
S
x2 x3
R R
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Missing Features (II) – Closed Worlds
flight(MAN,STR)flight(MAN,LHR)flight(MAN,FRA)flight(FRA,ZAG)
Question: is there a flight from MAN to MUC?
Open worlds (=OWL):Don’t know!
We did not specify thatwe know information about
all possible flights.
Closed worlds (=LP):No.
If we cannot prove something,it must be false.
• Partial solution: close off flight8 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 reasoningPerson v 9 father.Person Person(Peter) > v { Peter,Paul }
• CWA is available in various LP formalisms (e.g., ASP)
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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
• Constraints can be expressed in LP
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• “The heart is usually on the left, but in some cases it is on the right”
• Naïve approach: Human v HeartOnLeft
Dextrocardiac v HumanDextrocardiac v :HeartOnLeft
the class Dextrocardiac is unsatisfiable
“with no contrary evidence, the heart is on the left”
• Exceptions… …cannot be expressed in FOL
…can be expressed in ASP
Missing Features (IV)
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The Magic Formula
DLs (= taxonomical reasoning)
+
LP Rules (= relational expressivity + nonmonotonic inferences)
=
The Winning Combination!
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Contents
• Why Combine DLs with LP?
• Main Challenge: OWA vs. CVA
• Existing Approaches
• Minimal Knowledge and Negation as Failure
• MKNF Knowledge Bases
• Reasoning and Complexity
• Conclusion
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Open vs. Closed Worlds
• In DLs we derive Person(a)• The formula is equivalent to
8 x : [Father(x) ! Person(x)] eliminates all models in which x
is a father and not a person
• In LP, : is interpreted as default negation read as “is not provable”
• The example is unsatisfiable• Negation defined using
minimal knowledge
• “It is illegal to state that someone is a father without stating that he is a person”
8 x : [Father(x) Æ :Person(x) ! ?]
Father(a)
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Idea of Minimal Knowledge
• DLs • LP
Father(a)
M1
Father(a)
M2
Father(a), Person(a)
M Father(a)
8 x : [Father(x) Æ :Person(x) ! ?] kills all models in which the formula does not hold
All models are of equal “quality”.
This is the only minimal model.
(There is no model M’ ½ M.)
• We are left with models that contain Person(a)
• We are left with no model
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Minimal Knowledge and Negation
• DLs • Rules
Father(a)
8 x : [Father(x) Æ :Person(x) ! Cat(x)] esures Cat(x) in each model where x is a father and not a person
• Does not entail Cat(a) • Does entail Cat(a)
, Cat(a)M1
Father(a)
M2
Father(a), Person(a)
M Father(a), Cat(a)
Nonmonotonic semantics typically prefer certain models.
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Contents
• Why Combine DLs with LP?
• Main Challenge: OWA vs. CVA
• Existing Approaches
• Minimal Knowledge and Negation as Failure
• MKNF Knowledge Bases
• Reasoning and Complexity
• Conclusion
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First-Order Rule Formalisms
• First-order combinations of DLs and rules: SWRL, CARIN, AL-log, DL-safe rules
A1 Æ … Æ An à B1 Æ … Æ Bm
concepts (classes) = unary predicates roles (properties) = binary predicates interpreted as first-order clauses
• Semantics is standard first-order Woman(x) ! Person(x) and :Person(Lassie) imply :Woman(Lassie)
• Easily undecidable decidability achieved by syntactic restrictions; e.g., DL-safety
• Issues addressed: relational expressivity and polyadic predicates nonmonotonic features
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Loose Integration
• dl-programs [Eiter, Ianni, Lukasiewicz, Schindlauer, Tompits, AIJ 2008]
• A Ã B1 Æ … Æ Bm Æ not Bm+1 Æ … Æ not Bn
A and Bi are first-order atoms over non-DL-predicates
Bi can additionally be a query atom of the form
DL[ S1 [ p1, S2 [ p2, S3 Å p3; Q ]
Si – DL predicates
pi – non-DL-predicates
Q – a DL query understand as conditional queries over a DL ontology
• Rules are layered over a DL KB they do not contribute to DL consequences
+ - -
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Strong Integration
• A1 Ç … Ç Ak à B1 Æ … Æ Bm Æ not Bm+1 Æ … Æ not
Bn
• DL+log [Rosati, KR2006] DL-atoms cannot occur under negation as failure
semantics: DL-predicates interpreted under OWA
non-DL-predicates interpreted under CWA
no nonmonotonic reasoning over DL-predicates
• dl-programs [Lukasiewicz, ESWC2007] no classical negation cannot capture ASP
faithful extension of LP and DLs only w.r.t. entailment of positive ground atoms
unclear how to extend the semantics to make if faithful w.r.t. arbitrary consequences
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Autoepistemic Logics
• LP can be encoded into first-order AEL AEL by [Konolige, Fund. Inf. 1991]
Use AEL as a framework for integrating FOL and LP [de Bruijn, Eiter, Polleres, Tompits, IJCAI 2007]
• Various encodings proposed with different levels of faithfulness considers disjunctive datalog and not ASP
• No proof theory yet
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Contents
• Why Combine DLs with LP?
• Main Challenge: OWA vs. CVA
• Existing Approaches
• Minimal Knowledge and Negation as Failure
• MKNF Knowledge Bases
• Reasoning and Complexity
• Conclusion
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Knowledge Operator K
(Researcher t Programmer)(Boris)
Researcher v Employed
Programmer v Employed
² Employed(Boris)² Researcher(Boris)² Programmer(Boris)
• K allows us to reason about FO consequences KB ² K A iff KB ² A KB ² :K A iff KB ² A
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
• Used in an algebra-like query language EQL-Lite [Calvanese, De Giacomo, Lembo, Lenzerini, Rosati, IJCAI 2007]
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Default Negation Operator not
Bird(Tweety)
K Bird(Tweety) Æ not :Flies(Tweety) ! K Flies(Tweety)
• Interpreted as not consequence
• Read as: if
“Tweety is a bird” is a consequence
and “Tweety cannot fly” is not a consequence
then “Tweety can fly” should be a consequence
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Minimal Knowledge and Negation as Failure
• Satisfiability defined w.r.t. an MKNF structure (I,M,N) I – a FOL interpretation M and N – sets of FOL interpretations
• M is a model of if: (I,M,M) ² with I 2 M and for each M’ ¾ M, there is some I’ 2 M’ such that (I’,M’,M) ² [Lifschitz, IJCAI 91; Artificial Intelligence 94]
• MKNF explains many nonmonotonic formalisms
(I,M,N) ² A iif A is true in I
(I,M,N) ² : iif is false in I
(I,M,N) ² 1 Æ 2 iif both 1 and 2 are true in I
(I,M,N) ² K iif (J,M,N) ² for each J 2 M
(I,M,N) ² not iif (J,M,N) ² for some J 2 N
Gelfond-Lifschitz reduct!
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Contents
• Why Combine DLs with LP?
• Main Challenge: OWA vs. CVA
• Existing Approaches
• Minimal Knowledge and Negation as Failure
• MKNF Knowledge Bases
• Reasoning and Complexity
• Conclusion
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MKNF Knowledge Bases
• DL-safety: the rules are applicable only to explicitly named objects
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
• MKNF Rule:
• MKNF Knowledge BaseK = (O, P)• O – a FOL KB in some language DL• P – a finite set of MKNF rules
• Semantics by translation into MKNFK) = K (O) Æ Ær 2 P 8 x1,…,x
n : H1 Ç … Ç Hn ½ B1 Æ … Æ Bm
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Example (I)
• 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
default rule
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Example (II)
• 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
constraint
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Faithfulness
• MKNF KBs are fully faithful w.r.t. 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
• MKNF KBs are fully faithful w.r.t. ASP(;, P) ² (:)A iff P ² (:)A for A a ground atom
already shown by Lifschitz
• The combination seems quite intuitive
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Contents
• Why Combine DLs with LP?
• Main Challenge: OWA vs. CVA
• Existing Approaches
• Minimal Knowledge and Negation as Failure
• MKNF Knowledge Bases
• Reasoning and Complexity
• Conclusion
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How to Represent Models
• A MKNF model is a set of interpretations = typically infinite! we need a finite representation
• Idea: represent models by FOL formulae find a first-order formula such that
M = { I | I ² }
• We represent using K-atoms (P,N) – a partition of all K-atoms into positive and negative
defines the consequences that must hold in an MKNF model
objective knowledge:
obK,P = O [ { A | K A 2 P }
our main task is to find a partition (P,N) that defines a model
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Characterization of MKNF Models
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.
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Data Complexity
• Reasoning is undecidable without DL-safety different sources of undecidability than in the FO case
• 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 ground atom entailment:
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Contents
• Why Combine DLs with LP?
• Main Challenge: OWA vs. CVA
• Existing Approaches
• Minimal Knowledge and Negation as Failure
• MKNF Knowledge Bases
• Reasoning and Complexity
• Conclusion
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Conclusion
• MKNF rules… …generalize many known combinations of DLs and rules
…are fully compatible with both DLs and LP
…are intuitive think of K as consequence
think of not as not consequence
…have nice complexity defined by the DL and the LP fragment (in most cases)
• Future challenges implementation
use in applications
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