Semantics and Reasoning Algorithms for a Faithful Integration of Description Logics and Rules. Boris Motik, University of Oxford. Contents. Why Combine DLs with LP? Main Challenge: OWA vs. CVA Existing Approaches Minimal Knowledge and Negation as Failure MKNF Knowledge Bases
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Semantics and Reasoning Algorithmsfor a Faithful Integration ofDescription Logics and Rules
Boris Motik, University of Oxford
UK cities are in UK regions.
UKCity v9 isIn.UKRgn
8 x : UKCity(x) !9 y : isIn(x,y) Æ UKRgn(y)
UK regions are EU regions.
UKRgn v EURgn
8 x : UKRgn(x) ! EURgn(x)
8 x : [9 y : isIn(x,y) Æ EURgn(y)] ! EUPart(x)
Things in EU are parts of EU.
9 isIn.EURgn v EUPart
We can conclude:
8 x : UKCity(x) ! EUPart(x)
UK cities are parts of EU.
UKCity v EUPart
x
S
x1
R
R
x2
x3
9S.(9 R.C u9 R.D) v Q ,
8x:{[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)},
8x,x1,x2,x3:{ S(x,x1) Æ R(x1,x2) Æ C(x2) Æ R(x1,x3) Æ D(x3) ! Q(x) }
Question: is there a flight from MAN to MUC?
flight(MAN,STR)
flight(MAN,LHR)
flight(MAN,FRA)
flight(FRA,ZAG)
Open worlds (=OWL):
Don’t know!
We did not specify thatwe know information aboutall possible flights.
Closed worlds (=LP):
No.
If we cannot prove something,
it must be false.
8 x,y: flight(x,y) $ (x ¼ MAN Æ y ¼ STR) Ç (x ¼ MAN Æ y ¼ LHR) Ç …
Person u:(9 hasSSN.SSN) v?
Person v9 hasSSN.SSN
the class Dextrocardiac is unsatisfiable
DLs (= taxonomical reasoning)
+
LP Rules (= relational expressivity + nonmonotonic inferences)
=
The Winning Combination!
In DLs we derive Person(a)
The formula is equivalent to
8x : [Father(x) ! Person(x)]
eliminates all models in which x is a father and not a person
In LP, : is interpreted as defaultnegation
read as “is not provable”
The example is unsatisfiable
Negation defined using minimal knowledge
8x : [Father(x) Æ:Person(x) !?]
Father(a)
DLs
LP
Father(a)
This is the only minimal model.
(There is no model M’ ½ M.)
All models are of equal “quality”.
M1
Father(a)
M
Father(a)
M2
Father(a), Person(a)
8x : [Father(x) Æ:Person(x) !?]
DLs
Rules
Father(a)
M1
Father(a)
, Cat(a)
M
Father(a)
, Cat(a)
M2
Father(a), Person(a)
8x : [Father(x) Æ:Person(x) ! Cat(x)]
Nonmonotonic semantics typically prefer certain models.
DL[ S1[ p1, S2[ p2, S3Å p3; Q ]


+
no nonmonotonic reasoning over DLpredicates
Use AEL as a framework for integrating FOL and LP
(Researcher t Programmer)(Boris)
Researcher v Employed
Programmer v Employed
² Employed(Boris)
² Researcher(Boris)
² Programmer(Boris)
² K Employed(Boris)
² :KResearcher(Boris)
² :K Programmer(Boris)
Bird(Tweety)
K Bird(Tweety) Æ not:Flies(Tweety) !K Flies(Tweety)
GelfondLifschitz reduct!
H1Ç … Ç HnÃ B1, …, Bm
P(t1, …, tn) firstorder atom
K P(t1, …, tn) Katom
not P(t1, …, tn) notatom
K = (O, P)
(K) = K (O) ÆÆr 2 P8 x1,…,xn : H1Ç … Ç Hn½ B1Æ … Æ Bm
default rule
constraint
(O, ;) ² iff O²for any FOL formula
(;, P) ² (:)A iff P² (:)Afor A a ground atom
M = { I  I ² }
obK,P = O[ { A  K A 2 P }
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.