Description Logics Syntax, Semantics and Reasoning Services - Claudia d'Amato & Nicola Fanizzi
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Description Logics
Syntax, Semantics and Reasoning Services
Claudia d’Amato & Nicola Fanizzi
Machine Learning and the Semantic Web
PhD Programme in Computer Science and Mathematics
Università degli Studi di Bari ”Aldo Moro”
March 3, 2020Summary
1 Description Logics
The DL Famiy
Defining a DL Knowledge Base
Reasoning Services
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 2 / 28Agenda
1 Description Logics
The DL Famiy
Defining a DL Knowledge Base
Reasoning Services
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 3 / 28DLs: Characteristics
Description Logics:
family of logics of different expressive power
decidable fragment of FOL
grounded on Model Theoretic semantics
describe domains in terms of Concepts (classes), roles (relation),
individuals
endowed with several reasoning services (usually correct and complete)
theoretical and computational foundation of OWL
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 4 / 28DLs Architecture d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 5 / 28
DL: Basic Elements
Primitive Concepts NC = {C, D, . . .}: subsets of a domain
Primitive Roles NR = {R, S, . . .}: binary relations on the domain
Interpretation I = (∆I , ·I ) where
∆I : domain of the interpretation
·I : interpretation function
assigns to each primitive concept C a subset C I ⊆ ∆I
assigns to each primitive role R a binary relation RI ⊆ ∆I × ∆I
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 6 / 28The Basic Language: AL
AL = Attribute Language [Schmidt-Schauß, 1991]
Other languages obtained as extension of AL
Operators → Complex concept descriptions
Example.: Primitive Concepts: Person, Female → Person u Female;
Person u ¬Female
Primitive Role: hasChild → Person u ∀hasChild.Female
Person u ∃hasChild.>
Person u ∃hasChild.⊥
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 7 / 28AL: Semantics
Given the interpretation function, the interpretation of complex
descriptions is defined inductively
Two concepts C and D are said equivalent (C ≡ D) if for all
interpretations I results C I = DI
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 8 / 28Multiple Models and Single Model
A DLs Knowledge Base (KB) does not define a single model
a KB is a set of constraints defining a set of possible models
No constrains (== empty KB) ⇔ any model is possible
By increasing constraints decrease the possible models
A very high number of constraints may cause that any possible model
exists (inconsistent KB)
Differently, DB (and frame/rule KR systems) make assumptions so
that DB/KB define a single model
UNA == different names are interpreted as different individuals
Closed World Assumption == the domain is built by the only
individuals that are mentioned in DB/KB
Minimal Models == the extensions are as small as possible
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 9 / 28DLs: Constructs
Name Syntax Semantics
atomic negation ¬A, A ∈ NC A I ⊆ ∆I
full negation ¬C C I ⊆ ∆I
concept conj. CuD C I ∩ DI
concept disj. CtD C I ∪ DI
full exist. restr. ∃R.C {a ∈ ∆I | ∃b (a, b) ∈ RI ∧ b ∈ C I }
universal restr. ∀R.C {a ∈ ∆I | ∀b (a, b) ∈ RI → b ∈ C I }
at most restr. ≤ nR {a ∈ ∆I | | {b ∈ ∆I | (a, b) ∈ RI } |≤ n}
at least restr. ≥ nR {a ∈ ∆I | | {b ∈ ∆I | (a, b) ∈ RI } |≥ n}
qualif. at most r. ≤ nR.C {a ∈ ∆I | | {b ∈ ∆I | (a, b) ∈ RI ∧ b ∈ C I } |≤ n}
qualif. at least r. ≥ nR.C {a ∈ ∆I | | {b ∈ ∆I | (a, b) ∈ RI ∧ b ∈ C I } |≥ n}
one-of {a1 , a2 , ...an } {a ∈ ∆I | a = ai , 1 ≤ i ≤ n}
has value ∃R.{b} {a ∈ ∆I | (a, b) ∈ RI }
inverse of R− {(b, a) ∈ ∆I × ∆I | (a, b) ∈ RI }
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 10 / 28DLs: Constructors e Languages of the AL Family
DL Name Constructors
ALN AL ∪ {≤ nR, ≥ nR} (N )
ALE AL ∪ {∃R.C} (E)
ALEN ALE ∪ {≤ nR, ≥ nR}
ALC ALE ∪ {¬C, t} (C)
ALCN ALC ∪ {≤ nR, ≥ nR}
SHOIN ALC ∪ role transitivity (S)∪
∪role hierarchies (H)∪
∪ oneOf (i.e. nominals) (O)∪
∪{R− }(inverserole) (I) ∪ N
SHIQ SHI ∪ {≤ n R.C, ≥ n R.C} (Q)
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 11 / 28OWL e DLs
OWL DL is grounded on SHIQ e SHOIN (D)
OWL benefits of:
Reasoning algorithms and DLs Formal properties
algorithms complexity and decidability well understood
Reasoners implementing DLs services freely available
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 12 / 28KB & Subsumption
K = hT , Ai
TBox T – set of definitions C ≡ D equality axiom, meaning C I = DI
An equality axiom where C is an atomic concept is called definition
Es.: Mother ≡ Woman u ∃hasChild.Person
Other axioms: C v D inclusion axioms meaning C I ⊆ DI (risp.
RI ⊆ S I )
where C, D concepts and R, S roles;
ABox A – set of concept and role assertions e.g. C(a) e R(a, b),
meaning: aI ∈ C I e (aI , bI ) ∈ RI (risp.).
Subsumption
Let C and D two concept descriptions, C sussume D, denoted D v C, iff
for all interpretations I, results DI ⊆ C I
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 13 / 28TBox: Example
Primitive Concepts: NC = {Female, Male, Human}.
Primitive Roles: NR = {HasChild, HasParent, HasGrandParent, HasUncle}.
T = { Woman ≡ Human u Female; Man ≡ Human u Male
Parent ≡ Human u ∃HasChild.Human
Mother ≡ Woman u Parent
Father ≡ Man u Parent
Child ≡ Human u ∃HasParent.Parent
Grandparent ≡ Parent u ∃HasChild.( ∃ HasChild.Human)
Sibling ≡ Child u ∃HasParent.( ∃ HasChild ≥ 2)
Niece ≡ Human u (∃HasGrandParent.Parent t ∃HasUncle.Uncle)
Cousin ≡ Niece u ∃HasUncle.(∃ HasChild.Human)
}.
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 14 / 28ABox: Example
A = {Woman(Clara), Woman(Tiziana), Father(Leonardo), Father(Antonio),
Father(AntonioB),Mother(Maria), Mother(Giovanna), Child(Valentina), Sibling(Martina),
Sibling(Vito), HasParent(Clara,Giovanna), HasParent(Leonardo,AntonioB),
HasParent(Martina,Maria), HasParent(Giovanna,Antonio), HasParent(Vito,AntonioB),
HasParent(Tiziana,Giovanna), HasParent(Tiziana,Leonardo), HasParent(Valentina,Maria),
HasParent(Maria,Antonio), HasSibling(Leonardo,Vito), HasSibling(Martina,Valentina),
HasSibling(Giovanna,Maria), HasSibling(Vito,Leonardo), HasSibling(Tiziana,Clara),
HasSibling(Valentina,Martina), HasChild(Leonardo,Tiziana), HasChild(Antonio,Giovanna),
HasChild(Antonio,Maria), HasChild(Giovanna,Tiziana), HasChild(Giovanna,Clara),
HasChild(AntonioB,Leonardo), HasChild(Maria,Valentina), HasUncle(Martina,Giovanna),
HasUncle(Valentina,Giovanna)
}
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 15 / 28Subsumption: Example Given the definition: Father ≡ Male u ∃hasChild.Person ”a father is a male (person) having persons as children” Examples of assertions are: Male(Leonardo), Male(Vito), hasChild(Leonardo, Vito) Let us suppose that Male v Person: Person(Leonardo), Person(Vito) hence Father(Leonardo) Other definitions: Parent ≡ Person u ∃hasChild.Person and FatherWithoutSons ≡ Male u ∃hasChild.Person u ∀hasChild.(¬Male) It is straightforward to verify: Father v Parent and FatherWithoutSons v Father. d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 16 / 28
Reasoning on ontologies
Given an ontology ⇒ the implicit knowledge may be made explicit by
reasoning operators
Developed Reasoners:
FaCT
RACER
PELLET
Hermit
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 17 / 28TBox: Standard Reasoning Services
Concept Satisfiability: verify if a new added concept within an
ontology is coherent with its TBox or if it contradicts the TBox
Definition of Concept Satisfiability
Given a concept C it is satisfiable wrt the TBox T if there exists a model I
of T s.t. C I 6= ∅.
It is said that I is model of C
Example:
T = { Parent, Man, Woman ≡ ¬ Man, Mother ≡ Woman u
Parent}
added Mother v Man ⇒ new axiom unsatisfiable wrt the TBox
the disjointness axiom Woman ≡ ¬ Man violated
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 18 / 28TBox: Standard Reasoning Services [. . . cont.]
Concept Subsumption: verify if a se a concept C is more general
than another concept D
used for building the concept hierarchy
A concept C subsumes a concept D wrt TBox T if DI ⊆ C I for each model
I of T . Written as: D vT C or T |= D v C
Example:
T = { Parent, Man, Woman ≡ ¬ Man, Mother ≡ Woman u Parent} added
Father ≡ Man u Parent ⇒ Father v Parent and Father v Man
Two concepts C and D are equivalent wrt T if C I = DI for each model I
of T . It is written C ≡T D are T |= C ≡ D
Two concepts C are D are disjoint wrt T if C I ∩ DI = ∅ ∀ model I of T .
N.B.: the qualification "wrt T " can be omitted when T is clear from the context (or T is
empty)
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 19 / 28ABox: Standard Reasoning Services
ABox Consistency Check (wrt TBox): verify if a new (concept or
role) assertion in ABox A makes A inconsistent wrt TBox T or not
Example 1:
T = {Woman ≡ Person u Female, Man ≡ Person u ¬Female};
A = { Woman(MARY),Man(MARY)} ⇒ A is è inconsistent wrt T
Example 2: TBox T = {Woman, Man} ⇒ A consistent wrt T
there is no restriction on the interpretation of the concepts Woman and
Man
Instance Checking: assesses if an individual is an instance of a given
concept or not
Retrieval: returns all individuals that are instance of a concept
Example:
T = {Woman v Female }; A = { Female(Ann), Woman(Sara)}
⇒ Retrieval(Female) = {Ann, Sara}
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 20 / 28OWA vs. CWA...
Open World Assumption: typically mande in DLs
Missing information is interpreted as information not known
Knowledge in the KB is not considered as complete
Closed World Assumption: typically made in DB, ML
Missing information is interpreted as negative information
Knowledge in the KB is considered as complete
Example:
T = { Female,Woman};
A = { Female(Ann),Woman(Sara)}
CWA: q = Female(Sara) ? → N O
OWA: q = Female(Sara) ? → U N KN OW N
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 21 / 28OWA vs. CWA... [. . . cont.]
another example of the meaning of OWA: the scope of resource
descriptions is not the single OWL file
Example: Given a class C1 defined in the ontology O1 , C1 can be
extended in other ontologies
the consequences of the additional propositions concerning C1 are
MONOTONE ⇔ new information cannot retract previous inforamtion
New (also deduced) information can only be added, also if they
contradict existing knowledge
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 22 / 28Reasoning Algorithms
Almost all reasoning services may be reduced to a basic service: the
ABox Consistency Check
provided that the DL allows conjunction and (full) negation
Tableau based algorithms have been developed
Complexity of the algorithm: exponential in time and space
can be optimized → polynomial complexity in space
For DLs that do not allow negation
algorithms grounded on the comparison of the structure of
(normalized) concepts are considered (e.g. structural subsuption)
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 23 / 28Reduction of the standard inferences to the ABox Consistency Check Given the concepts C and D the following equivalence are valid Reduction to the Subsumption 1 C is unsatisfiable ⇔ C is subsumed by ⊥ 2 C ≡ D ⇔ C is subsumed by D and D is subsumed by C 3 C and D are disjoint ⇔ C u D is subsumed by ⊥ If the DL allows for negation, the following equivalence are valid: Reduction to Unsatisfiability 1 C is subsumed by D ⇔ C u ¬D is unsatisfiable 2 C ≡ D ⇔ both C u ¬D and ¬C u D unsatisfiable 3 C and D are disjoint ⇔ C u D is unsatisfiable N.B.: These propositions are verified also wrt TBox d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 24 / 28
Reduction of the standard inferences to the ABox
Consistency Check [. . . cont.]
Reduction of Unsatisfiability
Given the concept C, the following propositions are equivalent
1 C is unsatisfiable
2 C is subsumed by ⊥
3 C and ⊥ are equivalent
4 C and > are disjoint
Reduction of the Instance Check to the ABox Consistency Check
A |= C(a) iff A ∪ {¬C(a)} is inconsistent
where a is an arbitrary individual
Reduction of the Concept Satisfiability to the ABox Consistency Check
C is satisfiable iff {C(a)} is consistent
where a is an arbitrary individual
N.B.: These propositions are verified also wrt TBox
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 25 / 28Standard Reasoning Services in the Ontology
Lifecycle
Standard Reasoning Services can be exploited in different phases of the
Ontology Lifecycle
Design
Check for concepts satisfiability within the ontology and consequent
relationships (implicitly) implied
Ontology Merging/Alignment
Inter-ontology relationships
Captuiring/computing concept hierarchy among different ontologies
Ontology Deployment
Assess if a set of assertions is consistent with the ontology
Assess if a set of individuals are instances of one or more concepts
within the ontology
Query Inclusion / Classification-based querying
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 26 / 28Some Non-Standard Reasoning Services
least common subsumer is the most specific concept subsuming a given set
of concepts
A concept description E is the lcs of C1 , · · · , Cn iff
1 Ci v E ∀ i = 1, · · · , n and
2 if E 0 is a concept satisfying (1) then E v E 0
realization problem assesses the concepts to which an individual belongs
to. Among these concepts, the most specific concept (if it
exists) is particularly important and it is formally defined as:
most specific concept
Given an ABox A and an individual a, the most specific concept of a wrt A is
the concept C, denoted MSCA (a), s.t. A |= C(a) and ∀D s.t. A |= D(a),
C v D,.
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 27 / 28Bibliography
F. Baader et al. The Description Logic Handbook: Theory,
Implementation, Applications. Cambridge University Press (2003) Ch.
1,2,4 (2007 II Edition).
M. Schmidtand G. Smolka. Attributive concept descriptions with
complements. Artificial Intelligence, 48(1) pp. 1–26
Online material http : //dl.kr.org/courses.html
N. Guarino and P. Giaretta. Ontologies and Knowledge Bases:
Towards a Terminological Clarification. In Proc. of 2nd Int. Conf. on
Building and Sharing Very Large-Scale Knowledge Bases (1995).
d’Amato & Fanizzi (MLSW) Description Logics March 3, 2020 28 / 28You can also read
