Semantic Web technologies are a set of languages standardized by the World Wide Web Consortium (W3C) and designed to create a web of data that can be processed by machines. One of the core languages of the Semantic Web is Web Ontology Language (OWL), a family of knowledge representation languages for authoring ontologies or knowledge bases. The newest OWL is based on Description Logics (DL), a family of logics that are decidable fragments of first-order logic. leanCoR is a new description logic reasoner designed for experimenting with the new connection method algorithms and optimization techniques for DL. leanCoR is an extension of leanCoP, a compact automated theorem prover for classical first-order logic.

1. leanCoR: lean connection-based
DL Reasoner
By
Adriano S. T. de Melo
!
Supervisor
Frederico L. G. Freitas

2. Outline
Introduction
Background
Conversion to matrix
leanCoR
Conclusions
2

3. Introduction
3

4. –Tim Berners-Lee
“The semantic web is not a separate Web but an
extension of the current one, in which information is
given well-defined meaning, better enabling computers
and people to work in cooperation.”
4

5. Description Logics (DL)
• Logical formalism underpinning Web Ontology Language
(OWL), an ontology language for the Semantic Web;
• Family of logic-based knowledge representation languages,
• describe domain in terms of concepts, roles and
individuals;
• Characterized by formal semantics and provision of
inference services.
5

6. Reasoning over Description
Logics
• Draws conclusions that are not explicitly stated;
• Provides tools and services to help users:
• design and maintain high quality ontologies,
• answer queries over ontology classes and instances.
6 http://www.cs.man.ac.uk/~horrocks/Slides/lpar05.ppt

7. Problem
• Despite many optimization techniques implemented by DL
reasoners, it is easy to find ontologies where just few
reasoners can perform well.
• Major inefficiency sources can be due to:
• the high degree of non-determinism introduced by some
axioms,
• the interaction between language constructors,
• the construction of large models in memory.
7Baader et al, the description logic handbook

8. Connection Method (CM)
• Provides a general framework for automated deduction.
• CM has some interesting features for dealing with DL
knowledge bases:
• conservative use of memory,
• efficient search strategy,
• can utilize optimizations from tableau.
8W. Bibel, automated theorem proving

9. Goals
• Develop a practical implementation of a DL reasoner from
a first-order logic theorem prover based on the connection
method.
• Provide a mapping from DL axioms to the normal form used
by CM
• Translation rules: transformation from DL axioms to a
matricial representation
• Normalization: method to guarantee that a formula is in
the normal form
9

10. background
10

11. first-order logics
• Letters a,b,c or lower case: constants
• Letters x,y,z: variables
• Letters P,Q,R,S: predicate symbols
• Letters f,g: functions
• Letters s,t: terms (composed of constants, variables or
symbols)
• connectives →,∧,∨,¬ denote implication, conjunction,
disjunction and negation, respectively.
11

12. disjunctive normal form
• Atomic formulae or atoms are built from predicate symbols
and terms, e.g. P(a), P(s), Q(b).
• A literal, denoted by L, is either an atomic formula or a
negated atomic formula, e.g. P(a), ¬Q(b).
• A clause, denoted by C, takes the form L1 ⋀ … ⋀ Ln where
Li is a literal, e.g. P(a) ⋀ Q(b) ⋀ ¬Q(s).
• A formula in disjunctive normal form (DNF) or clausal form
has the form C1 ⋁ … ⋁ Cn where Ci is a clause.
• A formula in DNF can also be written as {C1,...,Cn} and is
called matrix.
12

13. Connection method (1/3)
• Given a formula in DNF,
!
!
!
• A path through a matrix M = {C1,...,Cn} is a set of occurrences of
literals, exactly one from each clause Ci ∈ M.
{P(x), Q(x), ¬P(c), R(c)}
{¬R(x), Q(x), ¬P(c), R(c)}
{P(x), ¬R(x), ¬P(c), R(c)}
{¬R(x), ¬R(x), ¬P(c), R(c)}
13
P(x)
¬R(x)
Q(x)
¬R(x)
¬P(c) R(c)

14. Connection method (2/3)
• A connection in a matrix is an unordered pair of occurrences of
complementary literals, i.e. {P(s1,...,sn),¬P(s1,...,sn)}. A path is said
to be complementary iff it contains a connection.
{P(x), Q(x), ¬P(c), R(c)}
{¬R(x), Q(x), ¬P(c), R(c)}
{P(x), ¬R(x), ¬P(c), R(c)}
{¬R(x), ¬R(x), ¬P(c), R(c)}
• A unification σ is a mapping from the set of variables to the set of
terms. In σ (L), all occurrences of a variable contained in the literal L
are substituted according to their mapping in σ.
σ (x) = c
14

15. Connection method (3/3)
• A formula is valid iff every path through its matrix
representation is complementary.
{P(x), Q(x), ¬P(c), R(c)}
{¬R(x), Q(x), ¬P(c), R(c)}
{P(x), ¬R(x), ¬P(c), R(c)}
{¬R(x), ¬R(x), ¬P(c), R(c)}
where σ (x) = c
15

16. Connection calculus (1/2)
16
Start rule (St) is the first rule to be applied in a proof search, then Reduction rule (Red)
and Extension Rule (Ext) are repeatedly applied until every branch reaches Axiom (Ax).
A formula M in disjunctive
normal form is valid in classical
logic iff there is a derivation for
“ε,M,ε” in the connection
calculus so all leaves are
axioms.

17. Connection calculus (2/2)
• Consider the formula M = {{P(x), ¬R(x)}, {Q(x), ¬R(x)},
{¬P(c)}, {R(c)}}. A proof for M in the connection calculus
with σ (x) = c is given below.
17

18. Connection calculus example
18

19. Connection calculus example
19

20. Connection calculus example
20

21. Connection calculus example
21

22. Connection calculus example
22

23. Connection calculus example
23

24. Description Logics (1/2)
• DL alphabet used in this presentation
• Letters A, B or camel case: Atomic concepts
• Letters C, D: Concept descriptions
• Letters a, b or lower case: Individuals
• Letters R,S or camel back case: Roles
• Concept descriptions are constructed as (limited to DL ALC):
C,D ::= A | ⊤ | ⊥ | C⨆D | C⨅D |¬C | ∃R.C | ∀R.C
• A TBox is a set of statements of the form C⊑D or C≡D, they are called general
concept inclusions (GCI).
• An ABox consists of statements of the form C(a) or R(a,b)
24

25. Description Logics (2/2)
ChildlessPerson ≡ Person ⨅ ¬Parent
(ChildlessPersons are exactly those who are persons and who are not parents)
Parent ≡ ∃hasChild.Person
(Parents are exactly those who have at least one child which is a Person)
25

26. DL Semantics (1/2)
• Semantics defined by interpretations
• An interpretation I = (ΔI, ∙I), where
• ΔI is the domain (a non-empty set)
• ∙I is an interpretation function that maps
• Atomic concept A to a set AI ⊆ ΔI
• Atomic role R to a binary relation RI ⊆ ΔI⨉ΔI
• Individual a to an element aI ⊆ ΔI
26

27. DL Semantics (2/2)
• Interpretation of complex concepts and roles follows from
the interpretation of the basic entities.
27
figure: examples of concept semantics
table 2.1

28. Reasoning tasks
• Subsumption checking: A concept C is subsumed by a
concept D (C⊑D) with respect toT if CI⊆DI for every model
I of T. In this case we write T ⊨ C⊑D.
• Classification: Use of subsumption checking to arrange
TBox concepts in a hierarchical way.
28more reasoning tasks can be found at dissertation

29. Conversion to matrix
29

30. Translation
• We call translation the process of transform a description
logic axiom or concept descriptions into a matricial form.
• Given a query KB ⊨ α, all formulas in KB are negated and α
is transformed in its positive form. We can express this
query as a first-order logic formula ¬KB ⋁ α.
30

31. 31
ChildlessPerson(x)
[¬Person(x) Parent(x)]
For instance, the axiom
ChildlessPerson ⊑ Person ⨅ ¬Parent
can be expressed in FOL as
∀x ChildlessPerson(x) ⟶ Person(x) ∧ ¬Parent(x)
if it is in KB, we are interested in its negation:
¬(∀x ChildlessPerson(x) ⟶ Person(x) ∧ ¬Parent(x))
∃x ChildlessPerson(x) ∧ (¬Person(x) ⋁ Parent(x))
and then, in a matricial form:

32. Translation Rules (axioms)
28 3.3. TRANSLATION RULES
Name DL formula FOL mapping
DNF Negated Matrix
mapping
Concept inclusion C v D 8x : C(x) ! D(x)

C(x)
¬D(x)
Concept equality C ⌘ D 8x : C(x) $ D(x)

C(x) ¬C(y)
¬D(x) D(y)
Concept disjointness C v ¬D 8x : C(x) ! ¬D(x)

C(x)
D(x)
Concept assertion C(a) C(a)
⇥
¬C(a)
⇤
Negative concept
assertion
¬C(a) ¬C(a)
⇥
C(a)
⇤
Role inclusion R(a,b) v S(x,y)
8x,y : r(x,y) !
s(x,y)

r(x,y)
¬s(x,y)
Role equality R ⌘ S
8x,y : r(x,y) $
s(x,y)

r(x,y) ¬r(z,w)
¬s(x,y) s(z,w)

33. Translation Rules (axioms)
Concept assertion C(a) C(a)
⇥
¬C(a)
⇤
Negative concept
assertion
¬C(a) ¬C(a)
⇥
C(a)
⇤
Role inclusion R(a,b) v S(x,y)
8x,y : r(x,y) !
s(x,y)

r(x,y)
¬s(x,y)
Role equality R ⌘ S
8x,y : r(x,y) $
s(x,y)

r(x,y) ¬r(z,w)
¬s(x,y) s(z,w)
Role assertion R(a,b) r(a,b)
⇥
¬r(a,b)
⇤
Negative role
assertion
¬R(a,b) ¬r(a,b)
⇥
r(a,b)
⇤
Role domain 9R.> v C 8x,y : r(x,y) ! C(x)

r(x,y)
¬C(x)
Role range > v 8R.C 8x,y : r(x,y) ! C(y)

r(x,y)
¬C(y)
Reflexive role > v 9R.Sel f 8x : r(x,x)
⇥
¬r(x,x)
⇤
Irreflexive role > v ¬9R.Sel f 8x : ¬r(x,x)
⇥
r(x,x)
⇤
Transitive role Trans(R)
8x,y,z : r(x,y)^
r(y,z) ! r(x,z)
2
4
r(x,y)
r(y,z)
¬r(x,z)
3
5
Table 3.1: Tranlation rules for axioms

34. Translation Rules (concept
descriptors)
Name DL formula FOL mapping DNF Negated Matrix
mapping
RHS Intersection A v B1 uB2
8x : A(x) !
B1(x)^B2(x)

A(x) A(x)
¬B1(x) ¬B2(x)
LHS Intersection A1 uA2 v B
8x : A1(x)^A2(x) !
B(x)
2
4
A1(x)
A2(x)
¬B(x)
3
5
RHS local
reflexivity
A v 9R.Sel f 8x : A(x) ! r(x,x)

A(x)
¬r(x,x)
LHS local
reflexivity
9R.Sel f v A 8x : r(x,x) ! A(x)

r(x,x)
¬A(x)
RHS existential
restriction
A v 9R.B
8x : A(x) ! (9y :
r(x,y)^B(y))

A(x) A(x)
¬r(x, f(x)) ¬B(f(x))
LHS existential
restriction
9R.A v B
8x : (9y :
r(x,y)^A(y)) ! B(x)
2
4
A(x)
r(x,y)
3
5

35. Translation Rules (concept
descriptors)
mapping
RHS Intersection A v B1 uB2
8x : A(x) !
B1(x)^B2(x)

A(x) A(x)
¬B1(x) ¬B2(x)
LHS Intersection A1 uA2 v B
8x : A1(x)^A2(x) !
B(x)
2
4
A1(x)
A2(x)
¬B(x)
3
5
RHS local
reflexivity
A v 9R.Sel f 8x : A(x) ! r(x,x)

A(x)
¬r(x,x)
LHS local
reflexivity
9R.Sel f v A 8x : r(x,x) ! A(x)

r(x,x)
¬A(x)
RHS existential
restriction
A v 9R.B
8x : A(x) ! (9y :
r(x,y)^B(y))

A(x) A(x)
¬r(x, f(x)) ¬B(f(x))
LHS existential
restriction
9R.A v B
8x : (9y :
r(x,y)^A(y)) ! B(x)
2
4
A(x)
r(x,y)
¬B(y)
3
5
RHS universal
restriction
A v 8R.B
8x : A(x) ! (8y :
r(x,y) ! B(y))
2
4
A(x)
r(x,y)
¬B(y)
3
5
LHS universal
restriction
8R.A v B
8x : (8y : r(x,y) !
A(y)) ! B(x)

¬r(x, f(x)) A(f(x))
¬B(x) ¬B(x)
Table 3.2: Tranlation rules for concept descriptors

36. Normalization (1/2)
• Some axioms can’t be direct mapped into clausal form using
the translation rules
• Normalization is the process of transforming a non-clausal
matrix into a clausal matrix
34

37. for instance, the axiom A⊑B⨆¬C⨆(¬D⨅E)⨆(F⨅¬G⨅H) can
generate the non-clausal matrix
A
¬B
C
[D ¬E]
[¬F G ¬H]
A
¬B
C
N1
N2
There are several approaches to normalize this matrix,
¬N1
D
¬N1
¬E
¬N2
¬F
¬N2
G
¬N2
¬H
A
¬B
C
D
¬F
A
¬B
C
¬E
G
A
¬B
C
D
¬H
A
¬B
C
¬E
¬F
A
¬B
C
D
G
A
¬B
C
¬E
¬H
35

38. leanCoR
36

39. Architecture
37

40. Command-line interface
38
Parameters are given by OWL Reasoner Evaluation
Workshop (ORE) command-line interface

41. Parser
• Takes an ontology and builds a data structure representing
the knowledge base
• Can be reused in other systems written in Prolog
• Can be extended to include new axioms
• Supports ontologies written in OWL 2 Functional-Style
syntax
39

42. Exemplo de ontologia
40
Prefix(:=<http://www.semanticweb.org/amelo/ontologies/2014/7/untitled-ontology-10#>)
Prefix(owl:=<http://www.w3.org/2002/07/owl#>)
Prefix(rdf:=<http://www.w3.org/1999/02/22-rdf-syntax-ns#>)
Prefix(xml:=<http://www.w3.org/XML/1998/namespace>)
Prefix(xsd:=<http://www.w3.org/2001/XMLSchema#>)
Prefix(rdfs:=<http://www.w3.org/2000/01/rdf-schema#>)
!
!
Ontology(<http://www.semanticweb.org/amelo/ontologies/2014/7/untitled-ontology-10>
!
Declaration(Class(:Parent))
Declaration(Class(:Person))
Declaration(ObjectProperty(:hasChild))
EquivalentClasses(:Parent ObjectSomeValuesFrom(:hasChild :Person))
)

43. Query resolver
• Assemble queries in the format KB ⊨ α to the theorem
prover, where α is a new axiom.
41
For instance, suppose the following KB:
Person⊑Man⨆Woman
Man⊑¬Woman
We want classify this KB. The new concept inclusions can be
obtained by trying to proof KB ⊨ α, with α ∈ {Man⊑Person,
Woman⊑Person, Man⊑Woman, Woman⊑Man, Person⊑Man,
Person⊑Woman}

44. Modified leanCoP
• We optimized the use of knowledge base database to reuse
all axioms in different queries;
• Start clause C is chosen from α, in KB ⊨ α, and the method
does not backtrack for the start clause.
42

45. Normalization
• Module responsible for guarantees that a formula is in
clausal form;
• We are using the second approach shown in slide 25.
43

46. Skolemization
• This module is responsible for adding skolem functions to
formulas.
• A function is generated for each skolem function and takes
the form fn(x), where n is an integer.
• For instance, the axiom A ⊑ ∃R.B ⨅ ∃S.C is mapped to:
44
A(x)
¬r(x, f1(x))
A(x)
¬B(f1(x))
A(x)
¬C(f2(x))
A(x)
¬s(x, f2(x))

47. Performance
• In order to compare our system to a state-of-the-art reasoner and evaluate
our results, we selected three ontologies to classify and compete against
ELK, a consequence-based reasoner winner of the ORE 2013 competition.
!
!
!
!
!
!
• The result of the benchmark shows that a naïve implementation without
optimizations can’t compete with a modern reasoner;
• New axioms derived from the classification are correct when them only need
axioms included in DL ALC.
45

48. Conclusions
46

49. Contributions
• Implementation of a DL reasoner based on connection
method that can be easily extended and modified for
experimenting reasoning over description logics.
• Normalization procedures independent of concept inclusion
side.
• Translation rules for all axioms of OWL 2 (DL SROIQ).
47

50. Open Work
• Finish the development of leanCoR:
• Execution of all reasoning tasks supported by modern
reasoners;
• Support more expressive DL languages, such as
SROIQ.
48

51. Future Work
• Incorporate all optimizations of the state-of-the-art
reasoners to truly compare the connection calculus
performance to other proof procedures.
• Develop implementations in languages with more features
for concurrency such as erlang and go.
49

52. leanCoR: lean connection-based
DL Reasoner
By
Adriano S. T. de Melo
!
Supervisor
Frederico L. G. Freitas