leanCoR: lean Connection-based DL Reasoner

leanCoR: lean connection-based DL Reasoner

By Adriano S. T. de Melo

!Supervisor

Frederico L. G. Freitas

Outline

Introduction

Background

Conversion to matrix

leanCoR

Conclusions

2

Introduction

3

–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

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

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

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

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

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

background

10

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

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

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)

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

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

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.

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

Connection calculus example

18

Connection calculus example

19

Connection calculus example

20

Connection calculus example

21

Connection calculus example

22

Connection calculus example

23

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

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

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 R I ⊆ ΔI⨉ΔI

• Individual a to an element aI ⊆ ΔI

26

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

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

Conversion to matrix

29

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

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:

Translation Rules (axioms)28 3.3. TRANSLATION RULES

Name DL formula FOL mapping DNF Negated Matrixmapping

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 conceptassertion

¬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 roleassertion

¬R(a,b) ¬r(a,b)⇥r(a,b)

⇤

Role domain 9R.>vC 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

4r(x,y)r(y,z)¬r(x,z)

3

5

Table 3.1: Tranlation rules for axioms

Translation Rules (axioms)

28 3.3. TRANSLATION RULES

Name DL formula FOL mapping DNF Negated Matrixmapping

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 conceptassertion

¬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 roleassertion

¬R(a,b) ¬r(a,b)⇥r(a,b)

⇤

Role domain 9R.>vC 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

4r(x,y)r(y,z)¬r(x,z)

3

5

Table 3.1: Tranlation rules for axioms

Translation Rules (concept descriptors)

29 3.3. TRANSLATION RULES

Name DL formula FOL mapping DNF Negated Matrixmapping

RHS Intersection A v B1 uB28x : 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

4A1(x)A2(x)¬B(x)

3

5

RHS localreflexivity A v 9R.Sel f 8x : A(x)! r(x,x)

A(x)

¬r(x,x)

�

LHS localreflexivity 9R.Sel f v A 8x : r(x,x)! A(x)

r(x,x)¬A(x)

�

RHS existentialrestriction

A v 9R.B8x : A(x)! (9y :

r(x,y)^B(y))

A(x) A(x)

¬r(x, f (x)) ¬B( f (x))

�

LHS existentialrestriction

9R.A v B8x : (9y :

r(x,y)^A(y))! B(x)

2

4A(x)

r(x,y)¬B(y)

3

5

RHS universalrestriction

A v 8R.B8x : A(x)! (8y :r(x,y)! B(y))

2

4A(x)

r(x,y)¬B(y)

3

5

LHS universalrestriction

8R.A v B8x : (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

Translation Rules (concept descriptors)

29 3.3. TRANSLATION RULES

Name DL formula FOL mapping DNF Negated Matrixmapping

RHS Intersection A v B1 uB28x : 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

4A1(x)A2(x)¬B(x)

3

5

RHS localreflexivity A v 9R.Sel f 8x : A(x)! r(x,x)

A(x)

¬r(x,x)

�

LHS localreflexivity 9R.Sel f v A 8x : r(x,x)! A(x)

r(x,x)¬A(x)

�

RHS existentialrestriction

A v 9R.B8x : A(x)! (9y :

r(x,y)^B(y))

A(x) A(x)

¬r(x, f (x)) ¬B( f (x))

�

LHS existentialrestriction

9R.A v B8x : (9y :

r(x,y)^A(y))! B(x)

2

4A(x)

r(x,y)¬B(y)

3

5

RHS universalrestriction

A v 8R.B8x : A(x)! (8y :r(x,y)! B(y))

2

4A(x)

r(x,y)¬B(y)

3

5

LHS universalrestriction

8R.A v B8x : (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

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

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

leanCoR

36

Architecture

37

Command-line interface

38

Parameters are given by OWL Reasoner Evaluation Workshop (ORE) command-line interface

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

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)) )

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}

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

Normalization

• Module responsible for guarantees that a formula is in clausal form;

• We are using the second approach shown in slide 25.

43

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))

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

Conclusions

46

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

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

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

leanCoR: lean connection-based DL Reasoner

By Adriano S. T. de Melo

!Supervisor

Frederico L. G. Freitas