Logical Aspects of Artificial Intelligence Introduction to - - PowerPoint PPT Presentation

Logical Aspects of Artificial Intelligence Introduction to Description Logics

St´ ephane Demri demri@lsv.fr December 9th, 2019

What is in this part of the course?

Introduction to Description Logics and Temporal Logics for Multi-Agent Aystems

◮ Today: Introduction to description logics. ◮ 16/12/2019: Tableaux calculi and complexity. ◮ 06/01/2020: Introduction to temporal logics for multi-agent

systems.

◮ 13/01/2020: 14h00–16h00 Exam on this part of the course

(slides allowed). but also first-order logic, modal logics, knowledge representation, etc... 2

What can you expect to learn?

◮ Basics of description logic including ALC as well as ATL

and variants.

◮ Tableaux for ALC, model-checking techniques for ATL-like

logics.

◮ Complexity, decidability, expressive power results for logics

dedicated to knowledge representation. 3

Background

• 1. Necessary background

◮ Basics of first-order logic. ◮ Basics of complexity theory.

• 2. Optional background

◮ Basics of modal logics, temporal logics ◮ Sequent-style proof systems. ◮ Basics of model-checking.

4

Course material

◮ Slides and exercises available on

https://wikimpri.dptinfo.ens-cachan.fr/doku. php?id=cours:c-1-39 http://www.lsv.fr/˜demri/notes-de-cours.html

◮ Slides available after each lecture. ◮ Do not hesitate to contact me (demri@lsv.fr).

5

Material mainly based on the following documents

◮ F

. Baader, I. Horrocks, C. Lutz and U. Sattler. An introduction to Description Logic. Cambridge University Press, 2017.

◮ Ivan Varzinczak’s slides (ESSLLI’18) ◮ Ulrike Sattler & Thomas Schneider’s slides (ESSLLI’15). ◮ S. Demri, V. Goranko, M. Lange.

Temporal Logics in Computer Science. Cambridge University Press, 2016. 6

Other (online) ressources

◮ Description Logic Complexity Navigator by Evgeny Zolin.

http://www.cs.man.ac.uk/˜ezolin/dl/

◮ Proceedings of the Description Logic Workshops

http://dl.kr.org/workshops/

◮ See also the proceedings of the international conferences:

◮ Int. Joint Conference on Artificial Intelligence.

(IJCAI)

◮ European Conference on Artificial Intelligence.

(ECAI)

◮ Int. Conference on Principles of Knowledge Representation

and Reasoning. (KR)

7

Plan of the lecture

◮ Knowledge representation. ◮ Basic description logic ALC. ◮ Several extensions of ALC. ◮ Relationships with first-order logic and modal logics. ◮ Exercises session.

8

Knowledge representation

9

DLs: where they come from

◮ First-order logic is not always the most natural language.

∀x ∃ y ∀z ((P(x, y) ∧ Q(y, z) ⇒ (¬Q(a, y) ∨ P(x, z))))

∀ x (Teacher(x) ⇔ Person(x)∧∃ y (Teaches(x, y)∧Course(y)))

◮ How to design user-friendly languages for knowledge

representation ?

◮ Concept definition from Description Logics.

Teacher ≡ Person ⊓ ∃Teaches.Course 10

Reasoning about . . .

◮ Knowledge

Epistemic Logics

◮ Rules and obligations

Deontic Logics

◮ Programs

Hoare Logics

◮ Time

Temporal Logics but also many-valued logics, non-monotonic logics, team logics, separation logics, etc. 11

Ontologies

◮ Formal specification of some domain with concepts,

• bjects, relationships between concepts, objects, etc.

◮ Backbone of ontologies includes:

◮ taxonomy (classification of objects), ◮ axioms (to constrain the models of the defined terms).

◮ Classification of medical terms: diseases, body parts,

drugs, etc.

◮ Well-known ontologies:

◮ Medical ontology SNOMED-CT formalised with description

logic EL + +.

◮ NCI Thesaurus (National Cancer Institute, USA). ◮ Gene ontology (world largest source of information on the

functions of genes).

◮ Free ontology editor Prot´

eg´ e http://protege.stanford.edu/ 12

The classical student ontology

◮ Natural language specification:

◮ Employed students are students and employees. ◮ Students are not taxpayers. ◮ Employed students are taxpayers. ◮ Employed students who are parents are not taxpayers. ◮ To work for is to be employed by. ◮ John is an employed student, John works for IBM.

◮ Classes/relations/individuals.

13

Main ingredients in formal ontologies

◮ Model of the world with classes within a domain,

relationships between classes and instantiations of classes.

◮ Formal: abstract model of some domain with

(mathematical) semantics and reasoning tasks.

◮ Classes or concepts: classes of objects with the domain of

• interest. (Employed student, Parent, Course)

◮ Relations or roles: relationships between concepts.

(being employed by, sibling-of)

◮ Instances of classes and relations.

◮ John is an employed student. ◮ Mary works for IBM.

14

Early KR formalisms

◮ Graphical formalisms easier to grasp and supposedly close

to how knowledge is represented by human beings.

◮ Large variety of semantical networks. ◮ Often, lack of formal semantics (see tentatives with the

knowledge representation system KL-ONE). 15

Why Description Logics?

◮ Formal languages for concepts, relations and instances. ◮ DLs have all one needs to formalise ontologies. ◮ Computational properties.

◮ Acceptable trade-off between expressivity and complexity. ◮ Decidability and often tractability. ◮ Implementation in tools of the main reasoning tasks.

◮ A remarkable suite of languages and tools. See e.g.

◮ OWL: Web Ontology Language. ◮ Prot´

eg´ e: ontology editor.

◮ FaCT++: DL reasoner supporting OWL DL.

16

Description Logics and Knowledge representation

◮ Description is a subfield of Knowledge Representation,

itself a subfield of Artificial Intelligence.

◮ Description Logic(s):

◮ a research field, ◮ a family of knowledge representation languages, ◮ a member of the family.

◮ Well-defined syntax with formal semantics, decision

problems, algorithms, etc. 17

Basic description logic ALC

18

DLs: the core

◮ Concept language.

Person ⊓ ∃Teaches.Course

◮ Syntactic ingredients of the concept language:

◮ Concept names for sets of elements, e.g. Person. ◮ Role names interpreted by binary relations between

• bjects, e.g. EmployedBy.

◮ Concept constructors to build complex concepts, e.g. ¬,

⊓, ⊔, ∃.

◮ Basic terminology stored in a TBox. ◮ Facts about individuals stored in an ABox.

19

Basic elements of the language

◮ Concept names.

NC

def

= {A1, A2, . . .} Examples: Parent, Sister, Student

◮ Role names.

NR

def

= {r1, r2, . . .} Examples: EmployedBy, MotherOf

◮ Individual names.

NI

def

= {a1, a2, . . .} Examples: Mary, Alice, John 20

Boolean constructors & role restrictions

◮ Boolean constructors.

◮ Concept negation ¬

(class complement)

◮ Concept conjunction ⊓

(class intersection)

◮ Concept disjunction ⊔

(class union)

◮ Role restrictions.

◮ Existential restriction ∃

(at least one related individual)

◮ Value restriction ∀

(all related individuals)

◮ Many more constructors exist, see forthcoming ALC

extensions.

◮ For modal logicians,

¬, ⊓, ⊔, ∃, ∀ ∼ ¬, ∧, ∨, ✸, ✷ 21

Complex concepts in ALC

◮ ALC: Attributive Concept Language with Complements. ◮ Complex concepts.

C ::= ⊤ | ⊥ | A | ¬C | C ⊓ C | C ⊔ C | ∃r.C | ∀r.C, where A ∈ NC and r ∈ NR.

◮ Examples of complex concepts:

◮ Student ⊓ ¬∃Pays.Tax ◮ ∃MotherOf.(∃MotherOf.A)

◮ Syntax errors in

Student ⊔ ∀¬Tax ∀∃MotherOf.Mary

◮ C ⇒ D

def

= ¬C ⊔ D. 22

Interpretation

◮ Concept/role/individual ∼ unary predicate/binary

predicate/constant.

◮ Interpretation I

def

= (∆I, ·I)

◮ ∆I: non-empty set (the domain). ◮ ·I: interpretation function such that

AI ⊆ ∆I r I ⊆ ∆I × ∆I aI ∈ ∆I

◮ A priori, ∆I is arbitrary and I can be viewed as a

first-order model for unary and binary predicate symbols and constants. 23

Semantics for complex concepts

⊤I

def

= ∆I ⊥I

def

= ∅ (¬C)I

def

= ∆I \ CI (C1 ⊔ C2)I

def

= CI

1 ∪ CI 2

(C1 ⊓ C2)I

def

= CI

1 ∩ CI 2

(∃r.C)I

def

= {a ∈ ∆I | r I(a) ∩ CI = ∅} (∀r.C)I

def

= {a ∈ ∆I | r I(a) ⊆ CI} (R(a)

def

= {b | (a, b) ∈ R})

◮ In modal logic lingua, a ∈ CI corresponds to I, a |

= C. 24

Graphical representation

a Person Teacher C1 Course C2 Course b Course Person Teaches Teaches Teaches

◮ ∆I = {a, b, C1, C2}. ◮ TeachesI = {(a, C1), (a, C2), (b, b)}. ◮ PersonI = {a, b}, CourseI = {C1, C2, b}. ◮ a ∈ (∀Teaches.Course)I.

25

Concept satisfiability problem

◮ Concept satisfiability problem:

Input: A (complex) concept C in ALC. Question: Is there an interpretation I = (∆I, ·I) such that CI = ∅?

◮ This corresponds to the standard formulation for the

satisfiability problem (in modal logics, temporal logics, etc.).

◮ The concept satisfiability problem for ALC is

PSPACE-complete.

◮ ALC has the finite interpretation property: every satisfiable

concept has an interpretation with a finite domain. 26

Statements

◮ Concept inclusion.

Teachers are persons. Employed students are employees.

◮ Concept and role membership.

Mary is a student. Alice is a teacher. Laura teaches the course “Automata Theory”.

◮ Statements are not concepts and express properties of

concepts, roles and individuals. 27

General concept inclusion (GCI)

◮ Expressions of the form

C ⊑ D

are called general concept inclusion.

◮ Intuitive meaning:

◮ D subsumes C. ◮ C is more specific than D.

◮ Example: Employee ⊑ ∃WorksFor.⊤. ◮ Satisfaction relation: I |

= C ⊑ D

def

⇔ CI ⊆ DI.

◮ C ⊑ D understood as a global statement about I.

28

Concept equivalence

◮ C ⊑ D and D ⊑ C abbreviated by

C ≡ D

called concept equivalence.

◮ Satisfaction relation: I |

= C ≡ D

def

⇔ CI = DI.

◮ Concept definition (A ∈ NC is a concept name)

A ≡ C

◮ ⊤ ≡ (¬Student ⊔ Student).

29

Subsumption problem

◮ Subsumption problem:

Input: A GCI C ⊑ D with C, D ∈ ALC. Question: Is it the case that for all interpretations I, we have I | = C ⊑ D?

◮ C ⊑ D is “not valid” iff C ⊓ ¬D is satisfiable. ◮ As coPSPACE =PSPACE, the subsumption problem for

ALC is PSPACE-complete too. 30

Assertions

◮ Concept assertion: stating that an individual is an

instance of a concept.

a : C

◮ Satisfaction relation: I |

= a : C

def

⇔ aI ∈ CI.

◮ Role assertion: stating that two individuals are in a

relation.

(a, b) : r

◮ Satisfaction relation: I |

= (a, b) : r

def

⇔ (aI, bI) ∈ r I.

◮ Examples:

◮ Alice : Student ⊓ ¬∃Pays.Tax. ◮ (Laura, CNRS) : WorksFor.

31

The validity problem

◮ Validity problem:

Input: A statement α in ALC. Question: Is the case that for all interpretations I, we have I | = α?

◮ Validity of α is written |

= α.

◮ Validity of ⊤ ⊑ C corresponds to the usual notion of validity

for C, i.e. for all interpretations I = (∆I, ·I), we have CI = ∆I.

◮ Examples of valid statements:

◮ |

= ∀r.C ⊓ D ⊑ ∀r.C.

◮ |

= a : C ⊔ ¬C.

◮ |

= ⊤ ⊑ (¬(C ⊓ D) ⊔ (C ⊔ D)).

◮ The validity problem for ALC is PSPACE-complete.

32

What is a knowledge base (a.k.a. ontology) ?

◮ Terminological Box (TBox) T : finite collection of GCIs.

◮ I.e., a finite set of subsumption statements. ◮ This provides definitions of concepts (a terminology).

◮ Assertional Box (ABox) A: finite collection of assertions.

◮ I.e., a finite set of concept and role assertions. ◮ This provides a partial view on the interpretations and can

be understood as a finite database.

◮ Knowledge base K is a pair (T , A). ◮ Knowledge bases are also called ontologies.

33

A knowledge base K⋆

◮ TBox T :

Course ⊑ ¬Person Teacher ⊑ Person ⊓ ∃Teaches.Course ∃Teaches.⊤ ⊑ Person Student ⊑ Person ⊓ ∃Attends.Course ∃Attends.⊤ ⊑ Person

◮ ABox A:

Mary : Person CS600 : Course Alice : Person ⊓ Teacher (Alice, CS600) : Teaches (Mary, CS600) : Attends 34

Consequences from knowledge bases

◮ Interpretation I = (∆I, ·I), knowledge base K = (T , A).

◮ I |

= A

def

⇔ for all α ∈ A, we have I | = α.

◮ I |

= T

def

⇔ for all α ∈ T , we have I | = α.

◮ I |

= K

def

⇔ I | = A and I | = T .

◮ K |

= α

def

⇔ for all interpretations I such that I | = K, we have I | = α.

◮ K⋆ |

= CS600 : ¬Person and K⋆ | = Alice : Teacher. 35

Decision problems relatively to a knowlegde base

◮ K is consistent

def

⇔ there is some I such that I | = K.

◮ C is satisfiable with respect to K

def

⇔ there is I such that I | = K and CI = ∅.

◮ C is subsumed by D with respect to K

def

⇔ K | = C ⊑ D.

◮ C and D are equivalent with respect to K

def

⇔ T | = C ≡ D.

◮ a is an instance of C with respect to K

def

⇔ K | = a : C.

◮ K |

= C ⊑ D also written C ⊑K D. K | = C ≡ D also written C ≡K D. 36

Relationships between reasoning problems

◮ C and D are equivalent w.r.t. K iff C is subsumed by D

w.r.t. K and D is subsumed by C w.r.t. K.

◮ C ⊑K D iff C ⊓ ¬D is not satisfiable w.r.t. K. ◮ C is satisfiable w.r.t. K iff C ⊑K⊥. ◮ C is satisfiable w.r.t. K iff (T , A ∪ {b : C}) is consistent.

(b is fresh)

◮ K |

= a : C iff (T , A ∪ {a : ¬C}) is not consistent. 37

C is satisfiable w.r.t. K iff (T , A ∪ {b : C}) is consistent

◮ Suppose that C is satisfiable w.r.t. K.

◮ There is I such that I |

= K and CI = ∅, say a ∈ CI.

◮ Let I′ be the variant of I such that bI′ def

= a.

◮ As b does not appear in K and C, we have I′ |

= K.

◮ Furthermore, I′ |

= b : C as CI = CI′.

◮ Consequently, I′ |

= (T , A ∪ {b : C}).

◮ Now, suppose that (T , A ∪ {b : C}) is consistent.

◮ There is I such that I |

= T , I | = A and I | = b : C.

◮ Consequently, bI ∈ CI. ◮ So, there is some I such that I |

= K and CI is non-empty.

38

Classification

◮ Deduce implicit knowledge from the explicitly represented

knowledge.

◮ For all A, B in K, check whether A ⊑K B. ◮ For all A in K, check whether A is satisfiable w.r.t. K.

If not for some B, a modelling error is probable.

◮ For all a and C in K, check whether K |

= a : C.

◮ Classifying a knowledge base K.

• 1. Check whether K is consistent, if yes, go 2.
• 2. For each pair A, B of concept names (plus ⊤, ⊥), check

whether K | = A ⊑ B.

• 3. For individual name a and concept C in K, check whether

K | = a : C.

leading to K’s inferred class hierarchy. 39

Complexity results for ALC

◮ Concept satisfiability and subsumption problems are

PSPACE-complete. (no knowledge base involved)

◮ Knowlegde base consistency problem is

EXPTIME-complete. NP ⊆ PSPACE ⊆ EXPTIME ⊂ 2EXPTIME ⊂ N2EXPTIME

◮ Recall that C ⊑K D iff (T , A ∪ {b : C ⊓ ¬D}) is not

consistent. 40

Several extensions of ALC

41

Extensions: a feature of DLs

◮ Concepts/assertions in ALC have a limited expressive

power.

◮ How to express simple arithmetical constraints such as

“Alice teaches at least three courses”?

◮ How to enforce constraints between roles?

For instance, r I = (sI)−1 or r I ⊆ sI.

◮ The expressive power of ALC concepts can be

characterised precisely, thanks to the notion of bisimulation (not presented today).

Trade-off between the expressive power and the

computational properties of the extensions.

◮ In the other direction: study of ALC fragments to reduce

the complexity while preserving the expression of interesting properties, see e.g. EL or FL0. 42

Inverse roles

Course ⊑ ¬Person Teacher ⊑ Person ⊓ ∃Teaches.Course ∃Teaches.⊤ ⊑ Person Student ⊑ Person ⊓ ∃Attends.Course ∃Attends.⊤ ⊑ Person Professor ⊑ Teacher Course ⊑ ∀TaughtBy.¬Professor

◮ Extending NR with inverse roles:

NR ∪ {r − | r ∈ NR}

◮ Given I

def

= (∆I, ·I), (r −)I def = (r I)−1 where R−1 def = {(b, a) | (a, b) ∈ R} 43

Elimination of the role name TaughtBy

◮ Back to the previous example.

Professor ⊑ Teacher Course ⊑ ∀Teaches−.¬Professor

◮ Given a logic L, LI is defined as L except that inverse

roles are added.

◮ Concept satisfiability for ALCI remains PSPACE-complete

and knowledge consistency remains EXPTIME-complete. 44

Number restrictions

◮ How to express in ALC that a student attends to at least

three courses? Student ⊑ ∃Attends.Course ⊓ A ∃Attends.Course ⊓ ¬A ⊓ B ∃Attends.Course ⊓ ¬A ⊓ ¬B

◮ Why isn’t it satisfactory ? ◮ How to express in ALC that a student attends to at most

10 courses?

◮ There is no concept C in ALC such that for all

interpretations I, for all a ∈ ∆I, a ∈ CI iff card({b | (a, b) ∈ AttendsI}) ≥ 3 45

(Unqualified) number restriction

◮ Extending the concepts with number restrictions (≤ n r)

and (≥ m r).

◮ Given I

def

= (∆I, ·I), (≤ n r)I def = {a ∈ ∆I | card({b | (a, b) ∈ r I}) ≤ n} (≥ m r)I def = {a ∈ ∆I | card({b | (a, b) ∈ r I}) ≥ m}

◮ Given a logic L, LN is defined as L except that

(unqualified) number restrictions are added.

◮ In ALCN, (≥ 3 Attends) ⊓ (≤ 10 Attends) does the job.

46

Qualified number restriction

◮ Generalising the number restrictions. ◮ Qualified number restrictions: (≤ n r · C), (≥ m r · C). ◮ Given I

def

= (∆I, ·I), (≤ n r·C)I def = {a ∈ ∆I | card({b | (a, b) ∈ r I and b ∈ CI}) ≤ n} (≥ m r·C)I def = {a ∈ ∆I | card({b | (a, b) ∈ r I and b ∈ CI}) ≥ m}

◮ (∼ n r) = (∼ n r · ⊤). ◮ Given a logic L, LQ is defined as L except that qualified

number restrictions are added.

◮ Concept satisfiability for ALCIQ is PSPACE-complete and

knowledge base consistency is EXPTIME-complete. 47

Naming individuals in complex concepts

◮ Concepts in ALC

C ::= ⊤ | ⊥ | A | ¬C | C ⊓ C | C ⊔ C | ∃r.C | ∀r.C

◮ . . . but individual names a (as in the concept assertion

a : C) are not concepts in ALC.

◮ How to express the concept “the courses taught by Alice” ?

Course ⊓ ∃Teaches−.Mary

◮ Nominals in hybrid (modal) logics: propositional variables

true in only one world of the domain.

◮ Nominals in DLs: individual names inside concept

descriptions, written {a}, where a ∈ NI with {a}I def = {aI}.

Syntactic trick {·}: Course ⊓ ∃Teaches−.{Mary}.

48

Nominals in DLs

◮ Given a logic L, LO is defined as L except that nominals

are added.

◮ Concept satisfiability for ALCOQ is PSPACE-complete and

knowledge base consistency is EXPTIME-complete.

◮ . . . but concept satisfiability for ALCOI is

EXPTIME-complete. 49

Role hierarchies

◮ ALC is not able to express complex role constraints such

that a relation is included in another relation.

◮ Typically, the interpretation of Attends should include the

interpretation of AttendsActively.

◮ Role inclusion axiom (RIA) of the form r ⊑ s with

I | = r ⊑ s

def

⇔ r I ⊆ sI

◮ Given a logic L, LH is defined as L except that role

inclusion axioms are added (in the TBox).

◮ Concept satisfiability for ALCH is PSPACE-complete and

knowledge base consistency is EXPTIME-complete. 50

Role value maps are concepts!

◮ A role value map is an atomic concept of the form r ⊑ s:

(r ⊑ s)I def = {a ∈ ∆I | r I(a) ⊆ sI(a)}

Role value maps are local variants of role inclusion axioms

(RIAs).

◮ The RIA r ⊑ s can be encoded by the GCI ⊤ ⊑ (r ⊑ s).

51

Transitive roles

◮ Many natural relations are transitive (AncestorOf,

HasPart, etc.) but this cannot be expressed in ALCH.

◮ Transitivity axioms are of the form Trans(r):

I | = Trans(r)

def

⇔ r I ◦ r I ⊆ r I

◮ Given L, its extension with transitivity axioms in TBoxes is

• btained by replacing ALC by S (new naming rule).

◮ Concept satisfiability for S is PSPACE-complete and

EXPTIME-complete for knowledge base consistency.

◮ Other properties are included in knowledge bases such as

reflexivity, irreflexivity, symmetry, functionality, . . . 52

A selection of complexity results

53

A queen logic SROIQ

◮ SROIQ agrees with the ontology language OWL 2 DL. ◮ SROIQ contains more than one might think from its

name.

◮ Its knowledge bases contain a role box (RBox) to specify

constraints about the interpretation of the role expressions.

◮ The set of roles R is made of role names r, its converses

r − and the universal role U with UI def = ∆I × ∆I.

◮ New atomic concept ∃R.Self with

(∃R.Self)I def = {a | (a, a) ∈ RI}

◮ Role assertions (a, b) : ¬R are allowed in the ABox. ◮ + ingredients the name (nominals, qualified number

restrictions, inverse). 54

Role axioms in the RBox

◮ Complex role inclusion axioms (CRIA) R1 ◦· · ·◦Rn ⊑ R.

A regularity constraint is required on the set of CRIAs

(unspecified here).

◮ Role axioms specifying disjointness, transitivity, reflexivity,

irreflexivity, symmetry, asymmetry.

◮ The knowledge base consistency problem for SROIQ is

N2EXPTIME-complete. 55

Semantic Web

◮ Semantic web:

◮ A vision of a computer-understandable Web. ◮ Distributed knowledge and data in reusable form. ◮ XML, RDF and OWL are part of the story.

◮ Principles towards a semantic Web of data

◮ Give a name to everything. ◮ Relationships form a graph between the entities. ◮ The names are addresses on the Web. ◮ Provide a formal semantics so that knowledge is encoded

in a machine interpretable way.

56

OWL based on description logics

◮ OWL: Web Ontology Language. ◮ Motivated by semantic web activities: add meaning to web

content by annotating it with terms defined in ontologies.

◮ It is a World Wide Web (W3C) standard. ◮ OWL has an explicit formal semantics. ◮ Supported by tools and infrastructures such as

development environments, reasoners and information systems.

◮ Based on description logics such as SROIQ.

57

DLs and ontology languages

◮ W3C’s OWL 2 is based on SROIQ. ◮ OWL 2 EL based on EL and OWL 2 QL based on DL-Lite. ◮ OWL was based on SHOIN. ◮ An OWL ontology is a mixed set containing TBox axioms

and ABox assertions.

◮ More on complexity/scalability:

◮ OWL (SHOIN) is NEXPTIME-complete. ◮ OWL 2 EL is PTIME-complete.

58

OWL RDF/XML exchange syntax

Person ⊓ ∀Teaches.Course 59

OWL reasoners and Prot´ eg´ e

◮ OWL reasoners: implement decision procedures for

consistency and ontology classification.

◮ Open-source ontology editor Prot´

eg´ e.

◮ Interaction with DL reasoners (FaCT++, Pellet, Racer) via

the OWL API.

◮ Show results about ontology classification. ◮ Helpful to work with toy ontologies.

60

Relationships with other logics

61

Foreign language for modal/classical logicians

◮ Description logics can be understood as fragments of

first-order logics.

◮ Similarly, reasoning tasks for description logics can be

understood as decision problems for modal logics.

◮ This can be made precise and sometimes results for

modal/first-oder logics can be used.

◮ Specificity of DLs: many fragments, many extensions and

numerous original reasoning tasks (non-exhaustive presentation in this course). 62

Description logics and its shared history

◮ Late 80s: description logics developped as logical

formalisms for semantics networks.

◮ In the 1990s: relationships with first-order logic, modal

logics, PDL-like logics, etc. (PDL = Propositional Dynamic Logic)

◮ Logical basis for the Web Ontology Language OWL. ◮ Analogies between ontologies and databases lead also to

relationships with query answering languages. 63

From concepts to first-order formulae

◮ Interpretations I = (∆I, ·I) understood as first-order

models.

◮ Translation of non-logical symbols.

Description logics First-order logic individual name a ∈ NI ≈ constant a concept name A ∈ NC ≈ unary predicate A role name r ∈ NR ≈ binary predicate r

◮ Translation of concepts, assertions and GCIs by

internalising the DL semantics. 64

Example of translation

◮ A small TBox and its small ABox:

∃Attends.⊤ ⊑ Person Teacher ≡ Person ⊓ ∃Teaches.Course Alice : Teacher

◮ Its translation in FOL:

∀ x (∃ y Attends(x, y) ⇒ Person(x)) ∧ ∀ x (Teacher(x) ⇔ (Person(x)∧∃ y(Teaches(x, y)∧Course(y)))) ∧ Teacher(Alice)

◮ Concepts can be understood as first-order formulae with

• ne free variable.

65

Internalisation of ALC semantics

t(A, x)

def

= A(x) t(⊤, x) / t(⊥, x)

def

= ⊤ / ⊥ t(¬C, x)

def

= ¬t(C, x) t(C1 ⊓ C2, x)

def

= t(C1, x) ∧ t(C2, x) t(C1 ⊔ C2, x)

def

= t(C1, x) ∨ t(C2, x) t(∃r.C, x)

def

= ∃ y r(x, y) ∧ t(C, y) t(∀r.C, x)

def

= ∀ y r(x, y) ⇒ t(C, y) where y is a fresh variable.

◮ Given I = (∆I, ·I), a ∈ CI iff I, ρ[x ← a] |

= t(C, x).

◮ Many-one reduction from ALC satisfiability problem to FOL

satisfiability. 66

Translating KBs

t(C ⊑ D)

def

= ∀ x t(C, x) ⇒ t(D, x) t(C ≡ D)

def

= ∀ x t(C, x) ⇔ t(D, x) t(a : C)

def

= t(C, x)[a/x] t((a, b) : r)

def

= r(a, b) where ϕ[a/x] (also written ϕ(a)) is the formula obtained from ϕ

• btained by replacing the free occurrences of x by a.

◮ t(K) is defined as the conjunction below:

• α∈T

t(α) ∧

• α∈A

t(α)

◮ Given I = (∆I, ·I), I |

= K iff I | = t(K). 67

Locating DLs within first-order logic fragments

◮ The definition of t(C, x) can be optimised to recycle

variables and to use only two variables x0 and x1. t(∃r.C, xi)

def

= ∃ x1−i r(xi, x1−i) ∧ t(C, x1−i) t(∀r.C, xi)

def

= ∀ x1−i r(xi, x1−i) ⇒ t(C, x1−i)

◮ FO2 (FOL restricted to two individual variables)

satisfiability is NEXPTIME-complete.

◮ Actually, recycling of variables leads to the guarded

fragment restricted to two variables GF2, whose satisfiability is EXPTIME-complete.

◮ Which additional DL features can be translated into FOL?

into a decidable fragment of FOL? 68

More translations into FOL

t(∃r −.C, x)

def

= ∃ y r(y, x) ∧ t(C, y) t({a}, x)

def

= a = x t((≥ n r · C), x)

def

= ∃≥n y r(x, y) ∧ t(C, y) t(∃r.Self, x)

def

= r(x, x) t(r ⊑ s)

def

= ∀ x, y r(x, y) ⇒ s(x, y) t(Trans(r))

def

= ∀x1, x2, x3 r(x1, x2) ∧ r(x2, x3) ⇒ r(x1, x3) t(r1 ◦ · · · ◦ rn ⊑ r)

def

= ∀x1, · · · , xn+1 r1(x1, x2) ∧ · · · ∧ rn(xn, xn+1) ⇒ r(x1, xn+1) ∃≥k x ϕ(x)

def

= ∃ x1, . . . , xk (

• i=j

xi = xj) ∧

• i

ϕ(xi) 69

Modal logics in a nutshell

◮ Formulae: ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | ✸ϕ | ✷ϕ. ◮ Kripke-style structures M = (W, R, V):

◮ W: non-empty set of worlds. ◮ R ⊆ W × W: accessibility relation. ◮ V : PROP → P(W): valuation.

w q p, q p q | = ✸✸p ∧ ✸✸¬p ∧ ✷¬p

◮ Satisfaction relation:

◮ M, w |

= p

def

⇔ w ∈ V(p).

◮ M, w |

= ✸ϕ

def

⇔ there is w′ s.t. (w, w′) ∈ R and M, w′ | = ϕ.

◮ M, w |

= ✷ϕ

def

⇔ for all w′ s.t. (w, w′) ∈ R, M, w′ | = ϕ.

70

Ubiquity of modal logics

◮ Satisfiability problem: given a formula ϕ, are there M, w

such that M, w | = ϕ?

◮ Plethora of modal logics depending on the frame

conditions:

◮ Modal logic S5: R is an equivalence relation (or

R = W × W).

◮ Modal logic K: R is arbitrary (or (W, R) is a finite tree). ◮ Modal logic S4: R is reflexive and transitive.

◮ Epistemic/temporal logics can be viewed as modal logics

with

◮ specific frame conditions (e.g., (W, R) is a tree), ◮ multiple modalities (e.g., [r] for r ∈ NR), ◮ modalities of arity > 1 (e.g., the until operator U).

ϕUψ, ϕ ϕ ϕ ψ . . .

71

Adding the universality modality and nominals

ϕ ::= A | a | ¬ϕ | ϕ ∧ ϕ | rϕ | [r]ϕ | [U]ϕ | Uϕ

◮ Models of the form M = (W, (Rr)r∈NR, V) with associated

modalities r and [r].

◮ University modality [U] such that M, w |

= [U]ϕ

def

⇔ for all w′, we have M, w′ | = ϕ.

◮ Propositional variables are now denoted by A, B to prepare

the encoding.

◮ Nominals in hybrid modal logics are propositional variables

holding for a unique world, here represented by a, b.

◮ Satisfiability problem for this logic is EXPTIME-complete.

72

Internalisation of the semantics

t(A)

def

= A t(⊤) / t(⊥)

def

= ⊤ / ⊥ t(¬C)

def

= ¬t(C) t(C1 ⊓ C2)

def

= t(C1) ∧ t(C2) t(C1 ⊔ C2)

def

= t(C1) ∨ t(C2) t(∃r.C)

def

= rt(C) t(∀r.C)

def

= [r]t(C)

◮ I = (∆I, ·I) → MI = (W, (Rr)r∈NR, V)

◮ W def

= ∆I and Rr = r I for all r.

◮ V(A) def

= AI for all A and V(a)

def

= {aI} for all a.

◮ Given I = (∆I, ·I), a ∈ CI iff MI, a |

= t(C).

◮ Many-one reduction from ALC satisfiability problem to

modal satisfiability. 73

Translating KBs

t(C ⊑ D)

def

= [U](t(C) ⇒ t(D)) t(C ≡ D)

def

= [U](t(C) ⇔ t(D)) t(a : C)

def

= U(a ∧ t(C)) t((a, b) : r)

def

= U(a ∧ rb)

◮ t(K) is defined as the conjunction below:

• α∈T

t(α) ∧

• α∈A

t(α)

◮ Given I = (∆I, ·I), I |

= K iff MI, a | = t(K) for all a ∈ ∆I. 74

Low hanging fruits

◮ Let ML (resp. ML+) be the multimodal logic into which ALC

concepts (resp. knowledge bases) are translated.

◮ ML satisfiability problem is PSPACE-complete, it has the

finite model property and the tree model property.

◮ ML+ satisfiability problem is EXPTIME-complete, it has the

finite model property and the tree model property.

◮ ALC satisfiability problem is PSPACE-complete and it has

the finite interpretation property.

◮ ALC knowledge base consistency problem is

EXPTIME-complete and it has the finite tree-shaped interpretation property. 75

Recapitulation

76

Recapitulation: concept and role constructors

Name Syntax Semantics Top ⊤ ∆I Bottom ⊥ ∅ Conjunction C ⊓ D CI ∩ DI Disjunction C ⊔ D CI ∪ DI Negation ¬C ∆I \ CI Existential restr. ∃r.C {a ∈ ∆I | r I(a) ∩ CI = ∅} Value restr. ∀r.C {a ∈ ∆I | r I(a) ⊆ CI}

• Unqual. nb. restr.

(≤ n r) {a ∈ ∆I | card({b | (a, b) ∈ r I} ≤ n)}

• Qual. nb. restr.

(≤ n r · C) {a ∈ ∆I | card({b ∈ CI | (a, b) ∈ r I} ≤ n)} Nominal {a} {aI} Role value map r ⊑ s {a ∈ ∆I | r I(a) ⊆ sI(a)} Inverse role r − {(b, a) | (a, b) ∈ r I} Role composition r ◦ s {(a, b) | ∃ a′ (a, a′) ∈ r I and (a′, b) ∈ sI}

77

Recapitulation: Terminological and assertional axioms

Name Syntax Semantics General inclusion axiom C ⊑ D CI ⊆ DI Concept definition A ≡ C AI = CI Role inclusion r ⊑ s r I ⊆ sI Role transitivity Trans(r) r I is transitive Concept assertion a : C aI ∈ CI Role assertion (a, b) : r (aI, bI) ∈ r I 78

Conclusion

◮ Lecture 1 (today): Introduction to description logics

◮ Getting familiar with DL terminology. ◮ Playing with formulae and decision problems.

◮ Lecture 2: Tableaux proof systems and complexity .

◮ Complete calculi for ALC and variants. ◮ Complexity results. ◮ Undecidability.

◮ Lecture 3: Introduction to temporal logics for multi-agents

systems. 79