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, objects, 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 objects, 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 . def = { A 1 , A 2 , . . . } N C Examples: Parent , Sister , Student ◮ Role names . def N R = { r 1 , r 2 , . . . } Examples: EmployedBy , MotherOf ◮ Individual names . def N I = { a 1 , a 2 , . . . } 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 ∈ N C and r ∈ N R . ◮ Examples of complex concepts: ◮ Student ⊓ ¬∃ Pays . Tax ◮ ∃ MotherOf . ( ∃ MotherOf . A ) ◮ Syntax errors in Student ⊔ ∀¬ Tax ∀∃ MotherOf . Mary def ◮ C ⇒ D = ¬ C ⊔ D . 22
Interpretation ◮ Concept/role/individual ∼ unary predicate/binary predicate/constant. def = (∆ I , · I ) ◮ Interpretation I ◮ ∆ I : non-empty set (the domain ). ◮ · I : interpretation function such that A I ⊆ ∆ I r I ⊆ ∆ I × ∆ I a I ∈ ∆ 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 def ⊤ I ∆ I = def ⊥ I = ∅ ∆ I \ C I def ( ¬ C ) I = def ( C 1 ⊔ C 2 ) I C I 1 ∪ C I = 2 def ( C 1 ⊓ C 2 ) I C I 1 ∩ C I = 2 { a ∈ ∆ I | r I ( a ) ∩ C I � = ∅} def ( ∃ r . C ) I = { a ∈ ∆ I | r I ( a ) ⊆ C I } ( ∀ r . C ) I def = def ( R ( a ) = { b | ( a , b ) ∈ R} ) ◮ In modal logic lingua, a ∈ C I corresponds to I , a | = C . 24
Graphical representation Teaches C2 b Course Course Person Teaches Teaches a C1 Person Course Teacher ◮ ∆ I = { a , b , C 1 , C 2 } . ◮ Teaches I = { ( a , C 1 ) , ( a , C 2 ) , ( b , b ) } . ◮ Person I = { a , b } , Course I = { C 1 , C 2 , 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 C I � = ∅ ? ◮ This corresponds to the standard formulation for the satisfiability problem (in modal logics, temporal logics, etc.). ◮ The concept satisfiability problem for ALC is PS PACE -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
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