Logical Aspects of Artificial Intelligence Introduction to Description Logics Stephane´ Demri [email protected] December 9th, 2019 What is in this part of the course? Introduction to Description Logics and Temporal Logics for Multi-Agent Aystems I Today: Introduction to description logics. I 16/12/2019: Tableaux calculi and complexity. I 06/01/2020: Introduction to temporal logics for multi-agent systems. I 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? I Basics of description logic including ALC as well as ATL and variants. I Tableaux for ALC, model-checking techniques for ATL-like logics. I Complexity, decidability, expressive power results for logics dedicated to knowledge representation. 3 Background 1. Necessary background I Basics of first-order logic. I Basics of complexity theory. 2. Optional background I Basics of modal logics, temporal logics I Sequent-style proof systems. I Basics of model-checking. 4 Course material I 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 I Slides available after each lecture. I Do not hesitate to contact me ([email protected]). 5 Material mainly based on the following documents I F. Baader, I. Horrocks, C. Lutz and U. Sattler. An introduction to Description Logic. Cambridge University Press, 2017. I Ivan Varzinczak’s slides (ESSLLI’18) I Ulrike Sattler & Thomas Schneider’s slides (ESSLLI’15). I S. Demri, V. Goranko, M. Lange. Temporal Logics in Computer Science. Cambridge University Press, 2016. 6 Other (online) ressources I Description Logic Complexity Navigator by Evgeny Zolin. http://www.cs.man.ac.uk/˜ezolin/dl/ I Proceedings of the Description Logic Workshops http://dl.kr.org/workshops/ I See also the proceedings of the international conferences: I Int. Joint Conference on Artificial Intelligence. (IJCAI) I European Conference on Artificial Intelligence. (ECAI) I Int. Conference on Principles of Knowledge Representation and Reasoning. (KR) 7 Plan of the lecture I Knowledge representation. I Basic description logic ALC. I Several extensions of ALC. I Relationships with first-order logic and modal logics. I Exercises session. 8 Knowledge representation 9 DLs: where they come from I First-order logic is not always the most natural language. 8x 9 y 8z ((P(x; y) ^ Q(y; z) ) (:Q(a; y) _ P(x; z)))) 8 x (Teacher(x) , Person(x)^9 y (Teaches(x; y)^Course(y))) I How to design user-friendly languages for knowledge representation ? I Concept definition from Description Logics. Teacher ≡ Person u 9Teaches:Course 10 Reasoning about . I Knowledge Epistemic Logics I Rules and obligations Deontic Logics I Programs Hoare Logics I Time Temporal Logics but also many-valued logics, non-monotonic logics, team logics, separation logics, etc. 11 Ontologies I Formal specification of some domain with concepts, objects, relationships between concepts, objects, etc. I Backbone of ontologies includes: I taxonomy (classification of objects), I axioms (to constrain the models of the defined terms). I Classification of medical terms: diseases, body parts, drugs, etc. I Well-known ontologies: I Medical ontology SNOMED-CT formalised with description logic EL + +. I NCI Thesaurus (National Cancer Institute, USA). I Gene ontology (world largest source of information on the functions of genes). I Free ontology editor Proteg´ e´ http://protege.stanford.edu/ 12 The classical student ontology I Natural language specification: I Employed students are students and employees. I Students are not taxpayers. I Employed students are taxpayers. I Employed students who are parents are not taxpayers. I To work for is to be employed by. I John is an employed student, John works for IBM. I Classes/relations/individuals. 13 Main ingredients in formal ontologies I Model of the world with classes within a domain, relationships between classes and instantiations of classes. I Formal: abstract model of some domain with (mathematical) semantics and reasoning tasks. I Classes or concepts: classes of objects with the domain of interest. (Employed student, Parent, Course) I Relations or roles: relationships between concepts. (being employed by, sibling-of) I Instances of classes and relations. I John is an employed student. I Mary works for IBM. 14 Early KR formalisms I Graphical formalisms easier to grasp and supposedly close to how knowledge is represented by human beings. I Large variety of semantical networks. I Often, lack of formal semantics (see tentatives with the knowledge representation system KL-ONE). 15 Why Description Logics? I Formal languages for concepts, relations and instances. I DLs have all one needs to formalise ontologies. I Computational properties. I Acceptable trade-off between expressivity and complexity. I Decidability and often tractability. I Implementation in tools of the main reasoning tasks. I A remarkable suite of languages and tools. See e.g. I OWL: Web Ontology Language. I Proteg´ e:´ ontology editor. I FaCT++: DL reasoner supporting OWL DL. 16 Description Logics and Knowledge representation I Description is a subfield of Knowledge Representation, itself a subfield of Artificial Intelligence. I Description Logic(s): I a research field, I a family of knowledge representation languages, I a member of the family. I Well-defined syntax with formal semantics, decision problems, algorithms, etc. 17 Basic description logic ALC 18 DLs: the core I Concept language. Person u 9Teaches:Course I Syntactic ingredients of the concept language: I Concept names for sets of elements, e.g. Person. I Role names interpreted by binary relations between objects, e.g. EmployedBy. I Concept constructors to build complex concepts, e.g. :, u, t, 9. I Basic terminology stored in a TBox. I Facts about individuals stored in an ABox. 19 Basic elements of the language I Concept names. def NC = fA1; A2;:::g Examples: Parent, Sister, Student I Role names. def NR = fr1; r2;:::g Examples: EmployedBy, MotherOf I Individual names. def NI = fa1; a2;:::g Examples: Mary, Alice, John 20 Boolean constructors & role restrictions I Boolean constructors. I Concept negation : (class complement) I Concept conjunction u (class intersection) I Concept disjunction t (class union) I Role restrictions. I Existential restriction 9 (at least one related individual) I Value restriction 8 (all related individuals) I Many more constructors exist, see forthcoming ALC extensions. I For modal logicians, :; u; t; 9; 8 ∼ :; ^; _; 3; 2 21 Complex concepts in ALC I ALC: Attributive Concept Language with Complements. I Complex concepts. C ::= > j ? j A j :C j C u C j C t C j 9r:C j 8r:C; where A 2 NC and r 2 NR. I Examples of complex concepts: I Student u :9Pays:Tax I 9MotherOf:(9MotherOf:A) I Syntax errors in Student t 8:Tax 89MotherOf:Mary def I C ) D = :C t D. 22 Interpretation I Concept/role/individual ∼ unary predicate/binary predicate/constant. def I I I Interpretation I = (∆ ; · ) I I ∆ : non-empty set (the domain). I I · : interpretation function such that AI ⊆ ∆I r I ⊆ ∆I × ∆I aI 2 ∆I I I A priori, ∆ 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 n CI I def I I (C1 t C2) = C1 [ C2 I def I I (C1 u C2) = C1 \ C2 (9r:C)I =def fa 2 ∆I j r I (a) \ CI 6= ;g (8r:C)I =def fa 2 ∆I j r I (a) ⊆ CI g (R(a) =def fb j (a; b) 2 Rg) I I In modal logic lingua, a 2 C corresponds to I; a j= C. 24 Graphical representation Teaches C2 b Course Course Person Teaches Teaches a C1 Person Course Teacher I I ∆ = fa; b; C1; C2g. I I Teaches = f(a; C1); (a; C2); (b; b)g. I I I Person = fa; bg, Course = fC1; C2; bg. I I a 2 (8Teaches:Course) . 25 Concept satisfiability problem I Concept satisfiability problem: Input: A (complex) concept C in ALC. Question: Is there an interpretation I = (∆I ; ·I ) such that CI 6= ;? I This corresponds to the standard formulation for the satisfiability problem (in modal logics, temporal logics, etc.). I The concept satisfiability problem for ALC is PSPACE-complete. I ALC has the finite interpretation property: every satisfiable concept has an interpretation with a finite domain. 26 Statements I Concept inclusion. Teachers are persons. Employed students are employees. I Concept and role membership. Mary is a student. Alice is a teacher. Laura teaches the course “Automata Theory”. I Statements are not concepts and express properties of concepts, roles and individuals. 27 General concept inclusion (GCI) I Expressions of the form C v D are called general concept inclusion. I Intuitive meaning: I D subsumes C. I C is more specific than D. I Example: Employee v 9WorksFor:>. def I I I Satisfaction relation: I j= C v D , C ⊆ D . I C v D understood as a global statement about I. 28 Concept equivalence I C v D and D v C abbreviated by C ≡ D called concept equivalence. def I I I Satisfaction relation: I j= C ≡ D , C = D . I Concept definition (A 2 NC is a concept name) A ≡ C I > ≡ (:Student t Student). 29 Subsumption problem I Subsumption problem: Input: A GCI C v D with C; D 2 ALC. Question: Is it the case that for all interpretations I, we have I j= C v D? I C v D is “not valid” iff C u :D is satisfiable. I As coPSPACE =PSPACE, the subsumption problem for ALC is PSPACE-complete too.