Ontologies and the Semantic Web

Transcription

1 Ontologies and the Semantic Web Dirk Pattinson Winter 2011/2012 Contents 1 Introduction and Basic Concepts 1 2 Syntax and Semantics of ALC 7 3 ABoxes and Knowledge Bases 16 4 Tableau Algorithms 19 5 Blocking: Non-Empty TBoxes 22 6 Extensions and Complexity of ALC 26 7 Description Logics vs. First-order Predicate Logic 34 8 Implementation Techniques 42 9 Ontologies and the Semantic Web Programming on the Semantic Web 63 1

2 1 Introduction and Basic Concepts Terminology Ontologies formal representation of knowledge as a set of concepts and their relationships used to reason about (represented) entities and to describe the domain Semantic Web encapsulates the idea of a web of data rather than documents describes W3C models and technologies to exchange and link datasets comprises W3C models and standars such as RDF, SPARQL, RDFS, OWL Course Content Ontologies Examples: ontologies in medicine Formalisation: the description logic ALC Reasoning: consistency, subsumption and other tasks Implementation techniques Semantic Web Data representation: the resource description framework (RDF) Description Logics and Distributed Data Ontologies by Example: SNOMED CT SNOMED CT: Systematized Nomenclature of Medicine: Clinical Terms created by the College of American Pathologists endorsed by the UK NHS defines 311,000 concepts using 1,360,000 semantic relationships used for clinical decision support, electronic health records Example 1. viral pneumonia is_a(n) infectious pneumonia (subclassing) viral pneumonia caused_by virus (semantic link) Pattern here: Subject, Predicate Object Subject (viral pneumonia) and object (virus, infectious pneumonia) are concepts Predicate (is_a, caused_by) establishes semantic relationship

3 Image courtesy of International Health Terminology Standards Development Organisation) Graphical Visualisation Mathematical Model caused_by induces a relation between nodes of a model is_a induces subset relationship Example Snomed Browser Challenges for Computer Science Questions. does the ontology make sense (consistency)? relationship between two concepts (subsumption)? consistency of hypotheses (satisfiability)? Example 2. A simple inconsistency A caused_by B B is_a C not (A caused_by C) In paractise inconsistencies can be spread over 100s of statements... need for efficient automated techniques. 2

4 Image courtesy of W3C More Examples Example 3. Diversity of Terminology finding synonyms (e.g. measles and rubeola) identifying equivalent concepts (subsumption) Example 4. Clinical Decision Support plausibility of diagnoses hypothesis checking (consistency) Semantic Web Example BBC Music Semantic Web Architecture Description Logic Approach Formalisation of Concepts language with formally defined syntax (here: ALC) ingredients: boolean operators, relational links ( eg. caused_by ) formally: a subset of first order logic over a relational signature Interpretation of Concepts set-theoretic model: individuals and relations concepts map to sets of individuals, subconcepts to subsets 3

5 relational links map to relations Reasoning Tasks connect concepts (syntax) with models (semantics) e.g. satisfiability: can there be a model of an ontology? e.g. subsumption: is A a subset of B in all models of an ontology? This course in detail Syntax and Semantics of ALC language and model, reasoning tasks, knowledge bases Reasoning concept satisfiability (empty / general knowledge base) from tableau calculi to reasoning algorithms Implementation propositional optimisation, rule invertibility backtracking vs caching Applications Reasoning over ontologies and the semantic web Course Structure Administrative 2 lectures and one tutorial / week tutorials partly lab-based Coursework two practical courseworks that implement reasoning tasks first coursework week 5: reasoning over the empty knowledge second coursework week 8: reasoning over general knowledge bases Your Feedback is Appreciated this course runs for the second time I ve changed many things... let me know if there s something you d like to know more about! 4

6 Example: The Pizza Ontology What we know about Pizzas every pizza has a topping and a base. concepts Pizza, PizzaTopping, PizzaBase semantic relationships hastopping, hasbase What we know about Toppings a topping can be cheese, vegetables or meat concepts CheeseTopping, VegetableTopping, MeatTopping subclass relationship And lots more... see today s (lab) tutorial where create and classify a pizza ontology Models of the Pizza Ontology Basic Ingredients a set, the domain of the model contains pizzas, toppings and pizza bases every concept defines a subset of can e.g. speak about the set of toppings semantic relationships define relations on e.g. hasbase is a relation between pizzas and pizza bases Reasoning Tasks consistency: is there a model of an ontology? subsumption: is there a subset relationship between concepts in all models? satisfiability: given a concept, is there a model where it is non-empty? Algorithmic Approach model searching is not sufficient (infinite supply of models) reduce question of model existence to formal provability replace model search by proof search 5

7 Further Reading Examples of Ontologies SNOMED CT in the UK: data/snomed internationally: Concept Broswer: Semantic Web The W3C s vision References, Continued The Pizza Ontology browsable at: to see the pizzas: [browse ontology] [classes] [DomainConcept] [Food] [Pizza] 6

8 2 Syntax and Semantics of ALC ALC: Attributive Language with Complements History in Brief introduced by Schmidt-Schauß and Smolka, early 1990s many predecessors, starting with KL-One, mid 1980s main advantage over ealier approaches: perspicuous semantics Syntax of ALC Definition 5. An ALC-signature is a pair (A, R) where A is a set (of atomic concept names) and R is a set (of role names). Signatures defined the basic vocabulary that we can use to express concepts. Definition 6. The set C(S) of ALC-concepts over an ALC-signature S = (A, R) is the least set such that, and all C A are concepts if C, D are concepts, then so are C D, C D and C if C is a concept and R R is a role name, then R.C and R.C are concepts Concepts are the sentences that we can formulate over a given signature. Reading and Derived Connectives Informal Interpretation Symbol Interpretation Reading universal concept top inconsistent (nonexisting) concept bottom conjunction (concept intersection) and disjunction (concept union) or concept negation not R._ universal restriction all R-sucessors... R._ existential restriction some R-sucessor... Derived Connectives Connective Definition Reading C D C D all C are D, concept subsumption C D (C D) (D C) C and D are equivalent Precedences Precedences use parenthesis (, ) to disambiguate whenever necessary we use the precedence order where stands for binds stronger than. Readability is more important than begin concise! 7

9 Pizza Examples Example 7. The Pizza signature has (at least) the following concept names: Pizza, Topping, Base, CheeseTopping, VegetableTopping, VegetarianPizza role names: hastopping, hasbase Example 8. Concepts over the Pizza signature: A pizza has a topping: Pizza hastopping.. Pizzas and pizza toppings are disjoint: Pizza Topping. Vegetarian pizzas have only cheese and vegetable toppings: VegetarianPizza hastopping.(cheesetopping VegetableTopping). Semantics of ALC Definition 9. A interpretation over a signature (A, R) is a pair (, I ) where is a set (the domain) and I is an interpretation function such that R I for all R R (role names are relations) C I for all C C(S) (concepts are subsets) such that the following holds: I =, I = (C D) I = C I D I, (C D) I = C I D I and ( C) I = C I ( R.C) I = {x y C I for all y R I (x)} ( R.C) I = {x y C I for some y R I (x)} where R I (x) = {y (x, y) R I } is the set of R I -successors of x. Examples Remark Interpretations are fully determined by a set that defines the domain of the interpretation an assignment R I for all roles R R an assignment C I for all atomic concepts C A and everything else is fixed by induction. Example 10. Choosing = defines an interpretation, the trivial interpretation. Example 11. Write R I (x) = {y (x, y) R} for R R. if R I (x) = for some x, then x ( R.C) I for any concept C if x ( R. ) I then R I (x) (and vice versa) 8

10 More Examples I More Examples II More Examples III 9

11 Terminological Axioms Definition 12. A concept inclusion over an ALC-signature S is an expression C D for concepts C, D C(S). A TBox is a finite set of concept inclusions. Remark We sometimes allow expressions of the form C D in TBoxes to abbreviate C D and D C. Definition 13. An interpretation I satisfies a concept inclusion C D if C I D I. We write I C D in this case. An interpretation I is a model of (or satisfies) T if I C D for all C D T, written I T. Informal Reading TBoxes assertions C D define subclass relationships do not necessarily consist of primitive concepts only used to formalise and aggregate background knowledge Interpretations think of C I as the set of individuals with property C interpretation I satisfies C D if I realises subclass relationship TBoxes and Interpretations Idea. TBoxes define interpretations that are consistent with background knowledge. 10

12 Reasoning Tasks Let T be a TBox over an ALC-signature S and C, D C(S). Definition 14. A concept C is satisfiable over T if there exists a model I of T with C I. A concept D subsumes a concept C over T if C I D I for all models I of T (also C is subsumed by D) Two concepts C, D are equivalent over T if C I = D I for all models I of T. Two concepts C, D are disjoint over T if C I D I = for all models I of T. Derived Notion A TBox T is satisfiable if is satisfiable over T. Examples Example 15. Consider the pizza signature and fix a TBox T that contains VegetableTopping Topping CheeseTopping Topping Topping Base. 1. clearly CheeseTopping subumes Topping over T. 2. the concept CheeseTopping Base is unsatisfiable over T. 3. the concept Pizza hastopping.cheesetopping is sat. over T. 4. is Base hastopping.cheesetopping satisfiable over T? Satisfiability is Basic Proposition 16. Let T is a TBox over an ALC-signature S. The following hold for all concepts C, D C(S): 1. C subsumes D iff D C is not satisfiable. 2. C and D are equivalent iff both C D and D C are unsatisfiable. 3. C and D are disjoint iff C D is unsatisfiable. Proof. Suppose C subsumes D. If I = (, I ) is an interpretation, then D I C I iff (D C) I = D I ( C I ) = iff D C is unsatisfiable. The remaining are analogous. Remark In the following, we therefore concentrate on satisfiability checking. 11

13 Relationship between Operators Lemma 17. Let C, D be a ALC-concepts over a signature S. Then ( ) I = I and I = I ( (C D)) I = ( C D) I ( (C D)) I = ( C D) I ( R.C) I = ( R. C) I ( R.C) I = ( R. C) I whenever I is an interpretation over S. Proof of Lemma Proof. Fix an interpretation I = (, I ). We show some of the cases. I = = = I = ( ) I. ( (C D)) I = (C D) I = (C I D I ) = ( C) I ( D) I = ( C D) I If R I (x) = {y (x, y) R I } we have: ( R.C) I = ( R.C) I = {x R I (x) C I } = {x R I (x) ( C I )} = ( R.C) I Negation Normal Form Definition 18. A concept C over an ALC-signature (A, R) is in negation normal form if negations only appear in front of atomic concepts. Example 19. In negation normal form: R. A R.( A S. B) Not in negation normal form: (A R.B) A R. (A B) where A, B are atomic concepts. Lemma 20. For every concept C there exists a concept D in negation normal form such that C I = D I for all interpretations I. Proof. Replace negated concepts by their equivalents as above. 12

14 From Protege to TBoxes In Protege: classes, properties, restrictions... can declare (sub)classes as disjoint can declare typed relations can declare defined classes In ALC: concepts, roles... only defines TBoxes Matching Protege against ALC Terminology Protege ALC class concept property roles restrictions quantifiers (, ) Basic Mechanism build up a TBox iteratively from Protege declarations every declaration adds to the TBox Disjointness and Subclassing Disjointness of C and D T = T {C D } Subclassing C D T = T {C D} 13

15 Object Restrictions Restriction of C Existential Restriction (C <property> some D) T = T {C R.D} Universal Restriction (C <property> all D): T = T {C R.D} Typed Relations Property R has domain C T = T { R. C} Property R has range D T = T { R.D} Defined Classes Define C by its properties T = T {D 1... D n C} if {D C D T } = {D 1,..., D n } Informally If D 1,..., D n are all the properties of C that are already defined (asserted in T ) then adding D 1... D n C asserts that everything that satisfies all D i also satisfies C. 14

16 Summary Mapping from Protege to ALC all Protege declarations can be expressed as a TBox many features mapped into a single concept Classification in Protege If an ontology induces TBox T and C, D are concepts, then C inferred subclass of D T C D giving a formal meaning to classification in Protege. scrutinise satisfiability ( subsumption) in ALC References F. Baader, W. Nutt. Basic Description Logics. In F. Baader, D. Calvanese, D.L. McGuinness, D. Nardi, P.F. Patel-Schneider, editors, The Description Logic Handbook, Cambridge University Press, 2002, pages F. Baader, I. Horrocks, and U. Sattler. Description Logics. In F. van Harmelen, V. Lifschitz, and B. Porter, editors, Handbook of Knowledge Representation, pages. Elsevier, BaHS07a.pdf 15

17 3 ABoxes and Knowledge Bases Concepts and Individuals Concepts and Interpretations Syntax. Language for describing concepts Semantics. Mapping of concepts to subsets of a domain Knowledge about Individuals Alice and Bob are Persons Alice likes Bob Not expressible as a concept: there cannot be a (possibly empty) set of things with the property of being Alice. We now extend the notion of signature and interpretation to accommodate knowledge about individuals. ABoxes Definition 21. ABoxes An (extended) ALCsignature is a triple (A, R, O) where (A, R) is an ALCsignature and O is a set of individual names.[1ex] A concept assertion is of the form x C where x O and C is an ALC-concept over (A, R). A role assertion is of the form (x, y) R where x, y O and R R.[1ex] A finite set consisting of concept and role assertions is called an ABox, and a Knowledge Base is a pair (T, A) where T is a Tbox and A an ABox (over the same ALC-signature). Example 22. Assume that likes R and Alice, Bob O and person is an (atomic) concept. Alice and Bob are persons: Alice person and Bob person. Alice likes Bob: (Alice, Bob) likes. ABoxes and Relational Databases Example: Relational Databases ISBN Author Title Publisher Pages 001 Joseph Conrad Heart of Darkness Wordsworth Adam Hochschild King Leopold s Ghost Macmillan 232 As triples: Book 001 is written by Joseph Conrad. Book 001 is titled Heart of Darkness.... Connection to Relational Databases we can think of a TBox as defining a database schema we can think of an ABox as collection of rows but (e.g.) instance checking allows us to derive new data 16

18 Models of Knowledge Bases Idea interpretations assign singletons to individuals x O open world assumption: different individuals x, y O may be mapped to the same element in the domain terminology: we do not have the unique name assumption. Definition 23. An Interpretation over an extended ALC-signature (A, R, O) is a pair I = (, I ) such that I is an interpretation over (A, R) and moreover: for all x O. Models of Knowledge Bases Idea x I interpretation needs to satisfy all concept assertions and role assertions in general, also constrained by TBox Definition 24. An interpretation I is a model of a Knowledge Base K = (T, A) if I T (all TBox axioms are satisfied) x I C I for all x C A (x I, y I ) R I for all (x, y) R (role assertions) We write I K in this case. Reasoning Tasks over Knowledge Bases Recall Reasoning Tasks over TBoxes Concept satisfiability, equivalence, subsumption, disjointness all reducible to concept satisfiability Reasoning Tasks over Knowledge Bases a knowledge base K is consistent if there exists an interpretation I = (, I ) such that I K and (Knowledge Base Consistency) a concept C issatisfiable over a knowledge base K if there exists an interpretation I with I K and C I (Concept Satisfiability) an individual x O is an instance of a concept C if x I C I for all interpretations I with I K (Instance Checking) Concept subsumption, equivalence, disjointness as before. 17

19 Examples Inconsistency between TBox and ABox Consider the TBox containing plane has.wing plane has.engine glider plane together with the ABox x glider x has. engine Reduction of Reasoning Tasks Proposition 25. Knowledge base consistency is Basic 1. a concept C is satisfiable over K = (T, A) iff the knowledge base (T, A {x C}) is consistent, where x is an individual not occurring in A. 2. an individual x O is an instance of a concept C iff the knowledge base (T, A {x C} is inconsistent. Proof. whiteboard / tutorial. Remark We focus on ABox consistency in the future. 18

20 4 Tableau Algorithms Overall Approach Recall A knowledge base K is consistent if there exists an interpretation I such that I K and. Goals. (Efficient) algorithm with input K that decides whether K is consistent or not, possibly producing a witnessing interpretation Approach We cannot check all possible interpretations infinitely many characterise satisfiability by means of tableau calculus decide satisfiability by analysing completion trees / tableaux. Tableau Calculi Tableau in General introduced by E. Beth in 1955 to characterise satisfiability the tableaux we use are sometimes called set-labelled most successful method, widely implemented (e.g. in Fact++) active research area (annual TABLEAUX conference) Main Idea start with the ABox iteratively expand the ABox by de-constructing a concept existential restriction generates new individuals every transformation step preserves consistency if no clash is found, this gives an interpretation if only clashes are found, the original ABox is inconsistent. 19

21 ABox Expansion Rules And-Expansion if x C D A and either x C A or x D A. Or-Expansion if x C D A and both x C A and x D A. -expansion (z fresh for A) A A {x C, x D} A A {x C} A {x D} A A {(x, z) R, z C} if x R.C A, no y satisfies (x, y) R A and y C in A. -expansion if (x, y) R A, x R.C A and y C A. Non-deterministic choice and freshness A A {y C} we say that z is fresh for A if z does not occur in A or-expansion is non-deterministic: we choose either of the alternatives non-deterministic algorithm Examples Example 26. Consider the ABox {x R. S.A B}. No expansion rule applies and indeed = {x}, R I =, A I = B I = provides a satisfying interpretation. Example 27. Consider the ABox {x R.(A A)}. We get {x R.(A A)} {x R.(A A), y A A}, R(x, y) {x R.(A A), y A Ay A, Y A, R(x, y)} which contains a clash hence the original ABox is not satisfiable. Some Terminology Definition 28. An ABox A is complete if none of the transformation rules applies to it. It contains a clash if x A or {x C, x C} A for an individual x and an atomic concept C. An ABox is called closed if it contains a clash, and open otherwise. Intuition consistent ABoxes can be expanded to complete and open ABoxes if all expansion of an ABox are closed, it is inconsistent Next Goal turn this into an algorithm establish that this algorithm decides ABox consistency 20

22 Correctness and Termination Lemma 29. There are no infinite sequences A 0 A n... of A-expansions. Proof. (Sketch) Every transformation rule only adds sub-concepts and the maximal number of new individuals added is bounded by the number of occurrences of in the original A. Lemma 30. Suppose that A A 1. Then A is consistent iff A 1 is consistent. If A A 1 A 2 then A is consistent iff A 1 or A 2 is consistent. Proof. (Sketch) Every model of the ABox on the left must satisfy all the constraints of one of the ABoxes on the right. Consistency Lemma 31. Every complete, open ABox is consistent. Proof. Construct an interpretation I = (, I ) where is the set of individuals in A and R I = {(x, y) (x, y) R A}. Lemma 32. Every closed ABox is inconsistent. Proof. (Sketch) Obvious. Algorithm for ABox consistency Algorithm: Decide the consistency of an ABox Algorithm consistent (ABox A) if A is closed then return false if A is complete then return true choose an expansion A -> A1 or A -> A1 A2 return consistent (A1) [ in the first case ] return consistent (A1) or consistent(a2) [in the second] Termination and Correctness Theorem 33. The algorithm above is terminating and correct. Proof. By putting together the earlier lemmas. Corollary 34. The consistency of a knowledge base with empty TBox is decidable. Remark complexity of this decision problem is PSPACE so far we assume that TBox is empty(!). 21

23 5 Blocking: Non-Empty TBoxes From Empty to Nonempty TBoxes So far: empty TBox convert ABox concepts to negation normal form four simple completion rules termination, correctness and completeness Idea For every concept inclusion C D convert to C D require that every individual satisfies C D add x C D to ABox for every indivudual x do the same for new individuals created by Naive Approach Example 35. Consider T = { R. } and A = {x }. [1ex] Step 1. Add x R. (or better, x R. ) [1ex] Step 2. Apply expansion rules: {x } T {..., x R. } {..., x R. } {... y, (x, y) R} T {..., y R. } {..., y R. } T {..., z R. } {..., z R. }... Problem. Non-termination: infinitely many individuals are created. Solution: Blocking Observation Even with K = ({ R. }, {x }) we have infinite branches: Idea: Re-use nominals that we have already created 22

24 Question. When can we bend arrows to avoid non-termination? Blocking Idea. An individual x in an ABox is blocked if there exists another indivdual y such that y is required to satisfy the same concepts as x. Definition 36. Consider a sequence A0 A1 A2 An of ABoxes such that Ai Ai+1 for all i = 1,..., n 1. An indiviual y is older than x if min{i y occurs in Ai } < min{j x occurs in Aj } i.e. y has been added earlier to An than x.[1ex] An individual x is blocked by an individual y in the sequnce A0 A1 An if (i) y is older than x and (ii) if x C An then y C An for all concepts C, i.e. y satisfies more concept assertions as x. The individual x is blocked if it is blocked by some y. Modification of the expansion Rules : if x not blocked, then A A {x C, x D} if x C D A and either x C A or x D A. : if x not blocked, then A A {x C} A {x D} if x C D A and both x C A and x D A. : if xnotblockedandz fresh A A {(x, z) R, z C} if x R.C A, no y satisfies (x, y) R A and y C in A. : if x not blocked, then: A A {y C} if (x, y) R A, x R.C A and y C A. TBox expansion if x is not blocked, C D T A T A {x E} if x E A and E is the negation normal form of the concept inclusion C D. 23

25 Example Example 37. Consider the knowledge base ({x }, { R. }). initialise x T x R. x R. y, (x, y) R T (y R. ) (note: y is blocked) complete y is blocked, no more rules are applicable. Model Construction x blocked by y identify x and y in a model: {x, x R., x R., (x, x) R} Knowledge Base consistency Algorithm 2: Knowledge Base consistency Suppose K = (T, A) is a knowledge base with A. Algorithm consistent (Knowledge Base (A, T)) 1. if A is closed then return false 2. if A is complete then return true 3. choose an expansion A -> A1 or A -> A1 A2 4. return consistent (A1) [ 1st case ] 5. return consistent (A1) or consistent(a2) [2nd case] where expansion rules refers to the modified expansion rules. Correctness and Termination Theorem Algorithm 2 always terminates on a finite knowledge base. 2. if it generates a complete and open ABox, then the knowledge base is consistent. 3. if it only generates ABoxes with clashes, then the knowledge base is unsatisfiable. Remark The last item can be equivalently stated as if the knowledge base is consistent, then Algorithm 2 generates a complete and open ABox. 24

26 Termination We show that the algorithm always terminates. 1. Let S be the set of all concepts that occur in the inital ABox and the set of negation normal forms of concepts C D in the TBox, together with the sub-concepts of these. 2. for every concept assertion x C we know that C S. 3. as the algorithm increases the size of the ABoxes, it increases the number of non-blocked individuals, which cannot grow beyond 2 size(s). Soundness Suppose Algorithm 2 has produced a clash-free, complete ABox A n. We show that A n is consistent. As A A n, consistency of A follows.[1ex] Define an interpretation I = {, I ) by the following data: = {x x occurs in A n and is not blocked} C I = {x x C A n } R I = {(x, y) 2 (x, y) R A n or (x, z) A n and y blocks z} We now have to show that if x C A n and x is not blocked, then x C I if C D T, then C I D I which follows from completeness of A n by induction on the structure of concepts. Completeness Suppose that A is consistent. We show that the algorithm produces an open and complete ABox. To do this, it suffices to show that if an ABox is consistent and not complete, then every applicable expansion rule produces (at least one) consistent ABox. this is obvious for, and T for, we can choose at least one consistent alternative for, consistency implies that a successor exists as required. Completeness now follows, as the algorithm always terminates. 25

27 6 Extensions and Complexity of ALC Recap on Blocking Recall In an ABox expansion series an individual y is blocked by x if A 0 A 1 A 2 A n x has been introduced earlier in the sequence for all concepts C, if y C A n, then x C A n Question. Why do we insist that individuals can only be blocked by older individuals? Wouldn t it be enough that they satisfy the same concepts, irrespective of age? Blocking and Age Answer We need a mechanism that ensures that individuals don t mutually block one another. Age is one such mechanism. Example 39. Consider T = { R.A A} and A = {x A, y A}. [2ex] If we would disregard the age requirment, x would block y and y would block x hence the ABox would be regarded as complete and open, and the completion algorithm would erroneously declare satisfiable. Remark One could declare that only newly introduced individuals can be blocked to invalidate this example can you give a general proof? Interlude: Complexity Question. How long does it take to check the satisfiabilty of a concept, measured in the size of the concept (and possibly the TBox)? Empty TBox Observation: No need for blocking we can construct ABox expansions one branch at a time we can discard unused concept/role assertions branches are polynomially long and take polynomial space Theorem: concept satisfiability (over the empty TBox) is in PSPACE. Acyclic TBox A TBox is acyclic if it does not contain cyclic definitions. If we have an acyclic TBox, then TBox axioms can be unfolded on the fly leading to PSPACE complexity (again). 26

28 More on Complexity General TBox Observation: need for blocking to guarantee termination blocking needs to remember previously constructed ABoxes need exponential amount of memory In fact, one can prove that concept satisfiability over general TBoxes is in EXPTIME, measured in the size of the concept plus the size of the TBox. The ABox expansion algorithm Observations ABoxes may be exponentially big exponentially many steps in the worst case choice points (due to ) are chosen non-deterministically One can show that the (worst case) complexity here is NEXPTIME. Non-Expressible Assertions Counting Every nuclear family has two parents, at least two children and a dog.[2ex] Concept Constructor: nr._ (at least n) Individuals Everybody loves Mary.[2ex] Concept Constuctor: {Mary} More Non-Expressible Assertions Properties of Relations Brothers of fathers are uncles.[2ex] Inverse Roles R and RBoxes R R, R R Composition of Relations My friends friends are also my friends.[2ex] Role composition R S Individuals: the Description Logic ALCO Terminology The letter O stands for the one of -constructor. Definition 40. The concepts of ALCO are the concepts of ALC, with one more formation rule: if o 1,..., o n are individuals, then {o 1,..., o n } is an ALCO-concept. Informal Reading x {o 1,..., o n } x is one of the o 1,..., o n. Caveat 27

29 we do not have the unique name assumption an individual may be known by more than one name in an interpretation, different individual names may be identified Examples Example 41. Everybody loves Mary.loves.Mary [2ex] If Homer is married to Marge, then Marge is married to Homer ({Homer marriedto.{marge}) ({Marge} marriedto.{homer}) Question. Can you express the set of all people who love Mary as a concept? The set of all people who only love Mary? The set of people who are not loved by anyone? Semantics of the one of Constructor Definition 42. If I = (, I ) is an interpretation, then {o 1,..., o n } I = {o I 1,..., o I n} where o 1,..., o n are individuals. [2ex] The notion of TBox, ABox, satisfiability etc. remain unchanged, except that we now consider ALCO-concepts. Definition 43. An ALCO-concept is in negation normal form, if the constructor { } is only applied to singletons, and negations only appear in front of singletons or atomic concepts. Rewriting into Negation Normal Form Lemma 44. Every ALCO-concept can be re-written into negation normal form by {o 1,..., o n } {o 1 }... {o n }. Example 28

30 D Þ* '-- ^\ v\ I' fd HI \ Ê--l Y u-\ e\ Ly ==-\)?. s-. 0 'a'- ñ ç. tl rì r^.\ o,\ :- ^ Þ i.i U ^-{-l ú \,)- \J Þ" -ì -- fit I (.. ( \ \ \ Reasoning with Individuals Completion Rules for Individuals Given an ABox A if A contains x {y} and not y {x}, then add y {x} if A contains x {y} and not y {x}, then add y {x} if A contains x C and x {y} and not y C, then add y C if A contains (x, y) R and y {z} and not (x, z) R then add (x, z) R For deciding the consistency of knowledge bases, these rules are only applied to non-blocked individuals x. An ABox A contains a clash, if x A or if x C and x C A for C atomic or C = {x}. Theorem 45. Adding these completion rules results in a terminating algorithm that decides ABox consistency (over knowledge bases). Inverse Roles: The Description Logic ALCI Terminology The I in ALCI stands for inverse roles. Idea. At the same time, we want to have access to parentof and childof haspart and ispartof that are each others relational converses. Definition 46. The concepts of ALCI are the concepts of ALC, with one more formation rule: if R is a role name and C is an ALCI-concept, then R.C and R.C are ALCI-concepts. 29

31 Examples Informal Reading x R.C x R.C x R.C x R.C x has an R-link to a C at least one C is R-linked to x all R-links from x are C s only C s are R-linked to x Example 47. Consider the TBox T = {A R A ( R. A S.B)}. Is the ABox A = {x A} consistent over T? Example 48. Consider the TBox T = {B R.B R. R. }. [2ex] Is the ABox A = {x B} satisfiable over T? Reasoning in ALCI Main Idea R -expansion: create a R-predecessor R -expansion: apply to all R-predecessors Additional Expansion Rules : if xnotblockedandz fresh A A {(z, x) R, z C} if x R.C A, no y satisfies (y, x) R A and y C in A. : if x not blocked, then: if (y, x) R A, x R.C A and y C A. Note The direction of role assertions is reversed Need to be careful with blocking! A A {y C} Blocking, Revisited Example 49. We expand x B with respect to the TBox {B R.B R. R. }. 1. x B 2. x R.B, x R. R. (by T,, ) 3. y B, (x, y) R (by y blocked by x?) If y were blocked by x, then this TBox would be open and complete the algorithm would declare satisfiable. Problem. If we use subset blocking, then inverse role restrictions are not unfolded. Solution. Use equality blocking instead and modify the expansion rules 30

32 Modified Expansion Rules for ALCI Idea Propagation of universal restrictions should always be possible Add concept assertions whenever the node being added to is not blocked : if y not blocked, then: A A {y C} if (x, y) R A, x R.C A and y C A. : if y not blocked, then: A A {y C} if (y, x) R A, x R.C A and y C A. (Note the change in the blocking requirement everything else is as before) Reasoning in ALCI Definition 50. Consider a sequence of ABox-expansions in the description logic ALCI A 0 A 1 A n and two individuals x, y. We say that y is equality-blocked by x, if x is older than y x and y satisfy the same concept assertions, i.e. {C x C A n } = {C y C A n } Theorem An ALCI-ABox is consistent over a TBox if there is a sequence of rule applications (using equality blocking) that creates an open and complete ABox. 2. If an ABox is open and complete (using equality blocking), then it is consistent. Reasoning Algorithm for ALCI Algorithm 3 Algorithm consistent (Knowledge Base (A, T)) 1. if A is closed then return false 2. if A is complete then return true 3. choose an expansion A -> A1 or A -> A1 A2 4. return consistent (A1) [ 1st case ] 5. return consistent (A1) or consistent(a2) [2nd case] where expansion refers to the ALCI expansion with equality blocking. Theorem Algorithm 3 always terminates on a finite knowledge base. 2. if it generates a complete and open ABox, then the knowledge base is consistent. 3. if it only generates ABoxes with clashes, then the knowledge base is unsatisfiable. 31

33 \ Counting: the Description Logic ALCQ Terminology The Q in ALCQ stands for qualified number restrictions. There s also a variant ALCN which only allows for unqualified number restrictions, i.e. assertions of the form nr. and nr.. Definition 53. The concepts of ALCQ are the concepts of ALC with two more formation rules: if C is an ALCQ concept and R is a role name, then nr.c is an ALCQ-concept, for all n 0. if C is an ALCQ concept and R is a role name, then nr.c is an ALCQ-concept, for all n 0. (Sometimes, one sees = nr.c for an abbreviation of nr.c nr.c.) Examples and Informal Reading Informal Reading x nr.c at most n R-successors of x are C s. [2ex] x nr.c at least n R-successors of x are C s. Example 54. Every Car has at least four wheels Car 4has.Wheel Every nuclear family has at least two parents, two children and a dog NuclearFamily 2has.Parent 2has.Child has.dog (Note that that all ALC-constructs are still valid, despite the fact that they could be encoded). Example 5" o ìq v1,-\ O -Ð ç ". ç _\ \\ \N,ç- tr.- >./ ii Ç^,Js (/) f -.) 0 -r) ro ç (9 -t,v\ \-\ '\-1 \rñ :U Ñ \P I \,-\ _f r, _í O -\) -v. () 1 Y\ (\, ' \\U - æ \-/ \-/, 1 v^? \_z \ \\ t -' ( 32

34 Semantics of Qualified Number Restrictions Definition 55. If I = (, I ) is an interpretation, then ( nr.c) I = {x {y C I (x, y) R I } n} ( nr.c) I = {x {y C I (x, y) R I } n} where S denotes the number of elements of a set S. [2ex] The notions of TBox, ABox, Knowledge base, consistency, subsumption are unchanged (for ALCQ-concepts). Negation Normal Form Question: Can we push the negation in nr.c or nr.c inside, so that only C is (possibly) negated? Negation Normal Form Lemma 56. If R is a role name and C is a concept, then for all interpretations I. ( nr.c) I = ( (n 1).C) I for n > 0 ( 0R.C) I = I ( nr.c) I = ( (n + 1).C) I Lemma 57. Every ALCQ-concept C can be re-written into an ALCQ-concept C in negation normal form, i.e. into a concept where negations only occur in front of atomic concepts. Reasoning with Number Restrictions Extended ABox expansion rules New expansion rules for nr.c and nr.c if there are not enough R-successors, then add the required number if there are too many R-successors, try and identify different successors Difficulty blocks may break due to identification of successors we need to know whether or not y C if (x, y) R and x nr.c is required (This can be done see the decription logic handbook for details) 33

35 xϕ xϕ 7 Description Logics vs. First-order Predicate Logic Mad Logics: Beyond ALC-Dome ALC R.E C D D? FOL φ ψ I = models M a set and predi- (, I ) a set and predicates/relations[.5cm] F ather haschild. cates/relations[.5cm] x.f ather(x) y.haschild(x, y) A Recap of First-order Logic (FOL) Powerful extension of propositional logic The most important logic of all (Ian Hodkinson) x(logic(x) x = fol lessimportant(x, fol)) xϕ χ Most of you should know this from 1st year logic course (these slides copy some slides from that course) Variables We have an infinite set of variables x, y, z, x 0, x 1,... Variables are needed for quantification Example: to say everything is a logic we can write ylogic(y) and read it as for all y, y is a logic FO-Signatures A FO-signature is a set of constants, and relation symbols with specific arities. Constants: pope, dirk,... Unary predicates: Logic( ), Human( ),... Binary relations: lessimportant(, ), hasmother(, ) We do not look at relations with higher arities here We do consider functions either Despite their names, these constants do not have a meaning yet 34

36 Terms and Formulae Fix a FO-signature L. A term is either a constant in L or a variable If t, t are terms, P is a predicate in L, and R is a relation in L then,, t = t, P (t), and R(t, t ) are atomic formulae If φ, ψ are formulae then so are φ, φ ψ, φ ψ, φ ψ, and φ ψ If φ is a formula and x is a variable then xφ and xψ are formulae Free and Bound Variables There are two ways in which a variable x can occur in a formula φ An occurrence of x in φ is bound if it is in the scope of a quantifier x or x If this is not the case then the occurrence of x is free in φ The free variables of φ are those with free occurrences in φ Example: x(r(x, y) R(y, z) z(s(x, z) R(z, y))) If we write φ(x, y) it means that at most the variables x and y are free in φ Structures Fix a FO-signature L. A structure is a pair M = (dom(m), M ) where: dom(m) is a set of objects called the domain M is an interpretation function mapping Assignments each constant c in L to an object c M in dom(m) each predicate P in L to a subset P M of dom(m) each relation R in L to a relation R M on dom(m) We want to define whether a formula φ is true in a given structure M, but... When φ has free variables, say φ = yr(x, y), we need to know something about these free variables. An assignment into M is a function mapping each variable to an object in dom(m) If g, h are assignments into M and x is a variable, we write g = x h if g(y) = h(y) for all variables y /= x. g agrees with h on all variables except possibly x 35

37 Evaluations Let L be a FO-signature, M be a structure, and h be an assignment into M For any term t, the value of t in M under h is: c M dom(m) if t is the constant c in L h(x) dom(m) if t is the variable x For a formula φ we write M, h φ if φ is true in M under h and M, h / φ if not M, h φ is defined inductively on the structure of φ Evaluations (cont.) M, h P (t) iff a P M where a is the value of t in M under h M, h R(t, t ) iff (a, a ) R M where a (a ) is the value of t (t ) in M under h M, h t = t iff t and t have the same value in M under h M, h and M, h / M, h φ ψ iff M, h φ and M, h ψ similar: φ, φ ψ, φ ψ, and φ ψ M, h φ iff M, g φ for some assignment g into M with g = x h M, h φ iff M, g φ for every assignment g into M with g = x h Notational Variations If φ(x 1,..., x n ) has at most x 1,..., x n as free variables then only the values of h(x 1 ),..., h(x n ) are important to determine M, h φ(x 1,..., x n ) In this case we also write M φ(a 1,..., a n ) where a i = h(x i ) Exercises Express each of the sentences below in FOL if possible. (These sentences are from the 3rd tutorial where you should express them in ALC. Some of the old solutions are given as help) Every person has a mother (Person has.mother) Penguins eat only fish (Penguin eats.fish). No smoker is a non-smoker (and vice versa) (Smoker Non Smoker ). Everybody loves Mary. Adam is not Eve (and vice versa). If Homer is married to Marge, then Marge is married to Homer. My friend s friends are also my friends. 36

38 ALC-Signatures vs. FO-Signatures An (extended) ALC-signature is a triple (A, R, O) where A is a set of atomic concept names and R is a set of role names and O is a set of individual names. A FO-signature is a set of constants, predicates, and relations. ALC-signatures and FO-signatures are essentially the same For didactic purposes, we write C f, R f, and o f for the elements of the FO-signature corresponding to an ALC-signature From now on, we fix an ALC-signature and its corresponding FO-signature (Basic) Interpretations vs. Structures An interpretation is a pair I = (, I ) where is a set (the domain) and I is an interpretation function. A structure is a pair M = (dom(m), M ) where dom(m) is a set of objects (the domain) and M is an interpretation function Again, interpretations and structures are essentially the same. Each interpretation corresponds uniquely to a structure and vice versa Concepts vs. Formulae ALC-concepts[0.3cm] and C C D C D R.C R.C Formulae[0.3cm] and φ φ ψ φ ψ x.φ x.φ Not exactly the same, but looks suspiciously similar. What about Variables? Assignments? Obviously, ALC has no variables, and therefore needs no assignments either ALC seems to quantify over predicates, something which is strictly forbidden in FOL If we want to relate ALC and FOL, we have to resolve this issue and we have to investigate how variables relate to ALC (Full) Interpretations vs. Structures Let I = (, I ) be an interpretation with matching model M In FOL: M φ(a 1,..., a n ) is either true or false (for a i dom(m)) In ALC: C I is a subset of Not really very similar yet, but... For d define: I C(d) iff d C I Now things start to look similar, provided φ(x) has at most one free variable So we might be able to relate ALC-concepts to formula with one free variable 37

39 The Standard Translation Basic idea: define a mapping from ALC-concepts to formulas with one free variable Of course, this translation should make sense The standard translation is also covered in the Modal Logic course in the 4th year There it is presented from a modal logic perspective The Formal Definition We define a function st x mapping an ALC-concept C to a formula, where x is a variable, by induction on the structure of C: C st x = C f (x), for C an atomic concept name st x = and st x = ( C) st x = (C st x ) (C D) st x = (C st x ) (D st x ) (C D) st x = (C st x ) (D st x ) ( R.C) st x = y(r f (x, y) (C st y )) where x /= y ( R.C) st x = y(r f (x, y) (C st y )) where x /= y Time for Some More Exercises Determine the standard translation of the following ALC-concepts: Person has.mother isson.( isson.oldperson isfriend. Sorcerer) Correctness Lemma So we have defined a function from ALC-concepts to formulae. Big whoop, there a many other such functions. But this one is special. Lemma 58. Let C be a concept. For all interpretations I = (, I ) and matching models M, and for all d, we have: d C I iff M C st x (d f ) Proof. Induction on the structure of C (whiteboard). Extensions to TBoxes and ABoxes So far, we have only translated ALC-concepts, but what about TBoxes? Straightforward: (C D) st x = x(cx st Dx st ) and Tx st = C D T (C D) st x And what about ABoxes? Also straightforward: (a C) st x = Ca st and ((a, b) R) st x = R f (a, b) and A st x = A A A st x Objects (and their counterparts constants in FOL) are only needed for translating ABoxes 38

40 Exercise: Extensions to More Expressive DLs Extend the standard translation to also cover the following concepts: Inverse roles. That is, extend the standard translation to cover the cases R.C and R.C and convince yourself (or your neighbour) that the correctness lemma still holds. Individuals (similar to the inverse role case). Quantified number restrictions (similar to the inverse role case). You might want to start with a concrete example, say 2R.P. Why Bother with DLs at all? If DLs are just a fragment of FOL, why do we consider them at all? xϕ FOL: powerful and expressive but undecidable DLs: decidable and (hopefully) still powerful enough Minor issue: a lot of people in industry seem to hate FOL so you have to disguise it to convince them to use it The Picture so far ALC FOL ( ) st x [.4cm] Remaining question: What parts of FOL are actually covered by the standard translation? Notation: We say that a FOL formula φ is covered iff there exists an ALC-concept C such that C st x = φ for some variable x More Than One Free Variable Surely all formulae with more than one free variable are not covered This is not surprising: the correctness lemma talks only about elements of the domain and not about pairs/triples/etc. of elements If we use more powerful description logics with operators on roles, we can also cover parts of FOL with two free variables Examples: role intersection ( R S.C), role union ( R S.C), role negation ( R.C), role composition ( R S.C), etc. 39

41 Syntactic Considerations It is easy to see that we cannot cover φ = y(p f (x) R f (x, y)) But we can cover ψ = y(r f (x, y) P f (x)) = ( R.P ) st x It is no big deal that we cannot cover φ because we can cover ψ, and φ and ψ are equivalent. Another example: R f (x, y) R f (x, y) which is equivalent to = st x So we should rather ask: for which formulae can we find an equivalent formula which is covered? Equivalent Formulae (Reminder) Two formulae φ(x) and ψ(y) are equivalent if the following holds:[.2cm] for all models M and all elements d dom(m), we have M φ(d) iff M ψ(d) What about R(x, x)? Can we find a formula which is covered and equivalent to R(x, x)? Lemma 59. There exists no ALC-concept C such that C st x is equivalent to R(x, x) Proof. Whiteboard Two Specific Interpretations I 1 and I 2 I 1 d I 2 R d R 1 d 2 [1cm] C I 1 = C I 2 = for all atomic concepts C R [1cm] Both models have The Missing Link We still have to show: Lemma 60. Let C be an ALC-concept. Then we have for all natural numbers i that d C I 1 iff d i C I 2. Proof. Induction on the structure of C. Last exercise of the day. Completing the Picture Lemma 61 (Van Benthem Characterisation Theorem). Let φ(x) be a formula. Then φ(x) is equivalent to the standard translation of an ALC-concept iff φ(x) is invariant for bisimulations. For a proof sketch and the exact definition of invariant for bisimulations see the 4th year course Modal Logic Our two interpretations I 1 and I 2 are bisimilar I 1 R f (d, d) but I 2 / R f (d 0, d 0 ) In particular R f (x, x) is not invariant for bisimulations 40

42 Conclusions ALC FOL ( ) st x [.4cm] DLs are not an island but are connected to other formalisms, for example FOL (but also modal logic) They can be seen as a decidable fragments of FOL 41

43 8 Implementation Techniques Optimisations in General What is an Optimisation? In general, optimisations do more work (and can be complicated to implement) in the hope that this makes the problem simpler. Observations. optimisations cannot change worst case complexity for all optimisations, there will be problems where they lead to huge improvements, and others where they lead to more overhead crucial: need to find the right balance Early Clash Detection Lemma 62. An ABox is unsatisfiable, if it contains both x C and x C for an arbitrary concept C. (Note: the ABox expansion algorithm just produces clashes on atomic concepts) Optimisation whenever x C is added to an ABox, check whether x nnf( C) is already present. If yes, declare unsatisfiable overhead: walk through the ABox after every expansion possible gain: avoid lots of ABox expansion steps Example: Early Clash Detection Example 63. Consider the ABox A = {x R.( C 1... C n, x R.C 1... C n }. We declare unsatisfiable after one expansion step using early clash detection. Example 64. Consider the A = {x R. S.A (B C}. This ABox is satisfiable (over the empty TBox) and early clash detection incurs computational overheads. Question. Are there examples where early clash detection also helps if an ABox is satisfiable? General Observation Recall. Two concepts C, D are called equivalent if C I = D I for all interpretations I. General Observation for satisfiability checking, we may always replace concepts by equivalent concepts many optimisations try and use that in a clever way. Example 65. R. and are equivalent C D and ( C D) are equivalent 42

44 Clashes on Equivalent Concepts Idea. an ABox containing both x C and x D for equivalent concepts C and D is unsatisfiable this happens for example if C = D Problem equivalent concepts are hard to recognise C and D are equivalent if both C D and D C are unsatisfiable Solution no need to recognise all equivalent concepts focus on those that are easy to recognise. Normalisation Idea. make equivalent concepts easy to recognise rewrite concepts into normal form and check for syntactic identity Basic Equivalences R.C R. C C D ( C D) C C C D D C Note: we can do without and try and recognise concepts that can be re-written into one another Caveat Re-written concepts are no longer in negation normal form. We need to adapt expansion rules accordingly! Normalisation, Formally Order in Conjunctions / Disjunctions Idea. the order of concepts in conjunctions and disjunctions doesn t matter represent conjunctions and disjunctions as sets of concepts (and similarly for disjunctions) C 1... C n {C 1,..., C n } advantage: comparing sets of concepts automatically factors out the order Change representation of concepts Represent C 1... C n by {C 1,..., C n }. 43

45 Simplification Idea. Apply a set of simple rules to normalise the way in which a concept is represented. Definition 66. The function simp on concepts is defined by the rules norm(a) = A (A atomic) norm( C) = simp( norm(c)) norm( {C 1,... C n }) = simp( {norm(c 1 ),..., norm(c n )}) norm( {C 1,..., C n }) = norm( { C 1,..., C n }) norm( R.C) = simp( R.norm(C)) norm( R.C) = norm( R. C) where conjunctions / disjunctions are represented using sets and simp simplifies a concept definition (next). Simplification Definition 67. The function simp is defined on atomic conceptsnegated concepts and universal restrictions as follows: simp(a) = A (A atomic) (if C = ) (if C = ) simp( C) = simp(d) (if C = D) simp(c) (otherwise) (if simp(c) = ) simp( R.C) = R.simp(C) (otherwise) (NB: simplification produces equivalent concepts!) Simplification, continued Definition 68. The function simp is defined on conjunctions as follows: (if S or C, C S (if S = ) simp( S) = simp( (S { })) ( if S) simp( ((S P) { P}) (if P S) S (otherwise) Note use of the conjunctions of sets -notation flattening conjunctions of conjunctions 44