1. Chapter 5A Semantic Web Primer 1 Chapter 5 Logic and Inference: Rules Grigoris Antoniou Paul Groth Frank van Harmelen Rinke Hoekstra

2. Chapter 5A Semantic Web Primer 2 Lecture Outline 1. Introduction 2. Example of Monotonic Rules: Family Relationships 3. Monotonic Rules: Syntax 4. OWL2 RL: Description Logic Meets Rules 5. Rule Interchange Format: RIF 6. Semantic Web Rules Language (SWRL) 7. Rules in SPARQL: SPIN 8. Nonmonotonic Rules: Motivation and Syntax 9. Example of Nonmonotonic Rules: Brokered Trade 10. Rule Markup Language (RuleML) 11. Summary

3. Chapter 5A Semantic Web Primer 3 Knowledge Representation The subjects presented so far were related to the representation of knowledge （知识的表示） – Knowledge about content of Web resources – Knowledge about concepts and their relationships Knowledge representation was studied long before the emergence of WWW in AI, or even earlier in philosophy (e.g. ancient Greek Aristotle, father of logic) Logic is still the foundation of KR, particularly in the form of predicate logic （谓词逻辑） (aka first-order logic ，一 阶逻辑 )

4. Chapter 5A Semantic Web Primer 4 The Importance of Logic High-level language for expressing knowledge in a transparent way High expressive power Well-understood formal semantics, assigning an unambiguous meaning to logical statements A precise notion of logical consequence （逻辑后承）, determining whether a statement follows semantically from a set of other statements (premises) – The primary motivation of logic was the study of objective laws （客观规律） of logical consequence

5. Chapter 5A Semantic Web Primer 5 The Importance of Logic Proof systems （证明系统） can automatically derive statements syntactically from a set of premises There exist proof systems for which semantic logical consequence coincides with （相符） syntactic derivation （语法推导） – Soundness （正确） : all derived statements semantically follow premises – Completeness （完备） : all logical consequences of premises can be derived

6. Chapter 5A Semantic Web Primer 6 The Importance of Logic Predicate logic （谓词逻辑） is unique in the sense that sound and complete proof systems do exist – Not for more expressive logics (higher-order logics) Because of the existence of proof systems, logic can provide explanations for answers by tracing a proof that leads to a logical consequence

7. Chapter 5A Semantic Web Primer 7 Specializations of Predicate Logic: RDF and OWL2 Profiles RDF and OWL2 profiles are specializations （特殊形 式） of predicate logic They provide a syntax that fits well with the intended use （预定需求） – Web languages based on tags They define reasonable subsets （可推理子集） of logic – Trade-off between the expressive power and the computational complexity – Most OWL variants correspond to a description logic, a subset of predicate logic

8. Chapter 5A Semantic Web Primer 8 Specializations of Predicate Logic: Horn Logic (Rule Systems) A rule has the form: A 1,..., A n  B – A i and B are atomic formulas ( 原子公式 ， a formula that contains no logical connectives or equivalently a formula that has no strict subformulas )logical connectivessubformulas There are 2 ways of reading such a rule: – Deductive rules （演绎规则） : If A 1,..., A n are known to be true, then B is also true – Reactive rules （反应式规则） : If the conditions A 1,..., A n are true, then carry out the action B

9. Chapter 5A Semantic Web Primer 9 Description Logics vs. Horn Logic Orthogonal （正交的） : neither of them is a subset of the other It is impossible to define the class of happy spouses as those who are married to their best friend in description logics – This can be done easily using rules: married(X, Y), bestFriend(X, Y)  happySpouse(X, Y)

10. Chapter 5A Semantic Web Primer 10 Description Logics vs. Horn Logic On the other hand, rules cannot (in the general case) assert – Negation/complement of classes – Disjunctive/union information (e.g. a person is either a man or a woman) – Existential quantification （存在量化） (e.g. all persons have a father)

11. Chapter 5A Semantic Web Primer 11 Monotonic vs. Non-monotonic Rules Example: An online vendor wants to give a special discount if it is a customer’s birthday Solution 1 R1: If birthday, then special discount R2: If not birthday, then not special discount But what happens if a customer refuses to provide his birthday due to privacy concerns?

12. Chapter 5A Semantic Web Primer 12 Monotonic vs. Non-monotonic Rules Solution 2 R1: If birthday, then special discount R2’: If birthday is not known, then not special discount Solves the problem but: – The premise of rule R2' is not within the expressive power of predicate logic – We need a new kind of rule system

13. Chapter 5A Semantic Web Primer 13 Monotonic vs. Non-monotonic Rules The solution with rules R1 and R2 works in case we have complete information about the situation The new kind of rule system will find application in cases where the available information is incomplete Predicate logic and its special cases are monotonic （单调的） in the following sense

14. Chapter 5A Semantic Web Primer 14 Monotonic vs. Non-monotonic Rules If a conclusion can be drawn, it remains valid even if new knowledge becomes available But, the conclusion “not special discount” in rule R2’ may become invalid if the customer’s birthday becomes known at a later stage R2’: If birthday is not known, then not special discount This behaviour is nonmonotonic （非单调的） because the addition of new information leads to a loss of consequence

15. Chapter 5A Semantic Web Primer 15 Rules on the Semantic Web Rule technology has been around for decades, finding extensive use in practice and reaching significant maturity – It is far more difficult to standardize this area in the context of the (Semantic) Web A W3C working group has developed the Rule Interchange Format (RIF) standard – was designed for the exchange of rules across different applications

16. Chapter 5A Semantic Web Primer 16 Rules on the Semantic Web Intended to serve as an interchange format among different rule systems, RIF combines many features, thus being quite complex There are various alternatives – Rules over RDF can be expressed using SPARQL constructs; one recent proposal is SPIN – SWRL couples OWL DL functionality with certain types of rules – OWL2 RL

17. Chapter 5A Semantic Web Primer 17 Lecture Outline 1. Introduction 2. Example of Monotonic Rules: Family Relationships 3. Monotonic Rules: Syntax 4. OWL2 RL: Description Logic Meets Rules 5. Rule Interchange Format: RIF 6. Semantic Web Rules Language (SWRL) 7. Rules in SPARQL: SPIN 8. Nonmonotonic Rules: Motivation and Syntax 9. Example of Nonmonotonic Rules: Brokered Trade 10. Rule Markup Language (RuleML) 11. Summary

18. Chapter 5A Semantic Web Primer 18 Family Relationships Facts in a database about relationships: – mother (X,Y), X is the mother of Y – father (X,Y)X is the father of Y – male(X)X is male – female(X), X is female Inferred relationships using rules. First we can define parent: a parent is either a father or a mother mother (X,Y)  parent (X,Y) father (X,Y)  parent (X,Y)

19. Chapter 5A Semantic Web Primer 19 Inferred Relationships We can also define a brother to be a male person sharing a parent: male (X), parent (P, X), parent (P, Y), notSame (X, Y)  brother (X, Y) Similarly, we can define sister as follows female(X), parent(P,X), parent(P,Y), notSame(X,Y)  sister(X,Y)

20. Chapter 5A Semantic Web Primer 20 Inferred Relationships An uncle is a brother of a parent: brother (X,P), parent (P,Y)  uncle (X,Y) A grandmother is the mother of a parent mother (X,P), parent (P,Y)  grandmother (X,Y) An ancestor is either a parent or an ancestor of a parent parent (X,Y)  ancestor (X,Y) ancestor (X,P), parent (P,Y)  ancestor (X,Y)

21. Chapter 5A Semantic Web Primer 21 Lecture Outline 1. Introduction 2. Example of Monotonic Rules: Family Relationships 3. Monotonic Rules: Syntax 4. OWL2 RL: Description Logic Meets Rules 5. Rule Interchange Format: RIF 6. Semantic Web Rules Language (SWRL) 7. Rules in SPARQL: SPIN 8. Nonmonotonic Rules: Motivation and Syntax 9. Example of Nonmonotonic Rules: Brokered Trade 10. Rule Markup Language (RuleML) 11. Summary

22. Chapter 5A Semantic Web Primer 22 Monotonic Rules – Syntax loyalCustomer (X), age(X) > 60  discount (X) We distinguish some ingredients of rules: – Variables which are placeholders for values: X – Constants denote fixed values: 60 – Predicates relate objects: loyalCustomer, discount – Function symbols which return a value for certain arguments: age

23. Chapter 5A Semantic Web Primer 23 Rules B 1,..., B n  A A, B 1,..., B n are atomic formulas A is the head of the rule B 1,..., B n are the premises (body of the rule) The commas in the rule body are read conjunctively （合 取） Variables may occur in A, B 1,..., B n – loyalCustomer (X), age(X) > 60  discount (X) (applied to any customer) – Implicitly universally quantified （默认全称量化）

24. Chapter 5A Semantic Web Primer 24 Rules In summary, a rule r B 1,..., B n  A is interpreted as the following formula, denoted by pl(r):  X 1 ···  X k ((B 1  ···  B n )  A) Or equivalently,  X 1 ···  X k (A  ¬ B 1 ...  ¬ B n ) 其中 pl(r) 为 r 的谓词逻辑解释

25. Chapter 5A Semantic Web Primer 25 Facts and Logic Programs A fact （事实） is an atomic formula, e.g. loyalCustomer(a345678) The variables of a fact are implicitly universally quantified A logic program （逻辑程序） P is a finite set of facts and rules Its predicate logic translation （谓词逻辑解释） pl(P) is the set of all predicate logic interpretations of rules and facts in P

26. Chapter 5A Semantic Web Primer 26 Goals A goal （目标） denotes a query G asked to a logic program The form: B 1,..., B n  If n = 0 we have the empty goal 

27. Chapter 5A Semantic Web Primer 27 Predicate Logic Interpretation of Goals  X 1 ···  X k (¬ B 1 ...  ¬ B n ) – Where X 1,..., X k are all variables occurring in B 1,..., B n – Same as pl(r), with the rule head omitted Equivalently: ¬  X 1...  X k (B 1 ...  B n ) – Suppose we know p(a) and we have the goal p(X)  – We want to know if there is a value for which p(X) is true – We expect a positive answer because of the fact p(a) – Thus p(X) is existentially quantified

28. Chapter 5A Semantic Web Primer 28 Why Negate the Formula? We use a proof technique from mathematics called proof by contradiction （反证法） : – Prove that A follows from B by assuming that A is false and deriving a contradiction, when combined with B In logic programming we prove that a goal can be answered positively by negating the goal and proving that we get a contradiction using the logic program – E.g., given the following logic program we get a logical contradiction

29. Chapter 5A Semantic Web Primer 29 An Example 逻辑程序： p(a) 否定目标： ¬  X p(X) The 2nd formula says that no element has the property p The 1st formula says that the value of a does have the property p Thus  Xp(X) follows from p(a)

30. Chapter 5A Semantic Web Primer 30 Lecture Outline 1. Introduction 2. Example of Monotonic Rules: Family Relationships 3. Monotonic Rules: Syntax 4. OWL2 RL: Description Logic Meets Rules 5. Rule Interchange Format: RIF 6. Semantic Web Rules Language (SWRL) 7. Rules in SPARQL: SPIN 8. Nonmonotonic Rules: Motivation and Syntax 9. Example of Nonmonotonic Rules: Brokered Trade 10. Rule Markup Language (RuleML) 11. Summary

31. Chapter 5A Semantic Web Primer 31 OWL2 RL: DL Meets Rules Description logics and horn logic are orthogonal To integrate both into one framework, the simplest approach is to consider the intersection of both logics The part of one language can be translated in a semantics-preserving way to the other―OWL2 RL seeks to capture this fragment of OWL

32. Chapter 5A Semantic Web Primer 32 OWL2 RL: DL Meets Rules Advantages: – From the modeler’s perspective, one can use either OWL or rules for modeling purposes – From the implementation perspective, either description logic reasoners or deductive rule systems can be used We will show how many constructs of RDFS and OWL2 RL can be expressed in Horn logic and some constructs that cannot be

33. RDF and RDF Schema A triple of the form (a,P,b) in RDF can be expressed as a fact P(a,b) An instance declaration of the form type(a,C) (stating a is an instance of class C) can be expressed as C(a) The fact that C is a subclass (or subproperty) of D can be expressed as C(X)  D(X) C is the domain of property P: P(X, Y)  C(X) Chapter 5A Semantic Web Primer 33

34. OWL equivalentClass(C,D) (or equivalentProperty) can be expressed by the pair of rules – C(X)  D(X) – D(X)  C(X) Transitivity of a property P can be expressed as – P(X,Y),P(Y,Z)  P(X,Z) Chapter 5A Semantic Web Primer 34

35. OWL The intersection of C 1 and C 2 is a subclass of D can be expressed as – C 1 (X), C 2 (X)  D(X) C is subclass of the intersection of D 1 and D 2 can be expressed as – C(X)  D 1 (X) – C(X)  D 2 (X) Chapter 5A Semantic Web Primer 35

36. OWL The union of C 1 and C 2 is a subclass of D can be expressed by the pair of rules – C 1 (X)  D (X) – C 2 (X)  D (X) The opposite direction cannot be expressed in Horn logic. There are cases where the translation is possible – For instance, D 1 is a subclass of D 2, and the rule C(X)  D 2 (X) is sufficient to express C is a subclass of union of D 1 and D 2 Chapter 5A Semantic Web Primer 36

37. Restrictions in OWL :C rdfs:subClassOf [rdf:type owl:Restriction; owl:onProperty :P; owl:allValuesFrom :D]. can be expressed in Horn logic as follows: C(X), P(X, Y)  D(Y) – The opposite direction cannot in general be expressed Chapter 5A Semantic Web Primer 37

38. Restrictions in OWL [rdf:type owl:Restriction; owl:onProperty :P; owl:someValuesFrom :D] rdfs:subClassOf :C. can be expressed in Horn logic as follows: – P(X, Y), D(Y)  C(X) – The opposite direction cannot in general be expressed – Also, cardinality constraints and complement classes cannot be expressed Chapter 5A Semantic Web Primer 38

39. Chapter 5A Semantic Web Primer 39 Lecture Outline 1. Introduction 2. Example of Monotonic Rules: Family Relationships 3. Monotonic Rules: Syntax 4. OWL2 RL: Description Logic Meets Rules 5. Rule Interchange Format: RIF 6. Semantic Web Rules Language (SWRL) 7. Rules in SPARQL: SPIN 8. Nonmonotonic Rules: Motivation and Syntax 9. Example of Nonmonotonic Rules: Brokered Trade 10. Rule Markup Language (RuleML) 11. Summary

40. Chapter 5A Semantic Web Primer 40 Rule Interchange Format: RIF Rule Technology have existed for decades, exhibiting a broad variety The aim of the W3C Rule Interchange Format Working Group was not to develop a new rule language that would fit all purposes Rather, it would focus on the interchange among various rule systems on the Web The approach was to develop a family of languages, called dialects: logic-based dialects and rules with actions

41. Chapter 5A Semantic Web Primer 41 Logic-based Dialects These are meant to include rule languages that are based on some form of logic, e.g. first-order logic and various logic programming approaches The concrete dialects developed so far under this branch are: – RIF Core, essentially corresponding to function-free Horn logic – RIF Basic Logic Dialect (BLD), essentially corresponding to Horn logic with equality

42. Chapter 5A Semantic Web Primer 42 Rules with Actions These are meant to include production systems （生产系统） and reactive rules （反应规则） The concrete dialect developed so far in this branch is – Production Rule Dialect (RIF-PRD)

43. Chapter 5A Semantic Web Primer 43 RIF-BLD Expected to support a uniform set of commonly used data types (integer, boolean, string, date), “built-in ” （ 内 建 ） predicates (numeric-greater than, starts-with), and functions (replace, numeric-subtract) ranging over these data types Suppose we wish to express the following rules: – An actor is a movie star if he has starred in more than three successful movies, produced in a span of at least five years – A film is successful if it has received critical acclaim （高度 评价） or was financially successful

44. Chapter 5A Semantic Web Primer 44 Rules Expressed in RIF-BLD Document ( Prefix(func Prefix(pred Prefix(rdfs Prefix(imdbrel Prefix(dbpedia Prefix(ibdbrel Group(

45. Chapter 5A Semantic Web Primer 45 Rules Expressed in RIF-BLD Forall ?Actor ?Film ?Year ( If And( dbpedia:starring(?Film ?Actor) dbpedia:dateOfFilm(?Film ?Year) Then dbpedia:starredInYear(?Film ?Actor ?Year) }

46. Chapter 5A Semantic Web Primer 46 Rules Expressed in RIF-BLD Forall ?Actor ( if (Exists ?Film1 ?Film2 ?Film3 ?Year1 ?Year2 ?Year3 And (dbpedia:starredInYear(?Film1 ?Actor ?Year1) dbpedia:starredInYear(?Film2 ?Actor ?Year2) dbpedia:starredInYear(?Film3 ?Actor ?Year3) External (pred:numeric-greater-than ( External(func:numeric-subtract ?Year1 ?Year3) 5)))

47. Chapter 5A Semantic Web Primer 47 Rules Expressed in RIF-BLD dbpedia:successful(?Film1) dbpedia:successful(?Film2) dbpedia:successful(?Film3) External (pred:literal-not-identical(?Film1 ?Film2)) External (pred:literal-not-identical(?Film1 ?Film3)) External (pred:literal-not-identical(?Film2 ?Film3)) } Then dbpedia:movieStar(?Actor) )

48. Chapter 5A Semantic Web Primer 48 Rules Expressed in RIF-BLD Forall ?Film ( If Or ( External(pred:numeric-greater-than( dbpedia:criticalRating(?Film 8)) External(pred:numeric-greater-than( dbpedia:boxOfficeGross(?Film) 100000000))) Then dbpedia:successful(?Film) )

49. Chapter 5A Semantic Web Primer 49 Rules Expressed in RIF-BLD The use of External in applying built-in predicate The use of Group to put together a number of rules The use of frames – The basic idea is to represent objects as frames and their properties as slots – Information expressed in RIF-BLD using the notation oid[slot1->value1…slotn->valuen]

50. Chapter 5A Semantic Web Primer 50 Compatibility with RDF and OWL A major feature of RIF is that it is compatible with the RDF and OWL standards – One can reason a combination of RIF, RDF, and OWL documents The basic idea of combining RIF with RDF is to represent RDF triples using RIF frame formulas – A triple s p o is represented as s[p -> o] – The semantic definitions are the triple is satisfied iff the corresponding RIF frame formula is satisfied, too

51. Chapter 5A Semantic Web Primer 51 Compatibility with RDF and OWL For example, if the RDF triple ex:GoneWithTheWind ex:FilmYear ex:1939 is true, then so is the RIF fact ex:GoneWithTheWind[ex:FilmYear->ex:1939]

52. Chapter 5A Semantic Web Primer 52 Compatibility between RDF and RIF Given the RIF rule (which states that the Hollywood Production Code was in place between 1930 and 1968) Group( Forall ?Film( If And( ?Film[ex:FilmYear->?Year] External(pred:dateGreaterThan(?Year 1930)) External(pred:dateGreaterThan(1968 ?Year))) Then ?Film[ex:HollywoodProductionCode->ex:True] ) ex:GoneWithTheWind[ex:HolywoodProductionCode->ex:True] ex:GoneWithTheWind ex:HolywoodProductionCode ex:True]

53. Chapter 5A Semantic Web Primer 53 Compatibility between OWL and RIF Similar techniques are used to achieve compatibility between OWL and RIF The main features are: – The semantics of OWL and RIF are compatible – One can infer conclusions from certain combinations of OWL axioms and RIF knowledge – OWL2 RL can be implemented in RIF

54. Chapter 5A Semantic Web Primer 54 OWL2 RL in RIF OWL2 RL is partially described by a set of first-order rules, thus forming the basis for implementation using rules OWL2 RL rules can be categorized in four categories: – Triple pattern （三元组模式） rules – Inconsistency （不一致） rules – List （列表） rules – Datatype （数据类型） rules

55. Chapter 5A Semantic Web Primer 55 Triple Pattern Rules Derive certain RDF triples from a conjunction of RDF triple patterns The triple pattern rules take the form: If then T(s 1, p 1, o 1 ) T(s, p, o)... T(s n, p n, o n ) where each argument to the T predicate may be a variable, an IRI(Internationalized Resource Identifier) or literal value

56. Chapter 5A Semantic Web Primer 56 Triple Pattern Rules Translation of these rules to RIF is straightforward: Group ( Forall ?v1... ?vn ( s[p->o] :- And( s 1 [p 1 ->o 1 ]... s n [p n ->o n ] )) )

57. Chapter 5A Semantic Web Primer 57 Inconsistency Rules Such rules indicate inconsistencies in the original RDF graph Rules can be easily represented in RIF as rules with conclusion rif:error Group ( Forall ?P1 ?P2 ?X ?Y ( rif:error :- And( ?P1[owl:propertyDisjointWith->?P2] ?X[?P1-> ->?Y] ?X[?P2->?Y])) )

58. Chapter 5A Semantic Web Primer 58 List Rules A number of OWL2 RL rules involve processing OWL expressions that include RDF lists Two approaches are possible to express these rules in RIF – One may use recursive （递归） rules to traverse RDF graphs at runtime – One may take a preprocessing approach in which rules are directly instantiated for the lists that occurs in the input RDF graph

59. Chapter 5A Semantic Web Primer 59 Datatype Rules Provide type checking （类型检查） and value equality/inequality checking for typed literals For example, such rules may derive owl:sameAs triples for literals with the same value (e.g. 1 and 1.0) An inconsistency occurs when a literal is specified to be an instance of a data type but its value is outside the value space of that data type

60. Chapter 5A Semantic Web Primer 60 Lecture Outline 1. Introduction 2. Example of Monotonic Rules: Family Relationships 3. Monotonic Rules: Syntax 4. OWL2 RL: Description Logic Meets Rules 5. Rule Interchange Format: RIF 6. Semantic Web Rules Language (SWRL) 7. Rules in SPARQL: SPIN 8. Nonmonotonic Rules: Motivation and Syntax 9. Example of Nonmonotonic Rules: Brokered Trade 10. Rule Markup Language (RuleML) 11. Summary

61. Semantic Web Rules Language A rule in SWRL has the form B 1, …, B n  A 1, …, A m – Commas denote conjunction on both sides – A 1, …, A m, B 1, …, B n can be of the form C(x), P(x,y), sameAs(x,y), or differentFrom(x,y) – where C is an OWL description, P is an OWL property, and x, y are Datalog variables, OWL individuals, or OWL data values Chapter 5A Semantic Web Primer 61

62. SWRL Properties If the head of a rule has more than one atom, the rule can be transformed to an equivalent set of rules with one atom in the head The main complexity: arbitrary OWL expressions, such as restrictions, can appear in the head or body of a rule This feature adds significant expressive power to OWL, but at the high price of undecidability （不可判 定性） Chapter 5A Semantic Web Primer 62

63. OWL2 RL vs. SWRL OWL2 RL uses a very conservative approach, trying to combine the advantages of both languages (description logics and function-free rules) in their common sublanguage SWRL takes a more maximalist approach and unites their respective expressivities – The challenge is to identify sublanguages of SWRL that find the right balance between expressive power and computational tractability Chapter 5A Semantic Web Primer 63

64. Chapter 5A Semantic Web Primer 64 Lecture Outline 1. Introduction 2. Example of Monotonic Rules: Family Relationships 3. Monotonic Rules: Syntax 4. OWL2 RL: Description Logic Meets Rules 5. Rule Interchange Format: RIF 6. Semantic Web Rules Language (SWRL) 7. Rules in SPARQL: SPIN 8. Nonmonotonic Rules: Motivation and Syntax 9. Example of Nonmonotonic Rules: Brokered Trade 10. Rule Markup Language (RuleML) 11. Summary

65. Chapter 5A Semantic Web Primer 65 Rules in SPARQL: SPIN Rules can be expressed in SPARQL using its CONSTRUCT feature For instance, the rule grandparent(X,Z)<-parent(Y,Z), parent(X,Y) can be expressed as CONSTRUCT { ?X grandParent ?Z. } WHERE { ?Y parent ?Z. ?X parent ?Y. }

66. Chapter 5A Semantic Web Primer 66 Ideas and Features of SPIN Uses ideas of object-oriented modeling in associating rules to classes – Rules may represent behavior of that class Expresses rules by SPARQL constructs CONSTRUCT, DELETE, and INSERT, and constraints using ASK Provides abstract mechanisms for rules using Templates, which encapsulates parameterized SPARQL queries

67. Chapter 5A Semantic Web Primer 67 OWL2 RL Rules Expressed in SPIN For example, the rule C 2 (X) <- C 1 (X), equivalentClass(C 1, C 2 ) can be expressed in SPARQL as CONSTRUCT { ?X rdf:type ?C2. } WHERE { ?X rdf:type ?C1. ?C1 equivalentClass ?C2. }

68. Chapter 5A Semantic Web Primer 68 Lecture Outline 1. Introduction 2. Example of Monotonic Rules: Family Relationships 3. Monotonic Rules: Syntax 4. OWL2 RL: Description Logic Meets Rules 5. Rule Interchange Format: RIF 6. Semantic Web Rules Language (SWRL) 7. Rules in SPARQL: SPIN 8. Nonmonotonic Rules: Motivation and Syntax 9. Example of Nonmonotonic Rules: Brokered Trade 10. Rule Markup Language (RuleML) 11. Summary

69. Chapter 5A Semantic Web Primer 69 Motivation – Negation in Rule Head In nonmonotonic rule systems, a rule may not be applied even if all premises are known because we have to consider contrary reasoning chains Now we consider defeasible rules （ 可废止的规则 ） that can be defeated by other rules Negated atoms （ 否定的原子公式 ） may occur in the head and the body of rules, to allow for conflicts – p(X)  q(X) – r(X)  ¬q(X)

70. Chapter 5A Semantic Web Primer 70 Defeasible Rules p(X)  q(X) r(X)  ¬q(X) Given also the facts p(a) and r(a), we conclude neither q(a) nor ¬q(a) – typical example of 2 rules blocking （阻塞） each other Conflict may be resolved using priorities among rules （规则间优先关系） Suppose we knew somehow that the 1st rule is stronger than the 2nd, then we could derive q(a)

71. Chapter 5A Semantic Web Primer 71 Origin of Rule Priorities Higher authority （ 更权威 ） – E.g. in law, federal law preempts state law – E.g. in business administration, higher management has more authority than middle management Recency （ 最近 ） Specificity （ 特殊 ） – A typical example is a general rule with some exceptions We abstract from the specific prioritization principle （ 优先原则 ） – We assume the existence of an external priority relation on the set of rules

72. Chapter 5A Semantic Web Primer 72 Rule Priorities r 1 : p(X)  q(X) r 2 : r(X)  ¬q(X) r 1 > r 2 Rules have a unique label The priority relation to be acyclic （ 无环的 ） – That is, it is impossible to have cycles of the form r 1 > r 2 > r 3… > r n >r 1

73. Chapter 5A Semantic Web Primer 73 Competing Rules In simple cases two rules are competing （冲突） only if one head is the negation （否定） of the other But in many cases once a predicate p is derived, some other predicates are excluded from holding – E.g., an investment consultant may base his recommendations on three levels of risk investors are willing to take: low, moderate, and high – Only one risk level per investor is allowed to hold

74. Chapter 5A Semantic Web Primer 74 Defeasible Rules: Syntax r : L 1,..., L n  L r is the label （标号） {L 1,..., L n } is the body (or premises) and L the head L, L 1,..., L n are positive or negative literals A literal is an atomic formula p ( t 1,..., t m ) or its negation ¬ p ( t 1,..., t m ) No function symbols may occur in the rule

75. Chapter 5A Semantic Web Primer 75 Defeasible Logic Programs A defeasible logic program （ 可废止逻辑程序 ） is a triple (F,R,>) consisting of – a set F of facts – a finite set R of defeasible rules – an acyclic binary relation > on R A set of pairs r > r' where r and r' are labels of rules in R

76. Chapter 5A Semantic Web Primer 76 Lecture Outline 1. Introduction 2. Example of Monotonic Rules: Family Relationships 3. Monotonic Rules: Syntax 4. OWL2 RL: Description Logic Meets Rules 5. Rule Interchange Format: RIF 6. Semantic Web Rules Language (SWRL) 7. Rules in SPARQL: SPIN 8. Nonmonotonic Rules: Motivation and Syntax 9. Example of Nonmonotonic Rules: Brokered Trade 10. Rule Markup Language (RuleML) 11. Summary

77. Chapter 5A Semantic Web Primer 77 Brokered Trade Brokered trades （有经纪人的交易） take place via an independent third party, the broker （经纪人） The broker matches the buyer’s requirements and the sellers’ capabilities, and proposes a transaction when both parties can be satisfied by the trade The application is apartment renting, an activity that is common and often tedious and time-consuming

78. Chapter 5A Semantic Web Primer 78 The Potential Buyer’s Requirements Carlos is looking for an apartment – At least 45 sq m with at least 2 bedrooms – Elevator if on 3rd floor or higher – Pet animals must be allowed Carlos is willing to pay: – $ 300 for a centrally located 45 sq m apartment – $ 250 for a similar flat in the suburbs – An extra $ 5 per square meter for a larger apartment – An extra $ 2 per square meter for a garden – He is unable to pay more than $ 400 in total If given the choice, he would go for the cheapest option His second priority is the presence of a garden His lowest priority is additional space

79. Chapter 5A Semantic Web Primer 79 Formalization of Carlos’s Requirements – Predicates Used apartment(x) stating x is an apartment size(x,y) y is the size of apartment x (in sq m) bedrooms(x,y) x has y bedrooms price(x,y) y is the price for x floor(x,y) x is on the y-th floor garden(x,y) x has a garden of size y

80. Chapter 5A Semantic Web Primer 80 Formalization of Carlos’s Requirements – Predicates Used elevator(x) there is an elevator in the house of x pets(x) pets are allowed in x central(x) x is centrally located acceptable(x), flat x satisfies Carlos’s requirements offer(x,y), Carlos is willing to pay $ y for flat x

81. Chapter 5A Semantic Web Primer 81 Formalization of Carlos’s Requirements – Rules r1: apartment(X)  acceptable(X) r2: bedrooms(X,Y), Y < 2  ¬acceptable(X) r3: size(X,Y), Y < 45  ¬acceptable(X) r4: ¬pets(X)  ¬acceptable(X) r5: floor(X,Y), Y > 2,¬elevator(X)  ¬acceptable(X) r6: price(X,Y), Y > 400  ¬acceptable(X) r2 > r1, r3 > r1, r4 > r1, r5 > r1, r6 > r1

82. Chapter 5A Semantic Web Primer 82 Formalization of Carlos’s Requirements – Rules r7: size(X,Y), Y ≥ 45, garden(X,Z), central(X)  offer(X, 300 + 2*Z + 5*(Y − 45)) r8: size(X,Y), Y ≥ 45, garden(X,Z), ¬central(X)  offer(X, 250 + 2*Z + 5(Y − 45)) r9: offer(X,Y), price(X,Z), Y < Z  ¬acceptable(X) r9 > r1

83. Chapter 5A Semantic Web Primer 83 Representation of Available Apartments bedrooms(a1,1) size(a1,50) central(a1) floor(a1,1) ¬elevator(a1) pets(a1) garden(a1,0) price(a1,300)

84. Chapter 5A Semantic Web Primer 84 Representation of Available Apartments FlatBedroomsSizeCentralFloorElevatorPetsGardenPrice a1150yes1noyes0300 a2245yes0noyes0335 a3265no2 yes0350 a4255no1yesno15330 a5355yes0noyes15350 a6260yes3no 0370 a7365yes1noyes12375

85. Chapter 5A Semantic Web Primer 85 Determining Acceptable Apartments If we match Carlos’s requirements and the available apartments, we see that – flat a1 is not acceptable because it has one bedroom only (rule r2) – flats a4 and a6 are unacceptable because pets are not allowed (rule r4) – for a2, Carlos is willing to pay $ 300, but the price is higher (rules r7 and r9) – flats a3, a5, and a7 are acceptable (rule r1)

86. Chapter 5A Semantic Web Primer 86 Selecting an Apartment r10: acceptable(X)  cheapest(X) r11: acceptable(X), price(X,Z), acceptable(Y), price(Y,W), W < Z  ¬cheapest(X) r12: cheapest(X)  largestGarden(X) r13: cheapest(X), garden(X,Z), cheapest(Y), garden(Y,W), W > Z  ¬largestGarden(X)

87. Selecting an Apartment r14: largestGarden(X)  rent(X) r15: largestGarden(X), size(X,Z), largestGarden(Y), size(Y,W), W > Z  ¬ rent(X) r11 > r10, r13 > r12, r15 > r14 The final selection is a5 and Carlos will soon move in Chapter 5A Semantic Web Primer 87

88. Chapter 5A Semantic Web Primer 88 Lecture Outline 1. Introduction 2. Example of Monotonic Rules: Family Relationships 3. Monotonic Rules: Syntax 4. OWL2 RL: Description Logic Meets Rules 5. Rule Interchange Format: RIF 6. Semantic Web Rules Language (SWRL) 7. Rules in SPARQL: SPIN 8. Nonmonotonic Rules: Motivation and Syntax 9. Example of Nonmonotonic Rules: Brokered Trade 10. Rule Markup Language (RuleML) 11. Summary

89. RuleML RuleML is a long-running effort to develop markup of rules on the Web – A family of rule markup languages, corresponding to different kinds of rule languages The RuleML family provides descriptions of rule markup language in XML Chapter 5A Semantic Web Primer 89

90. Example: Customer Discount The discount for a customer buying a product is 7.5 percent if the customer is premium and the product is luxury discount customer product 7.5 percent Chapter 5A Semantic Web Primer 90

91. Example: Customer Discount premioum customer luxury product Chapter 5A Semantic Web Primer 91

92. SWRL using RuleML SWRL is an extension of RuleML, and its use is straightforward For instance, we show the representation of the rule brother(X,Y), childOf(Z,Y)  uncle(X,Z) in the XML syntax of SWRL using RuleML 1.0 Chapter 5A Semantic Web Primer 92

93. Example: Uncle of brother(X,Y), childOf(Z,Y)  uncle(X,Z) <swrlx : individualPropertyAtom swrlx : property=“uncle”> X Z Chapter 5A Semantic Web Primer 93

94. Example: Uncle of (2) <swrlx : individualPropertyAtom swrlx : property=“brother”> X Y <swrlx : individualPropertyAtom swrlx : property=“childOf”> Chapter 5A Semantic Web Primer 94

95. Example: Uncle of (3) Z Y Chapter 5A Semantic Web Primer 95

96. Chapter 5A Semantic Web Primer 96 Lecture Outline 1. Introduction 2. Example of Monotonic Rules: Family Relationships 3. Monotonic Rules: Syntax 4. OWL2 RL: Description Logic Meets Rules 5. Rule Interchange Format: RIF 6. Semantic Web Rules Language (SWRL) 7. Rules in SPARQL: SPIN 8. Nonmonotonic Rules: Motivation and Syntax 9. Example of Nonmonotonic Rules: Brokered Trade 10. Rule Markup Language (RuleML) 11. Summary

97. Chapter 5A Semantic Web Primer 97 Summary Rules on the (semantic) Web form a very rich and heterogeneous landscape Horn logic is a subset of predicate logic that allows efficient reasoning, orthogonal to description logics Horn logic is the basis of monotonic rules RIF is a new standard for rules on the Web. Its logical dialect BLD is based on Horn logic

98. Summary OWL2 RL is essentially the intersection of DL and Horn logic and can be embedded in RIF SWRL is a much richer rule language, combining DL features with restricted types of rules Nonmonotonic rules are useful in situations where the available information is incomplete They are rules that may be overridden by contrary evidence Chapter 5A Semantic Web Primer 98

99. Summary Priorities are used to resolve some conflicts between nonmonotonic rules Representation of XML-like languages, such as those provided by RIF and RuleML, is straightforward Chapter 5A Semantic Web Primer 99