Inference Techniques

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1 Bernhard Schueler Inference Techniques with respect to applications in the Semantic Web

2 Overview Example [Decker98]: A A Query and Inference Service for RDF From RDF to logic Lots of logic: F-Logic, F DLs,, OIL, ALC, FOL, : Resolution, Magic sets, tableaux calculus 2

3 Example: Inferencing with RDF Give me all resources which are about In [Decker98] a program is described, which can produce this resources from a database containing RDF descriptions of Subjects of resources, Synonyms, Broader terms. It further contains rules to define broader terms as a transitive relation. 3

4 The Database RDF descriptions: <rdf:description about= "ftp://rtfm.mit.edu/.../environment/"> <s:subject resource=" </rdf:description> Can be translated/abbreviated (SiRPAC( SiRPAC) ) to: Subject("ftp://.../environment/"," broader_term(" broader_term(" synonym(" RDF triples resemble ground facts in a logic based language. 4

5 Rules FORALL O,V subject(o,v) <- O[" ->> V]. FORALL Concept1, Concept2, Concept3 broader_term(concept1,concept3) <- broader_term(concept1,concept2) AND broader_term(concept2,concept3). FORALL Resource, Concept about(resource,concept) <- subject(resource,concept) OR // "...or is about a synonym of that concept, " EXISTS X (subject(resource,x) AND synonym(x,concept)) OR (subject(resource,x) AND synonym(concept,x)). OR (broader_term(concept,x) AND subject(resource,x)). 5

6 Query Query: FORALL Classification <- about (" Results: Classification = RESOURCE-1 Classification = RESOURCE-2 Classification = RESOURCE-N The Query is processed by an inference engine, which can process Frame-logic and predicate logic queries. 6

7 Inference engine (SiLRI( SiLRI) All input is translated into the same format (predicate logic?)? 7

8 Overview Example [Decker98]: A A Query and Inference Service for RDF From RDF to logic Lots of logic: F-Logic, F DLs,, OIL, ALC, FOL, HOL, : Resolution, Magic sets, tableaux calculus 8

9 Translating RDF to F-logicF <rdf:rdf> <rdf:description about=" <s:creator> <rdf:description about=" <v:name>ora Lassila</v:Name> </rdf:description> </s:creator> </rdf:description> </rdf:rdf> F-logic: " Name->>"Ora Lassila"; ]]. 9

10 Translating RDF to logic <rdf:rdf> <rdf:description about=" <s:creator> <rdf:description about=" <v:name>ora Lassila</v:Name> </rdf:description> </s:creator> </rdf:description> </rdf:rdf> logic: s:creator(" ) v:name( "Ora Lasila") v: ( lassila@w3.org) 10

11 Translating RDFS to F-logicF <rdf:rdf <rdfs:class rdfs:id="employee"> <rdfs:subclassof rdf:resource=" </rdfs:class> <rdfs:class rdfs:id="researcher"> <rdfs:subclassof rdf:resource="#employee"/> </rdfs:class> <rdf:property ID="cooperatesWith"> <rdfs:domain rdf:resource="#researcher"/> <rdfs:range rdf:resource="#researcher"/> </rdf:property> </rdf:rdf> F-logic: logic: Employee :: Person. Researcher :: Employee[cooperatesWith=>>Researcher]. 11

12 Overview Example [Decker98]: A A Query and Inference Service for RDF From RDF to logic Lots of logic: F-Logic, F DLs,, OIL, ALC, FOL, : Resolution, Magic sets, tableaux calculus 12

13 Semantics of FOL (1) Example: F = x P(x,f(x)) Q(g(a,z)) Truth value of F is defined under a structure A=(U A,I A ). U A : I A : arbitrary non-empty set Mapping: k-ary predicate symbol P P A ={(x 1,,x k ) x i U A and } k-ary function symbol f f A : U Ak U A Variable x x A U A 13

14 Semantics of FOL (2) A structure that would be suitable for F = x P(x,f(x)) Q(g(a,z)): U A P A Q A = IN = {(m,n) m,n U A and m<n} = {n n U A and n is prime} f A (n) = n+1 g A (m,n) = n + m a A = 2, z A = 3 f A could be replaced using F A ={(m,n) m,n U A, n=m+1}: P(x,f(x)) P(x,y) F(x,y), if F is defined such that (x,f(x)) F. 14

15 Frame-Logic [Kifer90] A hierarchy: empl::person student::person faculty::empl john:student sheth:faculty Objects: mary[name -> Mary ;highestdegree -> ms; Classes: friends->>{bob,sally}] empl[affiliation=>dept;children=>>child(person); jointworks@empl=>>report] dept[assistants=>>(student,empl);mngr=>empl] Rules: E[boss->M]<- E:empl D:dept E[affiliation->D[mngr->M:empl]] 15

16 Semantics of Frame-logic F-structure I = (U, < U, U,I F, I ->, I ->>, I ->, I ->>, I =>, I =>> ) U = objects, attributes, classes < U = subclass relation (irreflexive partial order) U = class membership (if a U b and b U c then a U c) I F terms (like mary[ Mary,ms,] on last slide) (U i U) Elements of U are mapped to 4 sets of partial functions: I -> / I ->> : non-inheritable object attributes/set-valued attrs. I -> / I ->> : inheritable class attributes/set-valued attrs. I => /I =>> : signature of class attrs./set-valued class attrs. (defining types) 16

17 Remarks on Frame-logic Frame-logic has higher-order syntax, because variables are allowed in place of functions or predicates: Set-valued functions, Attributes and methods are viewed as objects. Semantics remain first-order: Object names U. Being first-order: Variables do not range over complex domains (functions, sets), Variables can range over individuals or over names of higher- order entities. 17

18 Translating F-Logic F to logic As used in Ontobroker [Onto]. F-logic C1::C2 O: C C 1 [A=>>C 2 ] O[A->>V] O1<: O2 Predicate logic sub(c 1,C 2 ) isa(o,c) att_type(c 1,A,C 2 ) att_val(o,a,v) part_of(o 1,O 2 ) 18

19 (F-)logic in Ontobroker Langu age Input F-logic Ontology, Query Predicate logic normal logic program F-logic to FOL Lloyd-Topor transformation Output Variable substitution in F-logic Variable substitutions to F-logic Fixpoint procedure Variable substitution 19

20 Why F-logic? F Ontobroker [Onto] uses a modified subset of F-logic, because: It provides for concepts as classes, is-a relationship, attributes, axioms. In contrast to most Descr. Logics variables can take the places of concept and attribute names. F-logic has a model theoretic semantics and a sound and complete resolution-based proof theory. [Kifer95] 20

21 DLs Knowledge base T-box (Concepts) Man = Human П Male DL Constructors include at least (ALC): Conjunction, disjunction, negation Restricted forms of quantification (, ) A-box (Individuals) John : Man 21

22 OIL adds some DL to RDFS class-def subclass-of slot-def subslot-of domain range RDFS OIL class-expressions AND, OR, NOT Slot-constraints has-value, value-type Cardinality Slot-properties Trans, symm 22

23 Overview Example [Decker98]: A A Query and Inference Service for RDF From RDF to logic Lots of logic: F-Logic, F DLs,, OIL, ALC, FOL, HOL, : Resolution, Magic sets, tableaux calculus 23

24 SLD Resolution, Magic sets, tableaux calculi Countless variations for improvements in specific applications Incomplete system (FLORID) for F-logic Translating into FOL and into Horn clauses popular: I will present some basic techniques as applied to FOL 24

25 Resolution (1) F n = {,{A,B},{ A,C}} = (A B) ( A C) {B,C} resolvent F n+1 = {,{A,B},{ A,C}, {B,C} } F n satisfiable iff F n+1 satisfiable : F n+1 satisfiable F n satisf., because F n+1 =F n (B C) F n satisf. B C satisf., (either A or A isn t) F n (B C) satisf. F n+1 satisf. 25

26 Resolution (2) F n = {,{A},{ A}} = (A) ( A) empty clause F n+1 = {,{A},{ A}, } All F i are unsatisfiable. 26

27 SLD Resolution in Prolog located_in(georgia,atlanta). located_in(florida,miami). located_in(usa,x):- located_in(georgia,x). located_in(usa,x):- located_in(florida,x).?- located_in(usa,miami). located_in(geogia,miami). located_in(florida,miami). No match. Back up. Match. 27

28 Magic sets has_changed(url) my_links(url,lastvisited) doc_mtime(url,modif) Modif>LastVisited URL LastVisited π URL URL Modif URL LastVi sited Modif 28

29 Tableaux calculus A B A B A B A B A A B A B B (A B) ( A B) A B A B The root is satisfiable if one of the leaves is. 29

30 Remarks Countless improvements of basic techniques have been proposed and used. They party exclude each other. [Brass95] seems to present a promising way to integrate SLD Resolution and Magic sets and get a lot of advantages over other improvements. (And the paper is readable ) 30

31 Reference [Decker98] S. Decker, D. Brickley, J. Saarela, J. Angele. A Query and Inference Service for RDF [Onto] S. Decker, D. Fensel, M. Erdmann, R. Studer. The Technical Core of Ontobroker. Draft, [Brass00] S. Brass. SLDMagic The Real Magic (With Applications to Web Queries). In Computational Logic, , URL: [Kifer90] M. Kifer, G. Lausen, J. Wu. Logical Foundations of Object- Oriented and Frame-based Languages. Technical Report 90/14, Dep. of Computer Sc., State Univ. of New York at Stony Brook (SUNY), URL: 31

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