FOUNDATIONS OF SEMANTIC WEB TECHNOLOGIES
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1 FOUNDATIONS OF SEMANTIC WEB TECHNOLOGIES Semantics of SPARQL Sebastian Rudolph Dresden, June 14
2 Content Overview & XML 9 APR DS2 Hypertableau II 7 JUN DS5 Introduction into RDF 9 APR DS3 Tutorial 5 11 JUN DS2 RDFS Syntax & Intuition 12 APR DS5 SPARQL Syntax & Intuition 11 JUN DS3 RDFS Semantics 16 APR DS2 SPARQL Semantics 14 JUN DS2 RDFS Rule-based Reasoning 16 APR DS3 SPARQL Algebra 14 JUN DS3 Tutorial 1 19 APR DS5 SPARQL JUN DS5 OWL Syntax & Intuition 26 APR DS5 Tutorial 6 18 JUN DS2 Tutorial 2 3 MAY DS5 SPARQL Entailment 18 JUN DS3 OWL Syntax & Intuition ctd. 7 MAY DS2 Ontology Editing 2 JUL DS2 OWL & Description Logics 7 MAY DS3 Ontology Engineering 2 JUL DS3 OWL 2 10 MAY DS5 Tutorial 7 9 JUL DS2 Tutorial 3 14 MAY DS2 Linked Data 9 JUL DS3 Tableau I 14 MAY DS3 Applications 12 JUL DS5 Tableau II 17 MAY DS5 Test Exam 16 JUL DS2 Tutorial 4 7 JUN DS2 Test Exam Evaluation 16 JUL DS3 Hypertableau I 7 JUN DS3 Q&A Session 19 JUL DS5 TU Dresden, June 14 Foundations of Semantic Web Technologies slide 2 of 67
3 The SPARQL Query Language TU Dresden, June 14 Foundations of Semantic Web Technologies slide 3 of 67
4 The SPARQL Query Language TU Dresden, June 14 Foundations of Semantic Web Technologies slide 4 of 67
5 Agenda 1 Recap 2 Output Formats 3 SPARQL Semantics 4 Transformation of Queries into Algebra Objects 5 Evaluation of the SPARQL Algebra 6 Summary TU Dresden, June 14 Foundations of Semantic Web Technologies slide 5 of 67
6 Agenda 1 Recap 2 Output Formats 3 SPARQL Semantics 4 Transformation of Queries into Algebra Objects 5 Evaluation of the SPARQL Algebra 6 Summary TU Dresden, June 14 Foundations of Semantic Web Technologies slide 6 of 67
7 Recap: Introduced SPARQL Features Basic Structure PREFIX WHERE Graph Patterns Basic Graph Patterns {... } OPTIONAL UNION Filter BOUND isuri isblank isliteral STR LANG DATATYPE sameterm langmatches REGEX Modifiers ORDER BY LIMIT OFFSET DISTINCT Output Formats SELECT TU Dresden, June 14 Foundations of Semantic Web Technologies slide 7 of 67
8 Agenda 1 Recap 2 Output Formats 3 SPARQL Semantics 4 Transformation of Queries into Algebra Objects 5 Evaluation of the SPARQL Algebra 6 Summary TU Dresden, June 14 Foundations of Semantic Web Technologies slide 8 of 67
9 Output Format SELECT So far all results have been tables (solution sequences): Output format SELECT Syntax: SELECT <VariableList> or SELECT * Advantage Simple sequential processing of the results Disadvantage Structure/relationships between the objects in the results is lost TU Dresden, June 14 Foundations of Semantic Web Technologies slide 9 of 67
10 Output Format CONSTRUCT CONSTRUCT creates an RDF graph for the results Example Query PREFIX ex: < CONSTRUCT {?person ex:mailbox? .?person ex:telephone?tel. } WHERE {?person ex: ? .?person ex:tel?tel. } TU Dresden, June 14 Foundations of Semantic Web Technologies slide 10 of 67
11 Output Format CONSTRUCT CONSTRUCT creates an RDF graph for the results Example Query PREFIX ex: < CONSTRUCT {?person ex:mailbox? .?person ex:telephone?tel. } WHERE {?person ex: ? .?person ex:tel?tel. } Advantage Structured result data with relationships between the elements Disadvantages Sequential processing of the results is harder No treatment of unbound variables (triples are omitted) TU Dresden, June 14 Foundations of Semantic Web Technologies slide 11 of 67
12 CONSTRUCT Templates with Blank Nodes foaf: < _:a foaf:firstname "Alice" ; foaf:surname "Hacker". _:b foaf:firstname "Bob" ; foaf:surname "Hacker". TU Dresden, June 14 Foundations of Semantic Web Technologies slide 12 of 67
13 CONSTRUCT Templates with Blank Nodes foaf: < _:a foaf:firstname "Alice" ; foaf:surname "Hacker". _:b foaf:firstname "Bob" ; foaf:surname "Hacker". Query PREFIX foaf: < PREFIX vcard: < CONSTRUCT {?x vcard:n _:v. _:v vcard:givenname?gname ; vcard:familyname?fname } WHERE {?x foaf:firstname?gname.?x foaf:surname?fname } TU Dresden, June 14 Foundations of Semantic Web Technologies slide 13 of 67
14 CONSTRUCT Templates with Blank Nodes Resulting RDF vcard: < _:v1 vcard:n _:x. _:x vcard:givenname "Alice" ; vcard:familyname "Hacker". _:v2 vcard:n _:z. _:z vcard:givenname "Bob" ; vcard:familyname "Hacker". TU Dresden, June 14 Foundations of Semantic Web Technologies slide 14 of 67
15 Further Output Formats: ASK & DESCRIBE SPARQL supports two additional output formats: ASK only checks whether the query has at least one answer (true/false result) DESCRIBE (informative) returns an RDF description for each resulting URI (application dependent) Example Query DESCRIBE?x WHERE {?x < "123" } Possible Result (prefix declarations omitted): _:a exorg:emplid "123" ; foaf:mbox_sha1sum "ABCD1234" ; vcard:n [ vcard:family "Smith" ; vcard:given "John" ]. foaf:mbox_sha1sum a owl:inversefunctionalproperty. TU Dresden, June 14 Foundations of Semantic Web Technologies slide 15 of 67
16 Agenda 1 Recap 2 Output Formats 3 SPARQL Semantics 4 Transformation of Queries into Algebra Objects 5 Evaluation of the SPARQL Algebra 6 Summary TU Dresden, June 14 Foundations of Semantic Web Technologies slide 16 of 67
17 Semantics of Query Languages So far only informal presentation of SPARQL features User: Which answers can I expect for my query? Developer: Which behaviour is expected from my SPARQL implementation? Marketing: Is our product already conformant with the SPARQL standard? Formal semantics should clarify these questions... TU Dresden, June 14 Foundations of Semantic Web Technologies slide 17 of 67
18 Logic-based Semantics Semantics of formal logics: Model-theoretic semantics: Which interpretations do satisfy my knowledge base? Proof-theoretic semantics: Which derivations can be build from my knowledge base?... TU Dresden, June 14 Foundations of Semantic Web Technologies slide 18 of 67
19 Logic-based Semantics Semantics of formal logics: Model-theoretic semantics: Which interpretations do satisfy my knowledge base? Proof-theoretic semantics: Which derivations can be build from my knowledge base?... TU Dresden, June 14 Foundations of Semantic Web Technologies slide 19 of 67
20 Semantics of Programming Languages Axiomatic semantics: Which logical statements hold for my program? Operational semantics: What happens during the processing of my program? Denotational semantics: How can we describe the input/output function of the program in an abstract way? TU Dresden, June 14 Foundations of Semantic Web Technologies slide 20 of 67
21 Semantics of Programming Languages Axiomatic semantics: Which logical statements hold for my program? Operational semantics: What happens during the processing of my program? Denotational semantics: How can we describe the input/output function of the program in an abstract way? What to do with query languages? TU Dresden, June 14 Foundations of Semantic Web Technologies slide 21 of 67
22 Semantics of Query Languages (1) Query Entailment Query as description of allowed results Data as set of logical assumptions (axiom set/theory) Results as logical entailment OWL DL and RDF(S) as query languages conjunctive queries TU Dresden, June 14 Foundations of Semantic Web Technologies slide 22 of 67
23 Semantics of Query Languages (2) Query Algebra Query as instruction for computing the results Queried data as input Results as output Relational algebra for SQL SPARQL Algebra TU Dresden, June 14 Foundations of Semantic Web Technologies slide 23 of 67
24 Agenda 1 Recap 2 Output Formats 3 SPARQL Semantics 4 Transformation of Queries into Algebra Objects 5 Evaluation of the SPARQL Algebra 6 Summary TU Dresden, June 14 Foundations of Semantic Web Technologies slide 24 of 67
25 Translation into SPARQL Algebra {?book ex:price?price. FILTER (?price < 15) OPTIONAL {?book ex:title?title } {?book ex:author ex:shakespeare } UNION {?book ex:author ex:marlowe } } Semantics of a SPARQL query: 1 Transformation of the query into an algebra expression 2 Evaluation of the algebra expression TU Dresden, June 14 Foundations of Semantic Web Technologies slide 25 of 67
26 Translation into SPARQL Algebra {?book ex:price?price FILTER (?price < 15) OPTIONAL {?book ex:title?title } {?book ex:author ex:shakespeare } UNION {?book ex:author ex:marlowe } } Attention: Filters apply to the whole group in which they occur TU Dresden, June 14 Foundations of Semantic Web Technologies slide 26 of 67
27 Translation into SPARQL Algebra {?book ex:price?price OPTIONAL {?book ex:title?title } {?book ex:author ex:shakespeare } UNION {?book ex:author ex:marlowe } FILTER (?price < 15) } 1 Expand abbreviated IRIs TU Dresden, June 14 Foundations of Semantic Web Technologies slide 27 of 67
28 Translation into SPARQL Algebra {?book < OPTIONAL {?book < } {?book < < } UNION {?book < < } FILTER (?price < 15) } TU Dresden, June 14 Foundations of Semantic Web Technologies slide 28 of 67
29 Translation into SPARQL Algebra {?book < OPTIONAL {?book < } {?book < < } UNION {?book < < } FILTER (?price < 15) } 2. Replace triple patterns with operator Bgp( ) TU Dresden, June 14 Foundations of Semantic Web Technologies slide 29 of 67
30 Translation into SPARQL Algebra { Bgp(?book < OPTIONAL {Bgp(?book < {Bgp(?book < < UNION {Bgp(?book < < FILTER (?price < 15) } TU Dresden, June 14 Foundations of Semantic Web Technologies slide 30 of 67
31 Translation into SPARQL Algebra { Bgp(?book < OPTIONAL {Bgp(?book < {Bgp(?book < < UNION {Bgp(?book < < FILTER (?price < 15) } 3. Introduce the LeftJoin( ) operator for optional parts TU Dresden, June 14 Foundations of Semantic Web Technologies slide 31 of 67
32 Translation into SPARQL Algebra { LeftJoin(Bgp(?book < Bgp(?book < true) {Bgp(?book < < UNION {Bgp(?book < < FILTER (?price < 15) } TU Dresden, June 14 Foundations of Semantic Web Technologies slide 32 of 67
33 Translation into SPARQL Algebra { LeftJoin(Bgp(?book < Bgp(?book < true) {Bgp(?book < < UNION {Bgp(?book < < FILTER (?price < 15) } 4. Combine alternative graph patterns with Union( ) operator Refers to neighbouring patterns and has higher precedence than conjunction (left associative) TU Dresden, June 14 Foundations of Semantic Web Technologies slide 33 of 67
34 Translation into SPARQL Algebra { LeftJoin(Bgp(?book < Bgp(?book < true) Union(Bgp(?book < < Bgp(?book < < FILTER (?price < 15) } TU Dresden, June 14 Foundations of Semantic Web Technologies slide 34 of 67
35 Translation into SPARQL Algebra { LeftJoin(Bgp(?book < Bgp(?book < true) Union(Bgp(?book < < Bgp(?book < < FILTER (?price < 15) } 5. Apply Join( ) operator to join non-filter elements TU Dresden, June 14 Foundations of Semantic Web Technologies slide 35 of 67
36 Translation into SPARQL Algebra { Join( LeftJoin(Bgp(?book < Bgp(?book < true), Union(Bgp(?book < < Bgp(?book < < FILTER (?price < 15) } TU Dresden, June 14 Foundations of Semantic Web Technologies slide 36 of 67
37 Translation into SPARQL Algebra { Join( LeftJoin(Bgp(?book < Bgp(?book < true), Union(Bgp(?book < < Bgp(?book < < FILTER (?price < 15) } 6. Translate a group with filters with the Filter( ) operator TU Dresden, June 14 Foundations of Semantic Web Technologies slide 37 of 67
38 Translation into SPARQL Algebra Filter(?price < 15, Join( LeftJoin(Bgp(?book < Bgp(?book < true), Union(Bgp(?book < < Bgp(?book < < TU Dresden, June 14 Foundations of Semantic Web Technologies slide 38 of 67
39 Translation into SPARQL Algebra Filter(?price < 15, Join( LeftJoin(Bgp(?book < Bgp(?book < true), Union(Bgp(?book < < Bgp(?book < < Online translation tool: TU Dresden, June 14 Foundations of Semantic Web Technologies slide 39 of 67
40 Agenda 1 Recap 2 Output Formats 3 SPARQL Semantics 4 Transformation of Queries into Algebra Objects 5 Evaluation of the SPARQL Algebra 6 Summary TU Dresden, June 14 Foundations of Semantic Web Technologies slide 40 of 67
41 Semantics of the SPARQL Algebra Operations Now we have an algebra object, but what do the algebra operations mean? Bgp(P) match/evaluate pattern P Join(M 1, M 2 ) conjunctive join of solutions M 1 and M 2 Union(M 1, M 2 ) union of solutions M 1 with M 2 LeftJoin(M 1, M 2, F) optional join of M 1 with M 2 with filter constraint F (true if no filter given) Filter(F, M) filter solutions M with constraint F Z empty pattern (identity for join) TU Dresden, June 14 Foundations of Semantic Web Technologies slide 41 of 67
42 Semantics of the SPARQL Algebra Operations Now we have an algebra object, but what do the algebra operations mean? Bgp(P) match/evaluate pattern P Join(M 1, M 2 ) conjunctive join of solutions M 1 and M 2 Union(M 1, M 2 ) union of solutions M 1 with M 2 LeftJoin(M 1, M 2, F) optional join of M 1 with M 2 with filter constraint F (true if no filter given) Filter(F, M) filter solutions M with constraint F Z empty pattern (identity for join) Only Bgp( ) matches or evaluates graph patterns We can use entailment checking rather than graph matching TU Dresden, June 14 Foundations of Semantic Web Technologies slide 42 of 67
43 Definition of the SPARQL Operators How can we define that more formally? Output: solution set (formatting irrelevant) Input: Queried (active) graph Partial results from previous evaluation steps Different parameters according to the operation How can we formally describe the results? TU Dresden, June 14 Foundations of Semantic Web Technologies slide 43 of 67
44 SPARQL Results Intuition: Results coded as tables of variable assignments Result: List of solutions (solution sequence) each solution corresponds to one table row TU Dresden, June 14 Foundations of Semantic Web Technologies slide 44 of 67
45 SPARQL Results Intuition: Results coded as tables of variable assignments Result: List of solutions (solution sequence) each solution corresponds to one table row Solution: Partial function Domain: relevant variables Range: IRIs blank nodes RDF literals Unbound variables are those that have no assigned value (partial function) TU Dresden, June 14 Foundations of Semantic Web Technologies slide 45 of 67
46 Evaluation of Basic Graph Patterns Definition (Solution) Let P be a basic graph pattern. A partial function µ is a solution for Bgp(P) over the queried (active) graph G if: 1 the domain of µ is exactly the set of variables in P, 2 there exists an assignment σ from blank nodes in P to IRIs, blank nodes, or RDF literals such that: 3 the RDF graph µ(σ(p)) is a subgraph of G. TU Dresden, June 14 Foundations of Semantic Web Technologies slide 46 of 67
47 Evaluation of Basic Graph Patterns The result of evaluating Bgp(P) over G is written Bgp(P) G The result is a multi set of solutions µ The multiplicity of each solution µ corresponds to the number of different assignments σ TU Dresden, June 14 Foundations of Semantic Web Technologies slide 47 of 67
48 Multi Sets Definition (Multi Set) A multi set over a set S is a total function M : S IN + {ω} IN + denotes the positive natural numbers ω > n for all n IN + M(s) is the multiplicity of s S ω: countably infinite number of occurrences We represent a multi se over the set S also with the set {(s, M(s)) s S} We write (s, n) M if M(s) = n We assume that M(s) = 0 if s / S Alternative notation: {a, b, b } corresponds to the multi set M over the set {a, b} with M(a) = 1 and M(b) = 2 TU Dresden, June 14 Foundations of Semantic Web Technologies slide 48 of 67
49 Solution Mapping Example ex:birte ex:gives [ a ex:lecture ; ex:hastopic "SPARQL" ]. ex:sebastian ex:gives [ a ex:lecture ; ex:hastopic "DLs and OWL" ]. Bgp(?who ex:gives _ :x. _ :x ex:hastopic?what) TU Dresden, June 14 Foundations of Semantic Web Technologies slide 49 of 67
50 Solution Mapping Example ex:birte ex:gives _ :a. _ :a rdf:type ex:lecture. _ :a ex:hastopic "SPARQL". ex:sebastian ex:gives _ :b. _ :b rdf:type ex:lecture. _ :b ex:hastopic "DLs and OWL". Bgp(?who ex:gives _ :x. _ :x ex:hastopic?what) TU Dresden, June 14 Foundations of Semantic Web Technologies slide 50 of 67
51 Solution Mapping Example ex:birte ex:gives _ :a. _ :a rdf:type ex:lecture. _ :a ex:hastopic "SPARQL". ex:sebastian ex:gives _ :b. _ :b rdf:type ex:lecture. _ :b ex:hastopic "DLs and OWL". Bgp(?who ex:gives _ :x. _ :x ex:hastopic?what) µ 1 :?who ex : Birte,?what "SPARQL" σ 1 : _ :x _ :a TU Dresden, June 14 Foundations of Semantic Web Technologies slide 51 of 67
52 Solution Mapping Example ex:birte ex:gives _ :a. _ :a rdf:type ex:lecture. _ :a ex:hastopic "SPARQL". ex:sebastian ex:gives _ :b. _ :b rdf:type ex:lecture. _ :b ex:hastopic "DLs and OWL". Bgp(?who ex:gives _ :x. _ :x ex:hastopic?what) µ 1 :?who ex : Birte,?what "SPARQL" σ 1 : _ :x _ :a µ 2 :?who ex : Sebastian,?what "DLs and OWL" σ 2 : _ :x _ :b TU Dresden, June 14 Foundations of Semantic Web Technologies slide 52 of 67
53 Solution Mapping Example ex:birte ex:gives _ :a. _ :a rdf:type ex:lecture. _ :a ex:hastopic "SPARQL". ex:sebastian ex:gives _ :b. _ :b rdf:type ex:lecture. _ :b ex:hastopic "DLs and OWL". Bgp(?who ex:gives _ :x. _ :x ex:hastopic?what) µ 1 :?who ex : Birte,?what "SPARQL" σ 1 : _ :x _ :a µ 2 :?who ex : Sebastian,?what "DLs and OWL" σ 2 : _ :x _ :b Two solutions each with multiplicity 1 TU Dresden, June 14 Foundations of Semantic Web Technologies slide 53 of 67
54 Exercise Solution Sets ex:birte ex:gives [ a ex:lecture ; ex:hastopic "SPARQL" ]. ex:birte ex:gives [ a ex:lecture ; ex:hastopic "SPARQL Algebra" ]. Bgp(?who ex:gives _ :x. _ :x ex:hastopic _ :y) TU Dresden, June 14 Foundations of Semantic Web Technologies slide 54 of 67
55 Solution TU Dresden, June 14 Foundations of Semantic Web Technologies slide 55 of 67
56 Solution ex:birte ex:gives _ :a. _ :a rdf:type ex:lecture. _ :a ex:hastopic "SPARQL". ex:birte ex:gives _ :b. _ :b rdf:type ex:lecture. _ :b ex:hastopic "SPARQL Algebra". Bgp(?who ex:gives _ :x. _ :x ex:hastopic _ :y) TU Dresden, June 14 Foundations of Semantic Web Technologies slide 56 of 67
57 Solution ex:birte ex:gives _ :a. _ :a rdf:type ex:lecture. _ :a ex:hastopic "SPARQL". ex:birte ex:gives _ :b. _ :b rdf:type ex:lecture. _ :b ex:hastopic "SPARQL Algebra". Bgp(?who ex:gives _ :x. _ :x ex:hastopic _ :y) µ 1 :?who ex : Birte, σ 1 : _ :x _ :a _ :y "SPARQL" TU Dresden, June 14 Foundations of Semantic Web Technologies slide 57 of 67
58 Solution ex:birte ex:gives _ :a. _ :a rdf:type ex:lecture. _ :a ex:hastopic "SPARQL". ex:birte ex:gives _ :b. _ :b rdf:type ex:lecture. _ :b ex:hastopic "SPARQL Algebra". Bgp(?who ex:gives _ :x. _ :x ex:hastopic _ :y) µ 1 :?who ex : Birte, σ 1 : _ :x _ :a _ :y "SPARQL" µ 2 :?who ex : Birte, σ 2 : _ :x _ :b _ :y "SPARQL Algebra" TU Dresden, June 14 Foundations of Semantic Web Technologies slide 58 of 67
59 Solution ex:birte ex:gives _ :a. _ :a rdf:type ex:lecture. _ :a ex:hastopic "SPARQL". ex:birte ex:gives _ :b. _ :b rdf:type ex:lecture. _ :b ex:hastopic "SPARQL Algebra". Bgp(?who ex:gives _ :x. _ :x ex:hastopic _ :y) µ 1 :?who ex : Birte, σ 1 : _ :x _ :a _ :y "SPARQL" µ 2 :?who ex : Birte, σ 2 : _ :x _ :b _ :y "SPARQL Algebra" One solution with multiplicity 2 TU Dresden, June 14 Foundations of Semantic Web Technologies slide 59 of 67
60 Union of Solutions (1) Definition (Compatibility) Two solutions µ 1 and µ 2 are compatible if µ 1 (x) = µ 2 (x) for all x, for which µ 1 and µ 2 are defined TU Dresden, June 14 Foundations of Semantic Web Technologies slide 60 of 67
61 Union of Solutions (1) Definition (Compatibility) Two solutions µ 1 and µ 2 are compatible if µ 1 (x) = µ 2 (x) for all x, for which µ 1 and µ 2 are defined µ 1 :?x ex : a,?y ex : b µ 2 :?y ex : b,?z ex : c TU Dresden, June 14 Foundations of Semantic Web Technologies slide 61 of 67
62 Union of Solutions (1) Definition (Compatibility) Two solutions µ 1 and µ 2 are compatible if µ 1 (x) = µ 2 (x) for all x, for which µ 1 and µ 2 are defined µ 1 :?x ex : a,?y ex : b µ 2 :?y ex : b,?z ex : c TU Dresden, June 14 Foundations of Semantic Web Technologies slide 62 of 67
63 Union of Solutions (1) Definition (Compatibility) Two solutions µ 1 and µ 2 are compatible if µ 1 (x) = µ 2 (x) for all x, for which µ 1 and µ 2 are defined µ 1 :?x ex : a,?y ex : b µ 2 :?y ex : b,?z ex : c µ 1 :?x ex : a,?y ex : b µ 2 :?x ex : b,?z ex : c TU Dresden, June 14 Foundations of Semantic Web Technologies slide 63 of 67
64 Union of Solutions (1) Definition (Compatibility) Two solutions µ 1 and µ 2 are compatible if µ 1 (x) = µ 2 (x) for all x, for which µ 1 and µ 2 are defined µ 1 :?x ex : a,?y ex : b µ 2 :?y ex : b,?z ex : c µ 1 :?x ex : a,?y ex : b µ 2 :?x ex : b,?z ex : c TU Dresden, June 14 Foundations of Semantic Web Technologies slide 64 of 67
65 Union of Solutions (1) Definition (Compatibility) Two solutions µ 1 and µ 2 are compatible if µ 1 (x) = µ 2 (x) for all x, for which µ 1 and µ 2 are defined µ 1 :?x ex : a,?y ex : b µ 2 :?y ex : b,?z ex : c µ 1 :?x ex : a,?y ex : b µ 2 :?x ex : b,?z ex : c µ 1 :?x ex : a µ 2 :?y ex : b TU Dresden, June 14 Foundations of Semantic Web Technologies slide 65 of 67
66 Union of Solutions (1) Definition (Compatibility) Two solutions µ 1 and µ 2 are compatible if µ 1 (x) = µ 2 (x) for all x, for which µ 1 and µ 2 are defined µ 1 :?x ex : a,?y ex : b µ 2 :?y ex : b,?z ex : c µ 1 :?x ex : a,?y ex : b µ 2 :?x ex : b,?z ex : c µ 1 :?x ex : a µ 2 :?y ex : b TU Dresden, June 14 Foundations of Semantic Web Technologies slide 66 of 67
67 Union of Solutions (2) Union of two compatible solutions µ 1 and µ 2 : { µ1 (x) if x dom(µ (µ 1 µ 2 )(x) = 1 ) µ 2 (x) otherwise simple intuition: union of matching table rows Next lecture: Evaluation of the main algebra operators TU Dresden, June 14 Foundations of Semantic Web Technologies slide 67 of 67
68 Agenda 1 Recap 2 Output Formats 3 SPARQL Semantics 4 Transformation of Queries into Algebra Objects 5 Evaluation of the SPARQL Algebra 6 Summary TU Dresden, June 14 Foundations of Semantic Web Technologies slide 68 of 67
69 Summary SPARQL queries are translated into algebra objects The BGPs generate solutions Other algebra operators combine solutions Details in the next lecture TU Dresden, June 14 Foundations of Semantic Web Technologies slide 69 of 67
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