Schema Mappings for Data Graphs

Transcription

1 Schema Mappings for Data Graphs Nadime Francis 1 Leonid Likin 2 1 Université Paris-Est Marne-la-Vallée 2 University of Edinurgh BDA 2017 Nancy, Novemer, 16th Results originally pulished in PODS / 22

2 Introduction 2 / 22

3 Data Exchange Legacy Dataase over Σ s M Schema Mapping from Σ s to Σ t { } Possile Solutions over Σ t 3 / 22

4 Data Exchange Legacy Dataase over Σ s M Schema Mapping from Σ s to Σ t { } Possile Solutions over Σ t Q Query over Σ t Certain answers 3 / 22

5 Data Exchange as Virtual Data Integration Σ 1 Σ 2 Σ 3 Source Dataases over Σ s = {Σ 1, Σ 2, Σ 3 } { } Σ 3 M Schema Mapping from Σ s to Σ t Virtual Dataase over mediated schema Σ t Q Query over Σ t Integrated answers 4 / 22

6 Key questions Solution Existence Input: A dataase S and a mapping M Question: Does there exist a solution T to S under M? 5 / 22

7 Key questions Solution Existence Input: A dataase S and a mapping M Question: Does there exist a solution T to S under M? Query Answering Input: A dataase S, a tuple v S, a mapping M, a query Q Question: Is v a certain answer to Q over the solution to S under M? 5 / 22

8 Why look at graph data? Data naturally presented as graphs: Social networks, Internet, iological data, Semantic We... Queries care aout the topology of the graph: Paths, cycles, connected components, graph patterns... Known techniques for relational dataases do not apply Person, Postdoc name : Nadime institute : UoE knows since : 2010 colleague since : 2015 Person, Professor name : Leonid institute : UoE 6 / 22

9 Property Graphs, Graph Dataases and Data Graphs Property graphs: real life model, used in Neo4j (Cypher), advocated y LDBC. Graph dataases: theoretical modelization as edge-laelled graph. Topology only. Does not capture real life scenarios. Data graphs: edge-laelled graphs with nodes carrying values from an infinite domain. Better astraction of real life cases. Topology and data. Person, Postdoc name : Nadime institute : UoE knows since : 2010 colleague since : 2015 Person, Professor name : Leonid institute : UoE 7 / 22

10 Property Graphs, Graph Dataases and Data Graphs Property graphs: real life model, used in Neo4j (Cypher), advocated y LDBC. Graph dataases: theoretical modelization as edge-laelled graph. Topology only. Does not capture real life scenarios. Data graphs: edge-laelled graphs with nodes carrying values from an infinite domain. Better astraction of real life cases. Topology and data. a a a a 7 / 22

11 Property Graphs, Graph Dataases and Data Graphs Property graphs: real life model, used in Neo4j (Cypher), advocated y LDBC. Graph dataases: theoretical modelization as edge-laelled graph. Topology only. Does not capture real life scenarios. Data graphs: edge-laelled graphs with nodes carrying values from an infinite domain. Better astraction of real life cases. Topology and data a a 3 2 a 4 a 5 7 / 22

12 Data Exchange and Query Answering 8 / 22

13 Known Results Data Exchange for Relational Dataases Extensively investigated over the last 20 years Book material : Foundations of Data Exchange [Arenas, Barceló, Likin, Murlak, 2014] Principles of Data Integration [Doan, Halevy, Ives, 2012] When and how solutions can e efficiently uilt and queried 9 / 22

14 Known Results Data Exchange for Relational Dataases Extensively investigated over the last 20 years Book material : Foundations of Data Exchange [Arenas, Barceló, Likin, Murlak, 2014] Principles of Data Integration [Doan, Halevy, Ives, 2012] When and how solutions can e efficiently uilt and queried Data Exchange for Graph Data Schema Mappings and Data Exchange for Graph Dataases [Barceló, Pérez, Reutter, 2013] Answering queries is typically conp, tractaility requires RPQs in mappings to e rigid (no, no ) 9 / 22

15 Known Results Data Exchange for Relational Dataases Extensively investigated over the last 20 years Book material : Foundations of Data Exchange [Arenas, Barceló, Likin, Murlak, 2014] Principles of Data Integration [Doan, Halevy, Ives, 2012] When and how solutions can e efficiently uilt and queried Data Exchange for Graph Data Schema Mappings and Data Exchange for Graph Dataases [Barceló, Pérez, Reutter, 2013] Answering queries is typically conp, tractaility requires RPQs in mappings to e rigid (no, no ) No literature for data graphs! 9 / 22

16 Chosen Setting Key parameters Source (G s ) and target (G t ) dataases: Data graphs Schema mapping (M): Pairs of RPQs Query (Q): Data RPQs 10 / 22

17 Chosen Setting Key parameters Source (G s ) and target (G t ) dataases: Data graphs Schema mapping (M): Pairs of RPQs Query (Q): Data RPQs 4 1 Source: data graph over Σ s and D Target: data graph over Σ t and D 2 1 a a 3 2 a 4 a 5 10 / 22

18 Chosen Setting Key parameters Source (G s ) and target (G t ) dataases: Data graphs Schema mapping (M): Pairs of RPQs Query (Q): Data RPQs M = {(q i, q i )} (G s, G t ) = M iff q i (G s ) q i (G t) Note: nodes of G s are exported together with their data values Ex: M = {(a +, c); (ac, a )} a a c 1 M c 1 2 a c 1 10 / 22

19 Chosen Setting Key parameters Source (G s ) and target (G t ) dataases: Data graphs Schema mapping (M): Pairs of RPQs Query (Q): Data RPQs M = {(q i, q i )} (G s, G t ) = M iff q i (G s ) q i (G t) Note: nodes of G s are exported together with their data values Ex: M = {(a +, c); (ac, a )} a 3 M a 1 2 a c c 1 c 10 / 22

20 Chosen Setting Key parameters Source (G s ) and target (G t ) dataases: Data graphs Schema mapping (M): Pairs of RPQs Query (Q): Data RPQs M = {(q i, q i )} (G s, G t ) = M iff q i (G s ) q i (G t) Note: nodes of G s are exported together with their data values Ex: M = {(a +, c); (ac, a )} a a c 1 M 5 2 a c 1 2 a c 1 10 / 22

21 Chosen Setting Key parameters Source (G s ) and target (G t ) dataases: Data graphs Schema mapping (M): Pairs of RPQs Query (Q): Data RPQs Most general: regular expressions with memory Ex: Q = Σ t x.(σ + t [x = ]) (Σ + t [x = ]) Σ t (a ) A data value repeats three times along a path finishing with a or. a c Q(G t ) = {(, ); (, ); (, ); (, )} 1 2 a c 1 10 / 22

22 Query Answering: Certain Answers Semantics Certain Answers Let G s e a source dataase, M e a schema mapping from Σ s to Σ t, and Q e a query over Σ t. Then: M (Q, G s ) = Q(G t ) G t (G s,g t) =M 11 / 22

23 Query Answering: Certain Answers Semantics Certain Answers Let G s e a source dataase, M e a schema mapping from Σ s to Σ t, and Q e a query over Σ t. Then: M (Q, G s ) = Q(G t ) G t (G s,g t) =M Prolem : Query Answering(M, Q) Data complexity Input : A data graph G s, a tuple v of nodes of G s a schema mapping M and a query Q Question : Is v in M (Q, G s )? 11 / 22

24 Query Answering: Certain Answers Semantics Certain Answers Let G s e a source dataase, M e a schema mapping from Σ s to Σ t, and Q e a query over Σ t. Then: M (Q, G s ) = Q(G t ) G t (G s,g t) =M Prolem : Query Answering(M, Q) Data complexity Input : A data graph G s, a tuple v of nodes of G s a schema mapping M and a query Q Question : Is v in M (Q, G s )? 11 / 22

25 Contriutions Undecidaility and Intractaility Query answering is undecidale for RPQ mappings and data RPQ queries. (Already true for very simple mappings) Query answering is conp-complete for word mappings and data RPQ queries. (Already true for paths with tests) Recovering Tractaility Query answering is in NLogSpace for word mappings and data RPQ queries under the presence of SQL nulls. Query answering is in NLogSpace for word mappings and data RPQ queries with equality only. 12 / 22

26 Undecidaility and Intractaility 13 / 22

27 Undecidaility Theorem There exist a very simple mapping M and an RPQ with equality Q such that Query Answering(M, Q) is undecidale. 14 / 22

28 Undecidaility Theorem There exist a very simple mapping M and an RPQ with equality Q such that Query Answering(M, Q) is undecidale. Very simple: Rules of M are of two possile forms: Either (a, ), with a Σ s and Σ t ; Or (a, Σ t ), with a Σ s. 14 / 22

29 Undecidaility Theorem There exist a very simple mapping M and an RPQ with equality Q such that Query Answering(M, Q) is undecidale. RPQ with equality: Query defined y an expression with e = and e suexpressions Ex: Σ (a + ) = Σ A data value occurs twice with only a s in etween. Strictly weaker than data RPQs. 14 / 22

30 Undecidaility Theorem There exist a very simple mapping M and an RPQ with equality Q such that Query Answering(M, Q) is undecidale. For these fixed (data complexity) M and Q, given: a dataase G s over Σ s, a pair of nodes (x, y) of G s, it is undecidale whether (x, y) M (Q, G s ). 14 / 22

31 Intractaility Theorem Let M e a relational mapping and a Q a data RPQ. Then Query Answering(M, Q) is conp-complete. 15 / 22

32 Intractaility Theorem Let M e a relational mapping and a Q a data RPQ. Then Query Answering(M, Q) is conp-complete. Relational mapping: Rules of M are of the form (q, w), where: q is an aritrary RPQ over Σ s ; w is a single word over Σ t. It is already conp-hard when q is a single symol a Σ s. 15 / 22

33 Intractaility Theorem Let M e a relational mapping and a Q a data RPQ. Then Query Answering(M, Q) is conp-complete. Data RPQ: Any query defined y a register automaton or a regular expression with memory. It is already hard if Q is a path with tests. Ex: ((a) = c). No disjunction, no transitive closure. 15 / 22

34 Intractaility Theorem Let M e a relational mapping and a Q a data RPQ. Then Query Answering(M, Q) is conp-complete. For these fixed (data complexity) M and Q, given: a dataase G s over Σ s, a pair of nodes (x, y) of G s, it is conp-complete to decide whether (x, y) M (Q, G s ). 15 / 22

35 Tractale Fragments 16 / 22

36 Introducing SQL Nulls SQL Behavior SQL: Missing values modelled with a single null value n. Three valued logic: x = y returns unknown if either x = n or y = n; Otherwise, x = y evaluates as usual. At query answering time, unknown collapses to false. 17 / 22

37 Introducing SQL Nulls SQL Behavior SQL: Missing values modelled with a single null value n. Three valued logic: x = y returns unknown if either x = n or y = n; Otherwise, x = y evaluates as usual. At query answering time, unknown collapses to false. Translating to data RPQs Replace D with D n = D {n}. Comparisons: as in SQL. 17 / 22

38 Query Answering Approximation via SQL Nulls Theorem Let M e a relational mapping and Q e a data RPQ. Then, given a source dataase G s and (x, y) in G s, it can e decided in NLogSpace whether (x, y) n M (Q, G s ). 18 / 22

39 Query Answering Approximation via SQL Nulls Theorem Let M e a relational mapping and Q e a data RPQ. Then, given a source dataase G s and (x, y) in G s, it can e decided in NLogSpace whether (x, y) n M (Q, G s ). n M (Q, G s ) = G t over D n (G s, G t ) = M Q(G t ) 18 / 22

40 Query Answering Approximation via SQL Nulls Theorem Let M e a relational mapping and Q e a data RPQ. Then, given a source dataase G s and (x, y) in G s, it can e decided in NLogSpace whether (x, y) n M (Q, G s ). n M (Q, G s ) = G t over D n (G s, G t ) = M Q(G t ) Motivations n M (Q, G s ) is an underapproximation of M (Q, G s ). It matches the implementation of traditional DBMSs, and thus the expected ehavior if G t is materialized as such. 18 / 22

41 Queries without Inequalities Data RPQs without Inequalities Expressions with memory: only [x = ]. Expressions with equality: only e =. 19 / 22

42 Queries without Inequalities Data RPQs without Inequalities Results Expressions with memory: only [x = ]. Expressions with equality: only e =. Query Answering(M, Q) is in NLogSpace for relational mappings and data RPQs without inequalities. Query Answering(M, Q) is in conp for RPQ mappings and RPQ with equalities and no inequalities. The question remains open when Q is an RPQ with memory and no inequalities. 19 / 22

43 Perspectives 20 / 22

44 Very first steps in Data Exchange for Data Graphs Early picture of decidaility / tractaility Other tasks (metadata management) Application to real life graph data Property graphs and real-life query languages Neo Technology: Neo4j and Cypher Back to relational data exchange Use of SQL nulls instead of marked nulls Get efficient approximations 21 / 22

45 Thank you! 22 / 22