1. A Logical Formulation of PRMs

2. Example Institute(InstId,Type) Researcher(RID,Area,Salary,InstID) Paper(PaperId,Topic) Author(RId,PaperID) Cites(PaperId1,PaperId2)

3. Types of Uncertainty intra-relational dependency –a researcher’s salary depends on their research area inter-relational dependency –a researcher’s salary depends on the type of institute they work at reference uncertainty –a paper’s author is more likely to be a research in the same area as the paper exists uncertainty –a citation between two papers is more likely to exist if they are on the same topic identity uncertainty –the authors of two distinct papers are more likely to be the same individual if the author names are similar and if the co-authors are the same

4. Keys

5. DependsOn Predicates Examples: –DependsOn 1 (Salary; Area)  Researcher(RId,Salary,Area,InstId) –DependsOn 1 (Salary; Area, Type)  Researcher(RId,Salary,Area,InstId), Institute(InstId,Type)

6. Rules for DependsOn The set of DependsOn predicates occur only in the heads of clauses The body of a DependsOn clause may contain extensional predicates, built-in predicates Every descriptive attribute A must appear as the first argument of a DependsOn predicate If there is more than one DependsOn predicate for a particular attribute, require for each corresponding key, only one DependsOn matches.

7. Aggregates A researcher’s salary depends on the number of publications they have: –Count RID Author;PaperId (RId,CntPapers) –this takes the Author relation, Author(RId,PaperId) groups by RId and takes the count Equivalent to –select RId,count(PaperId) as CntPapersfrom Author group by RId More general form: –Aggr key predicate;Aggr-Variable-List (Key,AggrVal)

8. Syntax Predicates – ordinary predicates, aggregates, DependsOn Clauses – Key Constraints, DependsOn Clauses CPDs

9. Semantics Attribute Uncertainty –Background theory provides instantiations for both the primary key and foreign keys through a set of partially instantiated extensional predicates

10. A Sample KB Researcher-Inst(101) Researcher-Inst(102) Researcher-Institute- Inst(101,201) Researcher-Institute- Inst(102,201) Institute-Inst(201) Paper-Inst(301) Author-Inst(101,301) Author-Inst(102,301) Paper-Inst(302) Author-Inst(101,302) Cites-Inst(301-302)

11. Intensional Predicates to Introduce RVs  Area, Salary Researcher(RId,Area,Salary,InstId)  Researcher-Institute-Inst(RId,InstId)  Type Institute(InstId,Type)  Institute-Inst(InstId)

12. Dependency Graph Convert each numbered DependsOn statement to a general binary relation  –DependsOn(A i ;….,A,,…)  … –Let V i and V, be instantiations – we add V i < V, We require < to be acyclic

13. Reference Uncertainty Paper(PaperId,Topic), Venue(VenueId,Area), PublishedIn(PaperId,VenueID) Paper-Inst(301), Paper-Inst(302), Venue- Inst(stoc), Venue-Inst(focs), Venue-Inst(icse), Venue-Inst(pldi), Venue-Inst(isca)  Venue PublishedIn(301,Venue)  Venue PublishedIn(302,Venue) VenueKeys = { VenueId |  VenueId Venue- Inst(VenueId)} VenueKeys = {stoc,focs,icse,pldi,isca}

14. FKDependsOn FKDependsOn(VenueId;Area;Topic)  PublishedIn(PaperId,VenueId), Paper(PaperId,Topic),Venue(VenueId,Area) General form: –FKDependsOn(variable; ; ) Once a partition is chosen for a variable, the key is chosen uniformly from that partition.

15. Ensuring Coherence We require that the parents and the variables that define the partition come before the fk variable in the dependency graph Also, any dependencies based on the fk must occur after the fk is determined.

16. Existence Uncertainty CiteExists(PaperId1,PaperId2,Exists) Cites(PaperId1,PaperId2)  CiteExists(PaperId1,PaperId2,True) DependsOn(Exists;Topic1,Topic2)  CiteExists(PaperId1,PaperId2,Exists), Paper(PaperId1,Topic1), Paper(PaperId2,Topic2)