1) The document discusses querying incomplete data using different semantics like OR-semantics and weak semantics to return maximal answers rather than complete answers. 2) It describes computing full disjunctions of relations as a special case of weak semantics and how full disjunctions generalize to allow non-equality join constraints. 3) The complexity of evaluating maximal weak matchings and maximal OR matchings for cyclic queries is polynomial in the size of the query, database, and result.

1. Computing Full Disjunctions Yaron Kanza Yehoshua Sagiv The Selim and Rachel Benin School of Engineering and Computer Science The Hebrew University of Jerusalem

6. 1 2 4 5 6 title language 7 3 year 8 director 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 English 1/12/1935 Woody Allen title year acted in acted in A Semistructured Database About Movies

7. v 1 v 2 w 1 v 3 title actor movie director acted in w 2 w 3 w 4 date of birth name language A Query Under complete semantics, the query returns actor-movie pairs, such that the actor played in the movie and was also the director of the movie

8. 1 2 4 5 6 title language 7 3 year 8 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 English 1/12/1935 Woody Allen title year acted in acted in v 1 v 2 w 1 v 3 title actor movie director acted in w 2 w 3 w 4 date of birth name language A complete matching of the query variables to database objects director 1 2 5 6 4 10 11

10. language 1 2 4 5 title 7 3 year 8 director 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 1/12/1935 Woody Allen title year acted in acted in Suppose that Node 6 is missing 6 English language 6 English

11. 1 2 4 5 title 7 3 year 8 director 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 1/12/1935 Woody Allen title year acted in acted in v 1 v 2 w 1 v 3 title actor movie director acted in w 2 w 3 w 4 date of birth name language An incomplete matching This matching is maximal 1 2 5 4 10 11 w 2 null

15. 1 2 4 5 title 7 3 year 8 director 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 1/12/1935 Woody Allen title year acted in acted in v 1 v 2 w 1 v 3 title actor movie director acted in w 2 w 3 w 4 date of birth name language A weak matching w 2 1 2 5 4 10 11 null

16. 1 2 4 5 title 7 3 year 8 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 1/12/1935 Woody Allen title year acted in acted in A Movie Database Consider the case where the director edge is missing director director

17. 1 2 4 5 title 7 3 year 8 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 1/12/1935 Woody Allen title year acted in acted in v 1 v 2 w 1 v 3 title actor movie director acted in w 2 w 3 w 4 date of birth name language An incomplete matching that is not a weak matching w 2 1 2 5 4 10 11 null There is an edge that is not weakly satisfied

20. More Examples

21. 1 2 4 5 6 title language 7 3 year 8 director 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 English 1/12/1935 Woody Allen title year acted in acted in The Movie Database Before the Removals

22. 1 2 4 5 6 title language 7 3 year 8 director 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 English 1/12/1935 Woody Allen title year acted in acted in v 1 v 2 w 1 v 3 title actor movie director acted in w 2 w 3 w 4 date of birth name language A complete matching It is also a maximal weak matching It is also a maximal OR-matching In the result, the actor must be both an actor in the movie and the director of the movie 1 2 5 6 4 10 11

23. 1 2 4 5 6 title language 7 3 year 8 director 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 English 1/12/1935 Woody Allen title year acted in acted in v 1 v 2 w 1 v 3 title actor movie director acted in w 2 w 3 w 4 date of birth name language A second maximal weak matching In the result, if the actor and the movie are assigned non-null values, then the actor must be both an actor in the movie and the director of the movie 1 8 3 null null null null

24. 1 2 4 5 6 title language 7 3 year 8 director 9 name 10 movie date of birth 11 1983 movie actor Zelig Antz 1998 English 1/12/1935 Woody Allen title year acted in acted in v 1 v 2 w 1 v 3 title actor movie director acted in w 2 w 3 w 4 date of birth name language A maximal OR-matching In the result, the actor either played in the movie, directed the movie, or is not related at all to the movie 1 8 3 4 10 11 null It is not a weak matching

25. Complexity of Evaluating Maximal Weak Matchings and Maximal OR Matchings

31. Full Disjunctions What is the full disjunction of a set of relations? How are full disjunctions related to queries with incomplete answers ?

32. Movies Actors Acted-in Actors-that-Directed The Full Disjunction of the Given Relations English 1998 Armageddon 3 English 1940 Fantasia 4 English 1998 Antz 2 English 1983 Zelig 1 language year title m-id 19/3/1955 Bruce Willis 2 28/10/1967 Julia Roberts 3 1/12/1935 Woody Allen 1 date-of-birth name a-id Z 2 1 Harry 3 2 Zelig 1 1 role m-id a-id 1 1 m-id a-id Harry 19/3/1955 Bruce Willis 2 English 1998 Armageddon 3     English 1940 Fantasia 4  Z Zelig role 28/10/1967 1/12/1935 1/12/1935 Date-of-birth Julia Roberts Woody Allen Woody Allen name 3 1 1 a-id     English 1998 Antz 2 English 1983 Zelig 1 language year title m-id

33. The Full Disjunction of the Given Relations The full disjunction does not include subsumed tuples Movies Harry 19/3/1955 Bruce Willis 2 English 1998 Armageddon 3     English 1940 Fantasia 4  Z Zelig role 28/10/1967 1/12/1935 1/12/1935 Date-of-birth Julia Roberts Woody Allen Woody Allen name 3 1 1 a-id     English 1998 Antz 2 English 1983 Zelig 1 language year title m-id  role  Date-of-birth  name  a-id English 1983 Zelig 1 language year title m-id English 1998 Armageddon 3 English 1940 Fantasia 4 English 1998 Antz 2 English 1983 Zelig 1 language year title m-id This tuple will not be in the full disjunction

34. Movies Actors Acted-in Actors-that-Directed The Full Disjunction of the Given Relations The full disjunction does not include tuples that are based on Cartesian Product rather than join English 1998 Armageddon 3 English 1940 Fantasia 4 English 1998 Antz 2 English 1983 Zelig 1 language year title m-id 19/3/1955 Bruce Willis 2 28/10/1967 Julia Roberts 3 1/12/1935 Woody Allen 1 date-of-birth name a-id Z 2 1 Harry 3 2 Zelig 1 1 role m-id a-id 1 1 m-id a-id Harry 19/3/1955 Bruce Willis 2 English 1998 Armageddon 3     English 1940 Fantasia 4  Z Zelig role 28/10/1967 1/12/1935 1/12/1935 Date-of-birth Julia Roberts Woody Allen Woody Allen name 3 1 1 a-id     English 1998 Antz 2 English 1983 Zelig 1 language year title m-id  role 28/10/1967 Date-of-birth Julia Roberts name 3 a-id English 1940 Fantasia 4 language year title m-id

35. In the Full Disjunction of a Given Set of Relations: Every tuple of the input is a part of at least one tuple of the output Tuples are joined as in a natural join, padded with null values The result includes only “ maximal connected portions”

39. Example Movies Actors Acted-in A node is created for each tuple Edges are added between connected tuples, in both directions A root is added, and edges are added from the root to every node Creating The Database We use colors instead of labels Armageddon 3 Fantasia 4 Antz 2 Zelig 1 title m-id Bruce Willis 2 Julia Roberts 3 Woody Allen 1 name a-id Z 2 1 Harry 3 2 Zelig 1 1 role m-id a-id r

40. Movies Actors Acted-in Creating The Queries Example A node is created for each relation schema Edges are added between connected schemas, in both directions r The number of queries is equal to the number of schemas In each query, the root is connected to a different schema Armageddon 3 Fantasia 4 Antz 2 Zelig 1 title m-id Bruce Willis 2 Julia Roberts 3 Woody Allen 1 name a-id Z 2 1 Harry 3 2 Zelig 1 1 role m-id a-id Movies Actors Acted-in r

41. Queries are Evaluated under Weak Semantics Movies Actors Acted-in Example r Movies Actors Acted-in r Armageddon 3 Fantasia 4 Antz 2 Zelig 1 title m-id Bruce Willis 2 Julia Roberts 3 Woody Allen 1 name a-id Z 2 1 Harry 3 2 Zelig 1 1 role m-id a-id Zelig role Woody Allen name 1 a-id Zelig 1 title m-id role name a-id title m-id

42. Movies Actors Acted-in Example r Movies Actors Acted-in r Queries are Evaluated under Weak Semantics Armageddon 3 Fantasia 4 Antz 2 Zelig 1 title m-id Bruce Willis 2 Julia Roberts 3 Woody Allen 1 name a-id Z 2 1 Harry 3 2 Zelig 1 1 role m-id a-id Zelig role Woody Allen name 1 a-id Zelig 1 title m-id Z Zelig role Woody Allen Woody Allen name 1 1 a-id Antz 2 Zelig 1 title m-id

43. Movies Actors Acted-in Example r Movies Actors Acted-in r Queries are Evaluated under Weak Semantics Armageddon 3 Fantasia 4 Antz 2 Zelig 1 title m-id Bruce Willis 2 Julia Roberts 3 Woody Allen 1 name a-id Z 2 1 Harry 3 2 Zelig 1 1 role m-id a-id Zelig role Woody Allen name 1 a-id Zelig 1 title m-id Z Zelig role Woody Allen Woody Allen name 1 1 a-id Antz 2 Zelig 1 title m-id Harry Bruce Willis 2 Armageddon 3 Z Zelig role Woody Allen Woody Allen name 1 1 a-id Antz 2 Zelig 1 title m-id

44. Movies Actors Acted-in Example r Movies Actors Acted-in r Queries are Evaluated under Weak Semantics Armageddon 3 Fantasia 4 Antz 2 Zelig 1 title m-id Bruce Willis 2 Julia Roberts 3 Woody Allen 1 name a-id Z 2 1 Harry 3 2 Zelig 1 1 role m-id a-id Harry Bruce Willis 2 Armageddon 3 Z Zelig role Woody Allen Woody Allen name 1 1 a-id Antz 2 Zelig 1 title m-id Harry Bruce Willis 2 Armageddon 3  Z Zelig role Julia Roberts Woody Allen Woody Allen name 3 1 1 a-id   Antz 2 Zelig 1 title m-id null null

45. Movies Actors Acted-in Example r Movies Actors Acted-in r Queries are Evaluated under Weak Semantics Armageddon 3 Fantasia 4 Antz 2 Zelig 1 title m-id Bruce Willis 2 Julia Roberts 3 Woody Allen 1 name a-id Z 2 1 Harry 3 2 Zelig 1 1 role m-id a-id Harry Bruce Willis 2 Armageddon 3 Z Zelig role Woody Allen Woody Allen name 1 1 a-id Antz 2 Zelig 1 title m-id Harry Bruce Willis 2 Armageddon 3  Z Zelig role Julia Roberts Woody Allen Woody Allen name 3 1 1 a-id   Antz 2 Zelig 1 title m-id

46. Movies Actors Acted-in Example r Movies Actors Acted-in r Armageddon 3 Fantasia 4 Antz 2 Zelig 1 title m-id Bruce Willis 2 Julia Roberts 3 Woody Allen 1 name a-id Z 2 1 Harry 3 2 Zelig 1 1 role m-id a-id Harry Bruce Willis 2 Armageddon 3 Z Zelig role Woody Allen Woody Allen name 1 1 a-id Antz 2 Zelig 1 title m-id Harry Bruce Willis 2 Armageddon 3  Z Zelig role Julia Roberts Woody Allen Woody Allen name 3 1 1 a-id   Antz 2 Zelig 1 title m-id null null Harry Bruce Willis 2 Armageddon 3  Julia Roberts 3    Z Zelig role  Woody Allen Woody Allen name  1 1 a-id Fantasia 4 Antz 2 Zelig 1 title m-id

47. The Algorithm Computes Full Disjunctions in Polynomial Time Under Input-Output Complexity Theorem: The full disjunction of relations r 1 , …, r n can be computed in O ( n 5 s 2 f 2 ) time, where n is the number of relations, s is the total size of all the relations and f is the size of the result

49. Example Movies ( m-id , title, year, language, location) Actors ( a-id , name, date-of-birth) Acted-in (a-id, m-id, role) Actors-that-Directed (a-id, m-id) Historical-Events ( name , date, description) Historical-Sites (Country, State, City, Site) The date of the historical event is a date in the year when the movie was released The filming location is near the historical site

52. Employees (e-id, ename, city, dept-no) Departments (dept-no, dname, building) Located-in (building, city, street) Example The Full Disjunction Employee: (007, James Bond, London, 6) Department: (6, MI-6, 10) Located-in: (10, Liverpool, King) 10  building Liverpool  city 10 10 building 6 6 dept -no King MI-6      MI-6 6 London James Bond 007 street dname dept -no city ename e-id

53. Employees (e-id, ename, city, dept-no) Departments (dept-no, dname, building) Located-in (building, city, street) Example The Full Disjunction under OR-Semantics Employee: (007, James Bond, London, 6) Department: (6, MI-6, 10) Located-in: (10, Liverpool, King) 10 building Liverpool city 10 building 6 dept -no King MI-6 6 London James Bond 007 street dname dept -no city ename e-id

54. The Projection Problem : Computing the projection of the full disjunction on a given set of attributes The Restriction Problem : Computing only those tuples of the full disjunction that are non-null on a given set of attributes Two Related Problems The projection problem and the restriction problem cannot be computed in polynomial time (under input-output complexity) unless P=NP

57. Thank You Questions?