1. What is a Quasi-Inverse? Jordan Johnson May 29, 2008 TWIGS at UCSC

2. Background: Data Exchange Given: Databases A, B Problem: Populate B with data from A AB

3. Background: Schemata A, B specified by schemata S, T Mappings relate the schemata S = { EmployeeDB: { HrlyEmp(id, name, wage), SalariedEmp(id, name, sal) } T = { PersonnelDB: { Emp(id, name, ssn) }

4. Background: A Mapping M = (S, T,  ) where  = {  (id, nm, w): HrlyEmp(id, nm, w)   (ssn): Emp(id, nm, ssn),  (id, nm, s): SalariedEmp(id, nm, s)   (ssn): Emp(id, nm, ssn) } are tuple generating dependencies specifying the mapping.

5. Mappings as data? Operations on mappings:  compose [Madhavan/Halevy, 2003]  match/diff/etc [Bernstein, 2003]  invert [Fagin, 2006]  Well, only sometimes… Background: what, more?

6. The problem with inverses… Original must be injective. S-T mapping  Defined w.r.t. schemata  Domain/range are DB instances

7. Our example again:  (id, nm, w): HrlyEmp(id, nm, w)   (ssn): Emp(id, nm, ssn),  (id, nm, s): SalariedEmp(id, nm, s)   (ssn): Emp(id, nm, ssn)

8. Other problematic mappings: Projection:  {P(x, y)  Q(x)} Union:  {P(x)  R(x), Q(x)  R(x)} Decomposition:  {P(x, y, z)  Q(x, y) ^ R(y, z)}

9. Quasi-Inverse (of a function) Given f : X -> Y, a function g is a quasi-inverse if:  g : Z -> X, where ran(f)  Z  Y, and  f  g  f = f

10. Quasi-Inverse (of a function) Examples:  f(x) = x 2  g 1 (x) = sqrt(x)  g 2 (x) = -sqrt(x)  f(x) =  x   g(x) = x + , where 0    1.

11. Features of Quasi-Inverses Weaker conditions than for inverse Still recovers some of the original If an inverse f exists:  a single quasi-inverse g exists  g = f

12. Quasi-Inverse (of a mapping) Problem:  What’s a good analogous quasi-inverse definition for schema mappings?

13. Quasi-Inverse (of a mapping) Idea:  Parameterize the idea of “inverse” w.r.t. equivalence relations ~ 1, ~ 2 on schema instances.

14. Quasi-Inverse: Preliminaries S´ is a replica of S. Instance I´ is a replica of I. For mapping M = (S, S´,  ),  Inst(M) = set of all (I 1, I 2 ) such that I 1 and I 2 are instances of S, S´ (resp.) and satisfy .  Id = (S, S´,  id ), where  id contains R(x) -> R(x) for all R in S.  Note: I 1 (Inst(Id)) I 2 if I 1 ´  I 2.

15. Quasi-Inverse: Preliminaries Note: M’ is an inverse of M if Inst(Id) = Inst(M’ · M). For a binary relation D, let I 1 D[~ 1, ~ 2 ] I 2 if there exist J 1, J 2 such that (I 1, I 2 ) ~ (1,2) (J 1, J 2 ) and J 1 D J 2.

16. Quasi-Inverse (of a mapping) Formal definition:  Given a mapping M = (S, T,  ), M´ = (T, S,  ´) is a (~ 1, ~ 2 )-inverse of M iff Inst(Id)[~ 1, ~ 2 ] = Inst(M’ · M)[~ 1, ~ 2 ].

17. Quasi-Inverse (of a mapping) Again, M = (S, S´,  ). Let Sol(M, I) be the set of solutions for instance I under M: all J such that (I, J) satisfy . Let (I 1 ~ M I 2 )  Sol(M, I 1 ) = Sol(M, I 2 ). M’ is a quasi-inverse of M means: M’ is a (~ M, ~ M )-inverse of M.

18. Example Quasi-Inverses Projection:  {P(x, y)  Q(x)}  QI: {Q(x)   z.P(x,z)} Union:  {P(x)  R(x), Q(x)  R(x)}  QI: {R(x)  P(x)  Q(x)}