1. CSE544
Wednesday, March 29, 2006

2. Theory Relational databases invented by a theoretician (Codd)
Fundamental principle: separate the WHAT from the HOW - data independence
WHAT: First Order Logic (FO)
HOW: Relational algebra (RA)

3. FO Syntax Given: A vocabulary: R1, …, Rk
An arity, ar(Ri), for each i=1,…,k
An infinite supply of variables x1, x2, x3, …
Constants: c1, c2, c3, ...

4. FO Syntax Terms (t) and FO formulas () are: t ::= x | c
::= R(t1, ..., tar(R)) | ti = tj
|   ’ |   ’ | ’
| x. | x.

5. FO Examples Most interesting case:
Vocabulary = one binary relation R (encodes a graph) 1 4 2 3 1 2 3 4 R=

6. FO Sentences Does there exists a loop in the graph ?
Are there paths of length >2 ?
Is there a “sink” node ?
  x.R(x,x)
  x.y.z.u.(R(x,y)  R(y,z)  R(z,u))
  x.y.R(x,y)

7. Semantics Given a vocabulary R1, …, Rk A model is D = (D, R1D, …, RkD)
D = a set, called domain, or universe
RiD  D  D  ...  D, (ar(Ri) times) i = 1,...,k

8. Semantics
Given:
A model D = (D, R1D, ..., RkD)
A formula 
A substitution s : {x1, x2, ...}  D
We define next the relation: meaning “D satisfies with s”
D |= [s]

9. Semantics D |= (R(t1, ..., tn)) [s] D |= (t= t’) [s]
If (s(t1), ..., s(tn))  RD
D |= (t= t’) [s]
If s(t) = s(t’)

10. Semantics D |= (  ’) [s] D |= (  ’) [s] D |= (’) [s]
If D |= ()[s] and D |= (’) [s]
D |= (  ’) [s]
If D |= ()[s] or D |= (’) [s]
D |= (’) [s]
If not D |= ()[s]

11. First Order Logic: Semantics
D |= (x.) [s]
If for all s’ s.t. s(y) = s’(y) for all
variables y other than x, D |= ()[s’]
D |= (x.) [s]
If for some s’ s.t. s(y) = s’(y) for all
variables y other than x, D |= ()[s’]

12. FO and Databases FO: a sentence  is true in D if D |= 
Databases: a formula  with free variables x1, ..., xn defines the query: (D) = {(s(x1), ..., s(xn)) | D |= [s]}

13. FO Queries Find all nodes connected by a path of length 2:
Find all nodes without outgoing edges:
(x,y)  u.(R(x,u)  R(u,y))
(x)  u.(R(u,x)  y.R(x,y)
These are open formulas

14. In Class Retrieve all nodes with at least two children
A node x is more important than y if every child of y is also a child of x. Retrieve all ‘most important nodes’ in the graph

15. FO in Databases FO Databases Vocabulary: R1, ..., Rn Database schema:
Model: D = (D, R1D, …, RkD)
Database instance: D = (D, R1D, …, RkD)
Sentences are
true or false
Formulas compute queries

16. FO Semantics In FO we express WHAT we want
Sometimes it’s even unclear HOW to get it
See accompanying slides on FO semantics
They explain HOW to get it, but it’s impractical

17. Relational Algebra An algebra over relations Five operators: Meaning:
, -, , s, P
Meaning:
R1  R2 = set union
R1 - R2 = set difference
R1  R2 = cartesian product
sc(R) = subset of tuples satisfying condition c
Pa(R) = projection on the attributes in a

18. FO  RA (x)  R(x,x)  P1(s1=2(R))
(x,y)  z.u.(R(x,z)  R(z,u)  R(u,y)) 
P16(s2=34=5 (R  R  R))
P16 ((R join2=1 R) join4=1 R)) ?
(x)  y.R(x,y) 
WHAT 
HOW

19. FO v.s. RA Theorem. Every query in RA can be expressed in FO Proof
This shows how to go from HOW to WHAT
not very interesting
What about the converse ?

20. The Drinkers/Beers Example
Vocabulary:
Find all drinkers that frequent some bar that serve some beer that they like:
Likes(drinker,beer), Serves(bar,beer), Frequents(drinker,bar)
(d)  ba. be.(F(d,ba)  L(d,be))

21. Lots of Fun Examples (in class)
Find drinkers that frequent some bar that serves only beer they like
Find drinkers that frequent only bars that serve some beer they like
Find drinkers that frequent only bars that serve only beer they like

22. Unsafe FO Queries
Find all nodes that are not in the graph: what’s wrong ?

23. Unsafe FO Queries
Find all nodes that are connected to “everything”: what’s wrong ?

24. Unsafe FO Queries
Find all pairs of employees or offices: what’s wrong ?
We don’t want such queries !

25. Safe Queries A model D = (D, R1D, …, RkD) In FO: In databases:
both D and R1D, …, RkD may be infinite
In databases:
D may infinite (int, string, etc)
R1D, …, RkD are always finite
We call this a finite model

26. Safe Queries
 is a finite query if for every finite model D, (D) is finite
 is safe, or domain independent, if for every two models D, D’ having the same relations: D = (D, R1D, …, RkD), D’ = (D’, R1D, …, RkD) we have (D) = (D’)
If  is safe then it is also finite (why ?)
Note: book has different but equivalent definition

27. Safe Queries
Definition. Given D = (D, R1D, …, RkD), the active domain is Da = the set of all constants in R1D, …, RkD
Example. Given a graph D = (D, R) Da = { x | y.R(x,y)  z.R(z,x)}
Property. If a query is safe, it suffices to range quantifiers only over the active domain (why ?)
Hence we can compute safe queries

28. Safe Queries
The safe relational calculus consists only of safe queries. However:
Theorem It is undecidable if a given a FO query is safe.
Need to write only safe queries, but how do we know how which queries are safe ?
Work around: write them in an obviously safe way
Range restricted queries - formally defined in [AHU]

29. FO v.s. RA Theorem. Every safe query in FO can be expressed in RA
Proof
From WHAT to HOW
this is really interesting and motivated the relational model

30. Limited Expressive Power
Vocabulary: binary relation R
The following queries cannot be expressed in FO:
Transitive closure:
x.y. there exists x1, ..., xn s.t. R(x,x1)  R(x1,x2)  ...  R(xn-1,xn)  R(xn,y)
Parity: the number of edges in R is even