1. Problem Solving with Constraints
CSPs & Relational DBs
Problem Solving with Constraints
CSCE421/821, Spring 2018
All questions Piazza
Berthe Y. Choueiry (Shu-we-ri)
Avery Hall, Room 360
Tel: +1(402)
CSPs and Relational DBs

2. CSPs and Relational DBs
Background
Strong historical & conceptual connections exist between:
Constraint Databases & Constraint Logic Programming
Query processing in Relational DBs & Solving CSPs
Indeed:
Constraint databases (deductive BD, Datalog) and constraint logic programming (CLP) share the representation language (restricted forms of FOL)
Relational databases and Constraint Satisfaction share computation mechanisms
CSPs and Relational DBs

3. CSPs and Relational DBs
Relations
Binary relation: Given two sets Da and Db, a set of any 2-tuples
< x, y> with  Da and y  Db defines a relation Rab
Ra,b = {(x, y)}  Da x Db
Function: (special binary relation)
For any element x in Da there is at most one tuple < x, ? >  Rab
Da is called the domain of the the function
Db is called the co-domain of the function
k-ary relation: Given k sets D1, D2, …, Dk, any set of k-tuples
< x1, x2, ..., xk > with x1  D1, x2  D2, …, xk  Dk defines a
k-ary relation: R1, 2, ..., k = {(x1, x2, ..., xk)}  D1 x D2 x … x Dk
CSPs and Relational DBs

4. Representation of relations
Binary arrays:
 2-dim binary array (i.e., bit matrix):
 more generally, k-dimensional binary arrays
Tables: A B V C1 D V C2 E F G C3 V
CSPs and Relational DBs

5. Comparison of terminology
DB terminology
CSP terminology
Table, relation
Constraint
Relation arity
Constraint arity
Attribute
CSP variable
Value of an attribute
Value of a variable
Domain of an attribute
Domain of a variable
Tuple in a table
Tuple in a constraint
Tuple allowed by an constraint
Tuple consistent with a constraint
Constraint relation (in constraint databases)
Constraint of linear (in)equality
CSPs and Relational DBs

6. Relational Algebra: operations on relations
Database:
Intersection
Union
Difference
Selection
Projection
Join (Cartesian product), etc.
CSP:
The above and composition (= combination of join and projection)
CSPs and Relational DBs

7. Operators in Relational Algebra
Selection, projection
unary operators, defined on one relation
Intersection, union, difference
binary operators
relations must have same scope
Join
binary operator
relations have different scopes
CSPs and Relational DBs

8. CSPs and Relational DBs
Intersection
Input: two relations of the same scope
Output: a new more restrictive relation with the same scope, made of tuples that are in all the input relations (simultaneously)
Bit-matrix operation: logical AND
R R' R’’
R  R’ ? R  R'’ ?
x + y > 10
x - y > 10
Okay
Not defined
CSPs and Relational DBs

9. CSPs and Relational DBs
Union
Input: two relations of the same scope
Output: a new less restrictive relation with the same scope made of tuples that are in any of the input relations
Bit-matrix operation: logical OR
R R' R’’
R  R'? R  R''?
Okay
Not defined
CSPs and Relational DBs

10. CSPs and Relational DBs
Difference
Input: two relations R and R' of the same scope
Output: a new more restrictive relation than R made of tuples that are in R but not in R'
Bit-matrix operation: Boolean difference
R R' R’’
R - R'? R - R''?
Okay
Not defined
CSPs and Relational DBs

11. CSPs and Relational DBs
Selection
Input: A relation R and some test/predicate on attributes of R
Output: A relation R', same scope as R but containing only a
subset of the tuples in R (those that satisfy the predicate)
Relation operation: row selection R
Select such that X1> X2, ?
CSPs and Relational DBs

12. CSPs and Relational DBs
Projection
Input: A relation R and a subset s of the scope (attributes)
Output: A relation R' of scope s with the tuples rewritten such that
positions not in s are removed
Relation operation: column elimination R
Project R on X1> X2, ?
CSPs and Relational DBs

13. CSPs and Relational DBs
Join (natural join)
Input: Two relations R and R'
Output: A relation R'', whose scope is union of scopes of R and R' and tuples satisfy both R and R'.
 R and R' have no attribute common: Cartesian product
 R and R' have an attribute in common, compute all possibilities
Operation: Compute all solutions to a CSP.
R R"
Join R and R'', R R''?
CSPs and Relational DBs

14. Composition of relations Montanari'74
Input: two binary relations Rab and Rbc with 1 variable in common.
Output: a new induced relation Rac (to be combined by intersection to a pre-existing relation between them, if any).
Bit-matrix operation: matrix multiplication
Note: - generalization as constraint synthesis [Freuder, 1978]
- Direct (explicit) vs. induced (implicit) relations
CSPs and Relational DBs

15. CSPs and Relational DBs
Questions
Given
two variables V1 and V2 and
two constraints C1 and C2 between them
How do the two expressions C1  C2 and C1 C2 relate?
three variables V1, V2, V3 and
the binary constraints CV1, V2 and CV2, V3
write the induced CV1, V3, in relational algebra
the binary constraints CV1, V2, CV1, V3, and CV2, V3,
write the new induced C’V1, V3 in relational algebra
CSPs and Relational DBs

16. Comparison of Terminology
Databases
CSPs
(Natural, inner) join
Synthesized constraint
Left/right outer join
Synthesized constraint including (some) inconsistent tuples
Projection of a join
Induced constraint
(Composition of two constraints)
Computing r1 r2 … ri
Finding all solutions to the conjunction of the constraints rk
CSPs and Relational DBs

17. CSPs and Relational DBs
DB versus CSP
DB:
Relations
Few relations in a query
Usually, high arity relations
Usually, selective relations
Domains
Large domains, many, many tuples
Mostly finite (CDB introduce continuous domains, restricted to linear constraints)
Seeking (almost) all tuples
Cost model: I/O disk access, memory size
CSPs:
Relations
Many, many relations
Mostly low-arity relations (binary)
Typically, much looser than in DB
Domains
Small-size domains
Finite (frequent), but also continuous (with arbitrary relations).
Seeking (in general) one solution
Cost model: computational cost
CSPs and Relational DBs