Problem Solving with Constraints CSPs & Relational DBs Problem Solving with Constraints CSCE421/821, Spring 2018 www.cse.unl.edu/~choueiry/S18-421-821/ All questions Piazza Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 360 Tel: +1(402)472-5444 CSPs and Relational DBs

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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