Relational Model Extensions: Simplifying Data Modeling and Enhancing Expressive Power, Slides of Introduction to Database Management Systems

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Extensions of the Relational Model

David Toman School of Computer ScienceUniversity of Waterloo Introduction to Databases CS

Why?

1 To simplify • Data Modeling representing complex values conveniently )records/sets/... as first-class citizens

• (^) avoiding need for (foreign) key invention 2 To enhance • Expressive Power enabling transitive closure, etc.

OO/Complex Object/N1NF Model

Idea

Allow non-atomic objects as values of attributes. Records

• • allow attribute values to be (other) records preferably w/o pointers Sets
• allow attribute values to be sets of (other) objects

Classes of Objects (and language extensions)

Structure Data: Classes C = fA 1 : C 1 ; : : : ; Ak : Ck g

Example

Emp = { name: String, salary: Int, dept: Dept } Dept = { name: String, boss: Emp } Query Extensions:

• (^) variables range over “objects”
• attributes can form path expressions

Example

SELECT e.name, e.dept.boss.name FROMWHERE Empe.dept = d, e.salary > e.dept.boss.salary e, Dept d

Records

Are Tuples Objects themselves?

• if so, what is their “id”? Problems with “oid” invention
• (^) when are two tuples equal?
• structural equality may not work (cycles!) ... in general, we can do without.

Sets

how are sets created?

• • as subsets of the universe/active domain as groups what can we do with them in queries?
• • (^) variables range over objects of particularadditional operations for/between such values: structure
• • (^) xf (^) x (^2) j' yg for set membershipfor set-value construction
• quantifiers can also range over sets

Nesting and Unnesting

What’s the problem with Powerset? ) too many new objects! Nesting/Unnesting

Idea

Only allow construction of sets for which we can create a name fromvalues we already have in the database. ) just like the GROUP BY clause.

Theorem

Queries over flat databases are equivalent to flat queries.

Adding Recursion & Datalog

Can we have some middle ground (PTIME)?

Idea

Allow inductive definitions of relations of the form

• (^) certain base tuples are in the relation
• if a tuple is in the relation then another is too
• (^) nothing else is in the relation Transitive closure revisited:

Example

tc tc((xx ;; yy)) EE ((xx ;; yz )); tc(z ; y)

What Happened to Negation?

Note

The inductive definition must provide a monotone functional We cannot use general negation

• (^) restrict to positive queries
• • restrict tomore complex definition of even number of negations minimal model
  • • well-founded semanticsstable-model semantics

Caveat Emptor in SQL

Note

The framework only works for set semantics. In SQL it breaks when:

• (^) you use duplicate semantics
• you use arithmetic/string functions/...
• • you use EXCEPT ...

Constraint Queries

• (^) for R(x ; y) a set of points; do all the points lie on a straight line? 9 a; b: 8 x ; y:R(x ; y)  y = a  x + b ) is this “safe”? How can we find out a and b?
• (^) do relations R(x ; y) and S (x ; y) intersect in the plane? 9 x ; y:R(x ; y) ^ S (x ; y) ) imagine R stores “objects” rather than points...

Summary

Extensions allow:

• (^) (more) convenient modeling of data
• • (more) expressive queries (more) compact representation

Note

Understanding why things work depends on understanding LOGICAL FOUNDATIONS Also helps with understanding properties,...