Relational Algebra & Calculus

Dr.S.Sivakumar,Principal

C.P.A College, Bodinayakanur

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Relational Query Languages

• Query languages: Allow manipulation and retrieval of data from a database.
• Relational model supports simple, powerful QLs:
• Strong formal foundation based on logic.
• Allows for much optimization.
• Query Languages != programming languages!
• QLs not expected to be “Turing complete”.
• QLs not intended to be used for complex calculations.
• QLs support easy, efficient access to large data sets.

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Formal Relational Query Languages

• Two mathematical Query Languages form the basis for “real” languages (e.g. SQL), and for implementation:
• Relational Algebra: More operational (procedural), very useful for representing execution plans.
• Relational Calculus: Lets users describe what they want, rather than how to compute it: Non-operational, declarative.

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Preliminaries

• A query is applied to relation instances, and the result of a query is also a relation instance.
• Schemas of input relations for a query are fixed.
• The schema for the result of a given query is also fixed! - determined by definition of query language constructs.
• Positional vs. named-field notation:
• Positional notation easier for formal definitions, named-field notation more readable.
• Both used in SQL

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Example Instances

• “Sailors” and “Reserves” relations for our examples.
• We’ll use positional or named field notation, assume that names of fields in query results are `inherited’ from names of fields in query input relations.

R1

S1

S2

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Relational Algebra

• Basic operations:
• Selection ( ) Selects a subset of rows from relation.
• Projection ( ) Deletes unwanted columns from relation.
• Cross-product ( ) Allows us to combine two relations.
• Set-difference ( ) Tuples in reln. 1, but not in reln. 2.
• Union ( ) Tuples in reln. 1 and in reln. 2.
• Additional operations:
• Intersection, join, division, renaming: Not essential, but (very!) useful.
• Since each operation returns a relation, operations can be composed: algebra is “closed”.

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Projection

• Deletes attributes that are not in projection list.
• Schema of result contains exactly the fields in the projection list, with the same names that they had in the input relation.
• Projection operator has to eliminate duplicates! Why?
• Note: real systems typically don’t do duplicate elimination unless the user explicitly asks for it (by DISTINCT). Why not?

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Selection

• Selects rows that satisfy selection condition.
• No duplicates in result! Why?
• Schema of result identical to schema of input relation.
• What is Operator composition?
• Selection is distributive over binary operators
• Selection is commutative

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Union, Intersection, Set-Difference

• All of these operations take two input relations, which must be union-compatible:
• Same number of fields.
• `Corresponding’ fields have the same type.
• What is the schema of result?

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Cross-Product (Cartesian Product)

• Each row of S1 is paired with each row of R1.
• Result schema has one field per field of S1 and R1, with field names `inherited’ if possible.
• Conflict: Both S1 and R1 have a field called sid.

• Renaming operator:

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Joins: used to combine relations

• Condition Join:

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• Result schema same as that of cross-product.
• Fewer tuples than cross-product, might be able to compute more efficiently
• Sometimes called a theta-join.

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Join

• Equi-Join: A special case of condition join where the condition c contains only equalities.

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• Result schema similar to cross-product, but only one copy of fields for which equality is specified.
• Natural Join: Equijoin on all common fields.

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Properties of join

• Selecting power: can join be used for selection?
• Is join commutative? = ?
• Is join associative?
• Join and projection perform complementary functions
• Lossless and lossy decomposition

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Division

• Not supported as a primitive operator, but useful for expressing queries like: Find sailors who have reserved all boats.
• Let A have 2 fields, x and y; B have only field y:
• A/B =
• i.e., A/B contains all x tuples (sailors) such that for every y tuple (boat) in B, there is an xy tuple in A.
• Or: If the set of y values (boats) associated with an x value (sailor) in A contains all y values in B, the x value is in A/B.
• In general, x and y can be any lists of fields; y is the list of fields in B, and x y is the list of fields of A.

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Examples of Division A/B

A

B1

B2

B3

A/B1

A/B2

A/B3

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Example of Division

• Find all customers who have an account at all branches located in Chville
• Branch (bname, assets, bcity)
• Account (bname, acct#, cname, balance)

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Example of Division

R1: Find all branches in Chville

R2: Find (bname, cname) pair from Account

R3: Customers in r2 with every branch name in r1

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Expressing A/B Using Basic Operators

• Division is not essential op; just a useful shorthand.
• Also true of joins, but joins are so common that systems implement joins specially.
• Idea: For A/B, compute all x values that are not `disqualified’ by some y value in B.
• x value is disqualified if by attaching y value from B, we obtain an xy tuple that is not in A.

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Disqualified x values:

A/B:

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Exercises

Given relational schema:

Sailors (sid, sname, rating, age)

Reservation (sid, bid, date)

Boats (bid, bname, color)

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1. Find names of sailors who’ve reserved boat #103
2. Find names of sailors who’ve reserved a red boat
3. Find sailors who’ve reserved a red or a green boat
4. Find sailors who’ve reserved a red and a green boat
5. Find the names of sailors who’ve reserved all boats

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Summary of Relational Algebra

• The relational model has rigorously defined query languages that are simple and powerful.
• Relational algebra is more operational; useful as internal representation for query evaluation plans.
• Several ways of expressing a given query; a query optimizer should choose the most efficient version.

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Relational Algebra & Calculus

Chapter 4, Part B (Relational Calculus)

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Relational Calculus

• Comes in two flavors: Tuple relational calculus (TRC) and Domain relational calculus (DRC).
• Calculus has variables, constants, comparison ops, logical connectives and quantifiers.
• TRC: Variables range over (i.e., get bound to) tuples.
• DRC: Variables range over domain elements (= field values).
• Both TRC and DRC are simple subsets of first-order logic.
• Expressions in the calculus are called formulas. An answer tuple is essentially an assignment of constants to variables that make the formula evaluate to true.

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Domain Relational Calculus

• Query has the form:

• Answer includes all tuples that

make the formula be true.

• Formula is recursively defined, starting with

simple atomic formulas (getting tuples from

relations or making comparisons of values),

and building bigger and better formulas using

the logical connectives.

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DRC Formulas

• Atomic formula:
• , or X op Y, or X op constant
• op is one of
• Formula:
• an atomic formula, or
• , where p and q are formulas, or
• , where X is a domain variable or
• , where X is a domain variable.
• The use of quantifiers and is said to bind X.

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Free and Bound Variables

• The use of quantifiers and in a formula is said to bind X.
• A variable that is not bound is free.
• Let us revisit the definition of a query:

• There is an important restriction: the variables x1, ..., xn that appear to the left of `|’ must be the only free variables in the formula p(...).

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Find all sailors with a rating above 7

• The condition ensures that the domain variables I, N, T and A are bound to fields of the same Sailors tuple.
• The term to the left of `|’ (which should be read as such that) says that every tuple that satisfies T>7 is in the answer.
• Modify this query to answer:
• Find sailors who are older than 18 or have a rating under 9, and are called ‘Joe’.

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Find sailors rated > 7 who have reserved boat #103

• We have used as a shorthand for

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• Note the use of to find a tuple in Reserves that `joins with’ the Sailors tuple under consideration.

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Find sailors rated > 7 who’ve reserved a red boat

• Observe how the parentheses control the scope of each quantifier’s binding.
• This may look cumbersome, but with a good user interface, it could be intuitive. (MS Access, QBE)

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Find sailors who’ve reserved all boats

• Find all sailors I such that for each 3-tuple either it is not a tuple in Boats or there is a tuple in Reserves showing that sailor I has reserved it.

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Find sailors who’ve reserved all boats (again!)

• Simpler notation, same query. (Much clearer!)
• To find sailors who’ve reserved all red boats:

.....

Any other way to specify it? Equivalence in logic

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Unsafe Queries, Expressive Power

• It is possible to write syntactically correct calculus queries that have an infinite number of answers! Such queries are called unsafe.
• e.g.,

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• It is known that every query that can be expressed in relational algebra can be expressed as a safe query in DRC / TRC; the converse is also true.
• Relational Completeness: Query language (e.g., SQL) can express every query that is expressible in relational algebra/calculus.

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Exercise of tuple calculus

Given relational schema:

Sailors (sid, sname, rating, age)

Reservation (sid, bid, date)

Boats (bid, bname, color)

1. Find all sialors with a rating above 7.
2. Find the names and ages of sailors with a rating above 7
3. Find the sailor name, boal id, and reservation date for each reservation
4. Find the names of the sailors who reserved all boats.

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Summary of Relational Calculus

• Relational calculus is non-operational, and users define queries in terms of what they want, not in terms of how to compute it. (Declarativeness.)
• Algebra and safe calculus have same expressive power, leading to the notion of relational completeness.

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