Relational algebra

In database theory, relational algebra is a theory that uses algebraic structures for modeling data and defining queries on it with well founded semantics. The theory was introduced by Edgar F. Codd.^[1]

The main application of relational algebra is to provide a theoretical foundation for relational databases, particularly query languages for such databases, chief among which is SQL. Relational databases store tabular data represented as relations. Queries over relational databases often likewise return tabular data represented as relations.

The main purpose of relational algebra is to define operators that transform one or more input relations to an output relation. Given that these operators accept relations as input and produce relations as output, they can be combined and used to express complex queries that transform multiple input relations (whose data are stored in the database) into a single output relation (the query results).

Unary operators accept a single relation as input. Examples include operators to filter certain attributes (columns) or tuples (rows) from an input relation. Binary operators accept two relations as input and combine them into a single output relation. For example, taking all tuples found in either relation (union), removing tuples from the first relation found in the second relation (difference), extending the tuples of the first relation with tuples in the second relation matching certain conditions, and so forth.

Introduction

Relational algebra received little attention outside of pure mathematics until the publication of E.F. Codd's relational model of data in 1970. Codd proposed such an algebra as a basis for database query languages.

Relational algebra operates on homogeneous sets of tuples $S=\{(s_{j1},s_{j2},...s_{jn})|j\in 1...m\}$ where we commonly interpret m to be the number of rows of tuples in a table and n to be the number of columns. All entries in each column have the same type.

A relation also has a unique tuple called the header which gives each column a unique name or attribute inside the relation). Attributes are used in projections and selections.

Set operators

The relational algebra uses set union, set difference, and Cartesian product from set theory, and adds additional constraints to these operators to create new ones.

For set union and set difference, the two relations involved must be union-compatible—that is, the two relations must have the same set of attributes. Because set intersection is defined in terms of set union and set difference, the two relations involved in set intersection must also be union-compatible.

For the Cartesian product to be defined, the two relations involved must have disjoint headers—that is, they must not have a common attribute name.

In addition, the Cartesian product is defined differently from the one in set theory in the sense that tuples are considered to be "shallow" for the purposes of the operation. That is, the Cartesian product of a set of n-tuples with a set of m-tuples yields a set of "flattened" $(n + m)$ -tuples (whereas basic set theory would have prescribed a set of 2-tuples, each containing an n-tuple and an m-tuple). More formally, R × S is defined as follows:

$R\times S:=\{(r_{1},r_{2},\dots ,r_{n},s_{1},s_{2},\dots ,s_{m})|(r_{1},r_{2},\dots ,r_{n})\in R,(s_{1},s_{2},\dots ,s_{m})\in S\}$

The cardinality of the Cartesian product is the product of the cardinalities of its factors, that is, |R × S| = |R| × |S|.

Projection

A projection ( $Π$ ) is a unary operation written as $\Pi _{a_{1},\ldots ,a_{n}}(R)$ where $a_{1},\ldots ,a_{n}$ is a set of attribute names. The result of such projection is defined as the set that is obtained when all tuples in R are restricted to the set $\{a_{1},\ldots ,a_{n}\}$ .

Note: when implemented in SQL standard the "default projection" returns a multiset instead of a set, and the $Π$ projection to eliminate duplicate data is obtained by the addition of the DISTINCT keyword.

Selection

A generalized selection (σ) is a unary operation written as $\sigma _{\varphi }(R)$ where $φ$ is a propositional formula that consists of atoms as allowed in the normal selection and the logical operators $\wedge$ (and), $\lor$ (or) and $\neg$ (negation). This selection selects all those tuples in R for which $φ$ holds.

To obtain a listing of all friends or business associates in an address book, the selection might be written as $\sigma _{{\text{isFriend = true}}\,\lor \,{\text{isBusinessContact = true}}}({\text{addressBook}})$ . The result would be a relation containing every attribute of every unique record where $isFriend$ is true or where $isBusinessContact$ is true.

Rename

A rename (ρ) is a unary operation written as $\rho _{a/b}(R)$ where the result is identical to R except that the b attribute in all tuples is renamed to an a attribute. This is commonly used to rename the attribute of a relation for the purpose of a join.

To rename the "isFriend" attribute to "isBusinessContact" in a relation, $\rho _{\text{isBusinessContact / isFriend}}({\text{addressBook}})$ might be used.

There is also the $\rho _{x(A_{1},\ldots ,A_{n})}(R)$ notation, where R is renamed to x and the attributes $\{a_{1},\ldots ,a_{n}\}$ are renamed to $\{A_{1},\ldots ,A_{n}\}$ .^[2]

Natural join

Natural join (⨝) is a binary operator that is written as (R ⨝ S) where R and S are relations.^[a] The result of the natural join is the set of all combinations of tuples in R and S that are equal on their common attribute names. For an example consider the tables Employee and Dept and their natural join:^{[citation needed]}

*Employee*
Name	EmpId	DeptName
Harry	3415	Finance
Sally	2241	Sales
George	3401	Finance
Harriet	2202	Sales
Mary	1257	Human Resources

*Dept*
DeptName	Manager
Finance	George
Sales	Harriet
Production	Charles

*Employee* ⨝ *Dept*
Name	EmpId	DeptName	Manager
Harry	3415	Finance	George
Sally	2241	Sales	Harriet
George	3401	Finance	George
Harriet	2202	Sales	Harriet

Note that neither the employee named Mary nor the Production department appear in the result.

This can also be used to define composition of relations. For example, the composition of Employee and Dept is their join as shown above, projected on all but the common attribute DeptName.

The natural join is arguably one of the most important operators since it is the relational counterpart of the logical AND operator. Note that if the same variable appears in each of two predicates that are connected by AND, then that variable stands for the same thing and both appearances must always be substituted by the same value (this is a consequence of the idempotence of the logical AND).

Formally the semantics of the natural join are defined as follows:

R\bowtie S=\left\{r\cup s\ \vert \ r\in R\ \land \ s\in S\ \land \ r_{A}=s_{A}\right\}

(1)

It is usually required that R and S must have at least one common attribute, but if this constraint is omitted, and R and S have no common attributes, then the natural join becomes the Cartesian product.

The natural join can be simulated with Codd's primitives as follows. Assume that c₁,...,c_m are the attribute names common to R and S, r₁,...,r_n are the attribute names unique to R and s₁,...,s_k are the attribute names unique to S. Furthermore, assume that the attribute names x₁,...,x_m are neither in R nor in S. In a first step the common attribute names in S can be renamed:

T=\rho _{x_{1}/c_{1},\ldots ,x_{m}/c_{m}}(S)=\rho _{x_{1}/c_{1}}(\rho _{x_{2}/c_{2}}(\ldots \rho _{x_{m}/c_{m}}(S)\ldots ))

(2)

Then we take the Cartesian product and select the tuples that are to be joined:

P=\sigma _{c_{1}=x_{1},\ldots ,c_{m}=x_{m}}(R\times T)=\sigma _{c_{1}=x_{1}}(\sigma _{c_{2}=x_{2}}(\ldots \sigma _{c_{m}=x_{m}}(R\times T)\ldots ))

(3)

Finally we take a projection to get rid of the renamed attributes:

U=\Pi _{r_{1},\ldots ,r_{n},c_{1},\ldots ,c_{m},s_{1},\ldots ,s_{k}}(P)

(4)

Common extensions

In practice the classical relational algebra described above is extended with various operations such as outer joins, aggregate functions and even transitive closure.^[3]

Outer joins

Whereas the result of a join (or inner join) consists of tuples formed by combining matching tuples in the two operands, an outer join contains those tuples and additionally some tuples formed by extending an unmatched tuple in one of the operands by "fill" values for each of the attributes of the other operand. Outer joins are not considered part of the classical relational algebra discussed so far.^[4]

The operators defined in this section assume the existence of a null value, ω, which we do not define, to be used for the fill values; in practice this corresponds to the NULL in SQL. In order to make subsequent selection operations on the resulting table meaningful, a semantic meaning needs to be assigned to nulls; in Codd's approach the propositional logic used by the selection is extended to a three-valued logic, although we elide those details in this article.

Three outer join operators are defined: left outer join, right outer join, and full outer join. (The word "outer" is sometimes omitted.)

Left outer join

The left outer join (⟕) is written as R ⟕ S where R and S are relations.^[b] The result of the left outer join is the set of all combinations of tuples in R and S that are equal on their common attribute names, in addition (loosely speaking) to tuples in R that have no matching tuples in S.^{[citation needed]}

For an example consider the tables Employee and Dept and their left outer join:

*Employee*
Name	EmpId	DeptName
Harry	3415	Finance
Sally	2241	Sales
George	3401	Finance
Harriet	2202	Sales
Tim	1123	Executive

*Dept*
DeptName	Manager
Sales	Harriet
Production	Charles

*Employee* ⟕ *Dept*
Name	EmpId	DeptName	Manager
Harry	3415	Finance	ω
Sally	2241	Sales	Harriet
George	3401	Finance	ω
Harriet	2202	Sales	Harriet
Tim	1123	Executive	ω

In the resulting relation, tuples in S which have no common values in common attribute names with tuples in R take a null value, ω.

Since there are no tuples in Dept with a DeptName of Finance or Executive, ωs occur in the resulting relation where tuples in Employee have a DeptName of Finance or Executive.

Let r₁, r₂, ..., r_n be the attributes of the relation R and let {(ω, ..., ω)} be the singleton relation on the attributes that are unique to the relation S (those that are not attributes of R). Then the left outer join can be described in terms of the natural join (and hence using basic operators) as follows:

(R\bowtie S)\cup ((R-\pi _{r_{1},r_{2},\dots ,r_{n}}(R\bowtie S))\times \{(\omega ,\dots ,\omega )\})

Right outer join

The right outer join (⟖) behaves almost identically to the left outer join, but the roles of the tables are switched.

The right outer join of relations R and S is written as R ⟖ S.^[c] The result of the right outer join is the set of all combinations of tuples in R and S that are equal on their common attribute names, in addition to tuples in S that have no matching tuples in R.^{[citation needed]}

For example, consider the tables Employee and Dept and their right outer join:

*Employee*
Name	EmpId	DeptName
Harry	3415	Finance
Sally	2241	Sales
George	3401	Finance
Harriet	2202	Sales
Tim	1123	Executive

*Dept*
DeptName	Manager
Sales	Harriet
Production	Charles

*Employee* ⟖ *Dept*
Name	EmpId	DeptName	Manager
Sally	2241	Sales	Harriet
Harriet	2202	Sales	Harriet
ω	ω	Production	Charles

In the resulting relation, tuples in R which have no common values in common attribute names with tuples in S take a null value, ω.

Since there are no tuples in Employee with a DeptName of Production, ωs occur in the Name and EmpId attributes of the resulting relation where tuples in Dept had DeptName of Production.

Let s₁, s₂, ..., s_n be the attributes of the relation S and let {(ω, ..., ω)} be the singleton relation on the attributes that are unique to the relation R (those that are not attributes of S). Then, as with the left outer join, the right outer join can be simulated using the natural join as follows:

(R\bowtie S)\cup (\{(\omega ,\dots ,\omega )\}\times (S-\pi _{s_{1},s_{2},\dots ,s_{n}}(R\bowtie S)))

Full outer join

The outer join (⟗) or full outer join in effect combines the results of the left and right outer joins.

The full outer join is written as R ⟗ S where R and S are relations.^[d] The result of the full outer join is the set of all combinations of tuples in R and S that are equal on their common attribute names, in addition to tuples in S that have no matching tuples in R and tuples in R that have no matching tuples in S in their common attribute names.^{[citation needed]}

For an example consider the tables Employee and Dept and their full outer join:

*Employee*
Name	EmpId	DeptName
Harry	3415	Finance
Sally	2241	Sales
George	3401	Finance
Harriet	2202	Sales
Tim	1123	Executive

*Dept*
DeptName	Manager
Sales	Harriet
Production	Charles

*Employee* ⟗ *Dept*
Name	EmpId	DeptName	Manager
Harry	3415	Finance	ω
Sally	2241	Sales	Harriet
George	3401	Finance	ω
Harriet	2202	Sales	Harriet
Tim	1123	Executive	ω
ω	ω	Production	Charles

In the resulting relation, tuples in R which have no common values in common attribute names with tuples in S take a null value, ω. Tuples in S which have no common values in common attribute names with tuples in R also take a null value, ω.

The full outer join can be simulated using the left and right outer joins (and hence the natural join and set union) as follows:

R ⟗ S = (R ⟕ S) ∪ (R ⟖ S)

Operations for domain computations

There is nothing in relational algebra introduced so far that would allow computations on the data domains (other than evaluation of propositional expressions involving equality). For example, it is not possible using only the algebra introduced so far to write an expression that would multiply the numbers from two columns, e.g. a unit price with a quantity to obtain a total price. Practical query languages have such facilities, e.g. the SQL SELECT allows arithmetic operations to define new columns in the result SELECT unit_price * quantity AS total_price FROM t, and a similar facility is provided more explicitly by Tutorial D's EXTEND keyword.^[5] In database theory, this is called extended projection.^[6]^: 213

Aggregation

Furthermore, computing various functions on a column, like the summing up of its elements, is also not possible using the relational algebra introduced so far. There are five aggregate functions that are included with most relational database systems. These operations are Sum, Count, Average, Maximum and Minimum. In relational algebra the aggregation operation over a schema (A₁, A₂, ... A_n) is written as follows:

G_{1},G_{2},\ldots ,G_{m}\ g_{f_{1}({A_{1}}'),f_{2}({A_{2}}'),\ldots ,f_{k}({A_{k}}')}\ (r)

where each A_j', 1 ≤ j ≤ k, is one of the original attributes A_i, 1 ≤ i ≤ n.

The attributes preceding the g are grouping attributes, which function like a "group by" clause in SQL. Then there are an arbitrary number of aggregation functions applied to individual attributes. The operation is applied to an arbitrary relation r. The grouping attributes are optional, and if they are not supplied, the aggregation functions are applied across the entire relation to which the operation is applied.

Let's assume that we have a table named Account with three columns, namely Account_Number, Branch_Name and Balance. We wish to find the maximum balance of each branch. This is accomplished by _{Branch_Name}G_Max(Balance)(Account). To find the highest balance of all accounts regardless of branch, we could simply write G_Max(Balance)(Account).

Grouping is often written as _{Branch_Name}ɣ_Max(Balance)(Account) instead.^[6]

Transitive closure

Although relational algebra seems powerful enough for most practical purposes, there are some simple and natural operators on relations that cannot be expressed by relational algebra. One of them is the transitive closure of a binary relation. Given a domain D, let binary relation R be a subset of D×D. The transitive closure R⁺ of R is the smallest subset of D×D that contains R and satisfies the following condition:

\forall x\forall y\forall z\left((x,y)\in R^{+}\wedge (y,z)\in R^{+}\Rightarrow (x,z)\in R^{+}\right)

It can be proved using the fact that there is no relational algebra expression E(R) taking R as a variable argument that produces R⁺.^[7]

SQL however officially supports such fixpoint queries since 1999, and it had vendor-specific extensions in this direction well before that.

Implementations

The first query language to be based on Codd's algebra was Alpha, developed by Dr. Codd himself. Subsequently, ISBL was created, and this pioneering work has been acclaimed by many authorities^[8] as having shown the way to make Codd's idea into a useful language. Business System 12 was a short-lived industry-strength relational DBMS that followed the ISBL example.

In 1998 Chris Date and Hugh Darwen proposed a language called Tutorial D intended for use in teaching relational database theory, and its query language also draws on ISBL's ideas.^[9] Rel is an implementation of Tutorial D. Bmg is an implementation of relational algebra in Ruby which closely follows the principles of Tutorial D and The Third Manifesto.^[10]

Even the query language of SQL is loosely based on a relational algebra, though the operands in SQL (tables) are not exactly relations and several useful theorems about the relational algebra do not hold in the SQL counterpart (arguably to the detriment of optimisers and/or users). The SQL table model is a bag (multiset), rather than a set. For example, the expression $(R\cup S)\setminus T=(R\setminus T)\cup (S\setminus T)$ is a theorem for relational algebra on sets, but not for relational algebra on bags.^[6]

Notes

^ In Unicode, the join symbol is ⨝ (U+2A1D), and the bowtie symbol, occasionally used instead, is ⋈ (U+22C8).
^ In Unicode, the Left outer join symbol is ⟕ (U+27D5).
^ In Unicode, the Right outer join symbol is ⟖ (U+27D6).
^ In Unicode, the Full Outer join symbol is ⟗ (U+27D7).

References

^ Codd, E.F. (1970). "A Relational Model of Data for Large Shared Data Banks". Communications of the ACM. 13 (6): 377–387. doi:10.1145/362384.362685. S2CID 207549016.
^ Silberschatz, Abraham; Henry F. Korth; S. Sudarshan (2020). Database system concepts (Seventh ed.). New York. p. 56. ISBN 978-0-07-802215-9. OCLC 1080554130.{{cite book}}: CS1 maint: location missing publisher (link)
^ M. Tamer Özsu; Patrick Valduriez (2011). Principles of Distributed Database Systems (3rd ed.). Springer. p. 46. ISBN 978-1-4419-8833-1.
^ Patrick O'Neil; Elizabeth O'Neil (2001). Database: Principles, Programming, and Performance, Second Edition. Morgan Kaufmann. p. 120. ISBN 978-1-55860-438-4.
^ C. J. Date (2011). SQL and Relational Theory: How to Write Accurate SQL Code. O'Reilly Media, Inc. pp. 133–135. ISBN 978-1-4493-1974-8.
^ ^a ^b ^c Hector Garcia-Molina; Jeffrey D. Ullman; Jennifer Widom (2009). Database systems: the complete book (2nd ed.). Pearson Prentice Hall. ISBN 978-0-13-187325-4.
^ Aho, Alfred V.; Jeffrey D. Ullman (1979). "Universality of data retrieval languages". Proceedings of the 6th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages: 110–119. doi:10.1145/567752.567763. S2CID 3242505.
^ C. J. Date. "Edgar F. Codd - A.M. Turing Award Laureate". amturing.acm.org. Retrieved 2020-12-27.
^ C. J. Date and Hugh Darwen. "Databases, Types, and the Relational model: The Third Manifesto" (PDF). Retrieved 2024-07-04.
^ "Bmg documentation". Retrieved 2024-07-04.

External links

RAT Relational Algebra Translator Free software to convert relational algebra to SQL
Lecture Videos: Relational Algebra Processing - An introduction to how database systems process relational algebra
Lecture Notes: Relational Algebra – A quick tutorial to adapt SQL queries into relational algebra
Relational – A graphic implementation of the relational algebra
Query Optimization This paper is an introduction into the use of the relational algebra in optimizing queries, and includes numerous citations for more in-depth study.
Relational Algebra System for Oracle and Microsoft SQL Server
Pireal – An experimental educational tool for working with Relational Algebra
DES – An educational tool for working with Relational Algebra and other formal languages
RelaX - Relational Algebra Calculator (open-source software available as an online service without registration)
RA: A Relational Algebra Interpreter
Translating SQL to Relational Algebra

[3] In Unicode, the join symbol is ⨝ (U+2A1D), and the bowtie symbol, occasionally used instead, is ⋈ (U+22C8).

[6] In Unicode, the Left outer join symbol is ⟕ (U+27D5).

[7] In Unicode, the Right outer join symbol is ⟖ (U+27D6).

[8] In Unicode, the Full Outer join symbol is ⟗ (U+27D7).

[Codd1970-1] Codd, E.F. (1970). "A Relational Model of Data for Large Shared Data Banks". Communications of the ACM. 13 (6): 377–387. doi:10.1145/362384.362685. S2CID 207549016.

[2] Silberschatz, Abraham; Henry F. Korth; S. Sudarshan (2020). Database system concepts (Seventh ed.). New York. p. 56. ISBN 978-0-07-802215-9. OCLC 1080554130.{{cite book}}: CS1 maint: location missing publisher (link)

[ÖzsuValduriez2011-4] M. Tamer Özsu; Patrick Valduriez (2011). Principles of Distributed Database Systems (3rd ed.). Springer. p. 46. ISBN 978-1-4419-8833-1.

[O'NeilO'Neil2001-5] Patrick O'Neil; Elizabeth O'Neil (2001). Database: Principles, Programming, and Performance, Second Edition. Morgan Kaufmann. p. 120. ISBN 978-1-55860-438-4.

[Date2011-9] C. J. Date (2011). SQL and Relational Theory: How to Write Accurate SQL Code. O'Reilly Media, Inc. pp. 133–135. ISBN 978-1-4493-1974-8.

[Garcia-MolinaUllman2009-10] Hector Garcia-Molina; Jeffrey D. Ullman; Jennifer Widom (2009). Database systems: the complete book (2nd ed.). Pearson Prentice Hall. ISBN 978-0-13-187325-4.

[11] Aho, Alfred V.; Jeffrey D. Ullman (1979). "Universality of data retrieval languages". Proceedings of the 6th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages: 110–119. doi:10.1145/567752.567763. S2CID 3242505.

[12] C. J. Date. "Edgar F. Codd - A.M. Turing Award Laureate". amturing.acm.org. Retrieved 2020-12-27.

[13] C. J. Date and Hugh Darwen. "Databases, Types, and the Relational model: The Third Manifesto" (PDF). Retrieved 2024-07-04.

[14] "Bmg documentation". Retrieved 2024-07-04.

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