Codd's theorem
Encyclopedia
Codd's theorem states that relational algebra
and the domain-independent relational calculus
queries, two well-known foundational query languages for the relational model, are precisely equivalent in expressive power. That is, a database query can be formulated in one language if and only if it can be expressed in the other.
The theorem is named after Edgar F. Codd
, the father of the relational model
for database management.
The domain independent relational calculus
queries are precisely those relational calculus queries that are invariant under choosing domains of values beyond those appearing in the database itself. That is, queries that may return different results for different domains are excluded. An example of such a forbidden query is the query "select all tuples other than those occurring in relation R", where R is a relation in the database. Assuming different domains, i.e., sets of atomic data items from which tuples can be constructed, this query returns different results and thus is clearly not domain independent.
Codd's Theorem is notable since it establishes the equivalence of two syntactically quite dissimilar languages: relational algebra
is an imperative, variable-free language, while relational calculus
is a logical language with variables and quantification
.
Relational calculus
is essentially equivalent to first-order logic, and indeed, Codd's Theorem was previously known to logicians since the late 1940s.
Query languages that are equivalent in expressive power to relational algebra were called relationally complete by Codd. By Codd's Theorem, this includes relational calculus. Relational completeness clearly does not imply that any interesting database query can be expressed in relationally complete languages. Well-known examples of inexpressible queries include simple aggregations (counting tuples, or summing up values occurring in tuples, which are operations expressible in SQL but not in relational algebra) and computing the transitive closure of a graph given by its binary edge relation (see also expressive power
). Nevertheless, relational completeness constitutes an important yardstick by which the expressive power of query languages can be compared.
Relational algebra
Relational algebra, an offshoot of first-order logic , deals with a set of finitary relations that is closed under certain operators. These operators operate on one or more relations to yield a relation...
and the domain-independent relational calculus
Relational calculus
Relational calculus consists of two calculi, the tuple relational calculus and the domain relational calculus, that are part of the relational model for databases and provide a declarative way to specify database queries...
queries, two well-known foundational query languages for the relational model, are precisely equivalent in expressive power. That is, a database query can be formulated in one language if and only if it can be expressed in the other.
The theorem is named after Edgar F. Codd
Edgar F. Codd
Edgar Frank "Ted" Codd was an English computer scientist who, while working for IBM, invented the relational model for database management, the theoretical basis for relational databases...
, the father of the relational model
Relational model
The relational model for database management is a database model based on first-order predicate logic, first formulated and proposed in 1969 by Edgar F...
for database management.
The domain independent relational calculus
Relational calculus
Relational calculus consists of two calculi, the tuple relational calculus and the domain relational calculus, that are part of the relational model for databases and provide a declarative way to specify database queries...
queries are precisely those relational calculus queries that are invariant under choosing domains of values beyond those appearing in the database itself. That is, queries that may return different results for different domains are excluded. An example of such a forbidden query is the query "select all tuples other than those occurring in relation R", where R is a relation in the database. Assuming different domains, i.e., sets of atomic data items from which tuples can be constructed, this query returns different results and thus is clearly not domain independent.
Codd's Theorem is notable since it establishes the equivalence of two syntactically quite dissimilar languages: relational algebra
Relational algebra
Relational algebra, an offshoot of first-order logic , deals with a set of finitary relations that is closed under certain operators. These operators operate on one or more relations to yield a relation...
is an imperative, variable-free language, while relational calculus
Relational calculus
Relational calculus consists of two calculi, the tuple relational calculus and the domain relational calculus, that are part of the relational model for databases and provide a declarative way to specify database queries...
is a logical language with variables and quantification
Quantification
Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method.In logic,...
.
Relational calculus
Relational calculus
Relational calculus consists of two calculi, the tuple relational calculus and the domain relational calculus, that are part of the relational model for databases and provide a declarative way to specify database queries...
is essentially equivalent to first-order logic, and indeed, Codd's Theorem was previously known to logicians since the late 1940s.
Query languages that are equivalent in expressive power to relational algebra were called relationally complete by Codd. By Codd's Theorem, this includes relational calculus. Relational completeness clearly does not imply that any interesting database query can be expressed in relationally complete languages. Well-known examples of inexpressible queries include simple aggregations (counting tuples, or summing up values occurring in tuples, which are operations expressible in SQL but not in relational algebra) and computing the transitive closure of a graph given by its binary edge relation (see also expressive power
Expressive power
In computer science, the expressive power of a language describes the ideas expressible in that language.For example, the Web Ontology Language expression language profile lacks ideas which can be expressed in OWL2 RL . OWL2 EL may therefore be said to have less expressive power than OWL2 RL...
). Nevertheless, relational completeness constitutes an important yardstick by which the expressive power of query languages can be compared.