Armstrong's axioms are a set of
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s (or, more precisely,
inference rule
Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the logical structure of valid arguments. If an argument with true premises follows a rule of inference then the co ...
s) used to infer all the
functional dependencies on a
relational database
A relational database (RDB) is a database based on the relational model of data, as proposed by E. F. Codd in 1970.
A Relational Database Management System (RDBMS) is a type of database management system that stores data in a structured for ...
. They were developed by
William W. Armstrong in his 1974 paper. The axioms are
sound
In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the br ...
in generating only functional dependencies in the
closure of a set of functional dependencies (denoted as
) when applied to that set (denoted as
). They are also
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
in that repeated application of these rules will generate all functional dependencies in the closure
.
More formally, let
denote a relational scheme over the set of attributes
with a set of functional dependencies
. We say that a functional dependency
is logically implied by
, and denote it with
if and only if for every instance
of
that satisfies the functional dependencies in
,
also satisfies
. We denote by
the set of all functional dependencies that are logically implied by
.
Furthermore, with respect to a set of inference rules
, we say that a functional dependency
is derivable from the functional dependencies in
by the set of inference rules
, and we denote it by
if and only if
is obtainable by means of repeatedly applying the inference rules in
to functional dependencies in
. We denote by
the set of all functional dependencies that are derivable from
by inference rules in
.
Then, a set of inference rules
is sound if and only if the following holds:
that is to say, we cannot derive by means of
functional dependencies that are not logically implied by
.
The set of inference rules
is said to be complete if the following holds:
more simply put, we are able to derive by
all the functional dependencies that are logically implied by
.
Axioms (primary rules)
Let
be a relation scheme over the set of attributes
. Henceforth we will denote by letters
,
,
any subset of
and, for short, the union of two sets of attributes
and
by
instead of the usual
; this notation is rather standard in
database theory
Database theory encapsulates a broad range of topics related to the study and research of the theoretical realm of databases and database management systems.
Theoretical aspects of data management include, among other areas, the foundations of q ...
when dealing with sets of attributes.
Axiom of reflexivity
If
is a set of attributes and
is a subset of
, then
holds
. Hereby,
holds
X \to Y">math>X \to Ymeans that
functionally determines
.
:If
then
.
Axiom of augmentation
If
holds
and
is a set of attributes, then
holds
. It means that attribute in dependencies does not change the basic dependencies.
:If
, then
for any
.
Axiom of transitivity
If
holds
and
holds
, then
holds
.
:If
and
, then
.
Additional rules (Secondary Rules)
These rules can be derived from the above axioms.
Decomposition
If
then
and
.
Proof
Composition
If
and
then
.
Proof
Union
If
and
then
.
Proof
Pseudo transitivity
If
and
then
.
Proof
Self determination
for any
. This follows directly from the axiom of reflexivity.
Extensivity
The following property is a special case of augmentation when
.
:If
, then
.
Extensivity can replace augmentation as axiom in the sense that augmentation can be proved from extensivity together with the other axioms.
Proof
Armstrong relation
Given a set of functional dependencies
, an Armstrong relation is a relation which satisfies all the functional dependencies in the closure
and only those dependencies. Unfortunately, the minimum-size Armstrong relation for a given set of dependencies can have a size which is an exponential function of the number of attributes in the dependencies considered.
References
External links
UMBC CMSC 461 Spring '99CS345 Lecture Notes from Stanford University
{{DEFAULTSORT:Armstrong's Axioms
Data modeling
Database management systems