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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 F^) when applied to that set (denoted as F). 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 F^+. More formally, let \langle R(U), F \rangle denote a relational scheme over the set of attributes U with a set of functional dependencies F. We say that a functional dependency f is logically implied by F, and denote it with F \models f if and only if for every instance r of R that satisfies the functional dependencies in F, r also satisfies f. We denote by F^ the set of all functional dependencies that are logically implied by F. Furthermore, with respect to a set of inference rules A, we say that a functional dependency f is derivable from the functional dependencies in F by the set of inference rules A, and we denote it by F \vdash _ f if and only if f is obtainable by means of repeatedly applying the inference rules in A to functional dependencies in F. We denote by F^_ the set of all functional dependencies that are derivable from F by inference rules in A. Then, a set of inference rules A is sound if and only if the following holds: F^_ \subseteq F^ that is to say, we cannot derive by means of A functional dependencies that are not logically implied by F. The set of inference rules A is said to be complete if the following holds: F^ \subseteq F^_ more simply put, we are able to derive by A all the functional dependencies that are logically implied by F.


Axioms (primary rules)

Let R(U) be a relation scheme over the set of attributes U. Henceforth we will denote by letters X, Y, Z any subset of U and, for short, the union of two sets of attributes X and Y by XY instead of the usual X \cup Y; 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 X is a set of attributes and Y is a subset of X, then X holds Y. Hereby, X holds Y math>X \to Ymeans that X functionally determines Y. :If Y \subseteq X then X \to Y.


Axiom of augmentation

If X holds Y and Z is a set of attributes, then X Z holds Y Z. It means that attribute in dependencies does not change the basic dependencies. :If X \to Y, then X Z \to Y Z for any Z.


Axiom of transitivity

If X holds Y and Y holds Z, then X holds Z. :If X \to Y and Y \to Z, then X \to Z.


Additional rules (Secondary Rules)

These rules can be derived from the above axioms.


Decomposition

If X \to Y Z then X \to Y and X \to Z.


Proof


Composition

If X \to Y and A \to B then X A \to Y B.


Proof


Union

If X \to Y and X \to Z then X \to YZ.


Proof


Pseudo transitivity

If X \to Y and Y Z \to W then X Z\to W.


Proof


Self determination

I \to I for any I. This follows directly from the axiom of reflexivity.


Extensivity

The following property is a special case of augmentation when Z=X. :If X \to Y, then X \to X Y. 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 F, an Armstrong relation is a relation which satisfies all the functional dependencies in the closure F^+ 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 '99

CS345 Lecture Notes from Stanford University
{{DEFAULTSORT:Armstrong's Axioms Data modeling Database management systems