Armstrong's Axioms

Armstrong's axioms are a set of axioms (or, more precisely, inference rules) used to infer all the functional dependencies on a relational database. They were developed by William W. Armstrong on his 1974 paper. The axioms are sound in that they generate only functional dependencies in the closure of a set of functional dependencies (denoted as F+) when applied to that set (denoted as ). They are also complete 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, r 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 .