Kleene Star

The Kleene star (or Kleene closure), denoted by the postfix operator * , is the fundamental operation of infinite iteration in formal language theory. Applied to a language L, the Kleene star L* is the set of all finite strings formed by concatenating zero or more strings from L, including the empty string. This operation, introduced by Stephen Kleene in 1956, is what transforms finite descriptions into infinite languages: the star of a single symbol {a} yields the infinite set {ε, a, aa, aaa, ...}. Without the star, regular expressions would describe only finite sets of strings; with it, they capture the full power of regular languages. The Kleene star is not merely a notational convenience. It is the algebraic mechanism by which finite automata, despite their bounded state spaces, recognize infinite sets of inputs.

See also: Regular Language, Regular Expression, Formal Language Theory, Stephen Kleene, Kleene's Recursion Theorem, Kleene Plus, Finite Automaton