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In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics it is more commonly known as the free monoid construction. The application of the Kleene star to a set ''V'' is written as ''V'' *. It is widely used for regular expressions, which is the context in which it was introduced by Stephen Kleene to characterise certain automata, where it means "zero or more". # If ''V'' is a set of strings then ''V'' * is defined as the smallest superset of ''V'' that contains the empty string ε and is closed under the string concatenation operation. # If ''V'' is a set of symbols or characters then ''V'' * is the set of all strings over symbols in ''V'', including the empty string ε. The set ''V'' * can also be described as the set of finite-length strings that can be generated by concatenating arbitrary elements of ''V'' allowing the use of the same element multiple times. If ''V'' is either the empty set ∅ or the singleton set , then ''V'' *=; if ''V'' is any other finite set, then ''V'' * is a countably infinite set. The operators are used in rewrite rules for generative grammars. == Definition and notation == Given a set ''V'' define :''V''0 = (the language consisting only of the empty string), :''V''1 = ''V'' and define recursively the set :''V''''i''+1 = for each ''i''>0. If ''V'' is a formal language, then ''V''''i'', the ''i''-th power of the set ''V'', is a shorthand for the concatenation of set ''V'' with itself ''i'' times. That is, ''V''''i'' can be understood to be the set of all strings that can be represented as the concatenation of ''i'' strings in ''V''. The definition of Kleene star on ''V'' is : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kleene star」の詳細全文を読む スポンサード リンク
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