Problem Detail: Let $Sigma$ be an alphabet. Have a look at following definitions frequently used in literature containing Kleene star and Kleene plus. $Sigma^* := Sigma^+ cup {varepsilon}$
$Sigma^+ := Sigma^* setminus {varepsilon}$ These definitions lead to different sets iff $varepsilon in Sigma_i$ for some $i$. What’s the proper one in general context? Is it similar to mathematical issues like multiplicative identity in rings or including zero in the set of natural numbers?
$Sigma^+ := Sigma^* setminus {varepsilon}$ These definitions lead to different sets iff $varepsilon in Sigma_i$ for some $i$. What’s the proper one in general context? Is it similar to mathematical issues like multiplicative identity in rings or including zero in the set of natural numbers?
Asked By : RomeoAndJuliet
Answered By : Shaull
$Sigma$ is a finite set of elements called letters. A word is a sequence of 0 or more letters. For every alphabet, there is a unique 0-letter word called the empty word, denoted $epsilon$. By definition, $epsilonnotin Sigma$, for eny $Sigma$. Thus, $Sigma^*$ and $Sigma^+$ are both well defined, and are used for different purposes. Mathematically, $Sigma^*$ with the concatenation operation is a monoid, with $epsilon$ being the identity element. $Sigma^+$ is thus a semi-group, with the concatenation operation, as it doesn’t have an identity element.
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Question Source : http://cs.stackexchange.com/questions/35600
