Kleene closure

Earth X

In the alternate reality of Earth X, everyone on Earth has been affected by the Terrigen Mists, granting everyone superpowers. Jameson turns into a humanoid donkey (a human body with an equine head). It is also revealed that after publishing information exposing Peter Parker as Spider-Man, his reputation was ruined, as no one trusted a man who had spent years paying off the very hero he called a threat, and the Daily Bugle has come out. in bankrupt. He has captured Jack Russell (Werewolf) in the old Bugle offices and brings him food, presumably mistaking Russell for his son John the werewolf.

G.I. Joe

Given

V0={λ λ !{displaystyle V_{0}={lambda },}

is defined recursively

Vi+1={wv:w한 한 Viandv한 한 V!{displaystyle V_{i+1}={wv:win V_{i}{mbox{ and }}vin V},} where i≥ ≥ 0.{displaystyle igeq 0,}

Yeah. V{displaystyle V} is a formal language, then the i{displaystyle i}- that's the power of V{displaystyle V} is the abbreviation of the concatenation V{displaystyle V} himself i{displaystyle i} Sometimes. This is, Vi{displaystyle V_{i}} can be understood as the set of all length chains i{displaystyle i}, formed from symbols on V{displaystyle V}.

The definition of star Kleene V{displaystyle V} That's it. V↓ ↓ = i한 한 NVi={λ λ ! V1 V2 V3 ...... .{displaystyle V^{*}=bigcup _{iin mathbb {N} }V_{i}=left{lambda right}cup V_{1}cup V_{2}cup V_{3}{3cupldots. !

That is, it is the collection of all possible finite length chains generated from symbols on V{displaystyle V}.

In some formal Language Studies, they use Kleene plus which is a variation of the star Kleene operation. Kleene plus omits the term V0{displaystyle V_{0}} in union. In other words, Kleene plus en V{displaystyle V} That's it. V+= i한 한 N⋆ ⋆ Vi=V1 V2 V3 ...... .{displaystyle V^{+}=bigcup _{iin mathbb {N} ^{star }}V_{i}=V_{1}cup V_{2}cup V_{3}cup ldots. !

Examples

Example of a Kleene closure applied to a character:

{a!↓ ↓ ={λ λ ,a,aa,aaa,aaaa,aaaaa,aaaaaa,...... !{displaystyle {a}^{*}={lambdaa,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa,dots }}}}

Example of Kleene closure applied to a set of strings:

{ab,c!↓ ↓ ={λ λ ,ab,c,abab,abc,cab,cc,ababab,ababc,abcab,abcc,cabab,cabc,ccab,ccc,...... !{displaystyle {ab,c}^{*}={lambdaab,c,abab,abc,cab,cc,ababab,ababc,abcab,abcc,cabab,cabc,ccab,ccc,dots }}}

Example of Kleene closure applied to a set of characters:

{a,b,c!↓ ↓ ={λ λ ,a,b,c,aa,ab,ac,ba,bb,bc,...... !{displaystyle {a,b,c}^{*}={lambdaa,b,c,aa,ab,ac,ba,bb,bc,dots }}}