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ä»»æã® ( α , β ) â R {\displaystyle (\alpha ,\beta )\in R} ã«ã€ã㊠P α β ( u , v ) = ( u α v , u β v ) {\displaystyle P_{\alpha \beta }(u,\,v)=(u\alpha v,\,u\beta v)} ãšãªããããªçæåå P α β : ( V ⪠Σ ) â à ( V ⪠Σ ) â â¶ ( V ⪠Σ ) â à ( V ⪠Σ ) â {\displaystyle P_{\alpha \beta }:(V\cup \Sigma )^{*}\times (V\cup \Sigma )^{*}\longrightarrow (V\cup \Sigma )^{*}\times (V\cup \Sigma )^{*}} ãååšããã
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é åºå¯Ÿ ( u α v , u β v ) {\displaystyle (u\alpha v,\,u\beta v)} ã G {\displaystyle G\,} ã®ãããã¯ã·ã§ã³(çæèŠå)ãšåŒã³ãäžè¬ã« u α v â u β v {\displaystyle u\alpha v\rightarrow u\beta v} ã®ããã«è¡šèšããã
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ä»»æã® u , v â ( V ⪠Σ ) â {\displaystyle u,v\in (V\cup \Sigma )^{*}} ã«ã€ããŠã u {\displaystyle u\,} ã v {\displaystyle v\,} ãçæããããšã u â v {\displaystyle u\Rightarrow v} ã§è¡šãããã ãã u = u 1 α u 2 {\displaystyle u\,=u_{1}\alpha u_{2}} ã〠v = u 1 β u 2 {\displaystyle v\,=u_{1}\beta u_{2}} ã§ â ( α , β ) â R , u 1 , u 2 â ( V ⪠Σ ) â {\displaystyle \exists (\alpha ,\beta )\in R,u_{1},u_{2}\in (V\cup \Sigma )^{*}} ãæãç«ããã°ãªããªãã
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ä»»æã® u , v â ( V ⪠Σ ) â {\displaystyle u,v\in (V\cup \Sigma )^{*}} ã«ã€ããŠã u â â v {\displaystyle u{\stackrel {*}{\Rightarrow }}v} (ããã㯠u ââ v {\displaystyle u\Rightarrow \Rightarrow v} )ã§ãããšã¯ã u â u 1 â u 2 ⯠â u k â v {\displaystyle u\Rightarrow u_{1}\Rightarrow u_{2}\cdots \Rightarrow u_{k}\Rightarrow v} ãšãªããã㪠â u 1 , u 2 , ⯠u k â ( V ⪠Σ ) â , k ⥠0 {\displaystyle \exists u_{1},u_{2},\cdots u_{k}\in (V\cup \Sigma )^{*},k\geq 0} ãæãç«ã€å Žåã§ããã
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L ( G ) = { w â Σ â : S â â w } {\displaystyle L(G)=\{w\in \Sigma ^{*}:S{\stackrel {*}{\Rightarrow }}w\}}
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