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‎tex/Multiple_Context-Free_Languages.tex‎

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@@ -113,92 +113,3 @@ \section{Разновидности MCFG}
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\item[\faTimes] $A(z_1,x_1,y_1,x_2,z_2,y_2,x_3,y_3) \leftarrow B(x_1,x_2,x_3),C(y_1,y_2,y_3),D(z_1,z_2)$
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\end{itemize}
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\begin{theorem}[Genaral MCFG]
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\begin{align*}
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&\forall L \in \text{m-MCFG } \exists n \geq 1 \ \underline{\boldsymbol{\exists} z} \in L (|z| \geq n) \\
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&\exists \text{ разбиение } z=u_1 v_1 w_1 s_1 u_2 \ldots u_m v_m w_m s_m u_{m+1}, \Sigma|v_js_j| \geq 1 \\
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&\forall i \geq 0: z_i = u_1 v_1^i w_1 s_1^i u_2 \ldots u_m v_m^i w_m s_m^i u_{m+1} \in L
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\end{align*}
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\end{theorem}
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\begin{theorem}[Well-nested MCFG]
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\begin{align*}
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&\forall L \in \text{m-wnMCFG } \exists n \geq 1 \ \underline{\boldsymbol{\forall} z} \in L (|z| \geq n) \\
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&\exists \text{ разбиение } z=u_1 v_1 w_1 s_1 u_2 \ldots u_m v_m w_m s_m u_{m+1}, \Sigma|v_js_j| \geq 1 \\
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&\forall i \geq 0: z_i = u_1 v_1^i w_1 s_1^i u_2 \ldots u_m v_m^i w_m s_m^i u_{m+1} \in L
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\end{align*}
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\end{theorem}
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\section{Иерархии внутри MCFL}
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\begin{theorem}
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$(m*(k-1))$-$MCFL(r-k) \subseteq m$-$MCFL(r) $ если $1 \leq k \leq r - 2$
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\end{theorem}
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\begin{theorem}[Seki et al]
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$L_{m+1} = \{a_1^nb_1^n\cdots a_{m+1}^n b_{m+1}^n \mid n\in \mathbb{N}\}$ является $(m+1)$-$MCFL(1)$, но не является $m$-$MCFL(r)$ ни для какого $r$
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\end{theorem}
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\begin{figure}
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\includegraphics[width=\textwidth]{figures/mcfg/mcfg.pdf}
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\label{fig:mcfg_hierarachy_1}
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\caption{Иерархия по $m$}
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\end{figure}
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Иерархия для $m=1$
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\begin{theorem}
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1-MCFL = CFL
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\end{theorem}
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\begin{theorem}
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1-MCFL(1) $\varsubsetneq$ 1-MCFL(2)
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\end{theorem}
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\begin{theorem}
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1-MCFL($r$) = 1-MCFL($r+1$), $r\geq2$
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\end{theorem}
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Иерархия для $m=2$
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\begin{theorem}[Ramow, Satta]
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2-MCFL(2) = 2-MCFL(3)
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\end{theorem}
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\begin{theorem}
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Если $m>2$ или $r>2$, то m-MCFL(r) $\varsubsetneq$ m-MCFL(r+1)
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\end{theorem}
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\begin{figure}
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\includegraphics[width=\textwidth]{figures/mcfg/mcfg_2.pdf}
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\label{fig:mcfg_hierarachy_2}
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\caption{Иерархия по $r$}
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\end{figure}
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Про MIX и $O_n$
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\begin{itemize}
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\item $mix = \{\omega \in \{a,b\}^* \mid |\omega|_a = |\omega|_b \}$ --- контекстно-свободный язык
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\item $MIX = \{\omega \in \{a,b,c\}^* \mid |\omega|_a = |\omega|_b = |\omega|_c\}$ --- MCFL? Хотелось верить, что нет
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\begin{itemize}
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\item \href{https://hal.inria.fr/inria-00564552/document}{MIX is a 2-MCFL and the word problem in $\mathbb{Z}^2$ is solved by a third-order collapsible pushdown automaton, Sylvain Salvati, 2011}~\cite{salvati:inria-00564552}
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\end{itemize}
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\item $O_2=\{\omega \in \{a,\overline{a},b,\overline{b}\}^* \mid |\omega|_a=|\omega|_{\overline{a}} \wedge |w|_b=|w|_{\overline{b}}\}$
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\item $O_n=\{\omega \in \{a_1,\overline{a_1},a_2,\overline{a_2},\ldots,a_n,\overline{a_n}\}^* \mid |\omega|_{a_1}=|\omega|_{\overline{a_1}} \wedge |w|_{a_2}=|w|_{\overline{a_2}} \wedge \cdots \wedge |w|_{a_n}=|w|_{\overline{a_n}}\}$
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\item $MIX_n = \{\omega \in \{a_1,\ldots,a_n\}^* \mid |\omega|_{a_1} = |\omega|_{a_2} =\cdots = |\omega|_{a_n}\}$
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\item $MIX_n$ регулярно эквивалентен $O_n$ (существует алгоритм построения грамматики одного языка по грамматике другого)
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\begin{itemize}
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\item \href{https://hal.archives-ouvertes.fr/hal-01771670/document}{$O_n$ is an n-MCFL, Sylvain Salvati, 2018}~\cite{GEBHARDT202241}
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\end{itemize}
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\end{itemize}
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\begin{itemize}
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\item Варианты леммы о накачке
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\item Представимость конкретных языков
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\begin{itemize}
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\item Многомерный язык Дика: \href{https://link.springer.com/chapter/10.1007/978-3-662-59620-3_5}{Towards a 2-Multiple Context-Free Grammar for the 3-Dimensional Dyck Language, Konstantinos Kogkalidis, Orestis Melkonian, 2019}~\cite{10.1007/978-3-662-59620-3_5}
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\item Шафл языков Дика: \href{https://dl.acm.org/doi/10.1145/3093333.3009848}{Context-sensitive data-dependence analysis via linear conjunctive language reachability, Qirun Zhang, Zhendong Su et al, 2017}~\cite{10.1145/3009837.3009848}
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\end{itemize}
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\end{itemize}

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