\begin{document}
-\begin{tikzpicture}[>=stealth]
+\begin{tikzpicture}[>=stealth, auto, sibling distance = 0.2in, inner sep = 1pt, outer sep = 0pt]
+%\begin{tikzpicture}[>=stealth]
\tikzstyle{every node}=[draw=none, shape=rectangle]
-% $(\epsilon|a^{0,\infty})(a|\epsilon)^{0,\infty}$
-
-\begin{scope}[xshift=0in, yshift=0in]
- \draw [draw=none] (-1in,0) rectangle (6.5in,0);
-\end{scope}
-
-\begin{scope}[xshift=4in, yshift=-0.4in]
- \node (a) {{
- $\begin{aligned}
- &\alpha
- = \Phi_0 (s)
-% = \Phi_0 \big(
-% {T}^{1} (
-% {T}^{2} (
-% {\varnothing}^{0},
-% {T}^{0}({a}^{0})
-% ), {T}^{3}(
-% {\varnothing}^{4}
-% )
-% )\big)
- &&\!\!\!\!=\;
- \overbracket {\Xl_1 \Xl_2 }%^{\alpha_0}
- \;a\;
- \overbracket { \Xr_1 \Xl_2 \Xm_2 \Xr_1 \Xr_0 }%^{\alpha_1}
- \\[-0.4em]
- &\beta
- = \Phi_0 (t)
-% = \Phi_0 \big(
-% {T}^{1} (
-% {T}^{2}(
-% {\varnothing}^{0},
-% {T}^{0}({\varnothing}^{0})
-% ),
-% {T}^{3}\big(
-% {T}^{4}({a}^{0},{\varnothing}^{0})
-% )
-% )\big)
- &&\!\!\!\!=\;
- \overbracket { \Xl_1 \Xl_2 \Xr_1 \Xl_2 \Xl_3 }%^{\beta_0}
- \;a\;
- \overbracket { \Xr_2 \Xr_1 \Xr_0 }%^{\beta_1}
- \\[-0.4em]
- &\gamma
- = \Phi_0 (u)
-% = \Phi_0 \big(
-% {T}^{1} (
-% {T}^{2}(
-% {\epsilon}^{0},
-% {\varnothing}^{0}
-% ),
-% {T}^{3} (
-% {T}^{4}({a}^{0},{\varnothing}^{0}),
-% {T}^{4}({\varnothing}^{0}, {\epsilon}^{0})
-% )
-% )\big)
- &&\!\!\!\!=\;
- \overbracket { \Xl_1 \Xl_2 \Xr_1 \Xl_2 \Xl_3 }%^{\gamma_0}
- \;a\;
- \overbracket { \Xr_2 \Xl_3 \Xr_2 \Xr_1 \Xr_0 }%^{\gamma_1}
- \end{aligned}$
- }};
-\end{scope}
-
-\setlength\tabcolsep{3pt}
-%\renewcommand{\arraystretch}{1.1}
-
-\begin{scope}[xshift=4in, yshift=-2.3in]
- \node (a) {
- $\begin{aligned}
- &\begin{tabular}{c|cc}
- $traces (\alpha, \beta)$ & frame 0 & frame 1 \\
- \hline \\[-1em]
- $\rho$ & 2 & 0 \\
- $\rho'$ & 1 & 0 \\
- \end{tabular}
- &&\quad
- \left[\begin{aligned}
- \rho_0 &= min (lasth (\Xl_1 \Xl_2), minh (\epsilon)) = min (2, \infty) = 2 \\[-0.2em]
- \rho'_0 &= min (lasth (\Xl_1 \Xl_2), minh (\Xr_1 \Xl_2 \Xl_3)) = min (2, 1) = 1 \\[-0.2em]
- \rho_1 &= min (\rho_0, minh (\Xr_1 \Xl_2 \Xm_2 \Xr_1 \Xr_0)) = min (2,0) = 0 \\[-0.2em]
- \rho'_1 &= min (\rho'_0, minh (\Xr_2 \Xr_1 \Xr_0)) = min (1,0) = 0
- \end{aligned}\right.
- \\[0.7em]
- &\begin{tabular}{c|cc}
- $traces (\beta, \gamma)$ & frame 0 & frame 1 \\
- \hline \\[-1em]
- $\rho$ & \!-1 & 0 \\
- $\rho'$ & \!-1 & 0 \\
- \end{tabular}
- &&\quad
- \left[\begin{aligned}
- \rho_0 &= -1 \\[-0.2em]
- \rho'_0 &= -1 \\[-0.2em]
- \rho_1 &= min (lasth (\Xr_2), minh (\Xr_1 \Xr_0)) = min (2,0) = 0 \\[-0.2em]
- \rho'_1 &= min (lasth (\Xr_2), minh (\Xl_3 \Xr_2 \Xr_1 \Xr_0)) = min (2,0) = 0
- \end{aligned}\right.
- \\[0.7em]
- &\begin{tabular}{c|cc}
- $traces (\alpha, \gamma)$ & frame 0 & frame 1 \\
- \hline \\[-1em]
- $\rho$ & 2 & 0 \\
- $\rho'$ & 1 & 0 \\
- \end{tabular}
- &&\quad
- \left[\begin{aligned}
- \rho_0 &= min (lasth (\Xl_1 \Xl_2), minh (\epsilon)) = min (2, \infty) = 2 \\[-0.2em]
- \rho'_0 &= min (lasth (\Xl_1 \Xl_2), minh (\Xr_1 \Xl_2 \Xl_3)) = min (2, 1) = 1 \\[-0.2em]
- \rho_1 &= min (\rho_0, minh (\Xr_1 \Xl_2 \Xm_2 \Xr_1 \Xr_0)) = min (2,0) = 0 \\[-0.2em]
- \rho'_1 &= min (\rho'_0, minh (\Xr_2 \Xl_3 \Xr_2 \Xr_1 \Xr_0)) = min (1,0) = 0
- \end{aligned}\right.
- \end{aligned}$
- };
-\end{scope}
-
-
\tikzstyle{styleA}=[draw=none
, shape=rectangle
- , rounded corners=3
- , level distance=0.35in
- , sibling distance=0.45in
, minimum size = 0.2in
+ , level distance=0.35in
+ , sibling distance=0.5in
, inner sep = 0pt
, outer sep = 0pt
]
-\tikzstyle{styleB}=[->
- %, color = lightgray
- , rounded corners=3
- %, line width = 0
- , dash pattern = on 1pt off 2.5pt
- ]
+\tikzstyle{styleB}=[->, rounded corners=3, dash pattern = on 1pt off 2.5pt]
\newcommand\w{\hspace{2em}}
-\small{
+{
\begin{scope}[xshift=0in, yshift=0in]
- \tikzstyle{every node}=[styleA]
- %\tikzstyle{every edge}=[draw=none]
- \node[xshift=-0.4in, draw=none] {$s:$};
+ \tikzstyle{every node}=[styleA, sibling distance = 0.4in]
+
+ \begin{scope}[xshift=0in, yshift=0in]
+ \node[xshift=0in, yshift=-1.25in, draw=none] {$s = T^1 (T^2 (\varnothing^0, T^0 (a^0, a^0)))$};
+ \small
\graph [tree layout, grow=down, fresh nodes] {
s1/"${T}^{1}$" -- {
s11/"${T}^{2}$" -- {
s111/"${\varnothing}^{0}$",
s112/"${T}^{0}$" -- {
- s1121/"${a}^{0}$"
+ s1121/"${a}^{0}$",
+ s1122/"${a}^{0}$"
}
- },
- s12/"${T}^{3}$" -- {
- s121/"${\varnothing}^{4}$"
}
}
};
- \node at (s1) {$\Xl_1 \w \Xr_0$};
- \node at (s11) {$\Xl_2 \w \Xr_1$};
- \node at (s12) {$\Xl_2 \w \Xr_1$};
- \node at (s121) {$\Xl_3 \w \Xr_2$};
- \draw [->, styleB] ($(s1.west)$)
+ \node at (s1) {$\Xl_1 \w \Xr_0$};
+ \node at (s11) {$\Xl_2 \w \Xr_1$};
+ \draw [styleB]
+ ($(s1.west)$)
-- ($(s11.west)$)
-- ($(s111.west)$)
-- ($(s111.south)$)
-- ($(s11.south)$)
-- ($(s112.west)$)
-- ($(s1121.west)$);
- \draw [->, styleB] ($(s1121.east)$)
+ \draw [styleB]
+ ($(s1121.east)$)
+ -- ($(s112.south)$)
+ -- ($(s1122.west)$);
+ \draw [styleB]
+ ($(s1122.east)$)
+ -- ($(s112.east)$)
+ -- ($(s11.east)$)
+ -- ($(s1.east)$);
+ \end{scope}
+
+ \begin{scope}[xshift=1.6in, yshift=0in]
+ \node[xshift=0in, yshift=-0.9in, draw=none] {$t = T^1 (T^2 (a^0, \varnothing^0), T^2 (a^0, \varnothing^0))$};
+ \small
+ \graph [tree layout, grow=down, fresh nodes] {
+ s1/"${T}^{1}$" -- {
+ s11/"${T}^{2}$" -- {
+ s111/"${a}^{0}$",
+ s112/"${\varnothing}^{0}$"
+ },
+ s12/"${T}^{2}$" -- {
+ s121/"${a}^{0}$",
+ s122/"${\varnothing}^{0}$"
+ }
+ }
+ };
+ \node at (s1) {$\Xl_1 \w \Xr_0$};
+ \node at (s11) {$\Xl_2 \w \Xr_1$};
+ \node at (s12) {$\Xl_2 \w \Xr_1$};
+ \draw [styleB]
+ ($(s1.west)$)
+ -- ($(s11.west)$)
+ -- ($(s111.west)$);
+ \draw [styleB]
+ ($(s111.east)$)
+ -- ($(s11.south)$)
+ -- ($(s112.west)$)
+ -- ($(s112.south)$)
-- ($(s112.east)$)
-- ($(s11.east)$)
-- ($(s1.south)$)
-- ($(s12.west)$)
- -- ($(s121.west)$)
- -- ($(s121.south)$)
- -- ($(s121.east)$)
+ -- ($(s121.west)$);
+ \draw [styleB]
+ ($(s121.east)$)
+ -- ($(s12.south)$)
+ -- ($(s122.west)$)
+ -- ($(s122.south)$)
+ -- ($(s122.east)$)
-- ($(s12.east)$)
-- ($(s1.east)$);
+ \end{scope}
+
+ \begin{scope}[xshift=4.5in, yshift=-0.65in]
+ \node (a) {{
+ $\begin{aligned}
+ &\begin{aligned}
+ \alpha = \Phi_0(s) &=
+ \overbracket {\Xl_1 \Xl_2 }
+ a
+ \overbracket {\vphantom{\Xm}}
+ a
+ \overbracket { \Xr_1 \Xr_0 }
+ \\[-0.4em]
+ \beta = \Phi_0(t) &=
+ \overbracket {\Xl_1 \Xl_2 }
+ a
+ \overbracket { \Xr_1 \Xl_2 }
+ a
+ \overbracket { \Xr_1 \Xr_0 }
+ \end{aligned}
+ \\
+ &traces (\alpha, \beta) =
+ \left[\begin{aligned}
+ \rho_0 &= -1 \\[-0.4em]
+ \rho'_0 &= -1 \\[-0.4em]
+ \rho_1 &= min (lasth(\Xl_1 \Xl_2), minh (\epsilon)) = min (2, \infty) = 2 \\[-0.4em]
+ \rho'_1 &= min (lasth(\Xl_1 \Xl_2), minh (\Xr_1 \Xl_2)) = min (2, 1) = 1 \\[-0.4em]
+ \rho_2 &= min (\rho_1, minh (\Xr_1 \Xr_0)) = min (2, 0) = 0 \\[-0.4em]
+ \rho'_2 &= min (\rho'_1, minh (\Xr_1 \Xr_0)) = min (1, 0) = 0
+ \end{aligned}\right.
+ \end{aligned}$
+ }};
+ \end{scope}
\end{scope}
+}
+\node (x1)
+ [ xshift=2.7in
+ , yshift=-1.6in
+ , draw=none
+ ] {(a) -- Rule 1: longest precedence, RE $(a|aa)^{0,\infty}$ and string $aa$.
+ Order on IPTs: $s <_1 t$ because $\|s\|^{Sub}_1 = 2 > 1 = \|t\|^{Sub}_1$
+ };
+\node (x2)
+ [ below of = x1
+ , yshift=0.225in
+ , draw=none
+ ] {
+ and $\|s\|^{Sub}_p = \|t\|^{Sub}_p \;\forall p < 1$.
+ Order on PEs: $\alpha < \beta$ because
+ $\rho_1 > \rho'_1 \;\wedge\; \rho_2 = \rho'_2 \;\Rightarrow\; \alpha \sqsubset \beta$.
+ };
+
-\begin{scope}[xshift=0in, yshift=-1.3in]
+{
+\begin{scope}[xshift=0in, yshift=-2.2in]
\tikzstyle{every node}=[styleA]
- \node[xshift=-0.4in, draw=none] {$t:$};
+
+ \begin{scope}[xshift=0in, yshift=0in]
+ \node[xshift=0in, yshift=-0.6in, draw=none] {$s = T^1 (a^2, \varnothing^3)$};
+ \small
+ \graph [tree layout, grow=down, fresh nodes] {
+ s1/"${T}^{1}$" -- {
+ s11/"${a}^{2}$",
+ s12/"${\varnothing}^{3}$"
+ }
+ };
+ \node at (s1) {$\Xl_1 \w \Xr_0$};
+ \node at (s11) {$\Xl_2 \w \Xr_1$};
+ \node at (s12) {$\Xl_2 \w \Xr_1$};
+ \draw [styleB]
+ ($(s1.west)$)
+ -- ($(s11.west)$);
+ \draw [styleB]
+ ($(s11.east)$)
+ -- ($(s1.south)$)
+ -- ($(s12.west)$)
+ -- ($(s12.south)$)
+ -- ($(s12.east)$)
+ -- ($(s1.east)$);
+ \end{scope}
+
+ \begin{scope}[xshift=1.6in, yshift=0in]
+ \node[xshift=0in, yshift=-0.6in, draw=none] {$t = T^1 (\varnothing^2, a^3)$};
+ \small
\graph [tree layout, grow=down, fresh nodes] {
t1/"${T}^{1}$" -- {
- t11/"${T}^{2}$" -- {
- t111/"${\varnothing}^{0}$",
- t112/"${T}^{0}$" -- {
- t1121/"${\varnothing}^{0}$"
- }
- },
- t12/"${T}^{3}$" -- {
- t121/"${T}^{4}$" -- {
- t1211/"${a}^{0}$",
- t1212/"${\varnothing}^{0}$"
- }
- }
+ t11/"${\varnothing}^{2}$",
+ t12/"${a}^{3}$"
}
};
- \node at (t1) {$\Xl_1 \w \Xr_0$};
- \node at (t11) {$\Xl_2 \w \Xr_1$};
- \node at (t12) {$\Xl_2 \w \Xr_1$};
- \node at (t121) {$\Xl_3 \w \Xr_2$};
- \draw [->, styleB] ($(t1.west)$)
+ \node at (t1) {$\Xl_1 \w \Xr_0$};
+ \node at (t11) {$\Xl_2 \w \Xr_1$};
+ \node at (t12) {$\Xl_2 \w \Xr_1$};
+ \draw [styleB]
+ ($(t1.west)$)
-- ($(t11.west)$)
- -- ($(t111.west)$)
- -- ($(t111.south)$)
- -- ($(t111.east)$)
-- ($(t11.south)$)
- -- ($(t112.west)$)
- -- ($(t1121.west)$)
- -- ($(t1121.south)$)
- -- ($(t1121.east)$)
- -- ($(t112.east)$)
-- ($(t11.east)$)
-- ($(t1.south)$)
- -- ($(t12.west)$)
- -- ($(t121.west)$)
- -- ($(t1211.west)$);
- \draw [->, styleB]
- ($(t1211.east)$)
- -- ($(t121.south)$)
- -- ($(t1212.west)$)
- -- ($(t1212.south)$)
- -- ($(t1212.east)$)
- -- ($(t121.east)$)
- -- ($(t12.east)$)
+ -- ($(t12.west)$);
+ \draw [styleB]
+ ($(t12.east)$)
-- ($(t1.east)$);
+ \end{scope}
+
+ \begin{scope}[xshift=4.5in, yshift=-0.4in]
+ \node (a) {{
+ $\begin{aligned}
+ &\begin{aligned}
+ \alpha = \Phi_0(s) &=
+ \overbracket {\Xl_1 \Xl_2 }
+ a
+ \overbracket { \Xr_1 \Xm_1 \Xr_0 }
+ \\[-0.4em]
+ \beta = \Phi_0(t) &=
+ \overbracket {\Xl_1 \Xm_1 \Xl_2 }
+ a
+ \overbracket { \Xr_1 \Xr_0 }
+ \end{aligned}
+ \\
+ &traces (\alpha, \beta) =
+ \left[\begin{aligned}
+ \rho_0 &= min (lasth (\Xl_1), minh (\Xl_2)) = min (1, 2) = 1 \\[-0.4em]
+ \rho'_0 &= min (lasth (\Xl_1), minh (\Xm_1 \Xl_2)) = min (1, 1) = 1 \\[-0.4em]
+ \rho_1 &= min (\rho_0, minh (\Xr_1 \Xm_1 \Xr_0)) = min (1, 0) = 0 \\[-0.4em]
+ \rho'_1 &= min (\rho'_0, minh (\Xr_1 \Xr_0)) = min (1, 0) = 0
+ \end{aligned}\right.
+ \end{aligned}$
+ }};
+ \end{scope}
+
\end{scope}
+}
+\node (y1)
+ [ xshift=2.7in
+ , yshift=-3.4in
+ , draw=none
+ ] {(b) -- Rule 2: leftmost precedence, RE $(a)|(a)$ and string $a$.
+ Order on IPTs: $s <_1 t$ because
+ $\|s\|^{Sub}_1 = 1 > -1 = \|t\|^{Sub}_1$
+ };
+\node (y2)
+ [ below of = y1
+ , yshift=0.225in
+ , draw=none
+ ] {
+ and $\|s\|^{Sub}_p = \|t\|^{Sub}_p \;\forall p < 1$.
+ Order on PEs: $\alpha < \beta$ because
+ $\rho_i = \rho'_i \;\forall i \;\Rightarrow\; \alpha \sim \beta$
+ and
+ $first(\alpha \backslash \beta) = \Xl < \Xm = first(\beta \backslash \alpha)
+ \;\Rightarrow\;
+ \alpha \subset \beta$.
+ };
-\begin{scope}[xshift=0in, yshift=-2.6in]
- \tikzstyle{every node}=[styleA]
- \node[xshift=-0.4in, draw=none] {$u:$};
+
+{
+\begin{scope}[xshift=0in, yshift=-3.9in]
+ \tikzstyle{every node}=[styleA, sibling distance = 0.4in]
+
+ \node[yshift=-0.95in, draw=none] {$s = T^1(T^2(a^0, \varnothing^0))$};
+ \begin{scope}[xshift=0in, yshift=0in]
+ \small
\graph [tree layout, grow=down, fresh nodes] {
- t1/"${T}^{1}$" -- {
- t11/"${T}^{2}$" -- {
- t111/"${\epsilon}^{0}$",
- t112/"${\varnothing}^{0}$"
+ s1/"${T}^{1}$" -- {
+ s11/"${T}^{2}$" -- {
+ s111/"${a}^{0}$",
+ s112/"${\varnothing}^{0}$"
+ }
+ }
+ };
+ \node at (s1) {$\Xl_1 \w \Xr_0$};
+ \node at (s11) {$\Xl_2 \w \Xr_1$};
+ \node at (s12) {$\Xl_2 \w \Xr_1$};
+ \draw [styleB]
+ ($(s1.west)$)
+ -- ($(s11.west)$)
+ -- ($(s111.west)$);
+ \draw [styleB]
+ ($(s111.east)$)
+ -- ($(s11.south)$)
+ -- ($(s112.west)$)
+ -- ($(s112.south)$)
+ -- ($(s112.east)$)
+ -- ($(s11.east)$)
+ -- ($(s1.east)$);
+ \end{scope}
+
+ \begin{scope}[xshift=1.6in, yshift=0in]
+ \node[xshift=0in, yshift=-0.95in, draw=none] {$t = T^1 (T^2 (a^0, \varnothing^0), T^2(\varnothing^0, \epsilon^0))$};
+ \small
+ \graph [tree layout, grow=down, fresh nodes] {
+ s1/"${T}^{1}$" -- {
+ s11/"${T}^{2}$" -- {
+ s111/"${a}^{0}$",
+ s112/"${\varnothing}^{0}$"
},
- t12/"${T}^{3}$" -- {
- t121/"${T}^{4}$" -- {
- t1211/"${a}^{0}$",
- t1212/"${\varnothing}^{0}$"
- },
- t122/"${T}^{4}$" -- {
- t1221/"${\varnothing}^{0}$",
- t1222/"${\epsilon}^{0}$"
- }
+ s12/"${T}^{2}$" -- {
+ s121/"${\varnothing}^{0}$",
+ s122/"${\epsilon}^{0}$"
+ }
+ }
+ };
+ \node at (s1) {$\Xl_1 \w \Xr_0$};
+ \node at (s11) {$\Xl_2 \w \Xr_1$};
+ \node at (s12) {$\Xl_2 \w \Xr_1$};
+ \draw [styleB]
+ ($(s1.west)$)
+ -- ($(s11.west)$)
+ -- ($(s111.west)$);
+ \draw [styleB]
+ ($(s111.east)$)
+ -- ($(s11.south)$)
+ -- ($(s112.west)$)
+ -- ($(s112.south)$)
+ -- ($(s112.east)$)
+ -- ($(s11.east)$)
+ -- ($(s1.south)$)
+ -- ($(s12.west)$)
+ -- ($(s121.west)$)
+ -- ($(s121.south)$)
+ -- ($(s121.east)$)
+ -- ($(s12.south)$)
+ -- ($(s122.west)$)
+ -- ($(s122.south)$)
+ -- ($(s122.east)$)
+ -- ($(s12.east)$)
+ -- ($(s1.east)$);
+ \end{scope}
+
+ \begin{scope}[xshift=4.5in, yshift=-0.45in]
+ \node (a) {{
+ $\begin{aligned}
+ &\begin{aligned}
+ \alpha = \Phi_0(s) &=
+ \overbracket {\Xl_1 \Xl_2 }
+ a
+ \overbracket { \Xr_1 \Xr_0 }
+ \\[-0.4em]
+ \beta = \Phi_0(t) &=
+ \overbracket { \Xl_1 \Xl_2 }
+ a
+ \overbracket { \Xr_1 \Xl_2 \Xr_1 \Xr_0 }
+ \end{aligned}
+ \\
+ &traces (\alpha, \beta) =
+ \left[\begin{aligned}
+ \rho_0 &= -1 \\[-0.4em]
+ \rho'_0 &= -1 \\[-0.4em]
+ \rho_1 &= min (lasth (\Xr_1), minh (\Xr_0)) = min (1, 0) = 0 \\[-0.4em]
+ \rho'_1 &= min (lasth (\Xr_1), minh (\Xl_2 \Xr_1 \Xr_0)) = min (1, 0) = 0
+ \end{aligned}\right.
+ \end{aligned}$
+ }};
+ \end{scope}
+
+\end{scope}
+}
+\node (z1)
+ [ xshift=2.7in
+ , yshift=-5.15in
+ , draw=none
+ ] {(c) -- Rule 3: absence of optional empty repetitions,
+ RE $(a|\epsilon)^{0,\infty}$ and string $a$.
+ Order on IPTs: $s <_1 t$ because
+ $\|s\|^{Sub}_2 = \infty > 0 = \|t\|^{Sub}_2$
+ };
+\node (z2)
+ [ below of = z1
+ , yshift=0.225in
+ , draw=none
+ ] {
+ and $\|s\|^{Sub}_p = \|t\|^{Sub}_p \;\forall p < 2$.
+ Order on PEs: $\alpha < \beta$ because
+ $\rho_i = \rho'_i \;\forall i \;\Rightarrow\; \alpha \sim \beta$
+ and
+ $first(\alpha \backslash \beta) = \Xr < \Xl = first(\beta \backslash \alpha)
+ \;\Rightarrow\;
+ \alpha \subset \beta$.
+ };
+
+
+
+\tikzstyle{every node}=[draw, shape = circle]
+
+\begin{scope}[xshift=-0.5in, yshift=-5.7in]
+
+\small{
+\begin{scope}[xshift=0in, yshift=-0in]
+ \tikzstyle{every node}=[draw, shape = circle]
+ \graph [tree layout, grow=down, fresh nodes, level distance = 0.2in] {
+ "$t_1$" -- {
+ "" -- {
+ "" -- { "$a$", "$a$", "$a$" }
+ }
+ }
+ , "$t_2$" -- {
+ "" -- {
+ "" -- { "$a$", "$a$" },
+ "" -- { "$a$" }
+ }
+ }
+ , t31/"$t_3$" -- {
+ t311/"" -- {
+ t3111/"" -- { t31111/"$a$" },
+ t3112/"" -- { t31121/"$a$", t31122/"$a$" }
+ }
+ }
+ , "$t_4$" -- {
+ "" -- {
+ "" -- { "$a$" },
+ "" -- { "$a$" },
+ "" -- { "$a$" }
+ }
+ }
+ , t51/"$t_5$" -- {
+ t511/"" -- {
+ t5111/"" -- { t51111/"$a$", t51112/"$a$" }
+ },
+ t512/"" -- {
+ t5121/"" -- { t51211/"$a$" }
+ }
+ }
+ , "$t_6$" -- {
+ "" -- {
+ "" -- { "$a$" }
+ },
+ "" -- {
+ "" -- { "$a$", "$a$" }
+ }
+ }
+ , "$t_7$" -- {
+ "" -- {
+ "" -- { "$a$"},
+ "" -- { "$a$"}
+ },
+ "" -- {
+ "" -- { "$a$" }
+ }
+ }
+ , "$t_8$" -- {
+ "" -- {
+ "" -- { "$a$"}
+ },
+ "" -- {
+ "" -- { "$a$"},
+ "" -- { "$a$" }
+ }
+ }
+ , "$t_9$" -- {
+ "" -- {
+ "" -- { "$a$" }
+ },
+ "" -- {
+ "" -- { "$a$" }
+ },
+ "" -- {
+ "" -- { "$a$" }
}
}
};
- \node at (t1) {$\Xl_1 \w \Xr_0$};
- \node at (t11) {$\Xl_2 \w \Xr_1$};
- \node at (t12) {$\Xl_2 \w \Xr_1$};
- \node at (t121) {$\Xl_3 \w \Xr_2$};
- \node at (t122) {$\Xl_3 \w \Xr_2$};
- \draw [->, styleB]
- ($(t1.west)$)
- -- ($(t11.west)$)
- -- ($(t111.west)$)
- -- ($(t111.south)$)
- -- ($(t111.east)$)
- -- ($(t11.south)$)
- -- ($(t112.west)$)
- -- ($(t112.south)$)
- -- ($(t112.east)$)
- -- ($(t11.east)$)
- -- ($(t1.south)$)
- -- ($(t12.west)$)
- -- ($(t121.west)$)
- -- ($(t1211.west)$);
- \draw [->, styleB]
- ($(t1211.east)$)
- -- ($(t121.south)$)
- -- ($(t1212.west)$)
- -- ($(t1212.south)$)
- -- ($(t1212.east)$)
- -- ($(t121.east)$)
- -- ($(t12.south)$)
- -- ($(t122.west)$)
- -- ($(t1221.west)$)
- -- ($(t1221.south)$)
- -- ($(t1221.east)$)
- -- ($(t122.south)$)
- -- ($(t1222.west)$)
- -- ($(t1222.south)$)
- -- ($(t1222.east)$)
- -- ($(t122.east)$)
- -- ($(t12.east)$)
- -- ($(t1.east)$);
\end{scope}
+
+\tikzstyle{styleA}=[draw=none
+ , shape=circle
+ , minimum size = 0.2in
+ , inner sep = 0pt
+ , outer sep = 0pt
+ ]
+
+\tikzstyle{styleB}=[->
+ , rounded corners=3.5
+ %, color = lightgray
+ , dash pattern = on 1pt off 2.5pt
+ ]
+
+\begin{scope}[xshift=0in, yshift=-2.5in]
+ \tikzstyle{every node}=[draw = none, shape = circle, inner sep = 0]
+ \node[xshift=-0.4in, draw=none] {$t_3:$};
+ \graph [tree layout, grow=down, fresh nodes, sibling distance = 0.45in, level distance = 0.35in] {
+ t31/"$T^1$" -- {
+ t311/"$T^2$" -- {
+ t3111/"$T^3$" -- { t31111/"$a^4$" },
+ t3112/"$T^5$" -- { t31121/"$a^6$", t31122/"$a^7$" }
+ }
+ }
+ };
+ \node at (t31) [styleA] {$\Xl_1 \w \Xr_0$};
+ \node at (t311) [styleA] {$\Xl_2 \w \Xr_1$};
+ \node at (t3111) [styleA] {$\Xl_3 \w \Xr_2$};
+ \node at (t3112) [styleA] {$\Xl_3 \w \Xr_2$};
+ \node at (t31111) [styleA] {$\Xl_4 \w \Xr_3$};
+ \node at (t31121) [styleA] {$\Xl_4 \w \Xr_3$};
+ \node at (t31122) [styleA] {$\Xl_4 \w \Xr_3$};
+ %
+ \node at (t31) [styleA] (tt31) {};
+ \node at (t311) [styleA] (tt311) {};
+ \node at (t3111) [styleA] (tt3111) {};
+ \node at (t3112) [styleA] (tt3112) {};
+ \node at (t31111) [styleA] (tt31111) {};
+ \node at (t31121) [styleA] (tt31121) {};
+ \node at (t31122) [styleA] (tt31122) {};
+ %
+ \draw [->, styleB] ($(tt31.west)$)
+ -- ($(tt311.west)$)
+ -- ($(tt3111.west)$)
+ -- ($(tt31111.west)$);
+ \draw [->, styleB] ($(tt31111.east)$)
+ -- ($(tt3111.east)$)
+ -- ($(tt311.south)$)
+ -- ($(tt3112.west)$)
+ -- ($(tt31121.west)$);
+ \draw [->, styleB] ($(tt31121.east)$)
+ -- ($(tt3112.south)$)
+ -- ($(tt31122.west)$);
+ \draw [->, styleB] ($(tt31122.east)$)
+ -- ($(tt3112.east)$)
+ -- ($(tt311.east)$)
+ -- ($(tt31.east)$);
+\end{scope}
+
+\begin{scope}[xshift=1.6in, yshift=-2.5in]
+ \tikzstyle{every node}=[draw = none, shape = circle, inner sep = 0]
+ \node[xshift=-0.4in, draw=none] {$t_5:$};
+ \graph [tree layout, grow=down, fresh nodes, sibling distance = 0.45in, level distance = 0.35in] {
+ t51/"$T^1$" -- {
+ t511/"$T^2$" -- {
+ t5111/"$T^3$" -- { t51111/"$a^4$", t51112/"$a^5$" }
+ },
+ t512/"$T^6$" -- {
+ t5121/"$T^7$" -- { t51211/"$a^8$" }
+ }
+ }
+ };
+ \node at (t51) [styleA] {$\Xl_1 \w \Xr_0$};
+ \node at (t511) [styleA] {$\Xl_2 \w \Xr_1$};
+ \node at (t5111) [styleA] {$\Xl_3 \w \Xr_2$};
+ \node at (t51111) [styleA] {$\Xl_4 \w \Xr_3$};
+ \node at (t51112) [styleA] {$\Xl_4 \w \Xr_3$};
+ \node at (t512) [styleA] {$\Xl_2 \w \Xr_1$};
+ \node at (t5121) [styleA] {$\Xl_3 \w \Xr_2$};
+ \node at (t51211) [styleA] {$\Xl_4 \w \Xr_3$};
+ %
+ \node at (t51) [styleA] (tt51) {};
+ \node at (t511) [styleA] (tt511) {};
+ \node at (t5111) [styleA] (tt5111) {};
+ \node at (t51111) [styleA] (tt51111) {};
+ \node at (t51112) [styleA] (tt51112) {};
+ \node at (t512) [styleA] (tt512) {};
+ \node at (t5121) [styleA] (tt5121) {};
+ \node at (t51211) [styleA] (tt51211) {};
+ %
+ \draw [->, styleB] ($(tt51.west)$)
+ -- ($(tt511.west)$)
+ -- ($(tt5111.west)$)
+ -- ($(tt51111.west)$);
+ \draw [->, styleB] ($(tt51111.east)$)
+ -- ($(tt5111.south)$)
+ -- ($(tt51112.west)$);
+ \draw [->, styleB] ($(tt51112.east)$)
+ -- ($(tt5111.east)$)
+ -- ($(tt511.east)$)
+ -- ($(tt51.south)$)
+ -- ($(tt512.west)$)
+ -- ($(tt5121.west)$)
+ -- ($(tt51211.west)$);
+ \draw [->, styleB] ($(tt51211.east)$)
+ -- ($(tt5121.east)$)
+ -- ($(tt512.east)$)
+ -- ($(tt51.east)$);
+\end{scope}
+
+
+\begin{scope}[xshift=5.7in, yshift=-2.3in]
+ \node [shape=rectangle, draw = none] (a) {
+ $\begin{aligned}
+ &\begin{aligned}
+ &\Phi_0(t_1) = \overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\
+ &\Phi_0(t_2) = \overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\
+ &\Phi_0(t_3) = \overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\
+ &\Phi_0(t_4) = \overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\
+ &\Phi_0(t_5) = \overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\
+ &\Phi_0(t_6) = \overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\
+ &\Phi_0(t_7) = \overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\
+ &\Phi_0(t_8) = \overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0} \\
+ &\Phi_0(t_9) = \overbracket {\Xl_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xl_2 \Xl_3 \Xl_4} a \overbracket {\Xr_3 \Xr_2 \Xr_1 \Xr_0}
+ \end{aligned}
+ \\[1em]
+ & \quad\quad t_1 < t_2 < t_3 < t_4 < t_5 < t_7 < t_6 < t_8 < t_9
+ \end{aligned}$
+ };
+\end{scope}
+
+\begin{scope}[xshift=2.1in, yshift=-2.2in]
+ \node [shape=rectangle, draw = none] (a) {
+ \setlength\tabcolsep{1.5pt}
+ \renewcommand{\arraystretch}{1.1}
+ $\begin{aligned}
+ &\begin{tabular}{c ccccccccc}
+ $t_2$ & $t_3$ & $t_4$ & $t_5$ & $t_6$ & $t_7$ & $t_8$ & $t_9$ \\
+ %
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & \!-1 & 3 & 0 \\[-3pt]
+ -1 & \!-1 & 2 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 3 & 0 \\[-3pt]
+ -1 & 2 & 2 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 3 & 0 \\[-3pt]
+ -1 & 2 & 2 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & \!-1 & 3 & 0 \\[-3pt]
+ -1 & \!-1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 3 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 3 & 0 \\[-3pt]
+ -1 & 2 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 3 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 3 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ & $t_1$
+ \\[1em]
+%
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 2 & 0 \\[-3pt]
+ -1 & 2 & 2 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 2 & 0 \\[-3pt]
+ -1 & 2 & 2 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & \!-1 & 2 & 0 \\[-3pt]
+ -1 & \!-1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 2 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 2 & 0 \\[-3pt]
+ -1 & 2 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 2 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 2 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ & $t_2$
+ \\[1em]
+%
+ & &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & \!-1 & 3 & 0 \\[-3pt]
+ -1 & \!-1 & 2 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \bf{
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 2 & 2 & 0 \\[-3pt]
+ -1 & 3 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ }
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 2 & 2 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & \!-1 & 3 & 0 \\[-3pt]
+ -1 & \!-1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 2 & 2 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 2 & 2 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ & $t_3$
+ \\[1em]
+%
+ & & &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 2 & 2 & 0 \\[-3pt]
+ -1 & 3 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 2 & 2 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & \!-1 & 2 & 0 \\[-3pt]
+ -1 & \!-1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 2 & 2 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 2 & 2 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ & $t_4$
+ \\[1em]
+%
+ & & & &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 1 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 1 & 0 \\[-3pt]
+ -1 & 2 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 1 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 3 & 1 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ & $t_5$
+ \\[1em]
+%
+ & & & & &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 1 & 1 & 0 \\[-3pt]
+ -1 & 2 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & \!-1 & 3 & 0 \\[-3pt]
+ -1 & \!-1 & 2 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & \!-1 & 3 & 0 \\[-3pt]
+ -1 & \!-1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ & $t_6$
+ \\[1em]
+%
+ & & & & & &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 2 & 1 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & 2 & 1 & 0 \\[-3pt]
+ -1 & 1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ & $t_7$
+ \\[1em]
+%
+ & & & & & & &
+ \begin{tabular}{|cccc|}
+ \hline
+ -1 & \!-1 & 2 & 0 \\[-3pt]
+ -1 & \!-1 & 1 & 0 \\
+ \hline
+ \end{tabular}
+ & $t_8$
+ \end{tabular}
+ \end{aligned}$
+ };
+\end{scope}
+}
+\normalsize{
+\node (w1)
+ [ xshift=3.3in
+ , yshift=-3.9in
+ , draw=none
+ , shape=rectangle
+ ] {(d) --
+ All possible IPTs for RE $(((a)^{1,3})^{1,3})^{1,3}$ and string $aaa$, the corresponding PEs,
+ and the table of $traces(\Phi_0(t_i), \Phi_0(t_j))$
+ };
+\node (w2)
+ [ below of = w1
+ , yshift=0.225in
+ , draw=none
+ , shape=rectangle
+ ] {
+ for each pair of PEs $t_i$, $t_j$.
+ IPTs $t_3$ and $t_5$ are shown in more detail
+ (their table entry is highlighted in bold).
+ };
}
+\end{scope}
+
\end{tikzpicture}
projects / re2c / commitdiff