$x_{t+1},\ldots x_{t+\ell}$

$O(|{\cal X}|^{2L})$

$\forall i=1,\ldots,L$

$1000$

${\cal M}_{b}$

$(X^{\tau},Z^{L-\tau})\sim{\cal M}_{b}^{L}.$

$(x^{i},y^{j})$

$x^{L}$

$\forall x\in{\cal X}$

$\forall x^{i_{0}}\in{\cal X}^{i_{0}}$

$X^{L}\sim{\cal M}_{s}^{L}$

$\mathbb{E}_{X^{L}\sim{\cal M}_{s}^{L}}[\tau]$

$(X^{L},Y^{L})$

$\ell=L$

$Z^{L-\tau}$

$x^{i}\in{\cal X}^{i}$

${\cal M}_{b}(\cdot|x^{n})=\text{Ber}(q),$

$\forall\ell\leq L$

$\tau,{\cal M}_{b,{\rm next}}$

$L-1$

$\beta(X^{L},Y^{L})=\tau$

$\displaystyle=\sum_{\ell=0}^{L}\ell\cdot\Pr{\left({\tau=\ell}\right)}=\sum_{% \ell=1}^{L}\sum_{x^{\ell}\in{\cal X}^{\ell}}\prod_{i=1}^{\ell}\min\{{\cal M}_{% b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\},$

$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\sum_{\ell\leq L}\sum_{x^{% \ell}}\min\left\{\text{Pr}\left(X^{1:\ell}=x^{\ell}\right),\text{Pr}\left(Y^{1% :\ell}=x^{\ell}\right)\right\}$

$\displaystyle=\Pr{\left({X^{i_{0}+1}=x^{i_{0}+1},\tau\geq i_{0}+1}\right)}+\Pr% {\left({Y^{i_{0}}=x^{i_{0}},\tau\leq i_{0}}\right)}\cdot p_{\rm res}^{x^{i_{0}% }}(x_{i_{0}+1})$

$Y_{\tau+1}\neq X_{\tau+1}$

$\displaystyle=\sum_{x^{\ell+1}\in{\cal X}^{\ell+1}}\prod_{i=1}^{\ell}\min\{{% \cal M}_{b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\}({\cal M}_{s}(% x_{\ell+1}\mid x^{\ell})-\min\{{\cal M}_{b}(x_{\ell+1}\mid x^{\ell}),{\cal M}_% {s}(x_{\ell+1}\mid x^{\ell})\})$

${\cal M}_{s}(\cdot\mid X^{i-1}),{\cal M}_{b}(\cdot\mid X^{i-1}),\forall i=0,% \ldots,L$

$L=8$

$\displaystyle\min\{{\cal M}_{s}(x^{\ell}),{\cal M}_{b}(x^{\ell})\}=\min\{\prod% _{i=1}^{\ell}{\cal M}_{s}(x_{i}\mid x^{i-1}),{\cal M}_{b}(x_{i}\mid x^{i-1})\},$

${\cal M}^{*}(\cdot\mid x^{t})$

$L-\beta(X^{L},Y^{L})$

$X_{1},\ldots,X_{L}\sim{\cal M}_{s}^{L}(\cdot)$

$\forall y^{L-\tau}\in{\cal X}^{{L-\tau}}$

$\Pr{\left({\tau=L}\right)}=\sum_{x^{L}\in{\cal X}^{L}}\prod_{i=1}^{L}\min\{{% \cal M}_{b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\}.$

$\eta_{i}\sim U(0,1)$

$\Pi({\cal M}_{s}^{L},{\cal M}_{b}^{L})$

$Y\sim{\cal M}_{b}$

$L=4$

$\displaystyle=\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau\geq\ell_{0}}\right% )}+\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau=\ell_{0}-1}\right)}$

$\displaystyle=\Pr{\left({Y^{i_{0}+1}=x^{i_{0}+1},\tau\geq i_{0}+1}\right)}+\Pr% {\left({Y^{i_{0}+1}=x^{i_{0}+1},\tau\leq i_{0}}\right)}$

$L=1$

$\ell=\ell_{0}-1$

$\displaystyle p_{\rm rej}(x^{i})$

$\displaystyle{:=}\sum_{x}{\left({{\cal M}_{b}(x^{i},x)-{\cal M}_{s}(x^{i},x)}% \right)}_{+},$

$\displaystyle\;\;\;\Pr{\left({\tau=\ell}\right)}$

$\displaystyle=\sum_{\ell=1}^{L}\Pr{\left({\tau\geq\ell}\right)}=\sum_{\ell=1}^% {L}\sum_{x^{\ell}\in{\cal X}^{\ell}}\Pr{\left({X^{\ell}=x^{\ell},\tau\geq\ell}% \right)}$

$\displaystyle=\min\{\sum_{x\in{\cal X}}\min\{{\cal M}_{b}(x^{\ell_{0}-1},x),{% \cal M}_{s}(x^{\ell_{0}-1},x)\}+p_{\rm remain}(x^{\ell_{0}-1}),$

$\ell>1$

$\forall\ell\leq L-1,x^{\ell}\in{\cal X}^{\ell}$

$x^{t}\in{\cal X}^{t}$

$\Pr{\left({X^{\ell}=x^{\ell},\tau\geq\ell}\right)}=\min\{{\cal M}_{b}(x^{\ell}% ),{\cal M}_{s}(x^{\ell})\}.$

$\displaystyle{:=}\sum_{x}{\left({{\cal M}_{s}(x^{i},x)-{\cal M}_{b}(x^{i},x)}% \right)}_{+}.$

$Y^{L}=(X^{\tau},Z^{L-\tau})$

$\displaystyle\sum_{y^{L}}\pi(x^{L},y^{L})$

$(x)_{+}=\max\{x,0\}.$

${\cal M}_{b}(\cdot\mid X^{\tau},Y)$

$\displaystyle={\cal M}_{s}(x^{L})\cdot\min\left\{1,\frac{{\cal M}_{b}(x^{L})}{% {\cal M}_{s}(x^{L})}\right\}$

${\cal M}^{L}_{s}(x^{L})$

$(x_{i},\ldots,x_{j})$

$0\leq\tau\leq L$

$\pi\in\Pi({\cal M}^{L}_{s},{\cal M}^{L}_{b})$

$\displaystyle=\sum_{\ell\leq L}\sum_{x^{\ell}}\text{Pr}_{X^{L},Y^{L}}\left(X^{% 1:\ell}=Y^{1:\ell}=x^{\ell},\beta(X^{L},Y^{L})\geq\ell\right)$

$X^{L}$

$Y_{\tau+1}$

$x^{L}\in{\cal X}^{L}$

$\textsc{Verify}_{\pi}$

${\cal M}_{b,{\rm next}}^{L-\tau}$

$\tau\leq L$

$\min\{{\cal M}_{b}(x^{L}),{\cal M}_{s}(x^{L})\}$

$\displaystyle=\prod_{i=1}^{\ell}\min\{1,\frac{{\cal M}_{b}(x_{i}\mid x^{i-1})}% {{\cal M}_{s}(x_{i}\mid x^{i-1})}\}{\left({1-\min\{1,\frac{{\cal M}_{b}(x_{% \ell+1}\mid x^{\ell})}{{\cal M}_{s}(x_{\ell+1}\mid x^{\ell})}\}}\right)}.$

$x^{t}$

$X_{i}\sim{\cal M}_{s}(\cdot\mid X^{i-1}).$

$\tau=L-i$

${\rm E.O.S}\in X^{\tau}$

$\displaystyle=\sum_{x^{L}\in{\cal X}^{L}}\Pr{\left({\tau=\ell,X^{L}=x^{L}}% \right)}$

$\pi_{\rm token}$

$\displaystyle=\sum_{x^{\ell+1}\in{\cal X}^{\ell+1}}\Pr{\left({\tau=\ell,X^{% \ell+1}=x^{\ell+1}}\right)}$

$Z\sim{\cal M}_{b,{\rm next}}(\cdot)$

$X^{\tau}$

$(a)$

$\ell

$\displaystyle=\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau\leq\ell_{0}-1}% \right)}\cdot\Pr{\left({X^{\ell_{0}-1}\text{ is accepted.}}\right)}$

$Y_{1}=X_{1}$

$\displaystyle=\min\left\{{\cal M}_{b}(x^{\ell_{0}-1}),{\cal M}_{s}(x^{\ell_{0}% -1})\right\},$

$\displaystyle=\text{Pr}_{\pi}{\left({\beta{{\color[rgb]{0,0,0}=\ell}}\mid X^{L% }}\right)}{:=}{{\color[rgb]{0,0,0}\frac{\sum_{\ell:\beta(X^{L},Y^{L})=\ell}\pi% (X^{L},Y^{L})}{\pi(X^{L})}}},$

$Y_{2}$

$\displaystyle\max_{\pi\in\Pi({\cal M}_{s}^{L},{\cal M}_{b}^{L})}\mathbb{E}_{X^% {L},Y^{L}\sim\pi}\left[\beta(X^{L},Y^{L})\right]\leq\sum_{\tau=1}^{L}\sum_{x^{% \tau}\in{\cal X}^{\tau}}\min\{{\cal M}_{s}(x^{\ell}),{\cal M}_{b}(x^{\ell})\}$

$\forall x^{n}$

$\eta_{0}\leq\min\left\{\frac{{\cal M}_{b}(X^{L})}{{\cal M}_{s}(X^{L})},1\right\}$

$O(|{\cal X}|^{L})$

$Z^{L-\tau}\sim{\cal M}_{b,{\rm next}}$

$L=6$

${\cal M}_{b}(Y_{3}\mid Y_{1},y)$

$Y^{L}=(X^{\tau},Z^{i-\tau})$

$\displaystyle=\sum_{\ell\leq L}\sum_{x^{\ell}}\min\{{\cal M}_{s}^{\ell}(x^{% \ell}),{\cal M}_{b}^{\ell}(x^{\ell})\}.$

$\displaystyle=\Pr{\left({X^{L}=x^{L}}\right)}\Pr{\left({\tau=L\mid X^{L}=x^{L}% }\right)}$

$p_{\rm res}(x)=\frac{{\left({{\cal M}_{b}(x\mid X^{\tau})-{\cal M}_{s}(x\mid X% ^{\tau})}\right)}_{+}}{\sum_{x^{\prime}}{\left({{\cal M}_{b}(x^{\prime}\mid X^% {\tau})-{\cal M}_{s}(x^{\prime}\mid X^{\tau})}\right)}_{+}},$

${\cal M}_{s}=\text{Ber}(1)$

$\frac{{\left({{\cal M}_{b}(x^{L})-{\cal M}_{s}(x^{L})}\right)}_{+}}{\sum_{x^{L% }}{\left({{\cal M}_{b}(x^{L})-{\cal M}_{s}(x^{L})}\right)}_{+}}.$

$i=0,1,\ldots,L$