LaTeX vs MathPix image — 2501.06662

\operatorname{Mag}(t \mathcal{M})=(t-1) \sum_{x \in \operatorname{ob}(\mathcal{M}) \backslash T(\perp)} H_{t}\left(p_{x}\right)+\#(T(\perp)), \quad t>0 .

\operatorname{Mag}(t\mathcal{M})=(t-1)\sum_{x\in\operatorname{ob}(\mathcal{M})\setminus T(\bot)}H_t(p_x) + \# (T(\bot)), \qquad t>0.

\frac{(t-1)\left(n^{1-t}-1\right)}{1-t} \cdot \#(\operatorname{ob}(\mathcal{M}) \backslash T(\perp))+\#(T(\perp))

\frac{(t-1)(n^{1-t}-1)}{1-t}\cdot \#(\text{ob}(\mathcal{M})\setminus T(\bot)) + \#(T(\bot))

\underbrace{\#(T(\perp))}_{\text {deterministic LM }} \leq \lim _{t \rightarrow \infty} \operatorname{Mag}(t \mathcal{M}) \leq \underbrace{\#(\operatorname{ob}(\mathcal{M}))}_{\text {maximally random LM }} .

\underbrace{\#(T(\bot))}_{\text{deterministic LM}} \;\leq \; \lim_{t\to \infty} \operatorname{Mag}(t\mathcal{M}) \; \leq \; \underbrace{\#(\text{ob}(\mathcal{M}))}_{\text{maximally random LM}}.

\operatorname{Mag}(t \mathcal{M})=\sum_{\ell} e^{-t \ell} \sum_{k \geq 0}(-1)^{k} \operatorname{rank}\left(H_{k, \ell}(\mathcal{M})\right),

\operatorname{Mag}(t\mathcal M) = \sum_{\ell } e^{-t\ell} \sum_{k\geq 0} (-1)^k \operatorname{rank}(H_{k,\ell}(\mathcal M)),

\operatorname{ob}(\mathrm{L})=\left\{\perp a: a \in A^{*} \text { and }|a| \leq N-1\right\} \sqcup\left\{\perp a \dagger: a \in A^{*} \text { and }|a|<N-1\right\} .

\operatorname{ob}(\mathsf{L})=\{\bot a : a\in A^* \text{ and } |a|\leq N-1\}\sqcup \{\bot a \dagger : a\in A^* \text{ and } |a|<N-1\}.

\begin{aligned} & T(x)=\left\{y \in \operatorname{ob}\left(\mathrm{~L}_{x}\right): y \text { is an unfinished text of length } N\right. \text { or } \\ & \qquad y \text { is a finished text such that }|y| \leq N\} . \end{aligned}

\begin{aligned} T(x) = \{ y \in \operatorname{ob}(\mathsf L_x) : y\text{ is an unfinished text of length }N \text{ or }\\ y \text{ is a finished text such that } |y|\leq N\}. \end{aligned}

p\left(a_{3} \mid \perp a\right) p\left(a_{4} \mid \perp a a_{3}\right) p\left(a_{5} \mid \perp a a_{3} a_{4}\right) p\left(\dagger \mid \perp a a_{3} a_{4} a_{5}\right) .

p(a_3|\bot a)p(a_4|\bot a a_3)p(a_5|\bot a a_3 a_4 )p(\dagger|\bot a a_3a_4a_5).

\pi(y \mid x):=\left\{\begin{array}{ll} 1 & \text { if } x=y \\ 0 & \text { if } x \nrightarrow y \\ \prod_{i=1}^{k} p\left(a_{t+i} \mid y_{<t+i}\right) & \text { if } x \rightarrow y \end{array} .\right.

\pi(y|x):= \begin{cases} 1 & \text{if } x=y\\[5pt] 0 & \text{if } x\not\to y\\[5pt] \prod_{i=1}^{k}p(a_{t+i}|y_{<t+i}) & \text{if } x\to y\\[5pt] \end{cases}.

\sum_{y \in T(x)} \pi(y \mid x)=\sum_{a \in A \cup\{\dagger\}} p_{x}(a)=1,

\sum_{y\in T(x)} \pi(y|x) = \sum_{a\in A\cup \{\dagger\}} p_x(a) =1,

\begin{aligned} \sum_{y \in T(x)} \pi(y \mid x) & =\sum_{i=0}^{m-1} \sum_{a^{\prime} \in A^{i}} \pi\left(x a^{\prime} \dagger \mid x\right)+\sum_{a \in A^{m}} \pi(x a \mid x) \\ & =\sum_{i=0}^{m-1} \sum_{a^{\prime} \in A^{i}} \pi\left(x a^{\prime} \dagger \mid x\right)+\sum_{\substack{a=a^{\prime} a^{\prime \prime} \\ a^{\prime} \in A^{m-1}, a^{\prime \prime} \in A}} p\left(a^{\prime \prime} \mid x a^{\prime}\right) \pi\left(x a^{\prime} \mid x\right) \\ & =\sum_{i=0}^{m-2} \sum_{a^{\prime} \in A^{i}} \pi\left(x a^{\prime} \dagger \mid x\right)+\sum_{a^{\prime} \in A^{m-1}} \pi\left(x a^{\prime} \mid x\right) \sum_{a^{\prime \prime} \in A \cup\{\dagger\}} p\left(a^{\prime \prime} \mid x a^{\prime}\right) \\ & =\sum_{i=0}^{m-2} \sum_{a^{\prime} \in A^{i}} \pi\left(x a^{\prime} \dagger \mid x\right)+\sum_{a^{\prime} \in A^{m-1}} \pi\left(x a^{\prime} \mid x\right) \end{aligned}

\begin{aligned} 1 & \leq \mathcal{C}(x, x) \\ \mathcal{C}(y, z) \otimes \mathcal{C}(x, y) & \leq \mathcal{C}(x, z) \end{aligned}

\begin{aligned} 1\leq \mathcal{C}(x,x) \\ \mathcal{C}(y,z)\otimes\mathcal{C}(x,y)\leq \mathcal{C}(x,z) \end{aligned}

\begin{aligned} & x=\perp a_{1} \cdots a_{t} \\ & y=\perp a_{1} \cdots a_{t} \cdots a_{t+k} \\ & z=\perp a_{1} \cdots a_{t} \cdots a_{t+k} \cdots a_{t+k+k^{\prime}} \end{aligned}

\begin{aligned} x&= \bot a_1\cdots a_t\\ y&=\bot a_1\cdots a_t\cdots a_{t+k}\\ z&=\bot a_1\cdots a_t\cdots a_{t+k}\cdots a_{t+k+k'}. \end{aligned}

\begin{aligned} \pi(y \mid x) \pi(z \mid y) & =\prod_{i=1}^{k} p\left(a_{t+i} \mid y_{<t+i}\right) \prod_{j=1}^{k^{\prime}} p\left(a_{t+k+j} \mid z_{<t+k+j}\right) \\ & =\prod_{i=1}^{k+k^{\prime}} p\left(a_{t+i} \mid z_{<t+i}\right) \\ & =\pi(z \mid x) \end{aligned}

\begin{aligned} \pi(y|x)\pi(z|y)&=\prod_{i=1}^{k}p(a_{t+i}|y_{<t+i})\prod_{j=1}^{k'}p(a_{t+k+j}|z_{<t+k+j})\\[5pt] &=\prod_{i=1}^{k+k'}p(a_{t+i}|z_{<t+i})\\[5pt] &=\pi(z|x). \end{aligned}

d(x, y):=-\ln \pi(y \mid x) .

d(x,y):=-\ln\pi(y|x).

\operatorname{Mag}(\mathcal{C})=\sum_{(x, y) \in \operatorname{ob}(\mathcal{C}) \times \operatorname{ob}(\mathcal{C})} \zeta_{\mathcal{C}}^{-1}(x, y) .

\operatorname{Mag}(\mathcal{C})=\sum_{(x,y)\in \operatorname{ob}(\mathcal{C})\times\operatorname{ob}(\mathcal{C})}\zeta_\mathcal{C}^{-1}(x,y).

(f * g)(s, u):=\sum_{s \leq t \leq u} f(s, t) g(t, u),

(f\ast g)(s,u):=\sum_{s\leq t \leq u}f(s,t)g(t,u),

\delta(s, u)= \begin{cases}1 & \text { if } s=u \\ 0 & \text { if } s \neq u\end{cases}

\delta(s,u)= \begin{cases} 1 &\text{if }s=u\\ 0 &\text{if }s\neq u \end{cases}

\zeta_{P}(s, u)=1, \quad \text { for all } s \leq u \text { in } P .

\zeta_P(s,u)=1, \quad \text{for all } s\leq u \text{ in } P.

\mu_{P}(s, u)= \begin{cases}1 & \text { if } s=u \\ -\sum_{s \leq t<u} \mu_{P}(s, t) & \text { if } s<u \\ 0 & \text { otherwise } .\end{cases}

\mu_P(s,u) = \begin{cases} 1 & \text{if } s=u\\[5pt] -\sum_{s\leq t < u}\mu_P(s,t) &\text{if } s< u\\[5pt] 0 &\text{otherwise}. \end{cases}

\begin{aligned} \mu_{\mathrm{L}}(x, x) & =1 \\ \mu_{\mathrm{L}}\left(x, x a_{1}\right) & =-\mu_{\mathrm{L}}(x, x)=-1 \\ \mu_{\mathrm{L}}\left(x, x a_{1} a_{2}\right) & =-\mu_{\mathrm{L}}(x, x)-\mu_{\mathrm{L}}\left(x, x a_{1}\right)=-1+1=0 \\ \mu_{\mathrm{L}}\left(x, x a_{1} a_{2} a_{3}\right) & =-\mu_{\mathrm{L}}(x, x)-\mu_{\mathrm{L}}\left(x, x a_{1}\right)-\mu_{\mathrm{L}}\left(x, x a_{1} a_{2}\right)=-1+1+0=0 . \end{aligned}

\begin{aligned} \mu_{\mathsf{L}}(x,x)&=1\\ \mu_{\mathsf{L}}(x,xa_1)&=-\mu_{\mathsf{L}}(x,x)=-1\\ \mu_{\mathsf{L}}(x,xa_1a_2)&=-\mu_{\mathsf{L}}(x,x)-\mu_{\mathsf{L}}(x,xa_1)=-1+1=0\\ \mu_{\mathsf{L}}(x,xa_1a_2a_3)&=-\mu_{\mathsf{L}}(x,x)-\mu_{\mathsf{L}}(x,xa_1) -\mu_{\mathsf{L}}(x,xa_1a_2)=-1+1+0=0. \end{aligned}

\begin{aligned} \operatorname{Mag}(\mathrm{L}) & =\sum_{x, y \in \mathrm{ob}(\mathrm{~L})} \zeta_{\mathrm{L}}^{-1}(x, y) \\ & =\sum_{x, y \in \mathrm{ob}(\mathrm{~L})} \mu_{\mathrm{L}}(x, y) \\ & =\sum_{x \in \mathrm{ob}(\mathrm{~L})} \mu_{\mathrm{L}}(x, x)+\sum_{\substack{x=\perp a \in \mathrm{ob}(\mathrm{~L}) \\ a \in \bigcup_{i=0}^{N-2} A^{i}}} \sum_{a_{1} \in A \cup\{\dagger\}} \mu_{\mathrm{L}}\left(x, x a_{1}\right) \\ & =\# \mathrm{ob}(\mathrm{~L})-\#\left\{x \in \mathrm{ob}(\mathrm{~L}): x=\perp a \text { with } a \in \bigcup_{i=0}^{N-2} A^{i}\right\}(\# A+1) \end{aligned}

\begin{aligned} \# \mathrm{ob}(\mathrm{~L}) & =1+\sum_{i=1}^{N-1} \# \mathrm{~L}^{(i)} \\ & =1+\sum_{i=1}^{N-1} \# A^{i}+\# A^{i-1}=\# A^{N-1}+2\left(\# A^{N-2}\right)+\cdots+2(\# A)+2 \end{aligned}

\begin{aligned} \#\operatorname{ob}(\mathsf{L}) &= 1 + \sum_{i=1}^{N-1}\#\mathsf{L}^{(i)} \\ &= 1+ \sum_{i=1}^{N-1} \#A^i + \#A^{i-1} = \#A^{N-1} + 2(\#A^{N-2}) + \cdots +2(\#A) + 2. \end{aligned}

\begin{aligned} \#\left\{x \in \mathrm{ob}(\mathrm{~L}): x=\perp a \text { with } a \in \bigcup_{i=0}^{N-2} A^{i}\right\} & (\# A+1)=\left(\sum_{i=0}^{N-2} \# A^{i}\right)(\# A+1) \\ & =\# A^{N-1}+2\left(\# A^{N-2}\right)+\cdots+2(\# A)+1 \end{aligned}

\begin{aligned} \#\left\{x\in\operatorname{ob}(\mathsf{L}):x=\bot a \text{ with }a\in \bigcup_{i=0}^{N-2} A^i\right\} (\#A+1)= \left( \sum_{i=0}^{N-2} \#A^i\right) (\#A+1) \\= \#A^{N-1} + 2(\#A^{N-2}) + \cdots + 2(\#A) + 1. \end{aligned}

\zeta_{\mathcal{M}}(x, y):=e^{-d(x, y)}=\pi(y \mid x),

\zeta_{\mathcal{M}}(x,y):=e^{-d(x,y)}=\pi(y|x),

\left(\zeta_{\mathcal{M}}\right)_{t}(x, y):=e^{-t d(x, y)}=\pi(y \mid x)^{t} .

(\zeta_\mathcal{M})_t(x,y):=e^{-td(x,y)}=\pi(y|x)^t.

\zeta_{t}^{-1}(x, y)=\sum_{k \geq 0} \sum_{\substack{\text { nondeg.paths } \\ x=y_{0} \rightarrow y_{1} \rightarrow \cdots \rightarrow y_{k}=y}}(-1)^{k} \prod_{i=1}^{k} \pi\left(y_{i} \mid y_{i-1}\right)^{t} .

\zeta_t^{-1}(x,y)= \sum_{k\geq 0}\sum_{\substack{\text{nondeg. paths}\\ x=y_0\to y_1\to\cdots \to y_k=y}\text{ in } \mathsf L}(-1)^k\prod_{i=1}^k\pi(y_i|y_{i-1})^t.

\zeta_{t}^{-1}=\sum_{k \geq 0}(-1)^{k}\left(\zeta_{t}-\delta\right)^{k}

\zeta_t^{-1} = \sum_{k\geq 0} (-1)^k (\zeta_t-\delta)^k,

\left(\zeta_{t}-\delta\right)(x, y)=\pi(x, y)^{t} \mathbf{1}_{(x, y) \in E},

(\zeta_t - \delta)(x,y) = \pi(x,y)^t \mathbf 1_{(x,y)\in E},

\left(\zeta_{t}-\delta\right)^{k}(x, y)=\sum_{\substack{\left(y_{0}, y_{1}, \ldots, y_{k-1}, y_{k}\right) \\ y_{0}=x, y_{k}=y \\\left(y_{i-1}, y_{i}\right) \in E \text { for } i=1, \ldots, k}} \prod_{i=1}^{k} \pi\left(y_{i} \mid y_{i-1}\right)^{t}

(\zeta_t - \delta)^k(x,y) = \sum_{\substack{(y_0,y_1,...,y_{k-1},y_k)\\ y_0=x,\, y_k = y \\ (y_{i-1}, y_{i})\in E \text{ for } i=1,...,k}} \prod_{i=1}^k\pi(y_i|y_{i-1})^t.

\zeta_{t}^{-1}(x, y)=\pi(y \mid x)^{t} \zeta_{\mathrm{L}}^{-1}(x, y) .

\zeta_t^{-1}(x,y)=\pi(y|x)^t\;\zeta_\mathsf{L}^{-1}(x,y).

\begin{aligned} \prod_{i=1}^{k} \pi\left(y_{i} \mid y_{i-1}\right)^{t} & =\pi\left(y_{1} \mid y_{0}\right)^{t} \pi\left(y_{2} \mid y_{1}\right)^{t} \pi\left(y_{3} \mid y_{2}\right)^{t} \cdots \pi\left(y_{k} \mid y_{k-1}\right)^{t} \\ & =\pi\left(y_{2} \mid y_{0}\right)^{t} \pi\left(y_{3} \mid y_{2}\right)^{t} \cdots \pi\left(y_{k} \mid y_{k-1}\right)^{t} \\ & =\quad \vdots \\ & =\pi\left(y_{k} \mid y_{0}\right)^{t} \end{aligned}

\begin{aligned} \prod_{i=1}^k\pi(y_i|y_{i-1})^t &= \pi(y_1|y_0)^t\pi(y_2|y_1)^t\pi(y_3|y_2)^t\cdots \pi(y_k|y_{k-1})^t\\ &= \pi(y_2|y_0)^t\pi(y_3|y_2)^t\cdots \pi(y_k|y_{k-1})^t\\ &=\quad \vdots\\ &= \pi(y_k|y_0)^t. \end{aligned}

\pi(y \mid x)^{t} \sum_{k \geq 0}(-1)^{k} \#\{\text { nondegenerate paths of length } k \text { from } x \text { to } y \text { in } \mathrm{L}\}

\pi(y|x)^t\sum_{k\geq 0}(-1)^k\#\{\text{nondegenerate paths of length $k$ from $x$ to $y$ in $\mathsf{L}$}\}

\zeta_{t}^{-1}(x, y)= \begin{cases}-\pi(y \mid x)^{t} & \text { if } y \in \mathrm{~L}_{x}^{(1)} \\ 1 & \text { if } y=x \\ 0 & \text { otherwise }\end{cases}

\zeta_t^{-1}(x,y)= \begin{cases} -\pi(y|x)^t &\text{if }y\in\mathsf{L}_x^{(1)}\\ 1 &\text{if }y=x\\ 0 &\text{otherwise}. \end{cases}

\operatorname{Mag}(t \mathcal{M}):=\sum_{x, y \in \operatorname{ob}(\mathcal{M})} \zeta_{t}^{-1}(x, y) .

\begin{aligned} \operatorname{Mag}(t\mathcal{M})&:=\sum_{x,y\in\operatorname{ob}(\mathcal{M})}\zeta_t^{-1}(x,y). \end{aligned}

H_{t}(p)=\frac{1}{t-1}\left(1-\sum_{i=1}^{n} p_{i}^{t}\right) .

H_t(p)=\frac{1}{t-1}\left(1-\sum_{i=1}^np_i^t\right).

\operatorname{Mag}(t \mathcal{M})=(t-1) \sum_{x \in \operatorname{ob}(\mathcal{M}) \backslash T(\perp)} H_{t}\left(p_{x}\right)+\#(T(\perp)), \quad t>0 .

\operatorname{Mag}(t\mathcal{M})=(t-1)\sum_{x\in\operatorname{ob}(\mathcal{M})\setminus T(\bot)}H_t(p_x) + \# (T(\bot)), \qquad t>0.

\begin{aligned} \operatorname{Mag}(t \mathcal{M}) & =\sum_{x, y \in \mathrm{ob}(\mathcal{M})} \zeta_{t}^{-1}(x, y) \\ & =\sum_{x \in \mathrm{ob}(\mathcal{M})} \zeta_{t}^{-1}(x, x)-\sum_{x \in \mathrm{ob}(\mathcal{M}) \backslash T(\perp)} \sum_{y \in \mathrm{~L}_{x}^{(1)}} \pi(y \mid x)^{t} \\ & =\sum_{x \in \mathrm{ob}(\mathcal{M}) \backslash T(\perp)}\left(1-\sum_{a \in A \cup\{\dagger\}} p_{x}(a)^{t}\right)+\#(T(\perp)) . \end{aligned}

\begin{aligned} \operatorname{Mag}(t\mathcal{M})&=\sum_{x,y\in\operatorname{ob}(\mathcal{M})}\zeta_t^{-1}(x,y) \\[5pt] &=\sum_{x\in\operatorname{ob}(\mathcal{M})} \zeta^{-1}_t(x,x) - \sum_{x\in\operatorname{ob}(\mathcal{M})\setminus T(\bot)}\sum_{y\in\mathsf{L}_x^{(1)}}\pi(y|x)^t \\[5pt] &=\sum_{x\in\operatorname{ob}(\mathcal{M})\setminus T(\bot)}\left(1-\sum_{a\in A\cup\{\dagger\}} p_x(a)^t\right) + \# (T(\bot)). \end{aligned}

\lim _{t \rightarrow 1} H_{t}(p)=\lim _{t \rightarrow 1} \frac{1}{t-1}\left(1-\sum_{s \in S} p(s)^{t}\right)=-\sum_{s \in S} p(s) \ln p(s)=: H(p) .

\lim_{t\to 1} H_t(p)= \lim_{t\to 1} \frac{1}{t-1} \left( 1 -\sum_{s\in S} p(s)^t\right) = -\sum_{s\in S} p(s)\ln p(s) =: H(p).

f^{\prime}(1)=\sum_{x \in \mathrm{ob}(\mathcal{M}) \backslash T(\perp)} H\left(p_{x}\right) .

f'(1) = \sum_{x\in\operatorname{ob}(\mathcal{M})\setminus T(\bot)}H(p_x).

f(t)=\#(\mathrm{ob}(\mathcal{M}))-\sum_{x \in \mathrm{ob}(\mathcal{M}) \backslash T(\perp)} Z_{x}(t),

f(t) = \# (\operatorname{ob}(\mathcal M)) - \sum_{x\in \operatorname{ob}(\mathcal M) \setminus T(\bot)} Z_x(t),

Z_{x}(t)=\sum_{a \in A \cup\{\dagger\}} p_{x}(a)^{t}=\sum_{a \in A \cup\{\dagger\}} e^{-t\left(-\ln p_{x}(a)\right)}

Z_x(t) = \sum_{a\in A\cup \{\dagger\}} p_x(a)^t= \sum_{a\in A\cup \{\dagger\}} e^{-t (-\ln p_x(a))}

H\left(p_{x}\right)=-\left.\frac{\mathrm{d}}{\mathrm{~d} t} Z_{x}(t)\right|_{t=1},

H(p_x) = -\left.\frac{\mathrm d}{\mathrm d t} Z_x(t)\right|_{t=1},

\mathbb{E}_{\rho}(E)=\sum_{s \in S} E(s) \frac{e^{-t E(s)}}{\tilde{Z}(t)}=-\frac{\mathrm{d}}{\mathrm{~d} t} \ln \tilde{Z}(t)=\frac{f^{\prime}(t)}{\tilde{Z}(t)},

\mathbb E_\rho(E) = \sum_{s\in S} E(s) \frac{e^{-tE(s)}}{\tilde Z(t)} = -\frac{\mathrm d}{\mathrm d t} \ln \tilde Z(t) = \frac{f'(t)}{\tilde Z(t)},

\zeta_{\mathcal{M}_{x}}=\left.\zeta_{\mathcal{M}}\right|_{\operatorname{ob}\left(\mathcal{M}_{x}\right) \times \operatorname{ob}\left(\mathcal{M}_{x}\right)}

\zeta_{\mathcal{M}_x}=\zeta_{\mathcal{M}}\big\vert_{\operatorname{ob}(\mathcal{M}_x)\times \operatorname{ob}(\mathcal{M}_x)}

\zeta_{\mathcal{M}_{x}}^{-1}=\left.\zeta_{\mathcal{M}}^{-1}\right|_{\operatorname{ob}\left(\mathcal{M}_{x}\right) \times \operatorname{ob}\left(\mathcal{M}_{x}\right)}

\zeta_{\mathcal{M}_x}^{-1}=\zeta_{\mathcal{M}}^{-1}\big\vert_{\operatorname{ob}(\mathcal{M}_x)\times \operatorname{ob}(\mathcal{M}_x)},

\begin{aligned} \operatorname{Mag}\left(t \mathcal{M}_{x}\right) & =\sum_{y, z \in \mathrm{ob}\left(\mathcal{M}_{x}\right)} \zeta_{\mathcal{M}_{x}}^{-1}(y, z) \\ & =\sum_{y \in \mathrm{ob}\left(\mathcal{M}_{x}\right)} \zeta_{\mathcal{M}_{x}}^{-1}(y, y)-\sum_{y \in \mathrm{ob}\left(\mathcal{M}_{x}\right) \backslash T(x)} \sum_{y \in \mathrm{~L}_{y}^{(1)}} \pi(z \mid y)^{t} \\ & =(t-1) \sum_{y \in \mathrm{ob}\left(\mathcal{M}_{x}\right) \backslash T(x)} H_{t}\left(p_{y}\right)+\#(T(x)) . \end{aligned}

G_{k, \ell}=\left\{\left(y_{0}, \ldots, y_{k}\right) \in M^{k+1} \mid \sum_{i=0}^{k-1} d\left(y_{i}, y_{i+1}\right)=\ell \text { and for all } i, y_{i} \neq y_{i+1}\right\} .

G_{k,\ell} = \left\{ ( y_0,\ldots , y_k ) \in M^{k+1} \mid \sum_{i=0}^{k-1} d(y_i,y_{i+1} ) = \ell \text{ and for all $i$, }y_i \neq y_{i+1}\right\}.

\partial_{k}=\sum_{i=0}^{k}(-1)^{i} \partial_{k}^{i}

\partial_k=\sum_{i=0}^k(-1)^i\partial_k^i

\begin{aligned} & \partial_{k}^{i}\left(y_{0}, \ldots, y_{k}\right)= \\ & \begin{cases}\left(y_{0}, \ldots, y_{i-1}, y_{i+1}, \ldots, y_{k}\right) & \text { if } d\left(y_{i-1}, y_{i}\right)+d\left(y_{i}, y_{i+1}\right)=d\left(y_{i-1}, y_{i+1}\right) \\ 0 & \text { otherwise }\end{cases} \end{aligned}

\begin{aligned} \partial_k^i(y_0,\ldots,y_k ) = \\ \begin{cases} (y_0,\ldots,y_{i-1},y_{i+1},\ldots,y_k ) &\text{if } d(y_{i-1},y_i)+d(y_i,y_{i+1})=d(y_{i-1},y_{i+1})\\ 0 &\text{otherwise} \end{cases} \end{aligned}

\operatorname{Mag}(t \mathcal{M})=\sum_{\ell} q^{\ell} \sum_{k \geq 0}(-1)^{k} \operatorname{rank}\left(H_{k, \ell}(\mathcal{M})\right)

\operatorname{Mag}(t\mathcal M) = \sum_{\ell } q^\ell \sum_{k\geq 0} (-1)^k \operatorname{rank}(H_{k,\ell}(\mathcal M))

\begin{aligned} \operatorname{Mag}(t \mathcal{M}) & =\sum_{x, y \in \mathrm{ob}(\mathcal{M})} \zeta_{t}^{-1}(x, y) \\ & =\sum_{x, y \in \mathrm{ob}(\mathcal{M})} \sum_{k \geq 0} \sum_{\substack{\text { nondeg.paths } \\ x=y_{0} \rightarrow y_{1} \rightarrow \cdots \rightarrow y_{k}=y}}(-1)^{k} \prod_{i=1}^{k} \pi\left(y_{i} \mid y_{i-1}\right)^{t} . \end{aligned}

\begin{aligned} \operatorname{Mag}(t\mathcal M)& =\sum_{x,y\in \operatorname{ob}(\mathcal M)} \zeta^{-1}_t(x,y) \\ &= \sum_{x,y\in \operatorname{ob}(\mathcal M)}\; \sum_{k\geq 0}\sum_{\substack{\text{nondeg. paths}\\ x=y_0\to y_1\to\cdots \to y_k=y}\text{ in } \mathsf L}(-1)^k\prod_{i=1}^k\pi(y_i|y_{i-1})^t. \end{aligned}

\operatorname{Mag}(t \mathcal{M})=\sum_{k \geq 0} \sum_{\ell \in[0, \infty)} \sum_{c \in G_{k, \ell}}(-1)^{k} e^{-t \ell} .

\operatorname{Mag}(t\mathcal M) = \sum_{k\geq 0} \sum_{\ell\in [0,\infty)} \sum_{c \in G_{k,\ell}} (-1)^k e^{-t\ell}.

\sum_{c \in G_{k, \ell}}(-1)^{k} e^{-t \ell}=(-1)^{k} q^{\ell} \# G_{k, \ell}=(-1)^{k} q^{\ell} \operatorname{rank}\left(M C_{k, \ell}(\mathcal{M})\right) .

\sum_{c\in G_{k,\ell} } (-1)^k e^{-t\ell} = (-1)^k q^\ell \# G_{k,\ell} = (-1)^k q^{\ell} \operatorname{rank}(MC_{k,\ell}(\mathcal M)) .

\operatorname{Mag}(t \mathcal{M})=\operatorname{rank} H_{0,0}(\mathcal{M})-\sum_{\ell \geq 0} q^{\ell} \operatorname{rank} H_{1, \ell}(\mathcal{M}) .

\operatorname{Mag}(t\mathcal M) = \operatorname{rank} H_{0,0}(\mathcal M) - \sum_{\ell\geq 0} q^\ell \operatorname{rank} H_{1,\ell}(\mathcal M).

P P L(x)=\exp \left\{-\frac{1}{n} \sum_{i=1}^{n} \ln p\left(a_{i} \mid y_{<i}\right)\right\}

PPL(x)=\exp\left\{-\frac{1}{n}\sum_{i=1}^n\ln p(a_i|y_{<i})\right\}

P P L(y)=1 / \zeta_{t}\left(a_{0}, y\right)=1 / e^{t \ln \pi\left(y \mid a_{0}\right)},

PPL(y)=1/\zeta_t(a_0,y)=1/e^{t\ln\pi(y|a_0)},

\theta(a, b)=0 \quad \text { if } \quad \mathrm{A}(a, b)=\varnothing .

\theta(a,b)=0 \quad \text{if } \quad \mathsf{A}(a,b)=\varnothing.

\begin{aligned} \mathscr{H}(\mathcal{M}, p, \zeta) & =-\sum_{y \in \mathrm{ob}(\mathcal{M})} p(y) \log \left(\sum_{z \in \mathrm{ob}(\mathcal{M})} \zeta_{t}(y, z) p(z)\right) \\ & =-\sum_{y \in \mathrm{ob}(\mathcal{M})} p(y) \log \left(\sum_{z \in \mathrm{ob}(\mathcal{M})} \pi(z \mid y)^{t} p(z)\right) . \end{aligned}

\begin{aligned} \mathscr{H}(\mathcal{M},p,\zeta)& = -\sum_{y\in\operatorname{ob}(\mathcal{M})}p(y)\log\left(\sum_{z\in\operatorname{ob}(\mathcal{M})}\zeta_t(y,z)p(z)\right)\\ &= - \sum_{y\in\operatorname{ob}(\mathcal{M})}p(y) \log\left(\sum_{z\in\operatorname{ob}(\mathcal{M})}\pi(z|y)^tp(z)\right). \end{aligned}

\begin{aligned} \lim _{n \rightarrow \infty} & \frac{\mathscr{H}\left(\mathcal{M},\left.\pi(-\mid \perp)\right|_{T(\perp)}, \zeta\right)}{n} \\ & =\lim _{n \rightarrow \infty}-\frac{1}{n} \sum_{\left(a_{1}, \ldots, a_{N-1}\right) \in A^{N-1}} \pi\left(\perp a_{1} \cdots a_{N-1} \mid \perp\right) \log \pi\left(\perp a_{1} \cdots a_{N-1} \mid \perp\right) \end{aligned}

\begin{aligned} \lim_{n\to \infty} \frac{\mathscr{H}(\mathcal{M},\pi(-|\bot)|_{T(\bot)},\zeta)}{n} \\ = \lim_{n\to\infty} -\frac 1n \sum_{(a_1,...,a_{N-1}) \in A^{N-1}} \pi(\bot a_1\cdots a_{N-1}|\bot) \log \pi(\bot a_1\cdots a_{N-1}|\bot) \end{aligned}

-\sum_{a \in A} \sum_{a^{\prime} \in A} p_{\perp}(a) P\left(a, a^{\prime}\right) \log P\left(a, a^{\prime}\right),

-\sum_{a\in A} \sum_{a'\in A} p_{\bot}(a) P(a,a') \log P(a,a'),