2501.06662: formula evidence

The rendered column is KaTeX, and KaTeX has no preamble. It cannot use \usepackage or this document’s own macros. Corpus-wide, \bm occurs 327 times, \Perp is defined by no package and spans 10 documents, and 11,088 of 11,624 undefined occurrences are the source document’s own macros. A row that looks wrong here may be correct, and one that looks right here may not compile. The LaTeX report renders through the document’s own preamble and is the surface to judge from.

385 rows

Inline formulas (first occurrence) (385)

The page, the confidence and the picture of an inline formula are its HOST LINE's --- a formula has none of its own. A line's confidence is not a formula's.

Identifier	Page	Conf.	LaTeX source
2501.06662_FO0001	1	1.000	t
2501.06662_FO0002	1	1.000	\pi(y \mid x) \in[0,1]
2501.06662_FO0003	1	1.000	x
2501.06662_FO0004	1	1.000	y
2501.06662_FO0005	2	1.000	\pi(y \mid x)
2501.06662_FO0006	2	1.000	n
2501.06662_FO0007	2	1.000	(n-1)
2501.06662_FO0008	2	0.528	\dagger
2501.06662_FO0009	2	1.000	a
2501.06662_FO0010	2	1.000	x=\perp a
2501.06662_FO0011	2	1.000	p_{x}
2501.06662_FO0012	2	0.844	a_{1}
2501.06662_FO0013	2	1.000	x a_{1}
2501.06662_FO0014	2	1.000	p_{x a}
2501.06662_FO0015	2	1.000	a_{2}
2501.06662_FO0016	2	1.000	a_{2}=\dagger
2501.06662_FO0017	2	1.000	a a_{1}
2501.06662_FO0018	2	1.000	\pi
2501.06662_FO0019	3	1.000	\pi(x \mid x)=1
2501.06662_FO0020	3	1.000	\pi(-\mid x)
2501.06662_FO0021	3	1.000	T(x)
2501.06662_FO0022	3	0.999	[0, \infty]
2501.06662_FO0023	3	0.999	-\ln \pi(y \mid x)
2501.06662_FO0024	3	0.999	([0, \infty], \geq,+, 0)
2501.06662_FO0025	3	1.000	\mathcal{M}
2501.06662_FO0026	3	1.000	x, y, \ldots
2501.06662_FO0027	3	1.000	d(x, y)=-\ln \pi(y \mid x)
2501.06662_FO0028	3	1.000	\operatorname{Mag}(t \mathcal{M})
2501.06662_FO0029	3	1.000	t>0
2501.06662_FO0030	3	0.984	\#(T(\perp))
2501.06662_FO0031	4	1.000	H_{t}
2501.06662_FO0032	4	1.000	H_{t}\left(p_{1}, \ldots, p_{m}\right)=\left(1-\sum_{i=1}^{m} p_{i}^{t}\right) /(t-1)
2501.06662_FO0033	4	1.000	\left(H_{t}\right)_{t \in(0, \infty) \backslash\{1\}}
2501.06662_FO0034	4	1.000	t>1
2501.06662_FO0035	4	1.000	H_{t}\left(p_{x}\right)
2501.06662_FO0036	4	1.000	\left(n^{1-t}-1\right) /(1-t)
2501.06662_FO0037	4	1.000	t \rightarrow \infty
2501.06662_FO0038	4	1.000	\mathcal{M}_{1}
2501.06662_FO0039	4	1.000	\mathcal{M}_{2}
2501.06662_FO0040	4	0.405	p_{\bullet}{ }^{(1)}
2501.06662_FO0041	4	0.405	p_{\bullet}^{(2)}
2501.06662_FO0042	4	0.405	x^{\prime}
2501.06662_FO0043	4	1.000	H_{t}\left(p_{x^{\prime}}^{(1)}\right)>H_{t}\left(p_{x}^{(2)}\right)
2501.06662_FO0044	4	1.000	\operatorname{Mag}\left(t \mathcal{M}_{1}\right)>\operatorname{Mag}\left(t \mathcal{M}_{2}\right)
2501.06662_FO0045	4	1.000	t<1
2501.06662_FO0046	4	0.999	\lim _{t \rightarrow 0} \operatorname{Mag}(t \mathcal{M})=(1-n) \cdot \#(\operatorname{ob}(\mathcal{M}) \backslash T(\perp))+\#(T(\perp))
2501.06662_FO0047	4	0.696	n>1
2501.06662_FO0048	4	0.999	t=1
2501.06662_FO0049	4	0.896	\sum_{x \in \operatorname{ob}(\mathcal{M}) \backslash T(\perp)} H\left(p_{x}\right)
2501.06662_FO0050	4	0.896	\mathcal{M}_{x}
2501.06662_FO0051	4	0.985	\left(\sum_{y} \mu(x, y)\right)_{x}
2501.06662_FO0052	5	1.000	\mu
2501.06662_FO0053	5	1.000	\ell
2501.06662_FO0054	5	0.985	\mathcal{L}
2501.06662_FO0055	5	0.977	-\ln :[0,1] \rightarrow[0, \infty]
2501.06662_FO0056	6	1.000	A
2501.06662_FO0057	6	1.000	a, b, \ldots
2501.06662_FO0058	6	1.000	A^{*}
2501.06662_FO0059	6	0.999	a \leq b
2501.06662_FO0060	6	0.999	b
2501.06662_FO0061	6	1.000	b=a a^{\prime}
2501.06662_FO0062	6	1.000	a^{\prime} \in A^{*}
2501.06662_FO0063	6	1.000	\leq
2501.06662_FO0064	6	1.000	\epsilon
2501.06662_FO0065	6	1.000	\|a\|
2501.06662_FO0066	6	1.000	a \in A^{*}
2501.06662_FO0067	6	1.000	(P, \leq)
2501.06662_FO0068	6	1.000	P
2501.06662_FO0069	6	1.000	x \rightarrow y
2501.06662_FO0070	6	1.000	x \leq y
2501.06662_FO0071	6	1.000	N \in \mathbb{N}
2501.06662_FO0072	6	1.000	\mathrm{L}:=\mathrm{L}^{\leq N}
2501.06662_FO0073	6	1.000	\left((A \cup\{\perp, \dagger\})^{*}, \leq\right)
2501.06662_FO0074	6	1.000	N-1
2501.06662_FO0075	6	0.803	x, y
2501.06662_FO0076	6	1.000	(A \cup\{\perp, \dagger\})^{*}
2501.06662_FO0077	6	0.918	\perp a
2501.06662_FO0078	6	0.918	\perp a \dagger
2501.06662_FO0079	6	0.999	\|x\|
2501.06662_FO0080	6	1.000	\|\perp\|=1
2501.06662_FO0081	6	0.981	\operatorname{ob}(\mathrm{L})
2501.06662_FO0082	6	0.981	\bigsqcup_{j=0}^{N-1} \mathrm{~L}^{(j)}
2501.06662_FO0083	6	0.981	\mathrm{L}^{(j)}
2501.06662_FO0084	6	0.998	1+j
2501.06662_FO0085	6	0.998	j
2501.06662_FO0086	6	0.990	L^{(j)}
2501.06662_FO0087	6	0.996	\mathrm{L}^{(i)}
2501.06662_FO0088	6	0.996	j>i
2501.06662_FO0089	6	0.881	\mathrm{L}_{x}
2501.06662_FO0090	6	0.881	y \in \mathrm{~L}
2501.06662_FO0091	6	0.993	\operatorname{ob}\left(\mathrm{L}_{x}\right)=\bigsqcup_{j=0}^{N-\|x\|} \mathrm{L}_{x}^{(j)}
2501.06662_FO0092	6	0.993	\mathrm{L}_{x}^{(j)}
2501.06662_FO0093	6	1.000	\|x\|+j
2501.06662_FO0094	6	1.000	\mathrm{L}_{x}^{(0)}=\{x\}
2501.06662_FO0095	6	1.000	\mathrm{L}_{x}^{(1)}=\left\{x a_{1}: a_{1} \in A \cup\{\dagger\}\right\}
2501.06662_FO0096	6	1.000	\mathrm{L}_{x}^{(2)}=\left\{x a_{1} a_{2}: a_{1} \in A, a_{2} \in A \cup\{\dagger\}\right\}
2501.06662_FO0097	6	1.000	\mathrm{L}_{\perp}^{(j)}=\mathrm{L}^{(j)}
2501.06662_FO0098	6	1.000	j \geq 0
2501.06662_FO0099	6	0.843	\top
2501.06662_FO0100	6	0.843	\mathbf{L}
2501.06662_FO0101	7	1.000	p_{x}:=p(-\mid x): A \cup\{\dagger\} \rightarrow[0,1]
2501.06662_FO0102	7	1.000	a_{1}=\dagger
2501.06662_FO0103	7	1.000	p_{\perp a a_{1}}
2501.06662_FO0104	7	1.000	A \cup\{\dagger\}
2501.06662_FO0105	7	0.827	p_{\perp a a_{1} a_{2}}
2501.06662_FO0106	7	1.000	y=x a_{1} \cdots a_{N-\|x\|}
2501.06662_FO0107	7	1.000	N
2501.06662_FO0108	7	1.000	a=a(x) \in A \cup\{\dagger\}
2501.06662_FO0109	7	1.000	p_{x}(a)=1
2501.06662_FO0110	7	1.000	p_{x}\left(a^{\prime}\right)=0
2501.06662_FO0111	7	1.000	\left.a^{\prime} \in(A \cup\{\dagger\}) \backslash\{a\}\right)
2501.06662_FO0112	7	0.720	p_{\perp a_{1} \cdots a_{n}}(-)
2501.06662_FO0113	7	0.720	a_{n}
2501.06662_FO0114	7	0.720	n \geq 1
2501.06662_FO0115	7	1.000	\tilde{A}=A \cup\{\dagger\}
2501.06662_FO0116	7	1.000	P_{n}\left(a_{n}, a_{n+1}\right):=p_{\perp a_{1} \cdots a_{n}}\left(a_{n+1}\right)
2501.06662_FO0117	7	1.000	P_{n}: \tilde{A} \times \tilde{A} \rightarrow[0,1]
2501.06662_FO0118	7	0.998	\sum_{a^{\prime} \in \tilde{A}} P_{n}\left(a, a^{\prime}\right)=1
2501.06662_FO0119	7	0.971	P_{n}(\dagger, \dagger)=1
2501.06662_FO0120	7	0.971	P_{n}(\dagger, a)=0
2501.06662_FO0121	7	0.971	a \in A
2501.06662_FO0122	7	1.000	n>2
2501.06662_FO0123	7	1.000	P_{n}=P_{1}
2501.06662_FO0124	7	1.000	a^{\prime} \in A
2501.06662_FO0125	7	1.000	P_{n}(a, \dagger)=0
2501.06662_FO0126	7	1.000	\left.P_{n}\right\|_{A \times A}
2501.06662_FO0127	7	1.000	\left(P_{n}\right)_{n \geq 1}
2501.06662_FO0128	7	1.000	q: A \rightarrow[0,1]
2501.06662_FO0129	7	1.000	\sum_{a^{\prime} \in A} q(a) P\left(a, a^{\prime}\right)=q\left(a^{\prime}\right)
2501.06662_FO0130	7	1.000	p_{\perp}(a)=q(a)
2501.06662_FO0131	8	1.000	n-1
2501.06662_FO0132	8	0.522	p_{\perp a_{1} \cdots a_{i}}
2501.06662_FO0133	8	0.522	\perp a_{1} \cdots a_{i}
2501.06662_FO0134	8	1.000	i<n-2
2501.06662_FO0135	8	1.000	p^{(\theta)}: A^{*} \rightarrow \Delta(A), \quad a \mapsto p_{\perp a}^{(\theta)}(-)
2501.06662_FO0136	8	1.000	\Delta(A)
2501.06662_FO0137	8	1.000	\theta
2501.06662_FO0138	8	0.998	\left(a_{1}, \ldots, a_{n-1}\right)
2501.06662_FO0139	8	0.998	\left(a_{1}, \ldots, a_{n}\right)
2501.06662_FO0140	8	0.998	C \subset A^{*}
2501.06662_FO0141	8	1.000	y=x a^{\prime}
2501.06662_FO0142	8	1.000	\left\|a^{\prime}\right\|=N-\|x\|
2501.06662_FO0143	8	1.000	a a^{\prime}
2501.06662_FO0144	8	1.000	y=x a^{\prime \prime} \dagger
2501.06662_FO0145	8	1.000	a^{\prime \prime} \in A^{*}
2501.06662_FO0146	8	1.000	\left\|a^{\prime \prime}\right\| \leq N-\|x\|-1
2501.06662_FO0147	8	1.000	a a^{\prime \prime}
2501.06662_FO0148	8	1.000	N>5
2501.06662_FO0149	8	1.000	b=
2501.06662_FO0150	8	1.000	a_{1} a_{2} a_{3} a_{4} a_{5} \in A^{*}
2501.06662_FO0151	8	1.000	a=a_{1} a_{2}
2501.06662_FO0152	8	1.000	f(x)
2501.06662_FO0153	8	1.000	\|f(x)\| \leq N-1
2501.06662_FO0154	8	1.000	p_{f(x)}
2501.06662_FO0155	9	1.000	x=\perp a_{1} \cdots a_{t}
2501.06662_FO0156	9	1.000	y=x a_{t+1} \cdots a_{t+k}
2501.06662_FO0157	9	1.000	k \geq 1
2501.06662_FO0158	9	0.977	\left(a_{i}\right)_{i=1}^{t+k-1} \subset A
2501.06662_FO0159	9	0.977	a_{t+k} \in A \cup\{\dagger\}
2501.06662_FO0160	9	0.977	y_{<t+i}
2501.06662_FO0161	9	0.977	\perp a_{1} \cdots a_{t+i-1}
2501.06662_FO0162	9	1.000	y_{<t+1}=x
2501.06662_FO0163	9	1.000	m=0
2501.06662_FO0164	9	1.000	T(x)=\{x\}
2501.06662_FO0165	9	1.000	m=1
2501.06662_FO0166	9	1.000	T(x)=\{x a \mid a \in A \cup\{\dagger\}\}
2501.06662_FO0167	9	1.000	m \geq 1
2501.06662_FO0168	9	1.000	a \in A^{m}
2501.06662_FO0169	9	1.000	\left(a^{\prime}, a^{\prime \prime}\right) \in A^{m-1} \times A
2501.06662_FO0170	9	1.000	i=m-1
2501.06662_FO0171	9	1.000	\pi\left(x a^{\prime} \mid x\right) p\left(\dagger \mid x a^{\prime}\right)
2501.06662_FO0172	9	1.000	a^{\prime} \in A^{m-1}
2501.06662_FO0174	10	0.561	(\mathcal{V}, \leq, \otimes, 1)
2501.06662_FO0175	10	1.000	(\mathcal{V}, \leq)
2501.06662_FO0176	10	1.000	(\mathcal{V}, \otimes, 1)
2501.06662_FO0177	10	1.000	x \otimes y \leq x^{\prime} \otimes y^{\prime}
2501.06662_FO0178	10	1.000	x \leq x^{\prime}
2501.06662_FO0179	10	1.000	y \leq y^{\prime}
2501.06662_FO0180	10	0.959	([0,1], \leq, \cdot, 1)
2501.06662_FO0181	10	0.554	a b:=a \cdot b
2501.06662_FO0182	10	0.554	a, b \in[0,1]
2501.06662_FO0183	10	0.842	[0, \infty], \geq,+, 0
2501.06662_FO0184	10	1.000	a+\infty:=\infty
2501.06662_FO0185	10	1.000	\infty+a:=\infty
2501.06662_FO0186	10	1.000	a \in[0, \infty]
2501.06662_FO0187	10	1.000	\mathcal{V}
2501.06662_FO0188	10	1.000	\mathcal{C}
2501.06662_FO0189	10	1.000	\operatorname{ob}(\mathcal{C})
2501.06662_FO0190	10	1.000	\mathcal{C}(x, y)
2501.06662_FO0191	10	0.998	x, y, z \in \operatorname{ob}(\mathcal{C})
2501.06662_FO0192	10	1.000	\pi(y \mid x)=0
2501.06662_FO0193	10	1.000	x=y
2501.06662_FO0194	10	0.803	\mathcal{L}(x, y):=\pi(y \mid x)
2501.06662_FO0195	11	0.939	x, y, z
2501.06662_FO0196	11	1.000	x \rightarrow y \rightarrow z
2501.06662_FO0197	11	1.000	x \neq y
2501.06662_FO0198	11	1.000	y \neq z
2501.06662_FO0199	11	1.000	\left(a_{i}\right)_{i=1}^{t+k+k^{\prime}-1} \subset A
2501.06662_FO0200	11	1.000	a_{t+k+k^{\prime}} \in A \cup\{\dagger\}
2501.06662_FO0201	11	1.000	\pi(y \mid x)=1
2501.06662_FO0202	11	1.000	\pi(z \mid y)=\pi(z \mid x)
2501.06662_FO0203	11	1.000	z=y
2501.06662_FO0204	11	0.593	z
2501.06662_FO0205	11	0.873	\pi(y \mid x) \pi(z \mid y) \leq \pi(z \mid x)
2501.06662_FO0206	11	0.873	\pi(y \mid x), \pi(z \mid x) \neq 0
2501.06662_FO0207	11	0.873	\pi(z \mid y)=0,{ }^{3}
2501.06662_FO0208	11	1.000	\pi(z \mid y), \pi(z \mid x) \neq 0
2501.06662_FO0209	11	1.000	\pi(y \mid x)=0 .{ }^{4}
2501.06662_FO0210	11	0.985	[0,1]
2501.06662_FO0211	11	1.000	\mathcal{L}(x, y)
2501.06662_FO0212	11	1.000	x=\perp
2501.06662_FO0213	11	1.000	y=\perp
2501.06662_FO0214	11	1.000	z=\perp
2501.06662_FO0215	12	1.000	d(x, y)+d(y, z) \geq d(x, z)
2501.06662_FO0216	12	1.000	d(x, x)=0
2501.06662_FO0217	12	0.444	\infty
2501.06662_FO0218	12	1.000	\mathcal{M}(x, y):=d(x, y)
2501.06662_FO0219	12	1.000	\mathrm{L}^{\leq N}
2501.06662_FO0220	12	1.000	R
2501.06662_FO0221	12	1.000	\\|-\\|: \operatorname{ob}(\mathcal{V}) \rightarrow R
2501.06662_FO0222	12	1.000	\\|v\\|=\\|w\\|
2501.06662_FO0223	12	0.981	v \cong w
2501.06662_FO0224	12	0.981	\\|1\\|=1
2501.06662_FO0225	12	0.981	\\|v \otimes w\\|=\\|v\\|\\|w\\|
2501.06662_FO0226	12	0.981	v, w \in \operatorname{ob}(\mathcal{V})
2501.06662_FO0227	12	1.000	\zeta_{\mathcal{C}}: \operatorname{ob}(\mathcal{C}) \times \operatorname{ob}(\mathcal{C}) \rightarrow R
2501.06662_FO0228	12	1.000	\zeta_{\mathcal{C}}(x, y):=\\|\mathcal{C}(x, y)\\|
2501.06662_FO0229	12	1.000	\zeta_{\mathcal{C}}
2501.06662_FO0230	12	1.000	\mu_{\mathcal{C}}:=\zeta_{\mathcal{C}}^{-1}
2501.06662_FO0231	12	1.000	\operatorname{Mag}(\mathcal{C})
2501.06662_FO0232	13	1.000	\mathcal{V}=[0, \infty]
2501.06662_FO0233	13	1.000	\\|x\\|_{t}=e^{-t x}
2501.06662_FO0234	13	1.000	t \in \mathbb{R} \cup\{\infty\}
2501.06662_FO0235	13	1.000	\\|-\\|_{1}
2501.06662_FO0236	13	1.000	f(t):=\operatorname{Mag}(t \mathcal{C})
2501.06662_FO0237	13	1.000	t \in(0, \infty)
2501.06662_FO0238	13	1.000	t \mathcal{C}
2501.06662_FO0239	13	1.000	[s, t]=\{u \in P: s \leq u \leq t\}
2501.06662_FO0240	13	1.000	s \leq t
2501.06662_FO0241	13	1.000	\mathbb{k}
2501.06662_FO0242	13	1.000	\operatorname{int}(P)
2501.06662_FO0243	13	1.000	I(P, \mathbb{k})
2501.06662_FO0244	13	1.000	f: \operatorname{int}(P) \rightarrow \mathbb{k}
2501.06662_FO0245	13	0.978	f(s, t)
2501.06662_FO0246	13	0.978	f([s, t])
2501.06662_FO0247	13	0.996	\delta: \operatorname{int}(P) \rightarrow \mathbb{k}
2501.06662_FO0248	13	1.000	f * \delta=\delta * f=f
2501.06662_FO0249	13	1.000	f \in I(P, \mathbb{k})
2501.06662_FO0250	13	1.000	\zeta_{P}
2501.06662_FO0251	13	0.754	\mathbf{2}:=\{
2501.06662_FO0252	13	0.754	\}
2501.06662_FO0253	13	1.000	s, u \in P
2501.06662_FO0254	13	1.000	s \leq u
2501.06662_FO0255	13	1.000	P(s, u)
2501.06662_FO0256	13	1.000	s \not \leq u
2501.06662_FO0257	13	0.999	\\|-\\|: \operatorname{ob}(\mathbf{2}) \rightarrow \mathbb{Z}
2501.06662_FO0258	13	0.999	\\|
2501.06662_FO0259	13	0.999	\\|=1
2501.06662_FO0260	13	0.999	\\|=0
2501.06662_FO0261	13	1.000	\zeta_{P}(s, u)=\\|P(s, u)\\|
2501.06662_FO0262	13	1.000	\mu_{P}
2501.06662_FO0263	14	0.576	\mu_{\mathrm{L}}
2501.06662_FO0264	14	0.997	x \in \operatorname{ob}(\mathrm{~L})
2501.06662_FO0265	14	0.997	a_{1}, a_{2}, \ldots \in A \cup\{\dagger\}
2501.06662_FO0266	14	1.000	\mu_{\mathrm{L}}\left(x, x a_{1} a_{2} \cdots a_{j}\right)=0
2501.06662_FO0267	14	1.000	j>1
2501.06662_FO0268	14	0.677	\# A
2501.06662_FO0269	14	0.677	\#
2501.06662_FO0270	14	0.677	(\mathrm{L})
2501.06662_FO0271	15	0.585	\mathrm{L}^{\leq N}=\bigsqcup_{i=0}^{N-1} \mathrm{~L}^{(i)}
2501.06662_FO0272	15	0.585	\mathrm{L}^{(0)}=\{\perp\}
2501.06662_FO0273	15	0.508	i \leq 1, \mathrm{~L}^{(i)}
2501.06662_FO0274	15	0.508	a \in A^{i}
2501.06662_FO0275	15	0.508	\perp a^{\prime} \dagger
2501.06662_FO0276	15	1.000	a^{\prime} \in A^{i-1}
2501.06662_FO0277	15	1.000	\# \mathrm{~L}^{(i)}=\# A^{i}+\# A^{i-1}
2501.06662_FO0278	15	0.998	\operatorname{Mag}(\mathrm{L})=1
2501.06662_FO0279	15	0.999	\operatorname{ob}(\mathcal{M})=\operatorname{ob}\left(\mathrm{L}^{\leq N}\right)
2501.06662_FO0280	15	1.000	\mathcal{M}(x, y)=d(x, y)=-\ln \pi(y \mid x)
2501.06662_FO0281	15	1.000	\zeta_{\mathcal{M}}: \operatorname{ob}(\mathcal{M}) \times \operatorname{ob}(\mathcal{M}) \rightarrow \mathbb{R}
2501.06662_FO0282	15	1.000	d(x, y)=\infty
2501.06662_FO0283	15	1.000	\zeta_{t}
2501.06662_FO0284	15	1.000	\left(\zeta_{\mathcal{M}}\right)_{t}
2501.06662_FO0285	15	1.000	\zeta_{t}^{-1}
2501.06662_FO0286	15	0.988	\operatorname{ob}(\mathcal{M})
2501.06662_FO0287	15	1.000	x=y_{0} \rightarrow y_{1} \rightarrow \cdots \rightarrow y_{k}=y
2501.06662_FO0288	16	1.000	\delta
2501.06662_FO0289	16	1.000	\delta(x, y)=1
2501.06662_FO0290	16	1.000	\delta(x, y)=0
2501.06662_FO0291	16	0.997	\zeta-\delta
2501.06662_FO0292	16	0.997	D
2501.06662_FO0293	16	0.412	E=\left\{\left(z, z^{\prime}\right) \mid z \rightarrow z^{\prime}\right.
2501.06662_FO0294	16	0.412	\left.\mathrm{L}, z \neq z^{\prime}\right\}
2501.06662_FO0295	16	1.000	\left(\zeta_{t}-\delta\right)^{k}
2501.06662_FO0296	16	0.672	k
2501.06662_FO0297	16	0.672	E
2501.06662_FO0298	16	0.990	(x, y)
2501.06662_FO0299	16	1.000	\zeta_{\mathrm{L}}^{-1}
2501.06662_FO0300	16	1.000	\zeta_{\mathrm{L}}
2501.06662_FO0301	17	0.998	k \geq 0
2501.06662_FO0302	17	1.000	\zeta_{t}^{-1}(x, y)
2501.06662_FO0303	17	0.999	\pi(y \mid x)^{t} \zeta_{\mathrm{L}}^{-1}(x, y)
2501.06662_FO0304	17	1.000	\zeta_{\mathrm{L}}^{-1}(x, y)=\mu_{\mathrm{L}}(x, y)
2501.06662_FO0305	17	0.988	f:(0, \infty) \rightarrow \mathbb{R}
2501.06662_FO0306	17	0.907	f(t)=\operatorname{Mag}(t \mathcal{M})
2501.06662_FO0307	17	1.000	p=\left(p_{1}, \ldots, p_{n}\right)
2501.06662_FO0308	17	0.885	\left(p_{x}: A \cup\{\dagger\} \rightarrow[0,1]\right)_{x \in \operatorname{ob}(\mathrm{~L})}
2501.06662_FO0309	17	1.000	T(\perp)
2501.06662_FO0310	17	0.998	\operatorname{Mag}(t \mathcal{M})=\sum_{x, y \in \operatorname{ob}(\mathcal{M})} \pi(y \mid x)^{t} \zeta_{\mathrm{L}}^{-1}(x, y)
2501.06662_FO0311	18	1.000	y \neq x
2501.06662_FO0312	18	1.000	\operatorname{ob}(\mathcal{M}) \backslash T(\perp)
2501.06662_FO0313	18	0.996	\operatorname{ob}(\mathcal{M})=T(\perp) \sqcup(\operatorname{ob}(\mathcal{M}) \backslash T(\perp))
2501.06662_FO0314	18	0.996	\zeta_{t}^{-1}(x, x)=1
2501.06662_FO0315	18	0.998	x \in \operatorname{ob}(\mathcal{M})
2501.06662_FO0316	18	1.000	t \neq 1
2501.06662_FO0317	18	1.000	(t-1) /(t-1)
2501.06662_FO0318	18	1.000	t \geq 1
2501.06662_FO0319	18	1.000	p: S \rightarrow[0,1]
2501.06662_FO0320	18	1.000	S
2501.06662_FO0321	18	1.000	H(p)
2501.06662_FO0322	18	1.000	p
2501.06662_FO0323	19	1.000	E_{x}(a)=
2501.06662_FO0324	19	1.000	-\ln p_{x}(a)=d(x, x a)
2501.06662_FO0325	19	1.000	f(t)=
2501.06662_FO0326	19	1.000	\#(\operatorname{ob}(\mathcal{M}))-\tilde{Z}(t)
2501.06662_FO0327	19	1.000	\tilde{Z}
2501.06662_FO0328	19	1.000	(x, a) \in
2501.06662_FO0329	19	0.884	S:=(\operatorname{ob}(\mathcal{M}) \backslash T(\perp)) \times(A \cup\{\dagger\})
2501.06662_FO0330	19	0.884	E(x, a)=d(x, a)=-\ln p_{x}(a)
2501.06662_FO0331	19	1.000	\rho
2501.06662_FO0332	19	1.000	\rho(s)=
2501.06662_FO0333	19	1.000	e^{-t E(s)} / \tilde{Z}(t)
2501.06662_FO0334	19	1.000	y, z \in \operatorname{ob}\left(\mathcal{M}_{x}\right)
2501.06662_FO0335	19	1.000	\mathcal{M}_{x}(y, z):=\mathcal{M}(y, z)=-\ln \pi(z \mid y)
2501.06662_FO0336	20	0.644	M, d
2501.06662_FO0337	20	1.000	\ell \in[0, \infty)
2501.06662_FO0338	20	1.000	\left(M C_{k, \ell}(M)\right)_{k \in \mathbb{N}}
2501.06662_FO0339	20	1.000	M C_{k, \ell}(M):=\mathbb{Z}\left[G_{k, \ell}\right]
2501.06662_FO0340	20	1.000	\partial_{k}: M C_{k, \ell}(M) \rightarrow M C_{k-1, \ell}(M)
2501.06662_FO0341	20	1.000	0<i<k
2501.06662_FO0342	20	0.704	\partial_{k}^{0}=\partial_{k}^{k}=0
2501.06662_FO0343	20	0.704	[0, \infty)
2501.06662_FO0344	20	0.704	\left(M C_{\bullet, \ell}(M), \partial_{\bullet}\right)
2501.06662_FO0345	20	1.000	M
2501.06662_FO0346	20	0.892	H_{\bullet \bullet}^{\bullet}(M)
2501.06662_FO0347	20	0.929	\left(M C_{\bullet}, \ell_{\ell}\right)_{\ell}
2501.06662_FO0348	21	1.000	q=e^{-t}
2501.06662_FO0349	21	1.000	\bigcup_{k \geq 0} M C_{k, \ell}(\mathcal{M}) \neq
2501.06662_FO0350	21	1.000	\emptyset
2501.06662_FO0351	21	1.000	x=y_{0} \rightarrow
2501.06662_FO0352	21	1.000	y_{1} \rightarrow \cdots \rightarrow y_{k}=y
2501.06662_FO0353	21	1.000	\prod_{i=1}^{k} \pi\left(y_{i} \mid y_{i-1}\right) \neq 0
2501.06662_FO0354	21	1.000	d\left(y_{i}, y_{i+1}\right)<\infty
2501.06662_FO0355	21	1.000	i=0, \ldots, k-1
2501.06662_FO0356	21	1.000	\ell:=\sum_{i=0}^{k-1} d\left(y_{i}, y_{i+1}\right)<\infty
2501.06662_FO0357	21	0.996	\left(y_{0}, \ldots, y_{k}\right)
2501.06662_FO0358	21	0.996	G_{k, \ell}
2501.06662_FO0359	21	0.996	\prod_{i=1}^{k} \pi\left(y_{i} \mid y_{i-1}\right)^{t}=\exp (-t \ell)
2501.06662_FO0360	21	0.996	M C_{k, \ell}
2501.06662_FO0361	21	0.996	\left(M C_{\bullet, \ell}\right)_{\ell}
2501.06662_FO0362	22	1.000	H_{0,0}(\mathcal{M})
2501.06662_FO0363	22	1.000	\ell>0
2501.06662_FO0364	22	1.000	H_{0, \ell}(\mathcal{M})=0
2501.06662_FO0365	22	1.000	H_{1, \ell}(\mathcal{M})
2501.06662_FO0366	22	1.000	d(x, y)=\ell
2501.06662_FO0367	22	1.000	y=a_{0} a_{1} \cdots a_{n}
2501.06662_FO0368	22	1.000	p\left(-\mid y_{<i}\right)
2501.06662_FO0369	22	1.000	y_{<i}
2501.06662_FO0370	22	1.000	\zeta_{t}\left(a_{0}, y\right)
2501.06662_FO0371	22	1.000	t=1 / n
2501.06662_FO0372	22	1.000	a_{0}
2501.06662_FO0373	22	1.000	\zeta_{t}\left(a_{0}, y\right)=1 / P P L(y)
2501.06662_FO0374	22	1.000	\zeta_{t}(x, y)
2501.06662_FO0375	22	1.000	\widehat{\mathcal{M}}:=[0, \infty]^{\mathcal{M}}
2501.06662_FO0376	22	1.000	f: \mathcal{M} \rightarrow[0, \infty]
2501.06662_FO0377	22	1.000	\max \{f(y)-f(x), 0\} \leq \mathcal{M}(x, y)
2501.06662_FO0378	22	1.000	\widehat{\mathcal{M}}
2501.06662_FO0379	23	0.916	p: \operatorname{ob}(\mathrm{A}) \rightarrow[0,1]
2501.06662_FO0380	23	0.916	\theta: \operatorname{ob}(\mathrm{A}) \times \operatorname{ob}(\mathrm{A}) \rightarrow[0, \infty)
2501.06662_FO0381	23	1.000	\theta=\zeta_{t}
2501.06662_FO0382	23	1.000	p=\left.\pi(-\mid x)\right\|_{T(x)}
2501.06662_FO0383	23	0.999	\left.\pi(-\mid x)\right\|_{T(x)}
2501.06662_FO0384	23	1.000	P: A \times A \rightarrow[0,1]
2501.06662_FO0385	23	1.000	p_{\perp}
2501.06662_FO0386	24	1.000	{ }^{\sim}

Identifier

Page

Conf.

LaTeX source

Rendered

Image

2501.06662_FO0001

1.000

2501.06662_FO0002

1.000

\pi(y \mid x) \in[0,1]

2501.06662_FO0003

1.000

2501.06662_FO0004

1.000

2501.06662_FO0005

1.000

\pi(y \mid x)

2501.06662_FO0006

1.000

2501.06662_FO0007

1.000

(n-1)

2501.06662_FO0008

0.528

\dagger

2501.06662_FO0009

1.000

2501.06662_FO0010

1.000

x=\perp a

2501.06662_FO0011

1.000

p_{x}

2501.06662_FO0012

0.844

a_{1}

2501.06662_FO0013

1.000

x a_{1}

2501.06662_FO0014

1.000

p_{x a}

2501.06662_FO0015

1.000

a_{2}

2501.06662_FO0016

1.000

a_{2}=\dagger

2501.06662_FO0017

1.000

a a_{1}

2501.06662_FO0018

1.000

\pi

2501.06662_FO0019

1.000

\pi(x \mid x)=1

2501.06662_FO0020

1.000

\pi(-\mid x)

2501.06662_FO0021

1.000

T(x)

2501.06662_FO0022

0.999

[0, \infty]

2501.06662_FO0023

0.999

-\ln \pi(y \mid x)

2501.06662_FO0024

0.999

([0, \infty], \geq,+, 0)

2501.06662_FO0025

1.000

\mathcal{M}

2501.06662_FO0026

1.000

x, y, \ldots

2501.06662_FO0027

1.000

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

2501.06662_FO0028

1.000

\operatorname{Mag}(t \mathcal{M})

2501.06662_FO0029

1.000

t>0

2501.06662_FO0030

0.984

\#(T(\perp))

2501.06662_FO0031

1.000

H_{t}

2501.06662_FO0032

1.000

H_{t}\left(p_{1}, \ldots, p_{m}\right)=\left(1-\sum_{i=1}^{m} p_{i}^{t}\right) /(t-1)

2501.06662_FO0033

1.000

\left(H_{t}\right)_{t \in(0, \infty) \backslash\{1\}}

2501.06662_FO0034

1.000

t>1

2501.06662_FO0035

1.000

H_{t}\left(p_{x}\right)

2501.06662_FO0036

1.000

\left(n^{1-t}-1\right) /(1-t)

2501.06662_FO0037

1.000

t \rightarrow \infty

2501.06662_FO0038

1.000

\mathcal{M}_{1}

2501.06662_FO0039

1.000

\mathcal{M}_{2}

2501.06662_FO0040

0.405

p_{\bullet}{ }^{(1)}

2501.06662_FO0041

0.405

p_{\bullet}^{(2)}

2501.06662_FO0042

0.405

x^{\prime}

2501.06662_FO0043

1.000

H_{t}\left(p_{x^{\prime}}^{(1)}\right)>H_{t}\left(p_{x}^{(2)}\right)

2501.06662_FO0044

1.000

\operatorname{Mag}\left(t \mathcal{M}_{1}\right)>\operatorname{Mag}\left(t \mathcal{M}_{2}\right)

2501.06662_FO0045

1.000

t<1

2501.06662_FO0046

0.999

\lim _{t \rightarrow 0} \operatorname{Mag}(t \mathcal{M})=(1-n) \cdot \#(\operatorname{ob}(\mathcal{M}) \backslash T(\perp))+\#(T(\perp))

2501.06662_FO0047

0.696

n>1

2501.06662_FO0048

0.999

t=1

2501.06662_FO0049

0.896

\sum_{x \in \operatorname{ob}(\mathcal{M}) \backslash T(\perp)} H\left(p_{x}\right)

2501.06662_FO0050

0.896

\mathcal{M}_{x}

2501.06662_FO0051

0.985

\left(\sum_{y} \mu(x, y)\right)_{x}

2501.06662_FO0052

1.000

\mu

2501.06662_FO0053

1.000

\ell

2501.06662_FO0054

0.985

\mathcal{L}

2501.06662_FO0055

0.977

-\ln :[0,1] \rightarrow[0, \infty]

2501.06662_FO0056

1.000

2501.06662_FO0057

1.000

a, b, \ldots

2501.06662_FO0058

1.000

A^{*}

2501.06662_FO0059

0.999

a \leq b

2501.06662_FO0060

0.999

2501.06662_FO0061

1.000

b=a a^{\prime}

2501.06662_FO0062

1.000

a^{\prime} \in A^{*}

2501.06662_FO0063

1.000

\leq

2501.06662_FO0064

1.000

\epsilon

2501.06662_FO0065

1.000

|a|

2501.06662_FO0066

1.000

a \in A^{*}

2501.06662_FO0067

1.000

(P, \leq)

2501.06662_FO0068

1.000

2501.06662_FO0069

1.000

x \rightarrow y

2501.06662_FO0070

1.000

x \leq y

2501.06662_FO0071

1.000

N \in \mathbb{N}

2501.06662_FO0072

1.000

\mathrm{L}:=\mathrm{L}^{\leq N}

2501.06662_FO0073

1.000

\left((A \cup\{\perp, \dagger\})^{*}, \leq\right)

2501.06662_FO0074

1.000

N-1

2501.06662_FO0075

0.803

x, y

2501.06662_FO0076

1.000

(A \cup\{\perp, \dagger\})^{*}

2501.06662_FO0077

0.918

\perp a

2501.06662_FO0078

0.918

\perp a \dagger

2501.06662_FO0079

0.999

|x|

2501.06662_FO0080

1.000

|\perp|=1

2501.06662_FO0081

0.981

\operatorname{ob}(\mathrm{L})

2501.06662_FO0082

0.981

\bigsqcup_{j=0}^{N-1} \mathrm{~L}^{(j)}

2501.06662_FO0083

0.981

\mathrm{L}^{(j)}

2501.06662_FO0084

0.998

1+j

2501.06662_FO0085

0.998

2501.06662_FO0086

0.990

L^{(j)}

2501.06662_FO0087

0.996

\mathrm{L}^{(i)}

2501.06662_FO0088

0.996

j>i

2501.06662_FO0089

0.881

\mathrm{L}_{x}

2501.06662_FO0090

0.881

y \in \mathrm{~L}

2501.06662_FO0091

0.993

\operatorname{ob}\left(\mathrm{L}_{x}\right)=\bigsqcup_{j=0}^{N-|x|} \mathrm{L}_{x}^{(j)}

2501.06662_FO0092

0.993

\mathrm{L}_{x}^{(j)}

2501.06662_FO0093

1.000

|x|+j

2501.06662_FO0094

1.000

\mathrm{L}_{x}^{(0)}=\{x\}

2501.06662_FO0095

1.000

\mathrm{L}_{x}^{(1)}=\left\{x a_{1}: a_{1} \in A \cup\{\dagger\}\right\}

2501.06662_FO0096

1.000

\mathrm{L}_{x}^{(2)}=\left\{x a_{1} a_{2}: a_{1} \in A, a_{2} \in A \cup\{\dagger\}\right\}

2501.06662_FO0097

1.000

\mathrm{L}_{\perp}^{(j)}=\mathrm{L}^{(j)}

2501.06662_FO0098

1.000

j \geq 0

2501.06662_FO0099

0.843

\top

2501.06662_FO0100

0.843

\mathbf{L}

2501.06662_FO0101

1.000

p_{x}:=p(-\mid x): A \cup\{\dagger\} \rightarrow[0,1]

2501.06662_FO0102

1.000

a_{1}=\dagger

2501.06662_FO0103

1.000

p_{\perp a a_{1}}

2501.06662_FO0104

1.000

A \cup\{\dagger\}

2501.06662_FO0105

0.827

p_{\perp a a_{1} a_{2}}

2501.06662_FO0106

1.000

y=x a_{1} \cdots a_{N-|x|}

2501.06662_FO0107

1.000

2501.06662_FO0108

1.000

a=a(x) \in A \cup\{\dagger\}

2501.06662_FO0109

1.000

p_{x}(a)=1

2501.06662_FO0110

1.000

p_{x}\left(a^{\prime}\right)=0

2501.06662_FO0111

1.000

\left.a^{\prime} \in(A \cup\{\dagger\}) \backslash\{a\}\right)

2501.06662_FO0112

0.720

p_{\perp a_{1} \cdots a_{n}}(-)

2501.06662_FO0113

0.720

a_{n}

2501.06662_FO0114

0.720

n \geq 1

2501.06662_FO0115

1.000

\tilde{A}=A \cup\{\dagger\}

2501.06662_FO0116

1.000

P_{n}\left(a_{n}, a_{n+1}\right):=p_{\perp a_{1} \cdots a_{n}}\left(a_{n+1}\right)

2501.06662_FO0117

1.000

P_{n}: \tilde{A} \times \tilde{A} \rightarrow[0,1]

2501.06662_FO0118

0.998

\sum_{a^{\prime} \in \tilde{A}} P_{n}\left(a, a^{\prime}\right)=1

2501.06662_FO0119

0.971

P_{n}(\dagger, \dagger)=1

2501.06662_FO0120

0.971

P_{n}(\dagger, a)=0

2501.06662_FO0121

0.971

a \in A

2501.06662_FO0122

1.000

n>2

2501.06662_FO0123

1.000

P_{n}=P_{1}

2501.06662_FO0124

1.000

a^{\prime} \in A

2501.06662_FO0125

1.000

P_{n}(a, \dagger)=0

2501.06662_FO0126

1.000

\left.P_{n}\right|_{A \times A}

2501.06662_FO0127

1.000

\left(P_{n}\right)_{n \geq 1}

2501.06662_FO0128

1.000

q: A \rightarrow[0,1]

2501.06662_FO0129

1.000

\sum_{a^{\prime} \in A} q(a) P\left(a, a^{\prime}\right)=q\left(a^{\prime}\right)

2501.06662_FO0130

1.000

p_{\perp}(a)=q(a)

2501.06662_FO0131

1.000

n-1

2501.06662_FO0132

0.522

p_{\perp a_{1} \cdots a_{i}}

2501.06662_FO0133

0.522

\perp a_{1} \cdots a_{i}

2501.06662_FO0134

1.000

i<n-2

2501.06662_FO0135

1.000

p^{(\theta)}: A^{*} \rightarrow \Delta(A), \quad a \mapsto p_{\perp a}^{(\theta)}(-)

2501.06662_FO0136

1.000

\Delta(A)

2501.06662_FO0137

1.000

\theta

2501.06662_FO0138

0.998

\left(a_{1}, \ldots, a_{n-1}\right)

2501.06662_FO0139

0.998

\left(a_{1}, \ldots, a_{n}\right)

2501.06662_FO0140

0.998

C \subset A^{*}

2501.06662_FO0141

1.000

y=x a^{\prime}

2501.06662_FO0142

1.000

\left|a^{\prime}\right|=N-|x|

2501.06662_FO0143

1.000

a a^{\prime}

2501.06662_FO0144

1.000

y=x a^{\prime \prime} \dagger

2501.06662_FO0145

1.000

a^{\prime \prime} \in A^{*}

2501.06662_FO0146

1.000

\left|a^{\prime \prime}\right| \leq N-|x|-1

2501.06662_FO0147

1.000

a a^{\prime \prime}

2501.06662_FO0148

1.000

N>5

2501.06662_FO0149

1.000

2501.06662_FO0150

1.000

a_{1} a_{2} a_{3} a_{4} a_{5} \in A^{*}

2501.06662_FO0151

1.000

a=a_{1} a_{2}

2501.06662_FO0152

1.000

f(x)

2501.06662_FO0153

1.000

|f(x)| \leq N-1

2501.06662_FO0154

1.000

p_{f(x)}

2501.06662_FO0155

1.000

x=\perp a_{1} \cdots a_{t}

2501.06662_FO0156

1.000

y=x a_{t+1} \cdots a_{t+k}

2501.06662_FO0157

1.000

k \geq 1

2501.06662_FO0158

0.977

\left(a_{i}\right)_{i=1}^{t+k-1} \subset A

2501.06662_FO0159

0.977

a_{t+k} \in A \cup\{\dagger\}

2501.06662_FO0160

0.977

y_{<t+i}

2501.06662_FO0161

0.977

\perp a_{1} \cdots a_{t+i-1}

2501.06662_FO0162

1.000

y_{<t+1}=x

2501.06662_FO0163

1.000

m=0

2501.06662_FO0164

1.000

T(x)=\{x\}

2501.06662_FO0165

1.000

m=1

2501.06662_FO0166

1.000

T(x)=\{x a \mid a \in A \cup\{\dagger\}\}

2501.06662_FO0167

1.000

m \geq 1

2501.06662_FO0168

1.000

a \in A^{m}

2501.06662_FO0169

1.000

\left(a^{\prime}, a^{\prime \prime}\right) \in A^{m-1} \times A

2501.06662_FO0170

1.000

i=m-1

2501.06662_FO0171

1.000

\pi\left(x a^{\prime} \mid x\right) p\left(\dagger \mid x a^{\prime}\right)

2501.06662_FO0172

1.000

a^{\prime} \in A^{m-1}

2501.06662_FO0174

0.561

(\mathcal{V}, \leq, \otimes, 1)

2501.06662_FO0175

1.000

(\mathcal{V}, \leq)

2501.06662_FO0176

1.000

(\mathcal{V}, \otimes, 1)

2501.06662_FO0177

1.000

x \otimes y \leq x^{\prime} \otimes y^{\prime}

2501.06662_FO0178

1.000

x \leq x^{\prime}

2501.06662_FO0179

1.000

y \leq y^{\prime}

2501.06662_FO0180

0.959

([0,1], \leq, \cdot, 1)

2501.06662_FO0181

0.554

a b:=a \cdot b

2501.06662_FO0182

0.554

a, b \in[0,1]

2501.06662_FO0183

0.842

[0, \infty], \geq,+, 0

2501.06662_FO0184

1.000

a+\infty:=\infty

2501.06662_FO0185

1.000

\infty+a:=\infty

2501.06662_FO0186

1.000

a \in[0, \infty]

2501.06662_FO0187

1.000

\mathcal{V}

2501.06662_FO0188

1.000

\mathcal{C}

2501.06662_FO0189

1.000

\operatorname{ob}(\mathcal{C})

2501.06662_FO0190

1.000

\mathcal{C}(x, y)

2501.06662_FO0191

0.998

x, y, z \in \operatorname{ob}(\mathcal{C})

2501.06662_FO0192

1.000

\pi(y \mid x)=0

2501.06662_FO0193

1.000

x=y

2501.06662_FO0194

0.803

\mathcal{L}(x, y):=\pi(y \mid x)

2501.06662_FO0195

0.939

x, y, z

2501.06662_FO0196

1.000

x \rightarrow y \rightarrow z

2501.06662_FO0197

1.000

x \neq y

2501.06662_FO0198

1.000

y \neq z

2501.06662_FO0199

1.000

\left(a_{i}\right)_{i=1}^{t+k+k^{\prime}-1} \subset A

2501.06662_FO0200

1.000

a_{t+k+k^{\prime}} \in A \cup\{\dagger\}

2501.06662_FO0201

1.000

\pi(y \mid x)=1

2501.06662_FO0202

1.000

\pi(z \mid y)=\pi(z \mid x)

2501.06662_FO0203

1.000

z=y

2501.06662_FO0204

0.593

2501.06662_FO0205

0.873

\pi(y \mid x) \pi(z \mid y) \leq \pi(z \mid x)

2501.06662_FO0206

0.873

\pi(y \mid x), \pi(z \mid x) \neq 0

2501.06662_FO0207

0.873

\pi(z \mid y)=0,{ }^{3}

2501.06662_FO0208

1.000

\pi(z \mid y), \pi(z \mid x) \neq 0

2501.06662_FO0209

1.000

\pi(y \mid x)=0 .{ }^{4}

2501.06662_FO0210

0.985

[0,1]

2501.06662_FO0211

1.000

\mathcal{L}(x, y)

2501.06662_FO0212

1.000

x=\perp

2501.06662_FO0213

1.000

y=\perp

2501.06662_FO0214

1.000

z=\perp

2501.06662_FO0215

1.000

d(x, y)+d(y, z) \geq d(x, z)

2501.06662_FO0216

1.000

d(x, x)=0

2501.06662_FO0217

0.444

\infty

2501.06662_FO0218

1.000

\mathcal{M}(x, y):=d(x, y)

2501.06662_FO0219

1.000

\mathrm{L}^{\leq N}

2501.06662_FO0220

1.000

2501.06662_FO0221

1.000

\|-\|: \operatorname{ob}(\mathcal{V}) \rightarrow R

2501.06662_FO0222

1.000

\|v\|=\|w\|

2501.06662_FO0223

0.981

v \cong w

2501.06662_FO0224

0.981

\|1\|=1

2501.06662_FO0225

0.981

\|v \otimes w\|=\|v\|\|w\|

2501.06662_FO0226

0.981

v, w \in \operatorname{ob}(\mathcal{V})

2501.06662_FO0227

1.000

\zeta_{\mathcal{C}}: \operatorname{ob}(\mathcal{C}) \times \operatorname{ob}(\mathcal{C}) \rightarrow R

2501.06662_FO0228

1.000

\zeta_{\mathcal{C}}(x, y):=\|\mathcal{C}(x, y)\|

2501.06662_FO0229

1.000

\zeta_{\mathcal{C}}

2501.06662_FO0230

1.000

\mu_{\mathcal{C}}:=\zeta_{\mathcal{C}}^{-1}

2501.06662_FO0231

1.000

\operatorname{Mag}(\mathcal{C})

2501.06662_FO0232

1.000

\mathcal{V}=[0, \infty]

2501.06662_FO0233

1.000

\|x\|_{t}=e^{-t x}

2501.06662_FO0234

1.000

t \in \mathbb{R} \cup\{\infty\}

2501.06662_FO0235

1.000

\|-\|_{1}

2501.06662_FO0236

1.000

f(t):=\operatorname{Mag}(t \mathcal{C})

2501.06662_FO0237

1.000

t \in(0, \infty)

2501.06662_FO0238

1.000

t \mathcal{C}

2501.06662_FO0239

1.000

[s, t]=\{u \in P: s \leq u \leq t\}

2501.06662_FO0240

1.000

s \leq t

2501.06662_FO0241

1.000

\mathbb{k}

2501.06662_FO0242

1.000

\operatorname{int}(P)

2501.06662_FO0243

1.000

I(P, \mathbb{k})

2501.06662_FO0244

1.000

f: \operatorname{int}(P) \rightarrow \mathbb{k}

2501.06662_FO0245

0.978

f(s, t)

2501.06662_FO0246

0.978

f([s, t])

2501.06662_FO0247

0.996

\delta: \operatorname{int}(P) \rightarrow \mathbb{k}

2501.06662_FO0248

1.000

f * \delta=\delta * f=f

2501.06662_FO0249

1.000

f \in I(P, \mathbb{k})

2501.06662_FO0250

1.000

\zeta_{P}

2501.06662_FO0251

0.754

\mathbf{2}:=\{

2501.06662_FO0252

0.754

2501.06662_FO0253

1.000

s, u \in P

2501.06662_FO0254

1.000

s \leq u

2501.06662_FO0255

1.000

P(s, u)

2501.06662_FO0256

1.000

s \not \leq u

2501.06662_FO0257

0.999

\|-\|: \operatorname{ob}(\mathbf{2}) \rightarrow \mathbb{Z}

2501.06662_FO0258

0.999

2501.06662_FO0259

0.999

\|=1

2501.06662_FO0260

0.999

\|=0

2501.06662_FO0261

1.000

\zeta_{P}(s, u)=\|P(s, u)\|

2501.06662_FO0262

1.000

\mu_{P}

2501.06662_FO0263

0.576

\mu_{\mathrm{L}}

2501.06662_FO0264

0.997

x \in \operatorname{ob}(\mathrm{~L})

2501.06662_FO0265

0.997

a_{1}, a_{2}, \ldots \in A \cup\{\dagger\}

2501.06662_FO0266

1.000

\mu_{\mathrm{L}}\left(x, x a_{1} a_{2} \cdots a_{j}\right)=0

2501.06662_FO0267

1.000

j>1

2501.06662_FO0268

0.677

\# A

2501.06662_FO0269

0.677

2501.06662_FO0270

0.677

(\mathrm{L})

2501.06662_FO0271

0.585

\mathrm{L}^{\leq N}=\bigsqcup_{i=0}^{N-1} \mathrm{~L}^{(i)}

2501.06662_FO0272

0.585

\mathrm{L}^{(0)}=\{\perp\}

2501.06662_FO0273

0.508

i \leq 1, \mathrm{~L}^{(i)}

2501.06662_FO0274

0.508

a \in A^{i}

2501.06662_FO0275

0.508

\perp a^{\prime} \dagger

2501.06662_FO0276

1.000

a^{\prime} \in A^{i-1}

2501.06662_FO0277

1.000

\# \mathrm{~L}^{(i)}=\# A^{i}+\# A^{i-1}

2501.06662_FO0278

0.998

\operatorname{Mag}(\mathrm{L})=1

2501.06662_FO0279

0.999

\operatorname{ob}(\mathcal{M})=\operatorname{ob}\left(\mathrm{L}^{\leq N}\right)

2501.06662_FO0280

1.000

\mathcal{M}(x, y)=d(x, y)=-\ln \pi(y \mid x)

2501.06662_FO0281

1.000

\zeta_{\mathcal{M}}: \operatorname{ob}(\mathcal{M}) \times \operatorname{ob}(\mathcal{M}) \rightarrow \mathbb{R}

2501.06662_FO0282

1.000

d(x, y)=\infty

2501.06662_FO0283

1.000

\zeta_{t}

2501.06662_FO0284

1.000

\left(\zeta_{\mathcal{M}}\right)_{t}

2501.06662_FO0285

1.000

\zeta_{t}^{-1}

2501.06662_FO0286

0.988

\operatorname{ob}(\mathcal{M})

2501.06662_FO0287

1.000

x=y_{0} \rightarrow y_{1} \rightarrow \cdots \rightarrow y_{k}=y

2501.06662_FO0288

1.000

\delta

2501.06662_FO0289

1.000

\delta(x, y)=1

2501.06662_FO0290

1.000

\delta(x, y)=0

2501.06662_FO0291

0.997

\zeta-\delta

2501.06662_FO0292

0.997

2501.06662_FO0293

0.412

E=\left\{\left(z, z^{\prime}\right) \mid z \rightarrow z^{\prime}\right.

2501.06662_FO0294

0.412

\left.\mathrm{L}, z \neq z^{\prime}\right\}

2501.06662_FO0295

1.000

\left(\zeta_{t}-\delta\right)^{k}

2501.06662_FO0296

0.672

2501.06662_FO0297

0.672

2501.06662_FO0298

0.990

(x, y)

2501.06662_FO0299

1.000

\zeta_{\mathrm{L}}^{-1}

2501.06662_FO0300

1.000

\zeta_{\mathrm{L}}

2501.06662_FO0301

0.998

k \geq 0

2501.06662_FO0302

1.000

\zeta_{t}^{-1}(x, y)

2501.06662_FO0303

0.999

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

2501.06662_FO0304

1.000

\zeta_{\mathrm{L}}^{-1}(x, y)=\mu_{\mathrm{L}}(x, y)

2501.06662_FO0305

0.988

f:(0, \infty) \rightarrow \mathbb{R}

2501.06662_FO0306

0.907

f(t)=\operatorname{Mag}(t \mathcal{M})

2501.06662_FO0307

1.000

p=\left(p_{1}, \ldots, p_{n}\right)

2501.06662_FO0308

0.885

\left(p_{x}: A \cup\{\dagger\} \rightarrow[0,1]\right)_{x \in \operatorname{ob}(\mathrm{~L})}

2501.06662_FO0309

1.000

T(\perp)

2501.06662_FO0310

0.998

\operatorname{Mag}(t \mathcal{M})=\sum_{x, y \in \operatorname{ob}(\mathcal{M})} \pi(y \mid x)^{t} \zeta_{\mathrm{L}}^{-1}(x, y)

2501.06662_FO0311

1.000

y \neq x

2501.06662_FO0312

1.000

\operatorname{ob}(\mathcal{M}) \backslash T(\perp)

2501.06662_FO0313

0.996

\operatorname{ob}(\mathcal{M})=T(\perp) \sqcup(\operatorname{ob}(\mathcal{M}) \backslash T(\perp))

2501.06662_FO0314

0.996

\zeta_{t}^{-1}(x, x)=1

2501.06662_FO0315

0.998

x \in \operatorname{ob}(\mathcal{M})

2501.06662_FO0316

1.000

t \neq 1

2501.06662_FO0317

1.000

(t-1) /(t-1)

2501.06662_FO0318

1.000

t \geq 1

2501.06662_FO0319

1.000

p: S \rightarrow[0,1]

2501.06662_FO0320

1.000

2501.06662_FO0321

1.000

H(p)

2501.06662_FO0322

1.000

2501.06662_FO0323

1.000

E_{x}(a)=

2501.06662_FO0324

1.000

-\ln p_{x}(a)=d(x, x a)

2501.06662_FO0325

1.000

f(t)=

2501.06662_FO0326

1.000

\#(\operatorname{ob}(\mathcal{M}))-\tilde{Z}(t)

2501.06662_FO0327

1.000

\tilde{Z}

2501.06662_FO0328

1.000

(x, a) \in

2501.06662_FO0329

0.884

S:=(\operatorname{ob}(\mathcal{M}) \backslash T(\perp)) \times(A \cup\{\dagger\})

2501.06662_FO0330

0.884

E(x, a)=d(x, a)=-\ln p_{x}(a)

2501.06662_FO0331

1.000

\rho

2501.06662_FO0332

1.000

\rho(s)=

2501.06662_FO0333

1.000

e^{-t E(s)} / \tilde{Z}(t)

2501.06662_FO0334

1.000

y, z \in \operatorname{ob}\left(\mathcal{M}_{x}\right)

2501.06662_FO0335

1.000

\mathcal{M}_{x}(y, z):=\mathcal{M}(y, z)=-\ln \pi(z \mid y)

2501.06662_FO0336

0.644

M, d

2501.06662_FO0337

1.000

\ell \in[0, \infty)

2501.06662_FO0338

1.000

\left(M C_{k, \ell}(M)\right)_{k \in \mathbb{N}}

2501.06662_FO0339

1.000

M C_{k, \ell}(M):=\mathbb{Z}\left[G_{k, \ell}\right]

2501.06662_FO0340

1.000

\partial_{k}: M C_{k, \ell}(M) \rightarrow M C_{k-1, \ell}(M)

2501.06662_FO0341

1.000

0<i<k

2501.06662_FO0342

0.704

\partial_{k}^{0}=\partial_{k}^{k}=0

2501.06662_FO0343

0.704

[0, \infty)

2501.06662_FO0344

0.704

\left(M C_{\bullet, \ell}(M), \partial_{\bullet}\right)

2501.06662_FO0345

1.000

2501.06662_FO0346

0.892

H_{\bullet \bullet}^{\bullet}(M)

2501.06662_FO0347

0.929

\left(M C_{\bullet}, \ell_{\ell}\right)_{\ell}

2501.06662_FO0348

1.000

q=e^{-t}

2501.06662_FO0349

1.000

\bigcup_{k \geq 0} M C_{k, \ell}(\mathcal{M}) \neq

2501.06662_FO0350

1.000

\emptyset

2501.06662_FO0351

1.000

x=y_{0} \rightarrow

2501.06662_FO0352

1.000

y_{1} \rightarrow \cdots \rightarrow y_{k}=y

2501.06662_FO0353

1.000

\prod_{i=1}^{k} \pi\left(y_{i} \mid y_{i-1}\right) \neq 0

2501.06662_FO0354

1.000

d\left(y_{i}, y_{i+1}\right)<\infty

2501.06662_FO0355

1.000

i=0, \ldots, k-1

2501.06662_FO0356

1.000

\ell:=\sum_{i=0}^{k-1} d\left(y_{i}, y_{i+1}\right)<\infty

2501.06662_FO0357

0.996

\left(y_{0}, \ldots, y_{k}\right)

2501.06662_FO0358

0.996

G_{k, \ell}

2501.06662_FO0359

0.996

\prod_{i=1}^{k} \pi\left(y_{i} \mid y_{i-1}\right)^{t}=\exp (-t \ell)

2501.06662_FO0360

0.996

M C_{k, \ell}

2501.06662_FO0361

0.996

\left(M C_{\bullet, \ell}\right)_{\ell}

2501.06662_FO0362

1.000

H_{0,0}(\mathcal{M})

2501.06662_FO0363

1.000

\ell>0

2501.06662_FO0364

1.000

H_{0, \ell}(\mathcal{M})=0

2501.06662_FO0365

1.000

H_{1, \ell}(\mathcal{M})

2501.06662_FO0366

1.000

d(x, y)=\ell

2501.06662_FO0367

1.000

y=a_{0} a_{1} \cdots a_{n}

2501.06662_FO0368

1.000

p\left(-\mid y_{<i}\right)

2501.06662_FO0369

1.000

y_{<i}

2501.06662_FO0370

1.000

\zeta_{t}\left(a_{0}, y\right)

2501.06662_FO0371

1.000

t=1 / n

2501.06662_FO0372

1.000

a_{0}

2501.06662_FO0373

1.000

\zeta_{t}\left(a_{0}, y\right)=1 / P P L(y)

2501.06662_FO0374

1.000

\zeta_{t}(x, y)

2501.06662_FO0375

1.000

\widehat{\mathcal{M}}:=[0, \infty]^{\mathcal{M}}

2501.06662_FO0376

1.000

f: \mathcal{M} \rightarrow[0, \infty]

2501.06662_FO0377

1.000

\max \{f(y)-f(x), 0\} \leq \mathcal{M}(x, y)

2501.06662_FO0378

1.000

\widehat{\mathcal{M}}

2501.06662_FO0379

0.916

p: \operatorname{ob}(\mathrm{A}) \rightarrow[0,1]

2501.06662_FO0380

0.916

\theta: \operatorname{ob}(\mathrm{A}) \times \operatorname{ob}(\mathrm{A}) \rightarrow[0, \infty)

2501.06662_FO0381

1.000

\theta=\zeta_{t}

2501.06662_FO0382

1.000

p=\left.\pi(-\mid x)\right|_{T(x)}

2501.06662_FO0383

0.999

\left.\pi(-\mid x)\right|_{T(x)}

2501.06662_FO0384

1.000

P: A \times A \rightarrow[0,1]

2501.06662_FO0385

1.000

p_{\perp}

2501.06662_FO0386

1.000

{ }^{\sim}