On the recognition problem for limits of entropy functions
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909775601598464 |
|---|---|
| author | Yashfe, Geva |
| author_facet | Yashfe, Geva |
| contents | We prove that there is no algorithm to decide whether a given integer vector is in the closure of the entropic cone $\overline{Γ_{n}^{*}}$. Equivalently, there is no decision procedure to determine whether a given integer-valued function $h:\mathcal{P}(\{1,\ldots,n\})\rightarrow\mathbb{Z}_{\ge 0}$ is a pointwise limit of joint entropy functions. In other words, given such an $h$, it is undecidable whether for all $\varepsilon > 0$ there exists a finite probability space $(Ω,P)$ with random variables $X_{1},\ldots,X_{n}$ such that their joint entropy $H$ satisfies $\max_{I\subseteq\{1,\ldots,n\}}\left|H\left(X_{I}\right)-h\left(I\right)\right|<\varepsilon$. This settles the last open case in a sequence of related undecidability results proved by L. Kühne and the author, with applications in algorithmic information theory. The main new tool is a Desargues'-type theorem for almost entropic polymatroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06302 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the recognition problem for limits of entropy functions Yashfe, Geva Combinatorics Information Theory 05B35, 68P30, 20F10 We prove that there is no algorithm to decide whether a given integer vector is in the closure of the entropic cone $\overline{Γ_{n}^{*}}$. Equivalently, there is no decision procedure to determine whether a given integer-valued function $h:\mathcal{P}(\{1,\ldots,n\})\rightarrow\mathbb{Z}_{\ge 0}$ is a pointwise limit of joint entropy functions. In other words, given such an $h$, it is undecidable whether for all $\varepsilon > 0$ there exists a finite probability space $(Ω,P)$ with random variables $X_{1},\ldots,X_{n}$ such that their joint entropy $H$ satisfies $\max_{I\subseteq\{1,\ldots,n\}}\left|H\left(X_{I}\right)-h\left(I\right)\right|<\varepsilon$. This settles the last open case in a sequence of related undecidability results proved by L. Kühne and the author, with applications in algorithmic information theory. The main new tool is a Desargues'-type theorem for almost entropic polymatroids. |
| title | On the recognition problem for limits of entropy functions |
| topic | Combinatorics Information Theory 05B35, 68P30, 20F10 |
| url | https://arxiv.org/abs/2509.06302 |