一句话：证明 Petri 网半线性 home-space 问题可判定，并给出 Ackermann 完全性。
本文研究 Petri 网的 home-space 问题：给定半线性配置集 X 与 H，判断从 X 可达的每个配置是否都能到达 H。此前仅在 X 为单点且 H 为具有相同周期的线性集有限并时已知可判定。作者证明一般半线性情形也可判定，核心是通过建立可达性问题与非 home-space 问题之间的对偶，对任意 Petri 网和半线性 H 有效构造一个半线性非可达性核 C，使得对任意 X，H 不是 X 的 home-space 当且仅当 C 从 X 可达。该归约进一步导出了半线性 home-space 问题的 Ackermann 完全性，并证明最小可达标识集可在 Ackermann 时间内构造。
A set of configurations $H$ is a home-space for a set of configurations $X$ of aPetri net if every configuration reachable from (any configuration in) $X$ can reach (some configuration in) $H$. The semilinear home-space problem for Petri nets asks, given a Petri net and semilinear sets of configurations $X$, $H$, if $H$ is a home-space for $X$. In 1989, David de Frutos Escrig and Colette Johnen proved that the problem is decidable when $X$ is a singleton and $H$ is a finite union of linear sets with the same periods. In this paper, we show that the general (semilinear) problem is decidable. This result is obtained by proving a duality between the reachability problem and the non-home-space problem. In particular, we prove that for any Petri net and any semilinear set of configurations $H$ we can effectively compute a semilinear set $C$ of configurations, called a non-reachability core for $H$, such that for every set $X$ the set $H$ is not a home-space for $X$ if, and only if, $C$ is reachable from $X$. We show that the established relation to the reachability problem yields the Ackermann-completeness of the (semilinear) home-space problem. For this we also show that, given a Petri net with an initial marking, the set of minimal reachable markings can be constructed in Ackermannian time.
@article{paperbot2699,
title = {On the Home-Space Problem for Petri Nets and its Ackermannian Complexity},
author = {Petr Jančar and Jérôme Leroux},
journal = {Logical Methods in Computer Science},
volume = {Volume 20, Issue 4},
year = {2024},
doi = {10.46298/lmcs-20(4:23)2024}
}