半线性集与Parikh自动机的可分离性复杂度
The complexity of separability for semilinear sets and Parikh automata
AI总结：
本文研究了半线性集和Parikh自动机语言的两种可分离性问题，证明了这两个问题均为coNP完全，并推广至有理子集可识别分离性与确定性Parikh自动机正则性的复杂度判定。
摘要：
在可分离性问题中，给定来自类$\mathcal{C}$的两个集合$K$和$L$，我们要判定是否存在来自类$\mathcal{S}$的集合$S$，使得$K\subseteq S$且$S\cap L=\emptyset$。此时，我们称之为用$\mathcal{S}$中的集合实现$\mathcal{C}$中集合的可分离性。我们研究了两种可分离性问题。首先，我们考虑半线性集（即对于某个$d$，$\mathbb{N}^d$的子集）由无量词一元Presburger公式（或等价地，$\mathbb{N}^d$的可识别子集）定义的集合的可分离性。这里，若每个原子至多使用一个变量，则公式为一元的。其次，我们考虑Parikh自动机的语言与正则语言的可分离性。Parikh自动机是一种可访问只能递增的计数器的机器，并且必须在运行结束时满足半线性约束。已知这两种可分离性问题都是可判定的，且具有基本复杂度。我们的主要结果是这两个问题都是coNP完全的。对于半线性集，无论输入集是由存在Presburger公式、无量词公式还是半线性表示指定，coNP完全性均成立。我们的结果意味着$Σ^*\times\mathbb{N}^d$的有理子集的可识别可分离性（由Choffrut和Grigorieff证明可判定）也是coNP完全的。另一个应用是确定性Parikh自动机（其中目标集使用无量词Presburger公式指定）的正则性也是coNP完全的。
英文摘要：
In a \emph{separability problem}, we are given two sets $K$ and $L$ from a class $\mathcal{C}$, and we want to decide whether there exists a set $S$ from a class $\mathcal{S}$ such that $K\subseteq S$ and $S\cap L=\emptyset$. In this case, we speak of \emph{separability of sets in $\mathcal{C}$ by sets in $\mathcal{S}$}. We study two types of separability problems. First, we consider separability of semilinear sets (i.e. subsets of $\mathbb{N}^d$ for some $d$) by sets definable by quantifier-free monadic Presburger formulas (or equivalently, the recognizable subsets of $\mathbb{N}^d$). Here, a formula is monadic if each atom uses at most one variable. Second, we consider separability of languages of Parikh automata by regular languages. A Parikh automaton is a machine with access to counters that can only be incremented, and have to meet a semilinear constraint at the end of the run. Both of these separability problems are known to be decidable with elementary complexity. Our main results are that both problems are coNP-complete. In the case of semilinear sets, coNP-completeness holds regardless of whether the input sets are specified by existential Presburger formulas, quantifier-free formulas, or semilinear representations. Our results imply that recognizable separability of rational subsets of $Σ^*\times\mathbb{N}^d$ (shown decidable by Choffrut and Grigorieff) is coNP-complete as well. Another application is that regularity of deterministic Parikh automata (where the target set is specified using a quantifier-free Presburger formula) is coNP-complete as well.
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