We consider an expansion of Presburger arithmetic which allows multiplication by $k$ parameters $t_1,\ldots,t_k$. A formula in this language defines a parametric set $S_\mathbf{t} \subseteq \mathbb{Z}^{d}$ as $\mathbf{t}$ varies in $\mathbb{Z}^k$, and we examine the counting function $|S_\mathbf{t}|$ as a function of $\mathbf{t}$. For a single parameter, it is known that $|S_t|$ can be expressed as an eventual quasi-polynomial (there is a period $m$ such that, for sufficiently large $t$, the function is polynomial on each of the residue classes mod $m$). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming \textbf{P} $\neq$ \textbf{NP}) we construct a parametric set $S_{t_1,t_2}$ such that $|S_{t_1, t_2}|$ is not even polynomial-time computable on input $(t_1,t_2)$. In contrast, for parametric sets $S_\mathbf{t} \subseteq \mathbb{Z}^d$ with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that $|S_\mathbf{t}|$ is always polynomial-time computable in the size of $\mathbf{t}$, and in fact can be represented using the gcd and similar functions.

中文翻译：

---

参数 Presburger 算法：计数和量词消除的复杂性

我们考虑 Presburger 算法的扩展，它允许乘以 $k$ 参数 $t_1,\ldots,t_k$。这种语言中的公式定义了一个参数集 $S_\mathbf{t} \subseteq \mathbb{Z}^{d}$ 因为 $\mathbf{t}$ 在 $\mathbb{Z}^k$ 中变化，我们检查计数函数 $|S_\mathbf{t}|$ 作为 $\mathbf{t}$ 的函数。对于单个参数，已知 $|S_t|$ 可以表示为最终的拟多项式（有一个周期 $m$，使得对于足够大的 $t$，该函数是每个残差上的多项式类 mod $m$）。我们表明，使用 2 个或更多参数时，这样一个好的表达式是不可能的。实际上（假设 \textbf{P} $\neq$ \textbf{NP}）我们构造了一个参数集 $S_{t_1,t_2}$ 这样 $|S_{t_1, t_2}|$ 甚至不是多项式时间可计算的在输入 $(t_1,t_2)$ 上。相比之下，