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- W2786244858 abstract "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." @default.
- W2786244858 created "2018-02-23" @default.
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- W2786244858 date "2018-02-03" @default.
- W2786244858 modified "2023-09-27" @default.
- W2786244858 title "Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination" @default.
- W2786244858 hasPublicationYear "2018" @default.
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