Parametric Presburger arithmetic: complexity of counting and quantifier elimination
We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1, horizontal ellipsis ,tk. A formula in this language defines a parametric set St subset of Zd as t varies in Zk, and we examine the counting function |St| as a function of t. For a single parameter, it is known that |St| 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 P not equal NP) we construct a parametric set St1,t2 such that |St1,t2| is not even polynomial-time computable on input (t1,t2). In contrast, for parametric sets St subset of Zd with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that |St| is always polynomial-time computable in the size of t, and in fact can be represented using the gcd and similar functions.
Bogart, Tristram, John Goodrick, Danny Nguyen, and Kevin Woods. 2019. "Parametric Presburger arithmetic: complexity of counting and quantifier elimination." Mathematical Logic Quarterly 65(2): 237-250.
Mathematical Logic Quarterly