Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1, . . . , tk. A formula in this language defines a parametric set St ⊆ Z d as t varies in Z, 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 6= 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 ⊆ Z 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.