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PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASIPOLYNOMIALS
"... Abstract. Presburger arithmetic is the firstorder theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric ..."
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Abstract. Presburger arithmetic is the firstorder theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, if p = (p1,..., pn) are a subset of the free variables in a Presburger formula, we can define a counting function g(p) to be the number of solutions to the formula, for a given p. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasipolynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions. ยง1. Introduction. A broad and interesting class of sets are those that can be defined over N = {0, 1, 2,...} with first order logic and addition. Definition 1.1. A Presburger formula is a firstorder formula in the language of addition, evaluated over the natural numbers. We will denote a generic Presburger formula as F (u), where u are the free variables (those not associated with