Counting the solutions of Presburger equations without enumerating them

The Number Decision Diagram (NDD) has recently been introduced as a powerful representation system for sets of integer vectors. NDDs can notably be used for handling sets defined by arbitrary Presburger formulas, which makes them well suited for representing the set of reachable states of finite-state systems extended with unbounded integer variables. In this paper, we address the problem of counting the number of distinct elements in a set of numbers or, more generally, of vectors, represented by an NDD. We give an algorithm that is able to produce an exact count without enumerating explicitly the vectors, which makes it capable of handling very large sets. As an auxiliary result, we also develop an efficient projection method that allows to construct efficiently NDDs from quantified formulas, and thus makes it possible to apply our counting technique to sets specified by formulas. Our algorithms have been implemented in the verification tool LASH, and applied successfully to various counting problems