Counting Affine Calculator and Applications

We present an interactive tool, called iscc, for manipulating sets and relations of integer tuples bounded by affine constraints over the set variables, parameters and existentially quantified variables. A distinguishing feature of iscc is that it provides a cardinality operation on sets and relations that computes a symbolic expression in terms of the parameters and domain variables for the number of elements in the set or the image of the relation. In particular, these expressions are piecewise quasipolynomials, which can be further manipulated in iscc. Besides basic operations on sets and piecewise quasipolynomials, iscc also provides an interface to code generation, lexicographic optimization, dependence analysis, transitive closures and the symbolic computation of upper bounds and sums of piecewise quasipolynomials over their domains.