Fractional Collections with Cardinality Bounds, and Mixed Linear Arithmetic with Stars

We present decision procedures for logical constraints involving collections such as sets, multisets, and fuzzy sets. Element membership in our collections is given by characteristic functions from a finite universe (of unknown size) to a user-defined subset of rational numbers. Our logic supports standard operators such as union, intersection, difference, or any operation defined pointwise using mixed linear integerrational arithmetic. Moreover, it supports the notion of cardinality of the collection, defined as the sum of occurrences of all elements. Deciding formulas in such logic has applications in software verification. Our decision procedure reduces satisfiability of formulas with collections to satisfiability of formulas in an extension of mixed linear integerrational arithmetic with a “star” operator. The star operator computes the integer cone (closure under vector addition) of the solution set of a given formula. We give an algorithm for eliminating the star operator, which reduces the problem to mixed linear integer-rational arithmetic. Star elimination combines naturally with quantifier elimination for mixed integer-rational arithmetic. Our decidability result subsumes previous special cases for sets and multisets. The extension with star is interesting in its own right because it can encode reachability problems for a simple class of transition systems.