Decision Procedures for Set-Valued Fields

A central feature of current object-oriented languages is the ability to dynamically instantiate user-defined container data structures such as lists, trees, and hash tables. Implementations of these data structures typically use references to dynamically allocated objects, which complicates reasoning about the resulting program. Reasoning is simplified if data structure operations are specified in terms of abstract sets of objects associated with each data structure. For example, an insertion into a data structure in this approach becomes simply an insertion into a dynamically changing set-valued field of an object, as opposed to a manipulation of a dynamically linked structure attached to the object. In this paper we explore reasoning techniques for programs that manipulate data structures specified using set-valued abstract fields associated with container objects. We compare the expressive power and the complexity of specification languages based on 1) decidable prefix vocabulary classes of first-order logic, 2) twovariable logic with counting, and 3) Nelson-Oppen combinations of multisorted theories. Such specification logics can be used for verification of object-oriented programs with supplied invariants. Moreover, by selecting an appropriate subset of properties expressible in such logic, the decision procedures for these logics enable automated computation of lattice operations in an abstract interpretation domain, as well as automated computation of abstract program semantics.