Automatic verification of programs manipulating arrays relies on specialised decision procedures. A methodology to classify the theories handled by these procedures is introduced. It is based on decomposition theorems in the style of Feferman and Vaught. The method is applied to obtain an extension of combinatory array logic that is closed under propositional operations and Hoare triples. A classification according to expressiveness of six different fragments studied in the literature is given.

We discuss two extensions to a recently introduced theory of arrays, which are based on considerations coming from the model theory of power structures. First, we discuss how the ordering relation on the index set can be expressed succinctly by referring to arbitrary Venn regions. Second, we show how to add general aggregators to the calculus. The result is a logic that subsumes four previous fragments discussed in the literature and is distinct from array fold logic, in that it can express summations, while its satisfiability problem remains in non-deterministic polynomial time .

Motivated by satisfiability of constraints with function symbols, we consider numerical inequalities on non-negative integers. The constraints we address are a conjunction of a linear system Ax = b and an arbitrary number of (reverse) convex constraints of the form xi ≥ xdj (xi ≤ xdj ). We show that the satisfiability of these constraints is NP-complete even if the solution to the linear part is given explicitly. As a consequence, we obtain NP- completeness for an extension of certain quantifier-free constraints on sets with cardinalities and function images.