  The aim of static analysis is to infer invariants about programs that are
precise enough to establish semantic properties, such as the absence of
run-time errors. Broadly speaking, there are two major branches of static
analysis for imperative programs. Pointer and shape analyses focus on inferring
properties of pointers, dynamically-allocated memory, and recursive data
structures, while numeric analyses seek to derive invariants on numeric values.
Although simultaneous inference of shape-numeric invariants is often needed,
this case is especially challenging and is not particularly well explored.
Notably, simultaneous shape-numeric inference raises complex issues in the
design of the static analyzer itself.
  In this paper, we study the construction of such shape-numeric, static
analyzers. We set up an abstract interpretation framework that allows us to
reason about simultaneous shape-numeric properties by combining shape and
numeric abstractions into a modular, expressive abstract domain. Such a modular
structure is highly desirable to make its formalization and implementation
easier to do and get correct. To achieve this, we choose a concrete semantics
that can be abstracted step-by-step, while preserving a high level of
expressiveness. The structure of abstract operations (i.e., transfer, join, and
comparison) follows the structure of this semantics. The advantage of this
construction is to divide the analyzer in modules and functors that implement
abstractions of distinct features.
