Frameworks for analyzing multi-threaded C

ion, in this case, means that we are looking for a mapping X so that for all v, S[v]⊆X[v]. This means that we allow to weaken the statement about program points. To compute such an over-approximation of (concrete) semantics, we start with picking a complete lattice (D,v) so that each element d ∈D is a finite representation of a possibly infinite set of states. This step is necessary as we obviously cannot compute using infinite sets directly. The relation between lattice elements and corresponding sets of states can be formalized using a description relation ∆ ⊆ S ×D as follows. The least element should not describe any state, i.e., there should be no s ∈ S so that s ∆ ⊥, and the greatest element in the lattice must describe the set of all states, i.e., ∀s ∈ S. s ∆ >. Additionally, the description relation must reflect the ordering of the lattice: s ∆ d1 ∧ d1 v d2 =⇒ s ∆ d2 . Sometimes, instead of a description relation, a concretization function γ ∈ D→ 2 is used instead. For each element of the lattice d ∈D, the set γ(d) is the set of (concrete) states that are described by d, i.e., γ(d) = { s ∈ S | s ∆ d } .