Constraint-Based Abstract Semantics for Temporal Logic: A Direct Approach to Design

interpretation and infinite state model checking In abstract interpretation we develop an abstract semantic function systematically from the standard (“concrete”) semantics. The formal framework is based on a Galois connection 〈L,vL〉 −−−→ ←−−− α γ 〈M,vM 〉 between the lattices 〈L,vL〉 and 〈M,vM 〉 which are the concrete and abstract semantic domains respectively. The functions α and γ are known as the abstraction and concretisation functions respectively. For our purposes we only consider abstractions based on Galois connections 〈2 ,⊆〉 −−−→ ←−−− α γ 〈2,⊆〉, where the concrete domain 2 consists of sets of concrete states and the abstract domain 2 consists of sets of abstract states. (In fact the abstract domain could be any lattice but the presentation become somewhat more complex). Given such a Galois connection, the concrete functions pre, p̃re, and states defined earlier are abstracted by their best possible counterparts over the abstract domain, yielding the following abstract functions. apre = α ◦ pre ◦ γ ãpre = α ◦ p̃re ◦ γ astates = α ◦ states We simply substitute apre, ãpre and astates for their concrete counterparts in the concrete semantic function to obtain abstract semantics [[.]] for the temporal logic. The properties of Galois connections ensure that α([[φ]]) ⊆ [[φ]] and γ([[φ]]) ⊇ [[φ]].