A CLP Approach to Modelling Systems

ion ICFEM’04, Seattle, 12/Nov/2004 Slide 53 (cs(· · ·), Ψ) abstracts to (cs(· · ·), Φ) where Ψ |= Φ Key Advantage: • In infinite state problems, can make it finite; • In traversing the program flow, can reduce the number of steps to a (post-)fixpoint. Key Problem: how to derermine the abstraction? Many verification systems use the methodology of Counter-example Guided Abstraction Refinement eg MAGIC, Murφ, SLAM, etc ... Goal-Directed Abstraction ICFEM’04, Seattle, 12/Nov/2004 Slide 54 To determine given a goal G = cs(· · ·), c1, · · · ,cn and a target constraint Ψ • if G |= Ψ, and • which of the constraints c1, · · · ,cn are redundant for this purpose? General method: let G (only) have descendants G1 and G2, and suppose • G1 |= Ψ1 with C1 ⊆ {c1, · · · ,cn} redundant • G2 |= Ψ1 with C2 ⊆ {c1, · · · ,cn} redundant Then: G |= Ψ1 ∨Ψ2 with C1 ∩C2 redundant GDA is complementary to Counterexample Guided Refinement Goal-Directed Abstraction ICFEM’04, Seattle, 12/Nov/2004 Slide 55 Example (initially x = 0) 〈0〉 if (p1) x++ else x := x + 2 〈1〉 if (p2) x++ else x := x + 2 ... 〈n-1〉 if (pn) x++ else x := x + 2 At the end, what is the maximum value of x? (Note that there are an exponential number of paths.) Goal-Directed Abstraction ICFEM’04, Seattle, 12/Nov/2004 Slide 56 〈0〉 if (p1) x++ else x := x + 2 〈1〉 if (p2) x++ else x := x + 2 〈2〉 if (p3) x++ else x := x + 2 (5) (5) (6) p3 ¬p3 p3 ¬p3 p3 ¬p3 p3 ¬p3 p2 ¬p2 p2 ¬p2 p1 ¬p1 X = 0