A Global Sparse Analysis Framework

Domain. We consider an analysis that over-approximates the reachable states for each control point: the abstract domain is a map from C → Ŝ, where C is the set of control points in the program and Ŝ is a non-relational abstract state such that P(S) −−−→ ←−−− αS γS Ŝ: Ŝ = L̂→ V̂ L̂ = Var V̂ = Ẑ× P̂ P̂ = P(L̂) Abstract state Ŝ is a map from abstract locations L̂ to abstract values V̂. An abstract location is a program variable. An abstract value is a pair of an abstract integer Ẑ and an abstract pointer P̂. A set of integers is abstracted into an abstract integer (P(Z) −−−→ ←−−− αZ γZstate Ŝ is a map from abstract locations L̂ to abstract values V̂. An abstract location is a program variable. An abstract value is a pair of an abstract integer Ẑ and an abstract pointer P̂. A set of integers is abstracted into an abstract integer (P(Z) −−−→ ←−−− αZ γZ Ẑ). Note that the abstraction is generic so we can choose any non-relational numeric domains of our interest, such as intervals ( Ẑ = {[l, u] | l, u ∈ Z∪{−∞,+∞}∧l ≤ u}∪{⊥}). For simplicity, we do not abstract pointers (because they are finite): pointer values are kept by a points-to set (P̂ = P(L̂)). Other pointer abstractions are also orthogonally applicable. ACM Transactions on Programming Languages and Systems, Vol. V, No. N, Article A, Publication date: January YYYY. Global Sparse Analysis Framework A:19 Abstract Semantics. The abstract semantics is defined by the least fixpoint of the following semantic function: F̂ ∈ (C→ Ŝ)→ (C→ Ŝ) F̂ (φ̂) = λi ∈ C.f̂i( ⊔Semantics. The abstract semantics is defined by the least fixpoint of the following semantic function: F̂ ∈ (C→ Ŝ)→ (C→ Ŝ) F̂ (φ̂) = λi ∈ C.f̂i( ⊔