Verifikation von C-Programmen Vorlesung 6 vom 04.12.2014: Abstract Interpretation

Verifikation von C-Programmen

Vorlesung 6 vom 04.12.2014: Abstract Interpretation

Christoph Lüth

Universität Bremen

Wintersemester 2014/15

Galois-Connections

Let L,M be lattices and

α:L→M γ :M →L

with α, γ monotone, thenhL, α, γ,Mi is a Galois connection if

γ.αwλl.l (1)

α.γ vλm.m (2)

Example of a Galois Connection

L=hP(Z),⊆i M =hInterval,vi γZI([a,b]) ={z ∈Z|a≤z ≤b}

α_ZI(Z) =

( ⊥ Z =∅ [inf(Z),sup(Z)] otherwise

Constructing Galois Connections

Let hL, α, β,Mi be a Galois connection, andS be a set. Then (i) S→L,S →M are lattices with functions ordered pointwise:

f vg ←→ ∀s.f svg s

(ii) hS →L, α⁰, γ⁰,S →Mi is a Galois connection with α⁰(f) =α.f

γ⁰(g) =γ.g

Generalised Monotone Framework

A Generalised Monotone Frameworkis given by

I a latticeL=hL,vi

I a finite flowF ⊆Lab×Lab

I a finite set of extremal labelsE vLab

I an extremal labelι∈Lab

I mappingsf fromlab(F) toL×Land lab(E) toL This gives a set of constraints

A◦(l)w^G{A_.(l⁰)|(l⁰,l)∈F} tι^l_E (3)

Correctness

Let R be a correctness relation R⊆V ×L, andhL, α, γ,Mi be a Galois connection, then we can construct a correctness relation S ⊆V ×M by

v S m←→v Rγ(m)

On the other hand, if B,M is a Generalised Monotone Framework, and hL, α, γ,Mi is a Galois connection, then a solution to the constraintsB^v is a solution toA^v.

This means: we can transfer the correctness problem from Lto M and solve it there.

An Example

The analysis SS is given by the latticeP(State),vand given a statement S∗:

I flow(S∗)

I extremal labels areE ={init(S∗)}

I the transfer functions (for Σ⊆State):

f_l^SS(Σ) ={σ[x 7→ A[[a]]σ]|σ∈Σ} if [x:=a]^l is inS∗

f_l^SS(Σ) = Σ if [skip]^l is inS∗

f_l^SS(Σ) = Σ if [b]^l is inS∗

Now use the Galois connection hP(State), α_ZI, γ_ZI,Intervali to construct a monotone framework with hInterval,vi, with in particular

What’s Missing?

I Fixpoints: Widening and narrowing.