This is the motivating (toy) example from the paper.
Below are the safety properties we want to verify, in negated form.
∃ z1,z2. z1≠z2 ∧ ( CacheState[z1] = E ∧ CacheState[z2] ≠ I )
All invariants are shown in their negated form, where
#1 and #2
are distinct existentially quantified variables.
Cache[#1] <> I && Shr[#1] = False Exg = False && Cache[#1] = E Cache[#1] = E && Shr[#2] = True
You can find the list of all invariants that can be extracted from BRAB here (also in negated form), this collection being inductive.
The algorithm starts from the formula located at the bottom,
inside a red
octagon. Variables #1, #2,
… that appear
in the nodes are distinct skolem variables so we show a formula
φ(#1, #2) as equivalent to ∃
z1 ≠ z2 ∧ φ(z1,
z2). Plain black edges represent
pre-image relations and are annotated by the transition instance that
was considered. Black circles denote nodes that were obtained by
pre-image computation and were not covered by already visited
nodes. The nodes circled in gray are the one that were not
useful because they were subsumed by formulas pointed by the
arrows. Approximations are shown
in blue rectangles. Each approximation originates from the node that
connects its rectangle with a bold dashed blue edge.