The Problem of "Weak Bisimulation up to"

"Bisimulation up to" is a technique for reducing the size of the relation needed to define a bisimulalion. It works smoothly in the strong case, where it was first introduced ([4]). But this does not directly generalize to the weak case, as erroneously reported in [4]. To overcome this problem, two new "up-to" techniques are proposed: They are respectively based on the use of expansion ([1]) and of almost-weak bisimulation. The second solution is more general than the first one, but expansion enjoys a nicer mathematical treatment. The usefulness and generality of the solutions is motivated with non-trivial examples: two different implementations of a sorting machine.