Separating Graph Logic from MSO

Graph logic (GL) is a spatial hoc for querying, graphs introduced by Cardelli et al. It has been observed that in terms of expressive power, this hoc is a fragment of Monadic Second Order Logic (MSO) with h quantification over sets of edges. We show that the containment is proper by exhibiting a property that is not GL definable but is definable in MSO, even in the absence of quantification over labels. Moreover, this holds when the graphs are restricted to be forests and thus strengthens in several ways a result of Marcinkowski. As a consequence we also obtain that Separation Logic, with a seperating conjunction but without the magic wand, is strictly weaker than MSO over memory heaps, settling an open question of Brochenin et al.