We define some new kinds of pseudoprimes to several bases, which generalize strong pseudoprimes. We call them Sylow p-pseudoprimes and elementary Abelian p-pseudoprimes. It turns out that every n < 1012, which is a strong pseudoprime to bases 2, 3 and 5, is not a Sylow p-pseudoprime to two of these bases for an appropriate prime p|n− 1. We also give examples of strong pseudoprimes to many bases...

We study BVASS (Branching VASS) which extend VASS (Vector Addition Systems with States) by allowing addition transitions that merge two configurations. Runs in BVASS are tree-like structures instead of linear ones as for VASS. We show that the construction of Karp-Miller trees for VASS can be extended to BVASS. This entails that the coverability set for BVASS is computable.