On symport/antiport P systems with a small number of objects

We look at the computational power of symport/antiport system (SA) acceptors and generators with small numbers of membranes and objects. We show that even with a single object and only three membranes, a SA acceptor can accept the non-semilinear set L={2 n | n≥0}. L can also be accepted with two objects and only one membrane. This latter model can accept all unary semilinear (i.e. regular) sets. We also show that, for any k≥1, the class of sets of k-tuples of non-negative integers accepted by partially blind (multi-) counter machines is a subclass of the class of sets of k-tuples accepted by one-object multi-membrane SA acceptors. Similarly, the class of sets of k-tuples of non-negative integers generated by partially blind counter machines is a subclass of the class of sets of k-tuples generated by one-object multi-membrane SA generators. As a corollary, the unary semilinear sets are a proper subclass of the unary sets of numbers accepted by SA acceptors with one object and five membranes. Whether or not one-object multi-membrane SA acceptors (resp. generators) are universal remains an interesting open question. However, allowing the use of priority relations makes them universal.