On small, reduced, and fast universal accepting networks of splicing processors

In this paper, we show that accepting networks of splicing processors (ANSPs) of size 2 are computationally complete. Since, by definition, an ANSP needs at least two nodes to perform non-trivial computations, this completely settles the question of designing complete ANSPs of minimal size. Also, we derive from this result the fact that all the languages in PSPACE can be accepted by ANSPs of size 2, having polynomial length complexity (the ANSP complexity measure for the space used in a computation). However, the construction that we propose, although efficient from the descriptional complexity and space complexity points of view, does not seem to have good properties from the time complexity point of view. In this respect, we prove that ANSPs of size three can decide all languages in NP in polynomial time. The previous lower bound on size for both completeness and efficient acceptance of NP-languages was seven. We also consider ANSPs with restricted features, proving the following normal forms: for any ANSP there exists an equivalent ANSP without input filters, and one without output filters. Finally, we show how to construct a small universal ANSP and make several considerations on the computational efficiency of universal ANSPs.