Arithmetically Controlled H Systems

We consider two classes of restricted H systems, both dealing with numbers associated to the terms of splicing operations. In one of them, these numbers indicate the age of the strings (the generation when the strings are produced), in the second one the numbers can be interpreted as valences of the strings. Restricting the splicing to strings of "a similar age", or accepting as complete splicing processes only those processes which produce strings with a null valence increase the generative power of H systems (with finite sets of rules)