There are several different (probably inequivalent) notions of computational universality (see for example the last couple pages of http://www.dna.caltech.edu/~woods/download/WoodsNearyTCS07-DRAFT.pdf) and there is no consensus among experts about which notions are most correct (see for example http://cs.nyu.edu/pipermail/fom/2007-October/012148.html).I'm trying to say something about a particular model of biomolecular computation.  I'd like to argue that it's more universal or more usefully universal than some other models, because you can construct a universal machine that runs a program and then deletes the input at the end and is ready to run another program.  Contrast this to, say, cellular automata, which can emulate any Turing machine, but then at the end of the computation, you've got a final, unchangeable configuration.  To emulate another TM, you need to define a completely separate CA.  So I'd like to say something is reusably universal if it behaves like your desktop, not a CA (i.e., can execute multiple programs without needing to recreate the universe).  Has this notion been formalized anywhere?
Is there an official name for a notion of reusably universal?
universal computation
As you mention in Automata Theory / Formal Language Thesis Topic my supervisors have at least some of the same intution about reusable universality as better than CA-style. Im not sure that a name is given though: http://www.diku.dk/~neil/blobentcs.pdfI havent focused much on that part, but as I see it, when going through biocomputing literature, the major difference lies in the meaning of the word programming/programmable, e.g. what is it in fact that is programmable? That, and the stored-program part as well but I appreciate the nuance posed by your questionI've got no readily available answer to what it is called though