Is a ‘Presburger set’ a subset of that can be described by a formula in Presburger arithmetic?

Exactly, and indeed Jérôme Leroux’ algorithm is extremely nice and simple. He recently gave a new proof of its correctness—which does not rely on the (now) classical decompositions by Mayr, Kosaraju, and Lambert—in

Jérôme Leroux, Vector addition system reachability problem: a short self-contained proof, POPL 2011, ACM Press.

]]>I have mystical reasons for wanting the reachability problem to only be solvable by an algorithm that’s not primitive recursive. Not very good reasons. I’ve gotten interested in the idea of problems that are decidable but only very slowly, or models of computation that fail to be Turing-complete but are still very powerful—I’d like to see results that probe the ‘borderline of computability’. I also like symmetric monoidal categories. So, I’m hoping they give natural examples of such results.

]]>This is very nice.

]]>There’s quite a lot of related work, of course, since the field goes back over 40 years, to E.J. Corey’s 1969 Science publication “Computer-Assisted Design of Complex Organic Syntheses”, available at http://www.sciencemag.org/content/166/3902/178.full.pdf

Although, perhaps, a review article or two would contextualize the work better. Cook et al.’s “Computer-aided synthesis design: 40 years on” (http://onlinelibrary.wiley.com/doi/10.1002/wcms.61/full) and Todd’s “Computer-aided organic synthesis” (http://pubs.rsc.org/en/content/articlepdf/2005/cs/b104620a) may help.

Also, if you’re interested in other modern formulations of chemistry by computer scientists for computer scientists, you might check out Masoumi and Soutchanski’s “Reasoning about Chemical Reactions using the Situation Calculus (http://www.cs.ryerson.ca/~mes/publications/MasoumiSoutchanski_2012FallAAAI-DiscoveryInformaticsSymposium_Nov2-4.pdf) or my “Construction of New Medicines via Game Proof Search” (http://www.aaai.org/ocs/index.php/AAAI/AAAI12/paper/view/4936).

Enjoy!

]]>Reachability sets in Petri nets go beyond what is expressible in Presburger arithmetic (Hopcroft and Pansiot, 1979, http://dx.doi.org/10.1016/0304-3975(79)90041-0). What Jérôme Leroux has shown is that, however, a proof of *non-reachability* can be expressed through a *Presburger invariant*, i.e. two disjoint Presburger sets: one containing the source configuration and all its reachable configurations, the other containing the target configuration and all the configurations that reach it. This yields a semi-algorithm for non-reachability: enumerate pairs of Presburger formulae, until you find a Presburger invariant; as there is a trivial smi-algorithm for reachability (enumerate paths until you find one from source to target), this means that the problem as a whole is decidable. Not a very constructive algorithm :)

Is there a series of reactions that gets from the first multiset to a *super-multiset* of the second one? For all I know there is a more efficient algorithm that solves this modified problem.

This is known as the **coverability** problem. Unlike the reachability problem, its complexity is well understood, as the ExpSpace lower bound of Lipton is matched by an ExpSpace upper bound due to Rackoff:

• Charles Rackoff, The covering and boundedness problems for vector addition systems, *Theoretical Computer Science* **6** (1978), 223–231.

• Petr Jančar, Bouziane’s transformation of the Petri net reachability problem and incorrectness of the related algorithm, *Information and Computation*, **206** (2008), 1259–1263.

At the moment there are *no* known upper complexity bounds for the problem (besides being decidable), not even Ackermannian ones (contrarily to what is stated in the post, or I would be happy to read a reference!).

An other simple idea: the human dna can have a Petri net representation, so that can be possible to represent the protein production in a mathematical form, to simplify the biological representation of the dna function.

Saluti

Domenico

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