related approaches II

Main graph transformation approaches part 2

I continue today with SPO and DPO and my personal view of them. I want to start by acknowledging the deep and clever ideas and results that have been derived so far. If you are DPO/SPO fan, please, comment!

In my humble opinion, there are some advantages and disadvantages to the way DPO and SPO are defined. The good news are that both can deal with labelled multigraphs (which are of great interest to applied computer scientists) in a quite abstract way. MGG also deals with them (more on this in future posts). However...

From a theoretical point of view, in DPO/SPO you can not really study a production alone but one has to consider where that production is going to be applied. Roughly speaking, if I think of a production as a function that transforms a small set of elements, I have to specify all the domain and not just the part the function acts on. More technically, you have to study direct derivations and not productions. In my personal opinion, productions are the "natural" object of study and not direct derivations. Of course this is arguable. My main point here is that I would like to consider the functions that make up my graph transformation system (these are productions) and apply them to different systems (initial state), but I do not want to specify the system along with the functions (these are direct derivations), i.e. I would like to have uncoupled the functions and the actual set they are going to be applied to. Otherwise stated, I want to be able to specify my algorithm independently of the concrete instance of the problem it is going to be applied to.

In DPO (not in SPO) a direct derivation may specify the simultaneous addition and deletion of one element, say a node of a graph. This is a quite counter intuitive fact to me. I would say that one function may perform at most one action on one element. This has several consequences, mostly related to dangling conditions (edges incident to non-existent nodes). I will not enter this topic for now and leave it for a future posts on matching.

Nevertheless, my main slight objection is the difficulty in using other branches from mathematics. Some examples follow:

1. MGG generalizes previous approaches to graph constraints and application conditions through so-called diagrams (see Ch. 7 in the MGG Book). In essence, they plug monadic second order logics into MGG. I think that a similar construction is a hard task for DPO/SPO not to say for adhesive HLR categories.
2. When one studies sequences of productions and sequential independence (to what extent the order inside the sequence matters) groups of permutations arise naturally. They are easily handled in MGG (some posts on this topic to come) but I do not see an easy way (nor a difficult one) to do the same in DPO. We are thus limited for the moment, as far as I know, to study pairs of productions through the critical pairs lemma, which is a clear limitation as the results of Ch. 6 in the MGG book show.
3. Some concepts - which I find natural - are not so straightforward to define. What is called the nihil matrix in MGG (Sec. 4.4 in the MGG book; basically elements that "should not be") does not have a clear counterpart. It is one of the fundamental pieces to build bridges between MGG and complex analysis (some preliminar results can be found in this paper).
4. Complexity theory is probably the most prominent area of modern computer science. How does one study DPO/SPO as model of computations? What are their relationships with other models such as Turing Machines and Boolean Circuits? Are there interesting submodels of computation? To the best of my knowledge there is no attempt to study this, in my opinion due to its inherent difficulty. I will dedicate several posts to this particular topic for MGG in the future.
5. Other promising directions of research are for example the introduction of harmonic analysis. Notice that productions have a kind of linear structure, which can be studied with the Fourier transform over locally compact Abelian groups. This is my current research topic. Due to the abstractness of the pushout construction, this kind of stuff seems to lie too far from DPO/SPO.
   

There are other examples but I think previous points illustrate my view. Please, do not misunderstand my criticism because I do acknowledge the importance of DPO and SPO (visit this webpage for its current state and available tools). I just try to highlight the potential benefits to moving (or at least complement) from a categorical perspective to a more purely algebraic point of view, i.e. MGG. The next post will not be so long. I will try to brush over other approaches to graph dynamics.


Another quote from Alfred North Whitehead which I absolutely subscribe: Fundamental progress has to do with the reinterpretation of basic ideas.

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