application conditions I

application conditions and graph constraints

Graph constraints and application conditions are restrictions set on the host graph and on the applicability of a production, respectively. Conceptually, what we try to address is the especification of properties. For example, is a (host) graph three-colorable?, or, is every node of type 1 linked to one of type 2? This topic is important both from the theoretical and the practical points of view. It si addressed in Chap. 7 of the MGG book in full detail.

If we try to describe properties of a single graph, they are known as graph constraints; if there is a production involved they are known as application conditions. Application conditions can be set on the left hand side of the production (precondition) or on the right hand side (postcondition), or on both.

The proposal in MGG generalizes any previous approach to the topic. In essence, a graph constraint consists of a diagram (made up of graphs and morphisms between them) and a logical formula (monadic second order, to be concrete). The diagram specifies the graphs that take part in the constraint and the formula whether it should be or it should not be found in the host graph. Also, the graphs can be existentialy or universaly quantified. An example is "whenever graph A is found, there must exist a graph B linked to it". Of course, much more complicated combinations can be defined.

Now notice that the left hand side L of a production is in fact a graph constraint or an application condition: It asks for the existence of L in the host graph (the graph the production is going to be applied to). In fact, in MGG, application conditions are a particular case of graph constraints. Notice that in order to apply a production we do not only ask for the existence of the LHS (left hand side, L) but also for the non-existence of the nihilation matrix K (see this post). So in fact we have a positive and a negative application condition.

The first thing we have to prove is that MGG is general enough to handle with positive and negative application conditions as well as with existential and universal quantifiers. Notice that the existence of the LHS of a production is an existential qunatifier, but universals are not considered, at least explicitely. Interestingly, Sec. 7.2 proves that it is not necessary to extend the theory in order to cope with application conditions.

I interpret the following quotation from Plato as saying that we have to be very careful when setting the initial framework: The beginning is the most important part of the work. In some sense this was the initial reason for this blog.

spreading mathematics

accessing mathematics and headaches

Studying mathematics and getting a headache need not be cause and effect. However, many times in seminars or reading some papers one gets the impression of being a little dumb. Well, I'm pretty sure you know those seminars in which everybody's constantly nodding or papers with obvious steps (once written they take three sheets, but they are obvious).

At times I think that we need to be admired for something and in mathematics this "something" is intelligence. The less people that understands something, the more difficult it is, the more intelligent the writer is. Something similar applies to speeches. The strange thing is that if you do not understand something in a seminar, you just keep quiet and spend (lose) one or two hours waiting for the storm to pass you by. Well one should stand up and quietly leave the room.

Our life is short. I should be able to quickly decide if something will be of my interest. If so, then I can dedicate time to it. For established theories there are plenty of tutorials on the web (god blesses the web) but not so for recent research papers or new theories. After all, it is not that difficult to include some examples to illustrate the concepts. Even in those papers submitted for publishing.

I agree in that mathematics are not easy, but most times we make them even more difficult. I think that mathematicians should be aware of the necessity of selling the product (I should include myself). The Bourbaki school which has had a big influence on the twentieth century belongs to the past. We should react and go to the opposite - as it is always the case: thesis, antithesis, synthesis - and try to make accessible even state-of-the-art papers. Good ideas are not difficult.

Another point is that technicalities prevail over conceptual research. In a "deep paper" one expects hard-to-follow equations just above obscure paragraphs. With minor exceptions, new approaches and concepts are not normally recognized though they are usually more influential as they allow us to think from a different perspective.

I confess I'm not a fan of B. Mandelbrot at all, but I do mind the importance of his work. Today's quote I agree, first, because it seems to support the thesis of this post and, second, because for me mathematics is nothing more (and nothing less) than a language: being a language, mathematics may be used not only to inform but also, among other things, to seduce.