I am faced with the following situation:
- I have a finite set of some $m$ positive integers $Q^m \in \mathbb{N}$
- These integers go through a series of $N$ possible black boxes that transform them. Let's call them $b_n$ ($n \in [1..N]$) and further constrain each element of $B$ to look like $b_n:\mathbb{N} \rightarrow \mathbb{N}$
So far so good, someone might even say that I have a finite state machine that can be described by a graph whose vertices are the positive integers and edges are the black boxes.
But, if I do this and then look at the nodes that are connected by some vertices, I can further decompose the series of black boxes down to a finite set of classes. In other words, imagine that each edge of the graph is coloured with some colour and same colour edges denote black boxes that have equal effect. So, if their functions have equal effect, they are effectively the same function. So, $B$ is actually smaller than it may appear to be initially.
This is where my knowledge stops.
I suspect that the graph defines an algebra and that the "class" of edges define the set of operators of this algebra. I further suspect that if I can show that some of "my" graphs are isomorphic to an established algebra, perhaps I can gain some insight into "my" graphs.
I am worried, as far as this approach is concerned, that "my" graphs look mostly like trees, but some of them can be dense enough in some areas to be seen as "local graphs" and therefore possibly look more like algebras (maybe over a specific field (?)). Otherwise, a tree would mean that you can apply operators conditionally or that certain chains of operator application are allowed but not all the times. It doesn't look like an algebra then (?) It has to be consistent like summation and multiplication are.
Provided that the above make sense, I would appreciate any sort of help with the following:
- Is there such a thing (proper term) as "discovering an algebra"? (discover its operators). If there is, where can I find out more about it?
- Is there such a thing (proper term) as "similarity of algebras"?. For instance, is there an algorithm or other method that can show (for example) that some physical phenomenon is actually an algebra on integers but only lacking an operator or only applicable over certain sets of numbers.
- Is it really going to be a problem that "my" algebra seems to be mostly tree-like? Should I be looking more into graph-like cases and treat the tree-like ones as special cases?
I suppose that reading up on prior work might make the above obvious but at the moment, I don't know where to go from here. I have been having a look at Universal Algebras and judging by this reference, they seem to be what I am after, but maybe the introduction is way off the main part (and more importantly, the practical part) that I am interested in.
EDIT:
Here is a tiny little toy example.
Because the nodes are integers, product (or division) wasn't sending them within the same Universe, so I opted for summation. This in turn revealed that the second operand of any given summation seemed to be coming from a finite subset which is why I went for some $f:\mathbb{N} \rightarrow \mathbb{N}$ instead of some $g:\mathbb{N}^2 \rightarrow \mathbb{N}$.
In the above figure, each node is a member of the Universe and each edge denotes transition from a node to another node via summation with another integer that happens to be "quantised". Therefore, what appear to be edge-labels in this graph are also members of the Universe.
One thing I can see here is patterns of successive consistent operator applications like $((6,2),9)$ (to mean either $6,9$ or $2,9$ that seem to be appearing in sequence. The trouble is of course that with such a tiny little graph of relationships it is difficult to spot such consistent relationships and this is why I am saying that such cases appear to be very special. As in, "It works only with some numbers but not all"...Which is not really how an algebra should work (?). Except if there is a way to then say "Well, the reason why it works with some specimens of the Universe is because the algebra that is depicted here uses a different underlying type system. Therefore, you can't expect summation to work consistently across all different types. Here $6-5=1$ it works in one way but here $[4,5,6] + [1] = [4,5,6,1]$ it works in another. One Universe contains only integers, the other also happens to contain the list-of-integers as well.".
I don't know, it's just an idea.
If I had to do it with my present knowledge, I would "unfold" an existing algebra and then use a graph matching algorithm to check its similarity with the "unknown" algebra. At least assuming a single type...possibly needs a few passes to remove layers of similarity one by one. But, this sounds like a dumb, primitive, coarse way of doing it. I suspect that some of these graphs could be subsets or supersets of something like Galois Fields. Small little objects with just a few valid operators, self contained.
Anyway, could there be something out there that encompasses what I am after?