Your problem requires the transformation of the input (ab,1) and output (bc) parameters into a suitable feature space in order for generic machine learning algorithms to work. The transformation itself would undeniably require some information regarding the nature of the input and output parameters. There isn't a generic algorithm for performing such transformations, which is why most of the machine learning research is focused on "feature learning" or "representation learning". People usually design feature spaces by employing some kind of expert knowledge. Even the most generic approaches to this problem (such as deep learning) have certain requirements for parameter tuning.  So, if you are willing to apply any kind of expert knowledge about your problem, there are many ways to solve it. Your focus should be to find the solution that requires least amount of expert knowledge. I would try my hand in text processing. 

Decision tree, random forest. And the latter one have a better performance.

The opening assertions aren't correct, there's been a lot of work done on understanding the generalization error and it's not true, either, that the associations are simply pairs. Triplets and p-tuples can be learned. Indeed, the most studied issue has been, to establish bounds on p, the number of associations.

Have you a considered have an evolutionary algorithm that could search for the best match. As long as you there a fitness evaluation to assess the quality of p-tuples. It has been done with timetables a few years ago. 

Thank you Stam, I am actually asking this question about the knowledge I lack so that people can point me to existing work. However, I still have doubts regarding systems that can be trained with p-tuples and that demonstrate a good generalization. I do not want any boundary on the "number of association" as what I want to learn are the rules that govern those associations... Somehow like a grammar : "John is timbulbing the morgul". You do not know those words, however you know the rule that governs them. And somehow you learnt that from a number of associations... I know a few solutions for this using SOM, but I would like more suggestions...

I'd recommend the paper of Mézard and Nadal on the tiling algorithm.

You could have a look at Fuzzy Inference Systems (cs.princeton.edu/courses/archive/fall07/cos436/HIDDEN/Knapp/fuzzy004.htm) whose aim is precisely to build a set of rules from input data.

I think what Patricia Ryser-Welch is alluding to (specifically) is genetic programming, and it seems to me like your problem should be an application of it.

If I understood your question correctly: I think both of your examples are the same. In a sense that in both cases you are trying to induce a hidden relation (aka function) from examples. It just happened that the second example has more complex structure compared to the first.

If you are concerned about theoretical guarantees to generalize for unseen examples and looking for a model selection method, then learning theory (VC dimension, structural risk minimization, minimum description length and Occam's principle ) and algorithms based on these concepts may be relevant to your work. 

Thanks for all the helpfull suggestion. The tiling algo and the fuzzy inference seem good choices for me...

Fuzzy Inference Systems (FIS) might be a very good choice....

URL: http://www.cs.princeton.edu/courses/archive/fall07/cos436/HIDDEN/Knapp/fuzzy004.htm

Your problem requires the transformation of the input (ab,1) and output (bc) parameters into a suitable feature space in order for generic machine learning algorithms to work. The transformation itself would undeniably require some information regarding the nature of the input and output parameters. There isn't a generic algorithm for performing such transformations, which is why most of the machine learning research is focused on "feature learning" or "representation learning". People usually design feature spaces by employing some kind of expert knowledge. Even the most generic approaches to this problem (such as deep learning) have certain requirements for parameter tuning.  So, if you are willing to apply any kind of expert knowledge about your problem, there are many ways to solve it. Your focus should be to find the solution that requires least amount of expert knowledge. I would try my hand in text processing. 

Hi Sarper, indeed tuning the algorithm to a specific problem, or injecting expert knowledge is what we want to avoid. We are working on an hippocampal model, so in theory the features are already extracted by the cortex and therefore we are given a "good representation" to learn on.

However you are right, our problem is pretty higly dependend of this input... In fact I am pretty much convinced that the hippocampus/cortex relationship is to define what is the set of features the most desirable.

Hi Stephane. If your main focus is a hippocampal model, I would suggest steering away from high level input-outputs like above, since we don't know how exactly the brain processes them. Just like you said, if hippocampus does a global memory encoding, we can assume that the information representation in hippocampus is also in a globally acceptable form (similar to the transformed features for classification algorithms).  

If I understand you right, you are looking for a decision tree algorithm. Have a look into WEKA, which comes with a big set of different kind of algorithms including finding rules or extracting rules from data sets.