One of the main objectives of cognitive science is to use abstraction to create
models that represent accurately the cognitive processes that constitute learning,
such as categorisation. Relational knowledge is important in this task, since
through the reasoning processes of induction and analogy over relations that the
mind "creates" categories (it later estabilishes causal relations between them by
using induction and abduction), and analogies exemplify crucial properties of
relational processing, like structure-consistent mapping[
Given the complexity of the task, no model today has accomplished it
completely. The associacionist/connectionist approach represents those processes
through associations between di erent informations. That is done by using arti
cial neural networks. However, it faces a great obstacle: the idea (called
propositional xation) that neural networks could not represent relational knowledge. A
recent attempt to tackle the symbolic extraction from arti cial neural networks
was proposed in [
The cognitive agent Amao uses a shared Neural Multi-Space (Shared NeMuS)
of coded rst-order expressions to model the various aspects of logical formulae
as separate spaces, with importance vectors of di erent sizes. Amao [
Whether a train is going east or west is determined by its properties. Using these trains, a simple base has been created, taking into account the size of the train wagons (short or not) and whether these wagons are closed or not. The number Copyright © 2017 for this paper by its authors. Copying permitted for private and academic purposes. of wheels, wagon format and other attributes have been ignored in order to make the base simpler.
All the eastbound trains have at least one wagon which is both short and closed. That is what determines whether a train is eastbound or westbound. The idea is to use the shared NeMuS structure to induce the rule eastbound knowing that t1 (the rst train) is going east. Having that information, we can directly get all predicate instances, called as bindings, which have t1 is an attribute.
They are the following: train(t1). car(t1, c1 t1). short(c1 t1). car(t1, c2 t1). closed(c1 t1). car(t1, c3 t1). car(t1, c4 t1).
The predicate car links t1 to all its wagons (or carriages), so car(t1, c1 t1) means that c1 t1 is a wagon that belongs to t1. Taking the rst instance of the predicate car, we now know that t1 has a wagon named c1 t1. Amao, through its shared NeMuS, accesses c1 t1's bindings and using a polynomial search, nds both occurrences of c1 t1 in short and closed, as seen above. This mechanism is called linkage pattern in Amao's learning mechanism.
At this point t1 is a train that has c1 t1 as a wagon, and this wagon is not closed. Amao also has the linkage predicate connecting both c1 t1 and t1. Thus, a candidate hypothesis generated would look like eastbound(X) car(X, Y) ^ short(Y) ^ closed(Y). However, this may not be the only possible hypothesis, so the other wagons being carried by t1 need to be considered.
short(c2 t1). short(c3 t1). short(c4 t1). closed(c2 t1. closed(c3 t1). closed(c4 t1).
Among the possible hypotheses that may de ne a train as being eastbound, we have: eastbound(X) eastbound(X) eastbound(X) car(X, Y) ^ short(Y) ^ closed(Y).
car(X, Y) ^ short(Y) ^ closed(Y).
car(X, Y) ^ short(Y) ^ closed(Y).
Adding negative examples, we can reduce the number of possible hypotheses.
In this case, the simplest way to do that is to use the 10th train t10 as a negative example. Using the same method as explained above, the structure can select all predicates that have t10 as an attribute: car(t10, c1 t10).
car(t10, c2 t10).
Then, all the predicates that have t10s wagons as attributes: short(c1 t10).
closed(c1 t10). short(c2 t10). closed(c2 t10).
Thus, the hypotheses that de nitely do not de ne a train as being eastbound are:
Both hypotheses are among the possible options de ned above. Excluding them, the correct option remains. The target eastbound(X) can be de ned by: eastbound(X)
car(X, Y) ^ short(Y) ^ closed(Y).
Formalizing what was explained above: 1. With the positive example ( t1), get all predicates (bindings) that have t1
as an attribute; 2. Access bindings of attributes linked to t1 using polynomial search (linkage pattern)
{ in this case, the attributes are c1 t1, c2 t1 and c3 t1 3. repeat the rst two steps for the negative example ( t10)]
{ in this case, the attributes linked to t10 are c1 t10 and c2 t10 4. if there are hypotheses generated by using the positive example that are repeated in the negative example, they will not be in the list of possible hypotheses.
{ some of the hypotheses generated by using only the positive example are: eastbound(X) car(X, Y) ^ short(Y) ^ closed(Y).
eastbound(X) car(X, Y) ^ short(Y) ^ closed(Y).
eastbound(X) car(X, Y) ^ short(Y) ^ closed(Y).
However, using only the negative example, the rst and third hypotheses would also be generated. By using both examples, these two don't make it into the list of possible hypotheses, and the correct one, which is eastbound(X) car(X, Y) ^ short(Y) ^ closed(Y), remains. 3
The knowledge base created is only a simpli cation of the original train problem. As explained before, many attributes such as number of wheels, wagon format, load shape and roof shape have been ignored. Had they been included, more hypotheses could have been generated through Amao's inductive learning mechanism over the shared NeMuS. One current limitation is not being able to deal with predicate invention, that would allow to automatically create categories by means of abstraction/new predicates.
One possible road to explore is to take advantage of shared NeMuS weights to integrate a neural network classi cation method to help identify categories. In the train set, we know which trains are eastbound, but whatever rule de nes the eastbound category is not known before using Amao to de ne it. Understanding what makes a train eastbound or not can help us categorize any train that might be added to the set in the future.
Another goal we aim to pursue is to make use of weights to implement neural mechanisms. We expect to envisage more e cient heuristics to guide hypotheses generation, improving Amao's learning mechanism.