In this section we illustrate the behavior of the model with a simple simulation. In this simulation a network learns both inter-MRU correlations across different dimensions and also simple relational categories. The simulation also illustrates how the ease of learning particular words can depend on the match between the words and the relational correlations that the system has already picked up on.
We first defined a set of correlations among non-linguistic dimensions and a set of correlations between the non-linguistic dimensions and words. There were two non-linguistic dimensions and four possible micro-relations within each of these dimensions. For example, micro-relation A represented a relation between a very low value for one object and a very high value for another object on Dimension 1, and micro-relation E represented a similar micro-relation on Dimension 2. The micro-relations within Dimension 1 are shown in Figure 18. Keep in mind that each micro-relation is a relation between values on a particular dimension for two different objects.
Across the dimensions there were correlations between particular micro-relations. That is, a pair of objects with a particular relation between their values on one dimension tended to have one or another of a set of relations between their values on the other. For example, micro-relation A on Dimension 1 correlated with micro-relations E and F, but not with micro-relations G and H, on Dimension 2. The relational correlations are shown in Figure 19a. Figure 19b illustrates one of the correlations, that between micro-relation A on Dimension 1 and micro-relation E on Dimension 2.
We defined two ``languages,'' an Easy language, which agrees with the non-linguistic correlations, and a Hard language, which disagrees with the non-linguistic correlations, as shown in Figure 20. Each language consists of two relational words. In the figure, the two words are indicated by the boxes with either solid (Word 1) or dashed (Word 2) borders. The two words in the Easy language are indicated by boxes with thick borders (the two vertical boxes); the two words in the Hard language are indicated by boxes with thin borders (the two diagonal boxes). All four words are relational; that is, they represents relations between two separate objects. Each word is associated with a cluster of four possible situations, each characterized by a pairing of a micro-relation on Dimension 1 and a micro-relation on Dimension 2. As can be seen in Figure 19a, each of these situations corresponds to an actual correlation of micro-relations in the real world. The two languages differ in terms of how these situations are grouped for the two words.
Consider first the choice of one word over the other in the Easy language. In this language, Word 1 (the vertical box on the left in Figure 20) is associated with situations AE, AF, BE, and BF, whereas Word 2 (the vertical box on the right) is associated with situations CG, CH, DG, and DH. On Dimension 1, distinct sets of micro-relations are associated with distinct words: A and B for Word 1, C and D for Word 2. Therefore, given a pair of related objects, we can select the appropriate word if we know their relation on Dimension 1 only; for example, objects related by A on Dimension 1 will be referred to with Word 1. The same holds for Dimension 2. E and F are associated with Word 1, G and H with Word 2. Given a pair of objects, we can select the appropriate word if we know their relation on Dimension 2 only. More importantly, each of the two words in this language agrees with the clusters of correlations in the world. Given these correlations and the relation for a pair of objects on Dimension 1, we know something about their relation on Dimension 2, and both of these relations are associated with the same word. For example, if we know that a pair of objects is associated by relation C on Dimension 1, the correlations in the world tell us that those objects are likely to be associated by relation G or H on Dimension 2 (Figure Figure 19a). But in the Easy language, both of these features of the objects (C on Dimension 1, G or H on Dimension 2) call forth the same word, Word 1. The same holds when we start with the Dimension 2 and predict the value on Dimension 1. Thus the words in the Easy language ``make sense.'' They should be relatively easy to learn because they are supported by the correlations in the world; inferences about uncertain values on one or the other dimension only help in the selection of the word.
Now consider the Hard language. Word 1 in the Hard language (the solid diagonal box in Figure 20), is associated with four possible pairings of relations on the two dimensions: AE, AF, DG, and DH. For Dimension 1, the situation is as with the Easy language. If we know that an input pair of objects has relation D on Dimension 1, we know that Word 1 is the appropriate word. However, unlike for the Easy language, Dimension 2 is of no help in selecting the word. Each word can take all four possible values on Dimension 2. In and of itself, the irrelevance of Dimension 2 for the Hard language does not make the language difficult. The learner could simply come to ignore that dimension and attend only to Dimension 1. On some accounts, this could actually make the learning task simpler. But consider how this language relates to what goes on in the world around the learner. Again, because of the correlations in the world, knowing the relation between a pair of objects on Dimension 1 allows one to make predictions about the relation on Dimension 2. But, unlike for the Easy language, this prediction is of no help; the relation on Dimension 2 is irrelevant for word selection. Similarly, knowing the relation between two objects on Dimension 2 allows predictions to made about Dimension 1. But for the Hard language, these predictions are of no use; the two relations on Dimension 1 correlated with a relation on Dimension 2 (for example, C and D with G) are associated with different words. Thus the Hard language doesn't make as much ``sense'' as the Easy language because it fails to capitalize on the clusters of correlations occurring in the world.
The architecture of the network used in the simulation is shown in Figure 21.
The goal of the experiment is to see how the different correlational patterns both between dimensions and with the words affect the difficulty of learning the two languages. The network was trained and tested on two different tasks. Training began with a Pre-Linguistic Phase in which the task was Non-Linguistic Pattern Completion. That is, for each trial the network was presented with a pattern on one of the visual dimensions, representing values for two different objects on that dimension, and expected to produce an appropriate pattern on the other, representing the values for those objects on that dimension. (Note that there are always two possibilities for the appropriate pattern.) The network can learn to solve this task using the connections joining the VISION and SPATIAL CONCEPTS layers or the connections between the two SPATIAL CONCEPTS layers. This phase continued for 30 repetitions of the relevant training patterns (epochs). Next, during a Linguistic Phase, Pattern Completion training was discontinued, and the networks were trained on Production for seven epochs. For this task, the network was presented with a pattern on the VISION layer, representing the values for two objects on both dimensions, and expected to output a word. Training in the Linguistic Phase began with weights of 0.0 connecting the SPATIAL CONCEPTS to the WORDS units, so the network was initially unable to produce any words.
We predicted that the Easy language would be learned faster than the Hard language during the Production phase because the Easy language categories agreed with the non-linguistic categories. That is, the correlations learned during the Pre-Linguistic Phase should support the selection of words in the Easy language but should provide no support for the selection of words in the Hard language.
During the Pre-linguistic Phase, the networks mastered the Pattern Completion task by learning weights between the two Hidden layers representing the non-linguistic correlations. Results for the Linguistic Phase are shown in Figure 22, starting with performance after one epoch.
The data were submitted to a 2(Language) 7(Epoch) analysis of variance for a mixed design. This analysis revealed a main effect of epoch, indicating that the networks get better as they receive more training. More importantly, as predicted, there is a main effect of language (p<.001). Thus, as predicted, the Easy language is learned faster than the Hard language, although by the end of the training the two networks have comparable performance. No interactions between language and epoch were found.