More complicated corpus

While Chalmers' corpus is useful to demonstrate that a connectionist architecture has interesting cognitive aspects, it misses many interesting points of natural language. For one thing, natural language utterances are well-formed and may refer to state of affairs^7.7. Thus, the corpus used in the experiment discussed in this section was modified and extended, as described in the following paragraphs.

In this more complex corpus, an additional conjunction and is introduced and full verb conjugations are taken into account. There are 240 sentences in total. The vocabulary is summarized in Table 7.2. There is considerable variation added to the corpus as compared to that used in [66]. For instance, the past participle of hit is hit, but that of love is loved. In addition, the conjunction and introduces the plurality of actor and recipient. A strategy that manipulates symbols simply on account of position will not work in this case. Moreover, hit (as past participle) and hit (as verb present plural) have the same eigenstate. In other words, they are indistinguishable according to the formulation operator $F$ of common English. As for the verb love, there are three eigenstates associated with it (loves, love, loved). Some examples are listed below.

    helen hits john <-> john is hit by helen
    helen loves john <-> john is loved by helen
    john kills diane and michael <-> diane and michael are killed by john

Table 7.2: Vocabulary used in the more complex syntax corpus. Words marked with * are homonyms that are represented by identical eigenstate in the vocabulary.

Category	Instances
Person	john michael helen diane
Action	kill love betray hit
Action Conjugated	kills loves betrays hits
Past Participle	killed loved betrayed hit*
Conjunction	and
Misc.	is are by

Fifty-six sentences (23% of the corpus) have been randomly chosen as the training set. The other 184 sentences are reserved as test. Using the same optimization algorithm as in the previous section, the quantum mechanical architecture can learn all the utterances in the training set. The architecture can generalize the task on all sentences in the test set (generalization rate is 100%). Given the complexity of the corpus (in comparison with that used in Chalmers' study) and the small size of the training set, this is a very encouraging result.

A typical training curve is shown in Figure 7.13. Using the conjugate gradient method, for around 100 epochs the architecture can learn all the instances in the training set. To visualize more details, the output of an utterance in the training set,

    helen and diane hit john -> john is hit by helen and diane

is shown in Figure 7.14, in which each component is illustrated as a vector on the complex plane. A comparison of the output to the target utterance is shown in Figure 7.15. The first two rows are the absolute squares (represented by the area of the black disks) of the targets and outputs respectively. As can be seen in the figure, seven eigenstates have the most significant coefficients. The lower two rows are the phases of target and output vector, respectively.

Figure 7.13: A typical training curve for the more complex syntax corpus.

Figure 7.14: An example of the training set shown as a series of vectors on a complex plane.

Figure 7.15: An example of the training set (the first and the second rows: absolute squares of the target and the output, respectively; the third and the fourth: the phases of the target and the output).

An output of an utterance in the test set,

    john kills diane and michael <-> diane and michael are killed by john

is shown in Figure 7.16. The quantum architecture has never seen the utterance, it is remarkable that the differences in absolute squares and phases are hardly noticeable.

Figure 7.16: An example of the test set (the first and the second rows: absolute squares of the target and the output, respectively; the third and the fourth: the phases of the target and the output).

Theoretically, a quantum mechanical architecture can perform the reverse computation if time is reversed. In this case, if the output state of affairs is subject to the inverse of the unitary operator using

$$U^{-1}=e^{iHt\over \hbar}$$

one should have the original input utterance at the input side. This is, however, the ideal case only if the output state of affairs is not formulated. If an orthographic output utterance is prepared according to the same procedure, there must be some minor difference between it and the genuine output state of affairs. This is shown in Figure 7.17. In this figure, the output utterance of same example of the training set above is prepared according to the standard procedure and then subject to the inverse of the unitary operator. The absolute squares and phases of the processed input are shown in the second and the fourth rows; that of the original input state of affairs is shown in the first and the third rows.

Figure 7.17: An example of the training set reverse in time (the first and the second rows: absolute squares of target and output, respectively; the third and the fourth: phases of target and output).

Figure 7.18: An example of an utterance which can not be transformed to passive form in the limited vocabulary of the language (absolute squares).

Figure 7.19: An example of an utterance which can not be transformed to passive form in the limited vocabulary of the language (arguments).

If, however, the input is a well-formed sentence but cannot be transformed to the passive form in the language, such as,

    john kills

the system can still arrive at some reasonable solution. This is shown in Figure 7.18 and Figure 7.19. As can be seen in Figure 7.18, four eigenstates (is, killed, by, john) have the most significant components in the end state of affairs vector. In a sense, it suggests a well-formed utterance:

    somebody is killed by john

However, in the miniature language we use here, it is not possible to identify who is killed.