Our first experiment is based on the argument form, traditionally called ``Barbara:''
all m are p and all s are m -> all s are p
The corpus is built by replacing the term m,p,s with three general symbols a,b,c and permuting the order of the three symbols. An additional term 
is reserved for testing purposes. Thus, there are 6 sentences in the corpus and the vocabulary of the corpus is (all, are, and, a, b, c, d). The corpus is then subject to the quantum architecture proposed above and trained using the conjugate gradient method starting with a small random parameter vector. In 20 experiments, run with different random start parameters, the architecture seems to have difficulty learning all these sentences. That is, the system cannot manage to generate all of the six sentences without error. There are always two sentences that cannot be learned, although we do not know in advance which two. This depends on the random initial vector in the optimization method. A closer look at the result reveals an interesting pattern. In fact, all the results agree with the following pattern:
all a are b and all c are a -> all c are b
all a are c and all b are a -> all b are c
all b are a and all c are b -> all are* ++
all b are c and all a are b -> all are* ++
all c are a and all b are c -> all b are a
all c are b and all a are c -> all a are b
in which the incorrect and missing words are marked with * and +, respectively. The absolute squares and phases are shown in Figure 7.4 and Figure 7.5. In Figure 7.4, the absolute square of each component is represented by the area of its corresponding black square. In Figure 7.5, the phase of each component is represented by the angle of the line in the circle (as the hand of a clock). The upper rows are the target sentences and the lower the actual outputs of the system.
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A closer look at the result reveals a not very surprising fact. If one replaces a,b,c above with concrete categories, one will notice that the only ``meaningful'' solution (one that makes sense to our intuitive common sense) is that a,b,c must be exact synonyms, for otherwise the following four conclusions:
all c are b
all b are c
all b are a
all a are b
cannot simultaneously be true. In this case, both ``all a are c'' and ``all c are a'' must be true. Indeed, if the threshold of the combinatorial optimizer is fixed to 0.05 (instead of subject to combinatorial optimization), the third and the fourth sentences in the corpus will be decoded as
all b are a and all c are b -> all a* are c*
all b are c and all a are b -> all c* are a*
where both confer meaningful state of affairs, although syntactically incorrect.
Observing this fact, another corpus is constructed where the symbols a,b,c,d are replaced by concrete categories (whales, dolphins, mammals, animals). Moreover, the sentences are arranged in such a way that they reflect the ``meaningful'' state of affairs, as far as our knowledge about the world is concerned. The corpus is shown below:
all whales are mammals and all mammals are animals -> all whales are animals all mammals are animals and all whales are mammals -> all whales are animals all dolphins are mammals and all mammals are animals -> all dolphins are animals all mammals are animals and all dolphins are mammals -> all dolphins are animals all dolphins are whales and all whales are animals -> all dolphins are animals all whales are animals and all dolphins are whales -> all dolphins are animals all dolphins are whales and all whales are mammals -> all dolphins are mammals all whales are mammals and all dolphins are whales -> all dolphins are mammals
Not surprisingly, this time the architecture can learn all the sentences. The absolute squares and phases are shown in Figure 7.6 and 7.7 respectively.
