List of Figures

1. Logical dependency of chapters in this thesis.
2. A spiral view of mental reality (cf. Bohm [1]).
3. A wave-particle duality of relationship between the mental and physical world.
4. Two-slit experiment of electron interference.
5. Elitzer-Vaidman bomb testing problem.
6. The deviation from the targets due to phase-difference in preparing the input state ( $\theta _1, \theta _2$)
7. The deviation of the output of $U$ from the target (the classical AND-function) w.r.t phase difference of inputs. The linear preparation function is trained for AND data.
8. The deviation of the output of $U$ to the target (the classical OR-function) w.r.t phase difference of inputs. The linear preparation function is trained for OR data.
9. The output of a totally undetermined input state w.r.t phase difference.
10. The relation of the error of the XOR function to the time at which the outcome is measured. The error is defined as in Equation 6.3.
11. The relation of the error of the XOR function to the time at which the outcome is measured. The relation of each input in the training set is shown here separately.
12. The relation of the error of the AND function to the time at which the outcome is measured.
13. The deviation from the targets due to phase-difference in preparing the input state ( $\theta _1, \theta _2$)
14. Relationship between the argument ($\theta$) / absolute value ($\rho$) of the refuted second antecedent and the output
15. Relationship between the argument ($\theta$) / absolute value ($\rho$) of the asserted second antecedent and the output
16. Relationship between the argument ($\theta$) / absolute value ($\rho$) of the second antecedent when it is both asserted and refuted at the same time, and the output
17. The probability of $s$ being asserted based on counterfactual situations where $(p,q,r)=(\mathbb T, \mathbb T, \mathbb T)$. The input states are prepared with different phase ( $\theta _1, \theta _2$)
18. The probability of $s$ being asserted based on counterfactual situations where $(p,q,r)=(\mathbb T, \mathbb F, \mathbb T)$. The input states are prepared with different phase ( $\theta _1, \theta _2$)
19. The probability of $s$ being asserted based on counterfactual situations where $p$ is partially asserted.
20. The probability of proposition $s$ being asserted based on counterfactual situations where $q$ is partially asserted.
21. The detailed relation between $\rho$ and proposition $s$'s being asserted when $\theta =\pi$.
22. Rationalist NLP.
23. Empiricist NLP (Application in Machine Translation).
24. Quantum theoretical NLP.
25. The absolute squares of the output trained with the ``Barbara'' corpus. The absolute square of each component is represented by the area of its corresponding black square.
26. The phases of the output trained with ``Barbara'' corpus. The phase of each component is represented by the angle of the line in the circle.
27. The absolute squares of the output trained with concrete ``Barbara'' corpus.
28. The phases of the output trained with concrete``Barbara'' corpus.
29. The absolute squares of the output trained with full syllogistic corpus. The absolute square of each component is represented by the area of its corresponding black square.
30. The phases of the output trained with full syllogistic corpus. The phase of each component is represented by the angle of the line in the circle.
31. An example of the training set shown as a series of vectors on complex plane.
32. An example of the training set.
33. An example of the test set.
34. A typical training curve for the more complex syntax corpus.
35. An example of the training set shown as a series of vectors on a complex plane.
36. 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).
37. 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).
38. 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).
39. An example of an utterance which can not be transformed to passive form in the limited vocabulary of the language (absolute squares).
40. An example of an utterance which can not be transformed to passive form in the limited vocabulary of the language (arguments).
41. An example of the training set in the bilingual corpus that is correctly learned.
42. An example of the testing set in the bilingual corpus which is not correctly decoded.
43. Yet another example of the test set in the bilingual corpus which is not correctly decoded. The error is mainly due to residue of irrelevant eigenstates.
44. A reverse translation task.
45. An English-German dictionary map.
46. A German-English dictionary map using the reasoning operator in time reversal mode.
47. The Hamiltonian (real part).
48. The Hamiltonian (imaginary part).
49. The Hamiltonian (real part).
50. The Hamiltonian (imaginary part).
51. The Hamiltonian (real part).
52. The Hamiltonian (imaginary part).