By A. S. Kechris, D. A. Martin, Y. N. Moschovakis

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**Example text**

A conversational model has several key properties: the system internal representation of the world and information must be modality free but able to be conveyed by way of any one of several modalities; the functions of the system must be modality free but able to be realized in any one of a number of diﬀerent surface behaviors in a number of diﬀerent modalities; the representations of conversation cannot be all symbolic because cultural, social and technical conventions might not be able to be captured in logical form; and co-occurrences of surface-level behaviors carry meaning over that carried by each of the constituent behaviors.

References 1. -S. R. -T. Sun, E. Mizutani, Neuro Fuzzy and Soft Computing, A Computational Approach to Learinig and Machine Intelligence Prentice Hall Upper Saddle River, NJ 07458. 2. E. C. Ezin, Neural Networks and Neural Fuzzy Systems for Speech Applications, PhD. Dissertation, Institut de Math´ematiques et de Sciences Physiques, Universit´e d’Abomey-Calavi, B´enin, March 2001. 3. A. Esposito, E. C. Ezin, C. Reyes-Garcia, Speech Noise Cancellation, Based on a Neuro-Fuzzy System: Experimental Results, in Proceedings of the IEEE on International Workshop on Intelligent Signal Processing, pp.

Deﬁnition 2 (optimality condition). A fuzzy interface Λ = Λ (Φ, ≺) is optimal if there exists a function: N : [0, 1] n → X (4) N (Λ (x)) = x (5) such that: that is, Λ is invertible. t. A (x) > 0 3 Bi-monotonic Fuzzy Sets The above mentioned properties are crucial for the design of interfaces that do not hamper the overall performance of the fuzzy model. As a consequence, an important issue concerns the study of the conditions that apply to families of fuzzy sets in order to induce optimal interfaces.

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