Abstract-Fuzzy automata, whose input alphabet is a set of numbers or symbols, are a formal model of computing with values. Motivated by Zadeh's paradigm of computing with words rather than numbers, Ying proposed a kind of fuzzy automata, whose input alphabet consists of all fuzzy subsets of a set of symbols, as a formal model of computing with all words. In this paper, we introduce a somewhat general formal model of computing with (some special) words. The new features of the model are that the input alphabet only comprises some (not necessarily all) fuzzy subsets of a set of symbols and the fuzzy transition function can be specified arbitrarily. By employing the methodology of fuzzy control, we establish a retraction principle from computing with words to computing with values for handling crisp inputs and a generalized extension principle from computing with words to computing with all words for handling fuzzy inputs. These principles show that computing with values and computing with all words can be respectively implemented by computing with words. Some algebraic properties of retractions and generalized extensions are addressed as well.

Abstract. The purpose of this study was to investigate the characteristics of ACERALIVEN ™ , a commercial tourmaline, and to test its effect on plant growth. According to our analysis, ACERALIVEN ™ belongs to the Si-Al-Mg tourmaline structure. It contains some trace metals such as zirconium, potassium, and iron and able to emit far-infrared energy in the emissivity of 0.829. Introducing the tourmaline into water changes the water to be more alkaline. Tourmaline also releases negative hydroxyl ions and dissolved oxygen creating what is called as hydrogen water. Mung beans submerged with ACERALIVEN ™ shows a longer lifetime than without submerging the tourmaline. Additionally, the tourmaline can promote plant growth by removing chlorine and releasing far-infrared which is beneficial for plant's metabolism.