Whole Cells - Molecular Theory of the Living Cell - page 467

by definition, it must not be a member of itself. If R is not a member of itself, again
by definition, it must be a member of itself. Thus R is both X and not-X at the same
time, violating the principle of crisp logic, the law of excluded middle.
Baez ( http://math.ucr.edu/home/baez/week73.html ) provides another useful
definition:
A category is something just as abstract as a set, but a bit more structured. It is not a mere
collection of objects; there are also morphisms between objects, in this case the functions
between sets . A category consists of a collection of “objects” and a collection of

“morphisms.” Every morphism f has a “source” object and a “target” object. If the source
of f is X and its target is Y, we write f: X ! Y. In addition, we have:
(a) Given a morphism f: X ! Y and a morphism g: Y ! Z, there is a morphism fg:
X ! Y, which we call the “composition” of f and g
(b) Composition is associative: (fg)h ¼ f(gh)
(c) For each object X there is a morphism 1 X :X ! X, called the “identity” of X. For any f:
X ! Y we have 1 X f ¼ f1 X ¼ f. (12.56)
The first two three rows of Table 12.19 may correspond to the O (i.e., objects)
and hom (i.e., morphism) components of a category defined in Statement 12.56.
(3) In analogy to the Turing machine, it may be convenient to refer to computing
machines based on words or “linguistic variables” as “Zadeh machine.” A linguistic
variable consists of a “linguistic term” and a positive real “number,” between 0 and
1, indicating a degree of membership to a fuzzy set (for a definition of fuzzy set, see
Sect. 5.2.5 ) (Kosko 1993; Zadeh 1996c). Frisco and Ji (2002, 2003) applied the
biological concepts of the conformon (Ji 1974b, 2000) and the cell membrane to
modeling computability. It is shown that this so-called conformons-P system
belongs to the universality class of the Turing machine. The suffix P stands for
the P-system, a biological membrane-inspired computational model developed by
G. Paun and his school in the 1990s (Paun 2002; Paun et al. 2002). The conformons-
P system can be viewed as a formal model of the computing aspect of the living cell
in contrast to the Bhopalator (Ji 1985a, b) which is its molecular model of the living
cell as a whole . We can alternatively refer to the conformons-P system as the
conformon-P machine , whenever convenient.
The conformon of the conformons-P system is a pair, [X, x], where X is the
“name” and x the “value” of the conformon. X and x thus defined are the
abstractions of their biological counterparts, information and energy , that constitute
the complementary pair of the original conformon . Conformon [X, x] is formally
identical with Zadeh's linguistic variable , if X and x are equated with the linguistic
term and the degree of membership to a fuzzy set , respectively. Consequently, the
conformon-P machine can be reduced to the Zadeh machine, as it can be reduced
(or related) to the Turing machine (Fig. 12.39 ).
(4) As pointed out by fuzzy theorists (Zadeh 1996a, b, c; Kosko 1993; Yen and
Langari 1999), the Turing machine is based on crisp sets , while Zadeh machine is
rooted in fuzzy sets . I here postulate that the set of molecules underlying the
conformon-P machine is both crisp (e.g., nucleotide sequences of DNA) and fuzzy
(e.g., ensemble of conformations belonging to a given amino acid sequence of a
protein) (Ji 2004a). Because of its “A AND not-A” nature (Kosko 1993), it may be
asserted that the Bhopalator can act as the ultimate source or ground for both the

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