Counting the contents of fuzzy membranes. . . and related problems

The content of a membrane in a configuration of an ‘exact’ P system is described by a multiset. Recall that a (crisp) multiset over a set of types X is simply a mapping d : X → N. The usual interpretation of a multiset d : X → N is that it describes a set consisting of d(x) “exact” copies of each type x ∈ X. In particular, it is assumed that the set described by the multiset does not contain any element that is not a copy of some x ∈ X, or rather that we do not care about these elements, and that an element of it cannot be a copy of two different types. Now, the uncertainty in an ‘inexact’ P system may arise at the level of the (lack of) crispness of its membranes’ contents, and this can be represented using fuzzy multisets of different kinds. For instance, we could understand that the objects are imperfect, approximate copies of the reactives purportedly involved in its reactions. This would lead us to multisets describing, for every reactive v and for every degree of approximation t, how many elements there are in the membrane that are approximate copies of v with (or within) degree of approximation t. We could also understand that our lack of knowledge of the system refers to the number of copies of the (now, exact) reactives in each membrane. This would lead us to multisets describing, for every reactive v and for every n ∈ N, the degree of certainty of there being n copies of v in the membrane. And so on. Even using these generalized kinds of multisets, the basic processes of P systems based on them would be still removing, creating and moving objects within the system, and the final result of a computation would be still obtained by counting (in some way) the objects in some membrane. This calls for the development and study of cardinalities to ‘count’ the kind of fuzzy multisets used in this context. We consider here two types of cardinalities: scalar, assigning to each multiset a positive real number, and fuzzy, assigning to each multiset a fuzzy natural number, a fuzzy subset of N with certain properties. Both may have their interest in different types of P systems. Fuzzy membrane systems with scalar cardinalities would produce computable (in the membrane sense) subsets of R, while fuzzy membrane systems using fuzzy cardinalities would produce computable sets of fuzzy natural numbers.