13th International Conference on Membrane Computing - MTA Sztaki

An analysis of correlative and quantitative causality in P systems
computing inspired by nature. Membrane computing abstracts computing models
from the architecture and the functioning of living cells, as well as from the
organization of cells in tissues, organs and, brain.
A transition P system is the simplest form of membrane system consisting
of a hierarchy of nested membranes, each membrane containing objects, rules
and possibly other membranes. The hierarchy of membranes models the compartments
of the biological pathway, the objects represent the species in each
compartments, and the rules correspond to the biochemical reactions forming
the pathways. The rules are considered to be applied in a maximally parallel
manner. The simplest form of transition P system is the one with only one membrane,
which basically consists of a set of rules and possibly an initial multiset
of objects.
A multiset w over a set A is a function w : A → N from A to the set of
natural numbers N; the multiplicity of an element a ∈ A is w(a). We denote
the empty multiset having multiplicity 0 for all a ∈ A by 0 A , or simply by 0 if
the set A is clear from the context. When describing a multiset characterized
by, for example, w(a) = 4, w(b) = 2 and w(c) = 0 for c ∈ A\{a, b}, we use the
representation 4a + 2b. For two multisets v, w over A we say that v is contained
in w if v(a) ≤ w(a) for all a ∈ A, and we denote this by v ≤ w. If v ≤ w, we
can define w − v by (w − v)(a) = w(a) − v(a). For two multisets v and w we
use the notation v ∩ w for the largest multiset contained in both v and w. In
other words, v ∩ w is defined by (v ∩ w)(a) = min{v(a), w(a)}, for all a ∈ A. We
denote by v\w the multiset v − v ∩ w. We sometimes use the notation a ∈ w to
denote the fact that w(a) > 0, i.e., the multiset w contains at least one a.
Formally, a transition P system with only one membrane is a tuple Π =
(O, R, u 0 ), where O is an alphabet of objects, R is a set of rules, while u 0 is a
multiset of objects which is initially in the membrane. Each rule r has the form
r : u → v, where u and v are multisets of objects and u is non-empty.
We use multisets of objects over O to represent resources available or being
produced inside the membrane. Then, u 0 evolves by applying the rules in R. We
use the notation lhs(r) for the left hand side u of a rule of form r : u → v and
similarly rhs(r) for the right hand side v. Therefore, by the application of the
rule r the lhs(r) is being subtracted from u 0 , if possible, and the rhs(r) is added.
In this way, the rules application models biological reactions. These notations
are extended naturally to multisets of rules.
We define causality directly, for a multiset of objects v. Note that this definition
differs from the presentation found in [2], where it has been obtained as a
theorem describing (global) causality. Here we present it directly as a definition,
in the interest of brevity.
Definition 1 (Quantitative Causality) A multiset of rules G is called a cause
for a multiset of objects v whenever the following hold:
– there is no rule r such that lhs(r) ≤ v\rhs(G);
– rhs(G) ∩ v > rhs(G − r) ∩ v for any rule r ∈ G.
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