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13th International Conference on Membrane Computing - MTA Sztaki

13th International Conference on Membrane Computing - MTA Sztaki

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M.A. Martínez-del-Amor, I. Pérez-Hurtado, M. García-Quism<strong>on</strong>do,<br />

L.F. Macías-Ramos, L. Valencia-Cabrera, A. Romero-Jiménez, C. Graciani,<br />

A. Riscos-Núñez, M.A. Colomer, M.J. Pérez-Jiménez<br />

– For each j (1 ≤ j ≤ m), M 1j , . . . , M qj are strings over Γ , describing<br />

the multisets of objects initially placed in the q regi<strong>on</strong>s of µ, within the<br />

envir<strong>on</strong>ment e j .<br />

In other words, a system as described in the previous definiti<strong>on</strong> can be<br />

viewed as a set of m envir<strong>on</strong>ments e 1 , . . . , e m linked between them such that<br />

they form a directed graph G. Each envir<strong>on</strong>ment e j c<strong>on</strong>tains a P system,<br />

Π j = (Γ, µ, R Πj , M 1j , . . . M q,j ), of degree q, where every rule r ∈ R has a<br />

computable functi<strong>on</strong> f r,j (specific for envir<strong>on</strong>ment j) associated with it. The set<br />

of rules r ∈ R of Π having included the functi<strong>on</strong>s f r,j is denoted by R Πj , for<br />

each envir<strong>on</strong>ment e j .<br />

A c<strong>on</strong>figurati<strong>on</strong> of the system at any instant t is a tuple of multisets of<br />

objects present in the m envir<strong>on</strong>ments and at each of the regi<strong>on</strong>s of each Π j ,<br />

together with the polarizati<strong>on</strong>s of the membranes in each P system. At the initial<br />

c<strong>on</strong>figurati<strong>on</strong> of the system we assume that all envir<strong>on</strong>ments are empty and all<br />

membranes have a neutral polarizati<strong>on</strong>.<br />

We assume that a global clock exists, marking the time for the whole system,<br />

that is, all membranes and the applicati<strong>on</strong> of all rules (from R E and all R Πj )<br />

are synchr<strong>on</strong>ized in all envir<strong>on</strong>ments.<br />

The P system can pass from <strong>on</strong>e c<strong>on</strong>figurati<strong>on</strong> to another by using the rules<br />

from ⋃ m<br />

j=1 R Π j<br />

∪ R E as follows: at each transiti<strong>on</strong> step, the rules to be applied<br />

are selected according to the probabilities assigned to them, and all applicable<br />

rules are simultaneously applied in a maximal way.<br />

p<br />

When a communicati<strong>on</strong> rule (x) ej −−−→(y 1 ) ej1 . . . (y h ) ejh between envir<strong>on</strong>ments<br />

is applied, object x passes from e j to e j1 , . . . , e jh possibly modified into<br />

objects y 1 , . . . , y h respectively. At any moment t (1 ≤ t ≤ T ) for each object<br />

x in envir<strong>on</strong>ment e j , if there exist communicati<strong>on</strong> rules whose left-hand side is<br />

(x) ej , then <strong>on</strong>e of these rules will be applied. If more than <strong>on</strong>e communicati<strong>on</strong><br />

rule can be applied to an object, the system randomly selects <strong>on</strong>e, according to<br />

their probability which is given by p(t).<br />

For each j (1 ≤ j ≤ m) there is just <strong>on</strong>e further restricti<strong>on</strong>, c<strong>on</strong>cerning the<br />

c<strong>on</strong>sistency of charges: in order to apply several rules of R Πj simultaneously to<br />

the same membrane, all the rules must have the same electrical charge <strong>on</strong> their<br />

right-hand side.<br />

3 Direct Distributi<strong>on</strong> Based <strong>on</strong> C<strong>on</strong>sistent Blocks<br />

Algorithm (DCBA)<br />

In this secti<strong>on</strong> we describe the Direct distributi<strong>on</strong> based <strong>on</strong> C<strong>on</strong>sistent Blocks<br />

Algorithm (DCBA), together with some auxiliary definiti<strong>on</strong>s and properties<br />

necessary for it. The DCBA is introduced in order to solve some distorti<strong>on</strong>s<br />

generated by the previous algorithm, DNDP. First, the number of applicati<strong>on</strong>s<br />

for competing rules (with overlapping left-hand sides) is proporti<strong>on</strong>ally<br />

distributed, avoiding the distorti<strong>on</strong> of using a random order over the rules,<br />

as made in the DNDP algorithm. Moreover, the management of c<strong>on</strong>sistency<br />

294

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