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13th International Conference on Membrane Computing - MTA Sztaki

13th International Conference on Membrane Computing - MTA Sztaki

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Sublinear-space P systems with active membranes<br />

Each instantaneous c<strong>on</strong>gurati<strong>on</strong> of a P system with active membranes is described<br />

by the current membrane structure, including the electrical charges, together<br />

with the multisets located in the corresp<strong>on</strong>ding regi<strong>on</strong>s. A computati<strong>on</strong><br />

step changes the current c<strong>on</strong>gurati<strong>on</strong> according to the following set of principles:<br />

Each object and membrane can be subject to at most <strong>on</strong>e rule per step,<br />

except for object evoluti<strong>on</strong> rules (inside each membrane several evoluti<strong>on</strong><br />

rules can be applied simultaneously).<br />

The applicati<strong>on</strong> of rules is maximally parallel : each object appearing <strong>on</strong> the<br />

left-hand side of evoluti<strong>on</strong>, communicati<strong>on</strong>, dissoluti<strong>on</strong> or elementary divisi<strong>on</strong><br />

rules must be subject to exactly <strong>on</strong>e of them (unless the current charge<br />

of the membrane prohibits it). The same principle applies to each membrane<br />

that can be involved to communicati<strong>on</strong>, dissoluti<strong>on</strong>, or elementary divisi<strong>on</strong><br />

rules. In other words, the <strong>on</strong>ly objects and membranes that do not evolve<br />

are those associated with no rule, or <strong>on</strong>ly to rules that are not applicable<br />

due to the electrical charges.<br />

When several c<strong>on</strong>icting rules can be applied at the same time, a n<strong>on</strong>deterministic<br />

choice is performed; this implies that, in general, multiple possible<br />

c<strong>on</strong>gurati<strong>on</strong>s can be reached after a computati<strong>on</strong> step.<br />

In each computati<strong>on</strong> step, all the chosen rules are applied simultaneously<br />

(in an atomic way). However, in order to clarify the operati<strong>on</strong>al semantics,<br />

each computati<strong>on</strong> step is c<strong>on</strong>venti<strong>on</strong>ally described as a sequence of microsteps<br />

as follows. First, all evoluti<strong>on</strong> rules are applied inside the elementary<br />

membranes, followed by all communicati<strong>on</strong>, dissoluti<strong>on</strong> and divisi<strong>on</strong> rules involving<br />

the membranes themselves; this process is then repeated to the membranes<br />

c<strong>on</strong>taining them, and so <strong>on</strong> towards the root (outermost membrane).<br />

In other words, the membranes evolve <strong>on</strong>ly after their internal c<strong>on</strong>gurati<strong>on</strong><br />

has been updated. For instance, before a membrane divisi<strong>on</strong> occurs, all chosen<br />

object evoluti<strong>on</strong> rules must be applied inside it; this way, the objects<br />

that are duplicated during the divisi<strong>on</strong> are already the nal <strong>on</strong>es.<br />

The outermost membrane cannot be divided or dissolved, and any object<br />

sent out from it cannot re-enter the system again.<br />

A halting computati<strong>on</strong> of the P system Π is a nite sequence of c<strong>on</strong>gurati<strong>on</strong>s<br />

C = (C 0 , . . . , C k ), where C 0 is the initial c<strong>on</strong>gurati<strong>on</strong>, every C i+1 is reachable<br />

by C i via a single computati<strong>on</strong> step, and no rules can be applied anymore in<br />

C k . A n<strong>on</strong>-halting computati<strong>on</strong> C = (C i : i ∈ N) c<strong>on</strong>sists of innitely many<br />

c<strong>on</strong>gurati<strong>on</strong>s, again starting from the initial <strong>on</strong>e and generated by successive<br />

computati<strong>on</strong> steps, where the applicable rules are never exhausted.<br />

P systems can be used as recognisers by employing two distinguished objects<br />

yes and no; exactly <strong>on</strong>e of these must be sent out from the outermost membrane<br />

during each computati<strong>on</strong>, in order to signal acceptance or rejecti<strong>on</strong> respectively;<br />

we also assume that all computati<strong>on</strong>s are halting. If all computati<strong>on</strong>s starting<br />

from the same initial c<strong>on</strong>gurati<strong>on</strong> are accepting, or all are rejecting, the P system<br />

is said to be c<strong>on</strong>uent. If this is not necessarily the case, then we have a<br />

371

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