LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

50 Francesco Bernardini, Marian Gheorghe, and Mike Holcombe
a sequential version and a parallel one, called EP systems and EPP systems,
respectively. In both variants, each transition has a specific set of evolution rules
acting upon the string objects contained in different regions of the membrane
system. The power of both variants working with string objects was investigated
as well as the suitability of EPP systems to solve hard problems. In this paper
multisets of symbol objects are considered and the corresponding of Eilenberg
P systems are called EOP systems and EOPP systems. The definition and the
behaviour of EOP and EOPP systems are very similar to those of EP and EPP
systems, respectively. More precisely, the system will start in a given state and
with an initial set of symbol objects. Given a state and a current multiset of
symbol objects, in the case of EOP systems, the machine will evolve by applying
rules associated with one of the transitions going out from the current state.
The system will resume from the destination state of the current transition. In
the parallel variant, instead of one state and a single multiset of symbol objects
we may have a number of states, called active states, that are able to trigger
outgoing transitions and such that each state hosts a different multiset of symbol
objects; all the transitions emerging from every active state may be triggered
once the rules associated with them may be applied; then the system will resume
from the next states, which then become active states. EOP systems are models
of cells evolving under various conditions when certain factors may inhibit
some evolution rules or some catalysts may activate other rules. Both variants
dealing with string objects and symbol objects have some similarities with the
grammar systems controlled by graphs [4], replacing a one-level structure, which
is the current sentential form, with a hierarchical structure defined by the membrane
system. On the other hand, these variants of P systems may be viewed
as Eilenberg machines [6] having sets of evolution rules as basic processing relationships.
EP and EOP systems share some similar behaviour with Eilenberg
machines based on distributed grammar systems [8].
Eilenberg machines, generally known under the name of X machines [6], have
been initially used as a software specification language [9], further on intensively
studied in connection with software testing [10]. Communicating X-machine systems
were also considered [2] as a model of parallel and communicating processes.
In this paper it is investigated the power of EOP and EOPP systems in
connection with three parameters: number of membranes, states and set of distributed
rules. It is proven that the family of Parikh sets of vectors of numbers
generated by EOP systems with one membrane, one state and one single set
of rules coincides with the family of Parikh sets of context-free languages. The
hierarchy collapses when at least one membrane, two states and four sets of
rules are used and in this case a characterization of the family of Parikh sets of
vectors associated with ET0L languages is obtained. It is also shown that every
EOP system may be simulated by an EOPP system and EOPP systems may
be used for solving NP-complete problems. In particular linear time solutions
are provided for the SAT problem. The last result relies heavily on similarities
between EOPP systems and P systems with replicated rewriting [11] and EPP
systems [1].