Abstract. Living cells are extremely well-organized autonomous systems, consisting of discrete interacting components. Key to understanding and modeling their behavior is modeling their system organization. Four distinct chemical toolkits (classes of macromolecules) have been characterized, each combinatorial in nature. Each toolkit consists of a small number of simple components that are assembled (polymerized) into complex structures that interact in rich ways. Each toolkit abstracts away from chemistry; it embodies an abstract machine with its own instruction set and its own peculiar interaction model. These interaction models are highly effective, but are not ones commonly used in computing: proteins stick together, genes have fixed output, membranes carry activity on their surfaces. Biologists have invented a number of notations attempting to describe these abstract machines and the processes they implement. Moving up from molecular biology, systems biology aims to understand how these interaction models work, separately and together. 1

P systems are parallel Molecular Computing models based on processing multisets of objects in cell-like membrane structures. Various variants were already shown to be computationally universal, equal in power to Turing machines. In this paper one proposes a class of P systems whose membranes are the main active components, in the sense that they directly mediate the evolution and the communication of objects. Moreover, the membranes can multiply themselves by division. We prove that this variant is not only computationally universal, but also able to solve NP complete problems in polynomial (actually, linear) time. We exemplify this assertion with the well-known SAT problem.

The P systems were recently introduced as distributed parallel computing models of a biochemical type. Multisets of objects are placed in a hierarchical structure of membranes and they evolve according to given rules, which are applied in a synchronous manner: at each step, all objects which can evolve, from all membranes, must evolve. We consider here the case when this restriction is removed. As expected, unsynchronized systems (even using catalysts) are weaker than the synchronized ones, providing that no priority relation among rules is considered. The power of P systems is not diminished when a priority is used and, moreover, the catalysts can change their states, among two possible states for each catalyst.

We look at 1-region membrane computing systems which only use rules of the form Ca Cv, where C is a catalyst anoncatalW:k and v is a(possiblW:kky string ofnoncatal sts. There are norulk of the form a v. Thus, we can think of these systems as"purelx catalxyMWe consider two types: (1) when theinitial configuration containsonl onecatalxkA and (2) when theinitial configuration contains mulains catalsy-" We show that systems of the first type are equivalyM to communication-free Petri nets, which are aly equivalyM to commutative context-free grammars. They defin eprecisel-kq semilel- sets. ThispartialkyM-:W"k an open question (in: WMC-CdeA'02, Lecture Notes in Computer Science,vol 2597, Springer,Berlge 2003, pp. 400 -- 409; Computational" universal P systems without priorities: two catal"k# are su#cient, availt,y at http://psystems.disco.unimib.it, 2003). Systems of the second type define exactl""# recursivelM-kq""yl sets oftupl# (i.e., Turing machinecomputablWk Weal" studyan extended model where therul- are of the form q :(p; Ca Cv) (where q and p are states), i.e., the appl":xyMk of therul- is guided bya #nite-statecontrol For thisgeneral"yM model type (1) aswel as type (2) with some restriction correspond to vector addition systems. Finally, we briefly investigate the closure properties of catalytic systems.

A variant of P systems, recently introduced, considers membranes which can multiply by division. Two types of division are considered: division for elementary membranes (i.e. membranes not containing other membranes inside) and division for non-elementary membranes. In two recent papers it is shown how to solve the Satisfiability problem and the Hamiltonian Path problem (two well known NP complete problems) in linear time with respect to the input length, using this variant of P systems. We show in this paper that P systems with division for elementary membranes only suffice to solve these two problems in linear time. What about the possibility of solving NP complete problems in polynomial time using P systems without membrane division? We show, moreover, that (if P 6= NP ) Deterministic P Systems without membrane division are not able to solve NP complete problems in polynomial time.

1. For 1-membrane catalytic systems (CS's), the sequential version is strictlyweaker than the parallel version in that the former defines (i.e. generates) exactly the semilinear sets, whereas the latter is known to define nonrecursivesets. 2. For 1-membrane communicating P systems (CPS's), the sequential versioncan only define a proper subclass of the semilinear sets, whereas the parallel version is known to define nonrecursive sets.3. Adding a new type of rule of the form: ab! axbyccomedcome to the CPS(a natural generalization of the rule ab! axbyccome in the original model),where x; y 2 fhere; outg, to the sequential 1-membrane CPS makes itequivalent to a vector addition system.

Summary. In this paper we study some computational properties of spiking neural P systems. In particular, we show that by using nondeterminism in a slightly extended version of spiking neural P systems it is possible to solve in constant time both the numerical NP–complete problem Subset Sum and the strongly NP–complete problem 3-SAT. Then, we show how to simulate a universal deterministic spiking neural P system with a deterministic Turing machine, in a time which is polynomial with respect to the execution time of the simulated system. Surprisingly, it turns out that the simulation can be performed in polynomial time with respect to the size of the description of the simulated system only if the regular expressions used in such a system are of a very restricted type. 1

Abstract. Membrane computing is a branch of molecular computing that aims to develop models and paradigms that are biologically motivated. It identifies an unconventional computing model, namely a P system, from natural phenomena of cell evolutions and chemical reactions. Because of the nature of maximal parallelism inherent in the model, P systems have a great potential for implementing massively concurrent systems in an efficient way that would allow us to solve currently intractable problems. In this paper, we look at various models of P systems and investigate their model-checking problems. We identify what is decidable (or undecidable) about model-checking these systems under extended logic formalisms of CTL. We also report on some experiments on whether existing conservative (symbolic) model-checking techniques can be practically applied to handle P systems with a reasonable size. 1

In this paper we propose a new way to represent P systems with active membranes based on Logic Programming techniques. This representation allows us to express the set of rules and the configuration of the P system in each step of the evolution as literals of an appropriate language of first order logic. We provide a Prolog program to simulate the evolution of these P systems and present some auxiliary tools to simulate the evolution of a P system with active membranes using 2-division which solves the SAT problem following the techniques presented in 10).

related. I am confident that at their interface great discoveries await those who seek them. ” (L.Adleman, [3]) 1. FOREWORD Natural computing is the field of research that investigates models and computational techniques inspired by nature and, dually, attempts to understand the world around us in terms of information processing. It is a highly interdisciplinary field that connects the natural sciences with computing science, both at the level of information technology and at the level of fundamental research, [98]. As a matter of fact, natural computing areas and topics come in many flavours, including pure theoretical research, algorithms and software applications, as well as biology, chemistry and physics experimental laboratory research. In this review we describe computing paradigms abstracted