From distribution to replication in cooperative systems with active membranes: A frontier of the efficiency

P systems with active membranes use evolution, communication, dissolution and division(or separation) rules. They do not use cooperation neither priorities, but they haveelectrical charges associated with membranes, which can be modified by rule applications.The inspiration comes from the behaviourof living cells, who “compute” with theirproteins in order to obtain energy, create components, send information to other cells,kill themselves (in a process called apoptosis), and so on. In these models, mitosisissimulated by divisionrules (for elementary and non-elementary membranes) and meiosis,that is, membrane fission inspiration, is captured in separationrules. The parent’s objectsare replicated into both child membranes when a division occurs, while in the caseof separation, objects are distributed (according to a prefixed partition). In both cases,active membranes have been proved to be too powerful for solving computationally hardproblems in an efficient way. Due to this, polarizationless P systems withactive membraneshave been widely studied from a complexity point of view. Evolution rules simulate the transformation of components in membranes, but it iswell known that in Biology elements interact with each other in order to obtain newcomponents. In this paper, (restricted) cooperation in object evolution rules is considered,and the efficiency of the corresponding models is studied

Polarizationless P Systems with Active Membranes: Computational Complexity Aspects

P systems with active membranes, in their classical definition, make use of noncooperative rules only. However, it is well known that in living cells, proteins interact among them yielding new products. Inspired by this biological phenomenon, the previous framework is reformulated in this paper, allowing cooperation in object evolution rules, while removing electrical charges associated with membranes. More precisely, minimal cooperation in object evolution rules is incorporated in polarizationless P systems with active membranes. In this paper, the term “minimal” means that the left-hand side of such rules consists of at most two symbols, and its length is greater than or equal to the corresponding right-hand side. The computational efficiency of this kind of P systems is studied by providing a uniform polynomial-time solution to SAT problem in such manner that only division rules for elementary membranes are used and dissolution rules are forbidden. Bearing in mind that only tractable problems can be efficiently solved by families of polarizationless P systems with active membranes and without dissolution rules, passing from non-cooperation to minimal cooperation in object evolution rules amounts passing from non-efficiency to efficiency in this framework. This frontier of efficiency provides, as any other borderline does, a possible way to address the P versus NP problem.National Natural Science Foundation of China No. 61033003National Natural Science Foundation of China No. 6132010600

Evaluating space measures in P systems

P systems with active membranes are a variant of P systems where membranes can be created by division of existing membranes, thus creating an exponential amount of resources in a polynomial number of steps. Time and space complexity classes for active membrane systems have been introduced, to characterize classes of problems that can be solved by different membrane systems making use of different resources. In particular, space complexity classes introduced initially considered a hypothetical real implementation by means of biochemical materials, assuming that every single object or membrane requires some constant physical space (corresponding to unary notation). A different approach considered implementation of P systems in silico, allowing to store the multiplicity of each object in each membrane using binary numbers. In both cases, the elements contributing to the definition of the space required by a system (namely, the total number of membranes, the total number of objects, the types of different membranes, and the types of different objects) was considered as a whole. In this paper, we consider a different definition for space complexity classes in the framework of P systems, where each of the previous elements is considered independently. We review the principal results related to the solution of different computationally hard problems presented in the literature, highlighting the requirement of every single resource in each solution. A discussion concerning possible alternative solutions requiring different resources is presented

Design Patterns for Efficient Solutions to NP-Complete Problems in Membrane Computing

Many variants of P systems have the ability to generate an exponential number of membranes in linear time. This feature has been exploited to elaborate (theoretical) efficient solutions to NP-complete, or even harder, problems. A thorough review of the existent solutions shows the utilization of common techniques and procedures. The abstraction of the latter into design patterns can serve to ease and accelerate the construction of efficient solutions to new hard problems.Ministerio de Economía y Competitividad TIN2017-89842-

Further Open Problems in Membrane Computing

A series of open problems and research topics in membrane com- puting are pointed out, most of them suggested by recent developments in this area. Many of these problems have several facets and branchings, and further facets and branchings can surely be found after addressing them in a more careful manner