This is a comprehensive (and friendly) introduction to membrane computing (MC), meant to offer both computer scientists and non-computer scientists an up-to-date overview of the field. That is why the set of notions introduced here is rather large, but the presentation is informal, without proofs and with rigorous definitions given only for the basic types of P systems — symbol object P systems with multiset rewriting rules, systems with symport/antiport rules, systems with string objects, tissue-like P systems, and neural-like P systems. Besides a list of (biologically inspired or mathematically motivated) ingredients/features which can be used in systems of these types, we also mention a series of results, as well as a series of research trends and topics.

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Introduction to Membrane Computing

P systems with active membranes are among the central ones in membrane computing, and they were shown to be both computationally universal (able to simulate Turing machines) and computationally efficient (able to solve hard problems in polynomial time). However, in all cases, these results were obtained by making use of several powerful features, such as membrane polarization, label changing, division of non-elementary membranes, priorities, or cooperative rules. This paper contributes to the research effort of introducing a class of P systems with active membranes having none of the features mentioned above, but still preserving the power and the efficiency. The additional feature we consider instead are the operations of endocytosis and exocytosis: moving a membrane inside a neighboring membrane, or outside the membrane where it is placed. We investigate the power and the efficiency of these systems (also using membrane division) by first proving that they can simulate (with a linear slowdown and without introducing non-determinism) rewriting P systems with 2-replication, for which the universality and the possibility of solving NP-complete problems in polynomial time are known. In this way, the universality and efficiency are also obtained for our systems. We also give a direct and simple proof for the universality result --- without using division rules (the proof uses nine membranes, but we do not know whether this number can be decreased).

P Systems with Mobile Membranes

rewriting systems on multisets (ARMS) are a simple construct in the context of Computational living systems, constituting a class of P systems that have been devised for modelling systems dynamics, with the belief that [177]: Environment + Computational living systems = Computation in agreement with the vision of living beings as computing entities whose computation purpose is keeping themselves alive. The model itself in its basic formulation is composed of a multiset of objects and a set of rewriting rules, while its intuitive meaning can be traced back to the study of abstract chemical machines [24]. As in classic P systems, the multiset of objects models the population of chemicals present into the system while rewriting rules are an abstraction of biochemical reactions. This is consistent with the idea of employing multisets as a representation formalism for undistinguishable floating chemical molecules uniformly distributed and randomly moving in a common environment. Rewriting rules model biochemical reactions between elements of the multiset defining the state of the system at each instant. A rewriting rule r over an alphabet A is a pair r = (α, β) where α, β ∈ A∗, it is denoted as α → β and its meaning is that α is replaced by β. Strings are a representation of multisets, thereby rewriting rules specified in this way can be thought as rewriting rules on multisets, but often strings α and β are substrings of the string representing the multiset encoding the state of the system. Following the terminology of [178], an ARMS is defined as a pair:

Membrane models of biological systems

Membrane computing is an emerging research field that belongs to the more general area of molecular computing, which deals with computational models inspired from bio-molecular processes. Membrane computing aims at defining models, called membrane systems or P systems, which abstract the functioning and structure of the cell. A membrane system consists of a hierarchical arrangement of membranes delimiting regions, which represent various compartments of a cell, and with each region containing bio-chemical elements of various types and having associated evolution rules, which represent bio-chemical processes taking place inside the cell. 
This work is a continuation of the investigations aiming to bridge membrane computing (where in a compartmental cell-like structure the chemicals to evolve are placed in compartments defined by membranes) and brane calculi (where one considers again a compartmental cell-like structure with the chemicals/proteins placed on the membranes themselves). We use objects both in compartments and on membranes (the latter are called proteins), with the objects from membranes evolving under the control of the proteins. Several possibilities are considered (objects only moved across membranes or also changed during this operation, with the proteins only assisting the move/change or also changing themselves). Somewhat expected, computational universality is obtained for several combinations of such possibilities. 
We also present a method for solving the NP-complete SAT problem using P systems with proteins on membranes. The SAT problem is solved in O(nm) time, where n is the number of boolean variables and m is the number of clauses for an instance written in conjunctive normal form. Thus, we can say that the solution for each given instance is obtained in linear time. We succeeded in solving SAT by a uniform construction of a deterministic P system which uses rules involving objects in regions, proteins on membranes, and membrane division. 
Then, we investigate the computational power of P systems with proteins on membranes in some particular cases: when only one protein is placed on a membrane, when the systems have a minimal number of rules, when the computation evolves in accepting or computing mode, etc. 
This dissertation introduces also another new variant of membrane systems that uses context-free rewriting rules for the evolution of objects placed inside compartments of a cell, and symport rules for communication between membranes. The strings circulate across membranes depending on their membership to regular languages given by means of regular expressions. We prove that these rewriting-symport P systems generate all recursively enumerable languages. We investigate the computational power of these newly introduced P systems for three particular forms of the regular expressions that are used by the symport rules. A characterization of ET0L languages is obtained in this context.

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Membrane systems with limited parallelism

In this paper contextual string grammar model is extended to a two dimensional array grammar model, called contextual iso-triangular array grammar. Membrane computing is a computational model and in which P system is its device. It is a cell like structure model. We continue the study of linking these two areas by introducing contextual iso triangular array P system and we investigate its power in generating the hexagonal picture languages. The main interesting study is that makes use of contextual iso triangular array erasing rules of contextual iso triangular array P system, the star picture is generated.

https://ieeexplore.ieee.org/iel7/7203316/7238030/07238531.pdf

Contextual iso-triangular array P system and hexagonal picture languages

In this paper, we continue the study of contextual and rewriting P systems. In contextual P systems, we improve a universality result avoiding the extended feature and by using rules of small weight. In rewriting P systems, we have two (rather surprising) universality results, both of them using three membranes, for non-extended systems with replicated rewriting and with leftmost rewriting, respectively.

Further Results on Contextual and Rewriting P Systems

In this paper, we continue the study of Bio-Turing machines introduced in [1]. It was proved in [1] that using two differentiated cells, and using antiport rules of weight 2, one can recognize the family 1RE. We show here that with just one differentiated cell, 1RE can be characterized, by using antiport rules of weight 2, or by using symport rules of weight 3. We also prove that RE can be characterized using arbitrary alphabets, using 2 differentiated cells, and antiport rules of weight 2. Finally, we examine the computational power when there are no differentiated cells and show that non-regular languages can be accepted.

On the power of bio-turing machines

We consider here a variant of rewriting P systems [1], where communication is controlled by the contents of the strings, not by the evolution rules used for obtaining these strings. Some new characterizations of recursively enumerable languages are obtained by means of P systems with a small number of membranes, which improves some of the known results from [1] and [4].

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Rewriting P Systems with Conditional Communication: Improved Hierarchies

In [3], P systems with active membranes were used to generate languages, in the sense of languages associated with the structure of membrane systems. Here, we analyze the power of P systems with membrane creation and dissolution restricted to elementary membranes, P systems without membrane dissolution operating according to certain output modes. This leads us to characterizations of recursively enumerable languages.

H. Ramesh

Papers

Further Results on Contextual and Rewriting P Systems

On the power of bio-turing machines

Rewriting P Systems with Conditional Communication: Improved Hierarchies

Further Results on Languages of Membrane Structures