Mobile Membranes

Mobile membranes represent a model of computation inspired from the biological movement provided by endocytosis and exocytosis in the living cells. This paper presents a survey of the results treating the computational power of the mobile membranes, their efficiency in solving NP-complete problems, and connections with other formal approaches able to handle mobility.


I. INTRODUCTION
Taking inspiration from the communication and mobility in cells, new computer architectures performing parallel computations are defined and studied. Bio-inspired computing has the possibility to use features as massive parallelism, reversible computations and non-determinism. The architecture and the behaviour of living cells motivate the membrane systems (also called P systems) which are defined in [57] and [59] by using formal languages theory, computability theory and complexity theory. Membrane systems can appear as a tree (cell-like [59]) or as a graph (tissuelike [53] and neural-like [41]). The membrane systems with a cell-like structure have the following characteristics: (i) an hierarchical structure of membranes (nested or disjoint) with a skin provided by the unique outermost membrane; an elementary membrane does not contain other membranes, while a composite one does; (ii) each membrane may contain multisets of objects and some rules that manipulate objects and membranes. The main research directions considered for membrane systems deal with the computational power with respect to the Turing machine, and with the efficient algorithms to solve NP-complete problems in polynomial time (and exponential space). Another research direction is the connections with other approaches (e.g., process calculi) in which the concept of mobility is central.
The fluid-mosaic model [65] is the currently accepted model for the structure of a cell membrane. The increased fluidity of the outer membrane allowed the development of two mechanisms, called endocytosis and exocytosis. Endocytosis and exocytosis are complementary operation The associate editor coordinating the review of this manuscript and approving it for publication was Eduardo Rosa-Molinar . that allow substances to enter (endocytosis) or exit (exocytosis) the cell through membrane-bounded vesicles. P systems with mobile membranes [51] are the first class of membrane systems to consider the operations of exocytosis and endocytosis.
After presenting some notions used throughout the paper (Section II) and existing variants of mobile membranes (Section III), the paper proceeds by presenting a survey of the results treating their efficiency in solving NP-complete problems (Section IV), computational power of the mobile membranes (Section V), and connections with other formal approaches able to handle mobility (Section VI).

II. PRELIMINARIES A. FORMAL LANGUAGES
Some notions from the theory of formal languages [35], [64] are presented in what follows. Consider O = {a 1 , . . . , a n } an alphabet, and λ an empty string. The set of all strings over the alphabet O is denoted by O * . The set O * with concatenation and unit λ is a monoid. Given x ∈ O * , the number of occurrences of symbol a in x is denoted by |x| a . A string over O (modulo permutation) identifies a multiset. The Parikh vector over O is defined as ψ O : O * → N n where ψ O (x) = (|x| a 1 , . . . , |x| a n ) for all x ∈ O * . In a similar manner, ψ V (L) = {ψ O (x) | x ∈ L} and PsFL = {ψ O (L) | L ∈ FL} denote the Parikh vector over a language L and a family FL of languages, respectively. RE and PsRE denote the family of recursively enumerable languages and its Parikh image (Turing computable sets of vectors of natural numbers), respectively.
An E0L system is a tuple G = (O, T , ω, R) such that O and T ⊆ O are the alphabet and terminal alphabet, respectively, VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ ω ∈ O * represents the initial axiom, and } is a finite set of rules. The set of rules is constructed such that for all a ∈ O it exists a → v ∈ R. Given w 1 , w 2 ∈ O * with w 1 = a 1 . . . a n and w 2 = v 1 . . . v n , then w 2 can be obtained from w 1 (denoted by w 1 ⇒ w 2 ) if exists a i → v i ∈ R, 1 i n. The set L(G) = {x ∈ T * | ω ⇒ * x} represents the generated language.
An ET 0L systems is a tuple G = (O, T , ω, R 1 , . . . R n ) if for all i such that 1 i n it holds that O, T , ω and R i form an E0L system. Each set R i with 1 i n is called a table. The set L(G) = {x ∈ T * | ω ⇒ R j 1 · · · ⇒ R jm w m = x} with m 0, 1 j i n and 1 i m represents the generated language. PsET 0L denotes the Parikh image of the ET 0L systems.
A context-free matrix grammar without appearance checking is denoted by G = (N , T , S, M ), where N and T with N ∩ T = ∅ are non-terminals and terminals alphabets (respectively), S ∈ N is the initial axiom, } is a finite set of matrices formed from context-free rules. Given a string w ∈ (N ∪ T ) * , a matrix m : (r 1 , . . . , r n ) is executed by applying the rules r 1 , . . ., r n in the given order. An execution denoted by w → m z happens if it exists m : | S ⇒ * x} is the language generated by G, and MAT is the family of languages generated by the context-free matrix grammars.
Other useful notations: N denotes the set of natural numbers, NRE the family of recursively enumerable sets, NCS the family of context-sensitive sets, NO λ the family of ordered sets, NMAT the family of sets of matrix grammars without appearance checking, and NFIN the family of finite sets of natural numbers. According to [35], we have: NMAT λ , NMAT ac , NCS NO λ ⊂ NRE.

B. BRANE CALCULI
The biological operations of exocytosis, endocytosis and mitosis inspired the creation of two brane calculi called PEP (standing for Pino/Exo/ Phago) and MBD (standing for Mate/Bud/Drip). As shown in [27], PEP can simulate MBD. An empty membrane with patch ρ can be created by using the action pino(ρ). The complementary exo actions n and n are used to merge two nested membranes. The complementary phago actions n and n (ρ) are used to model a membrane ''eating'' another membrane.
A PEP system consists of composed nested membranes made from patches σ . The composition of two system is denoted by P • Q, while a system composed from a patch σ and a system P is denoted by σ (P). The term a.σ denotes a brane able to execute a, and then behave as σ . We use shorthand notions: a, (P) and σ (), for a.0, 0(P) and σ ( ), respectively. The set of systems presented in Table 1 is denoted by P. The evolution of PEP systems takes place by using the rules given in Table 2.  Used to rearrange some parts of a PEP system, the structural congruence ≡ b is presented in Table 3.

C. SAFE AMBIENTS
A variant of mobile ambients [28] is represented by safe ambients [52]  open n | open n; Process 0 does nothing. The process C. A provided a movement by consuming the capability C, and then continues by the execution of A. An ambient n[ A ] denotes a process A that can be executed inside the bounded place with name n. A | B is a parallel composition. Processes can be rearranged by using the structural congruence ≡ amb constructed such that (A, |, 0) is a commutative monoid. The evolution of the safe ambients takes place by using the axioms and rules presented in what follows: Axioms:

Rules:
The transitive and reflexive closure of the relation ⇒ amb is denoted by ⇒ * amb .

D. COLOURED PETRI NETS
Coloured Petri nets [42] [42]: A non-hierarchical Coloured Petri Net is a tuple CPN = (P, T , A, , X , C, G, E, I ), where: • P and T , with P ∩ T = ∅, are finite sets of places and transitions: is a finite set of directed arcs; • is a non-empty finite set of colour sets; • X is a finite set of typed variables withType[x] ∈ for all x ∈ X ; • C : P → is a colour set function assigning to each place a colour set; • G : T → EXPR X is a guard function assigning to each transition t a guard with Type[G(t)] = Bool; • E : A → EXPR X is an arc expression function assigning to each arc a connected to a place p a guard with Type[E(a)] = C(p) MS ; • I : P → EXPR ∅ is an initialization function assigning to each place p an initialization expression Type[I (p)] = C(p) MS . A marking is given by a distribution of tokens over the places of a net. Given two markings m and m and a set of transitions U , the notation m[U m means that m leads to m by applying U .

III. MOBILE MEMBRANES
Inspired by the biological movements of cell membranes, the following variants of mobile membrane systems are defined: simple, enhanced and mutual mobile membranes, as well as mutual mobile membranes with objects on surface.

A. SIMPLE MOBILE MEMBRANES
Inspired also by the process calculus of mobile ambients [28], mobile P systems were introduced in [60]. Several years later, a variant of P systems with active membranes [57] called P system with mobile membranes was defined in [51]. We use this variant as simple mobile membrane.
Definition 2 [51]: A simple mobile membrane is a tuple = (O, H , µ, w 1 , . . . , w n , R), such that : • n 1 is the degree of the system; • O is a finite alphabet of objects; • H is a finite set of labels for membranes; • µ ⊂ H × H is the membrane structure; a pair (i, j) ∈ µ with j = i marks the fact that the membrane with label i contains a membrane with label j; • w i ∈ O * , with 1 i n, describes the multiset of objects placed in the membrane with label i; Inside a membrane with label m, an object a is rewritten to a multiset of objects v.
An object a is rewritten to a multiset of objects w, while the surrounding membrane with label h is moved into an adjacent membrane with label m without any change of the membrane labels.
An object a is rewritten to a multiset of objects w, while the surrounding membrane with label h is moved out of a membrane with label m without any change of the membrane labels. In [50], it is added the restriction |w| = 1 in the rules (b) and (c), and so the operations endo and exo are replaced by rendo and rexo, where r stands for restricted. Moreover, two rules are added for a higher control of the mobility: If a membrane with label m does not contain any object from a set S of inhibitors, then a membrane with label h and an object a inside can enter the membrane with label m without any change of the membrane labels or their objects.
inhibitive exocytosis (iexo) If a membrane with label m does not contain any object from a set S of inhibitors, then a membrane with label h and an object a inside can exit the membrane with label m without any change of the membrane labels or their objects. The rules are applied by using the principles listed below: • A maximal parallel manner is used, namely no more rules can be added to the applicable multiset of rules.
• The moving membrane with label h is called active, while the membrane with label m and that does not contain the set of inhibitors S is called passive. In a maximal multiset of rules to be applied in a computation VOLUME 8, 2020 step, the objects and active membranes can be appear only in one rule, while the passive membrane can appear in several rules.
• All objects appearing in an elementary active membranes need to evolve first before the membrane is moved by exocytosis or endocytosis.
• Once a membrane exits the skin membrane it will not be implied in further application of rules.
• The objects not rewritten in a computational step can be used in the subsequent steps.

B. ENHANCED MOBILE MEMBRANES
The simple mobile membranes were extended to enhanced mobile membranes in [5] to be able to describe some biological processes of the immune system [55]. Inspired by the approach presented in [5], the article [61] uses concepts from membrane computing, but also Bio-PEPA and CARMA for modelling the immune system. Definition 3 [5]: An enhanced mobile membrane is a tuple = (O, H , µ, w 1 , . . . , w n , R), such that: The local character of the rule is given by the fact that a multiset of objects u is rewritten into a multiset of object v only if it is place in a membrane with label m having as a parent a membrane with label h. If membrane h is not required in applying the rule, then this is a global evolution rule as in Definition 2.
The multisets of objects uv and v placed in the membranes with labels h and m, respectively, are rewritten into the multisets of objects w and w placed in the membranes with labels h and m, respectively. At the same time, the elementary membrane with label h enters the adjacent membrane with label m without any change of the membrane labels.
The multisets of objects uv and v placed in the membranes with labels h and m, respectively, are rewritten into the multisets of objects w and w placed in the membranes with labels h and m, respectively. At the same time, the elementary membrane with label h exits the membrane with label m without any change of the membrane labels.
Similar form as rule (c) except multiset u is not placed in the moving membrane with label h. The rules of enhanced mobile membranes are applied using the principles given in Subsection III-A for simple mobile membranes.
In [23] it is added the restriction |w + w | = 1 in the rules (d) and (e), and so the operations fendo and fexo are replaced by rfendo and rfexo, where r stands for restricted.

C. MUTUAL MOBILE MEMBRANES
It is worth noting that the rules of the enhanced mobile membranes allow a membrane to move without any permission from the other involved membrane. In contrast with such an approach, we introduce the mutual mobile membranes [6] where the movement is performed only if there is a mutual agreement between membranes. This means to use multisets of objects u and u where the multiset u marks the membrane that initializes the move, while a multiset of u marks the membrane that accepts the movement. Due to the equality u = u, fewer rules are needed (than for enhanced mobile membranes).
Definition 4 [6]: A mutual mobile membrane is a tuple = (O, H , µ, w 1 , . . . , w n , R), such that : • n, O, H , µ, w 1 , . . . , w n are similar to those from Definition 2; The multisets of objects uv and uv placed in the membranes with labels h and m, respectively, are rewritten into the multisets of objects w and w placed in the membranes with labels h and m, respectively. At the same time, the elementary membrane with label h enters the adjacent membrane with label m without any change of the membrane labels.
The multisets of objects uv and uv placed in the membranes with labels h and m, respectively, are rewritten into the multisets of objects w and w placed in the membranes with labels h and m, respectively. At the same time, the elementary membrane with label h exits the membrane with label m without any change of the membrane labels. The rules of mutual mobile membranes are applied using the principles given in Subsection III-A for simple mobile membranes.
Considering also timers for the membranes and objects of the mutual mobile membranes, it was obtained a new system which is defined and studied in [7].

D. MEMBRANES WITH OBJECTS ON SURFACE
The biological operations of pinocytosis and phagocytosis represent the inspiration for defining the following class of mobile membranes.
Definition 5 [13]: A mutual mobile membranes with objects on surface is a tuple = (O, µ, w 1 , . . . , w n , R), such that : The rules of the mutual mobile membranes with objects on surface are applied using the principles given in Subsection III-A for simple mobile membranes.
In [29], the authors consider slightly different versions of the above rules together with the rules inspired by the operations mate and drip from the brane calculus [27]: These operations are used in [40] to describe mitochondrial fusion in the membrane automata starting from a similar approach done in BioAmbients [1]. Other extensions of these operations are defined in [48].

IV. EFFICIENCY IN SOLVING NP-COMPLETE PROBLEMS
Based on the size of their input, the NP-complete problems can be classified into two categories: weak (e.g., Subset Sum, Knapsack, Partition) and strong (e.g., SAT, Bin Packing, Clique, Common Algorithmic Problem) [39]. A comprehensive survey presenting solutions to NP-complete problems obtained in polynomial-time is presented in [62]. The solutions are achieved by using P systems with active membranes with electrical charges together with rules modelling membrane creation and membrane division.
To obtain such solutions, the membrane systems need to respect the conditions: (i) all computations reach a halting configuration; (ii) beside the working alphabet also two objects yes and no need to be added to mark the response given by the algorithm; only one of these two objects can appear in the halting configuration; (iii) in the halting configuration the presence of the object yes marks a successful computation), while the presence of the object no marks an unsuccessful computation.
In addition to using mobility rules, also elementary division rules are added for generating the working space needed to tackle NP-complete problems: An object a is rewritten to the multisets of objects w and w . The surrounding elementary membrane with label h is divided into two copies with the same label h such that each new membrane with label h contains a copy of each object from the initial membrane with label h and one of the multisets of objects w and w .
In what follows we present some NP-complete problems and the results claiming that they can be solved efficiently by using mobile membrane systems. The k-Partition problem asks to decide whether or not there exists a partition of given a finite set A into k subsets such that they have the same weight according to a given weight function g : A → N.
Theorem 1 [14]: By using mobile membrane systems with the rules mendo, mexo and ediv, the 2-Partition problem is solvable in a polynomial number of steps.
The Subset Sum problem asks to determine whether or not there exists a non-empty subset B of a given a finite set A such that g(B) = s for a given a weight function g : A → N and constant s. Theorem 2 [16]: By using mobile membrane systems with the rules mendo, mexo and ediv, the Subset Sum problem can be solved in a polynomial number of steps.
The Bin Packing problem asks to decide whether or not there exists a partition of a given a finite set A = {a 1 , . . . , a n } into b subsets such that their weights do not exceed c for a given weight function g : A → N and two constants b, c ∈ N.
Theorem 3 [22]: By using mobile membrane systems with the rules mendo, mexo and ediv, the Bin Packing problem can be solved in a polynomial number of steps.
The Knapsack problem asks to decide whether or not there exists a subset of a given finite set A = {a 1 , . . . , a n } such that its weight does not exceed k and its value is greater than or equal to l for a given weight function g : A → N, value function r : A → N and constant l ∈ N.
Theorem 4 [16]: By using mobile membrane systems with the rules mendo, mexo and ediv, the Knapsack problem can be solved in a polynomial number of steps.
The SAT problem ask if a propositional logic formula written in conjunctive normal form (CNF) is satisfiable. A formula ϕ = C 1 ∧C 2 ∧· · ·∧C m is in CNF if C i = y 1 ∨y 2 ∨· · ·∨y r , for 1 i m and r n, and eithery j = x k or y j = ¬x k , for a given set of propositional variables X = {x 1 , x 2 , . . . , x n }.
Theorem 5 [19]: By using mobile membrane systems with the rules mendo, mexo and ediv, SAT can be solved in a polynomial number of steps.
Theorem 6 [50]: By using mobile membrane systems with the rules rendo, rexo and ediv, SAT can be solved in a polynomial number of steps.
The kQBF problem asks if a quantified Boolean formulas with k alternations of quantifiers is satisfiable. This means to check if there exists an assignment for a set of propositional variables X = X 1 ∪ · · · ∪ X k to obtain the satisfiability of the formula ϕ = ∃X 1 ∀X 2 ∃X 3 . . . ψ, where ψ is in CNF and the set of variables X is partitioned into k sets X 1 , . . . , X k .
Theorem 7 [16]: By using mobile membrane systems with the rules mendo, mexo and ediv, 2QBF can be solved in a polynomial number of steps.
Theorem 8 [50]: By using mobile membrane systems with the rules rendo, rexo, iendo and ediv, 2QBF can be solved in a polynomial number of steps.
Theorem 9 [23]: By using mobile membrane systems with the rules rendo, rexo, rfendo, rfexo, iendo and ediv, 4QBF can be solved in a polynomial number of steps.

V. COMPUTABILITY POWER
In this section we present some existing results related to the computational power of the different variants of mobile mem-branes: simple, enhanced, mutual and mutual with objects on surface.

A. SIMPLE MOBILE MEMBRANES
PsSMM n (α) denotes the family of sets Ps( ) generated by systems with at most n membranes and evolving by rules of the type α ⊆ {gevol, levol, exo, endo}. The subscript n can be replaced by * if the number of membranes is not fixed but finite. The rules of type α do not lead to an increase in the number of membranes during the computation, but as membranes can exit the skin membrane the number of membranes can decrease during computation.
Theorem 10 [51]: Corollary 1 [51]: When the global evolution rules are used, the computational universality is obtain with only four membranes. Theorem 11 [46]: By using the local evolution rules, an improvement is obtained as the number of membranes decreases to three (the minimal number for mobility). Theorem 12 [9]:
Theorem 13 [32]: The result is improved by obtaining the same power with nine membranes. Theorem 14 [9]: When using only the global evolution rules without any mobility rules, only three membranes are enough to obtain the computational universality. Theorem 15 [32]: An interesting aspect is that the systems with three membranes using either the pairs (exo, endo) or (fexo, fendo) have the same computational power. Theorem 16 [32]: Systems with eight membranes subsume PsET 0L, while those with seven membranes subsume PsE0L. Theorem 17 [32]: PsET 0L ⊆ PsEMM 8 (exo, fexo, endo, fendo).

C. MUTUAL MOBILE MEMBRANES
PsMMM n (α) denotes the family of all sets Ps( ) generated by systems with at most n membranes and evolving by rules of the type α ⊆ {mexo, mendo}. Due to the addition of co-objects, fewer membranes are needed (than for enhanced mobile membranes) in order to obtain computational universality. Theorem 27 [9]: This result was improved significantly such that only three membranes suffice. Theorem 28 [8]: Moreover, systems with three membranes subsume PsET 0L. Proposition 1 [8]: PsET 0L ⊂ PsMMM 3 (mexo, mendo).
In [7] was proven also that adding timers to mutual mobile membranes does not lead to an increase of the computational power.

D. MUTUAL MOBILE MEMBRANES WITH OBJECTS ON SURFACE
PsM 3 OS n (r 1 (s 1 ), r 2 (s 2 )) denotes the family of all sets Ps( ) generated by systems with at most n membranes and evolving by rules of the type r 1 ∈ {pino, pino i , pino e , phago, mate} and r 2 ∈ {exo, exo i , exo e , drip} of weight at most s 1 and s 2 , respectively. The weight of a rule is given by the number of objects of its right-hand side. The computational universality of the pair of operations (pino, exo) where the weight of pino is at most four and the weight of exo is at most three is obtained by using at most eight membranes.
By allowing an increase of the weight in the pino rules from four to five, and an increase of the weight in the exo rules from three to four, the number of membranes is reduced to only three. Theorem 30 [15]: PsRE = PsM 3 OS m (pino(s 1 ), exo(s 2 )), for all m 3, s 1 5, s 2 4.
By allowing a decrease of the weight in the pino rules from five to four, the number of membranes remains the same (namely, three). Theorem 31 [13]: The computational universality of the pair of operations (phago, exo) where the weight of phago is at most five and the weight of exo is at most two is obtained by using at most nine membranes. Theorem 32 [47]: exo(s 2 )), for all m 9, s 1 5, s 2 2.
A decrease by one of the weight for phago leads to an increase by one of the weight for exo in order to preserve the number of membranes. Theorem 33 [47]: , exo(s 2 )), for all m 9, s 1 4, s 2 3.
An important reduction on the number of membranes is obtained by preserving the weight of exo operations and increasing the weight of the phago operation from four to six. Theorem 34 [15]: for all m 4, s 1 6, s 2 3.
The same number of membranes is obtained while the weights of both phago and exo operations are decreased by one to five and two, respectively. Theorem 35 [13]: When using the operations mate and drop and the rules presented in [29], the following results are obtained. Lemma 1 [29]: PsM 3 OS m (mate(s 1 ),drip(s 2 )) ⊆PsM 3 OS m (mate(s 1 ),drip(s 2 )), for all m m , s 1 s 1 and s 2 s 2 .
Some related results are presented in [48] by using some variants of the mobility rules in which the objects are placed inside membranes.

VI. RELATIONSHIPS WITH OTHER APPROACHES
One of the research line in membrane systems is to connect mobile membranes with process calculi; e.g., [31], Petri nets [2], [45], cellular automata [34] and communicating X-machines [44]. Several studies focused on the relationships between these models in order to use their techniques and results: e.g., model checking of process calculi, invariants of Petri nets, testing methods of X-Machines. In what follows we present some connections following this line of research.

A. BRANE CALCULI AS MUTUAL MOBILE MEMBRANES WITH OBJECTS ON SURFACE
There exist articles treating similar aspects existing in both membrane systems and brane calculus: [24]- [26], [29], [30], [47]. Related to this research line, we present how to use mutual mobile membranes with objects on surface to encode the PEP fragment of brane calculus.
When studying the evolution of mobile membranes, the notion of membrane configuration is central. Considering membranes with objects from O and rules from R, by M → m N is denoted the reduction of a configuration M to a configuration N using a rule from the set R. The evolution is given by the following rules: where the membranes can be rearranged by using the structural congruence ≡ m presented in Table 4. Using the above membrane configurations and the evolution rules for mobile membranes, an encoding of the PEP fragment into the mutual mobile membranes with objects on surface can be achieved. For this purpose, we define a translation function that transforms the brane processes of the set P into membrane configurations from the set M.
Definition 6 [11]: A translation function T : P → M is defined as The translation function T can be used to transform the first three rules of Table 2 into the following rules: It holds that two structurally congruent PEP systems are translated into two congruent configurations of mutual mobile membranes with objects on surface. Proposition 2 [11]: Proposition 3 [11]: It holds also that if a PEP system reduces to another one by applying a rule, the membrane configurations obtained by translation can evolve one into another by using the translated rule. Proposition 4 [11]: If M = T (P), there is N such that M → m N and N = T (Q) whenever P → b Q.
Proposition 5 [11]: If M = T (P), there is Q such that

B. MOBILE AMBIENTS AS MOBILE MEMBRANES
The relationship between mobile membranes and mobile ambients is studied in [4], [31]. This relationship is achieved by translating the pure safe ambients into mobile membranes. Definition 7 [11]: The unique object dlock is used to control the evolution by blocking the application of rules corresponding to a mobile ambient that cannot evolve.
The structural congruence is preserved by this translation. Proposition 6 [31]: If M = dlock M 1 and N = dlock N 1 are two membrane configurations such that M 1 and N 1 do not contain the dlock object, then M ⇒ mem N denotes the application of certain rules r 1 , . . . , r n to obtain the membrane configuration N starting from the membrane configuration M . Proposition 7 [11]: Theorem 38 [31]: 2) If T (A) ⇒ mem M , then exists B such that A ⇒ amb B and M = T (B). These results facilitate the use for membrane systems of some existing tools for mobile ambients. As a consequence, checking if a mobile membrane configuration can be obtained from another membrane configuration (the reachability problem) was treated in [3]. It was proven that the problem is decidable (by using the decidability achieved in a variant of the ambient calculus).
Theorem 39 [3]: It is decidable whether an arbitrary mobile membrane M 1 reduces to another mobile membrane M 2 .
Reachability aspects were also studied in [36] in the framework of BioAmbients [63], a biological inspired version of mobile ambients. We consider that a uniform analysis of BioAmbients and membrane systems having now different notions of reachability represents and interesting topic to approach.
Inspired by the relationship between process calculi and membrane systems, some observational equivalences were defined and studied in the context of mobile membranes with lifetimes [20], [21]. These observational equivalences correspond to the ability of observing combinations of mobility, timing and positions of the membranes. Similar equivalences were studied in [37], [38] for the bio-inspired formalism Bio-PEPA presented in [33].

C. ENHANCED MOBILE MEMBRANES AS COLOURED PETRI NETS
In what follows we use lhs(r) and rhs(r) to represent the left-hand side and right-hand side of a rule r, respectively. A coloured Petri net CPN = (P, T , A, , X , C, G, E, I ) can be constructed from a given system of enhanced mobile membranes = (O, H , µ, w 1 , . . . , w n , R, i) as described in [10], [17]: • the set A of directed arcs contains: input arcs (P × T ) from the place structure and from places corresponding to the membranes from lhs(r k ) to the transition t k , output arcs (T × P) from the transition t k to the place structure and to the places corresponding to the membranes from rhs(r k ); y, z, . . .}, a finite set of variables used for checking the structure µ; if t simulates an endo, fendo or levol rule, true otherwise; • The set E is constructed as follows: on the arc from the place structure to a transition t k , add the pairs (i, j) ∈ µ describing the membrane structure appearing in lhs(r k ); on the arc from a place that represents a membrane containing an object a in lhs(r k ) to the transition t k , add the object a; on the arc from a transition t k to the place structure, add all the pairs (i, j) describing the membrane structure appearing in rhs(r k ); on the arc from a transition t k to a place representing a membrane containing a multiset of objects w appearing in rhs(r k ), add the multiset of objects w; • I (p) = µ if p = structure, w p if p ∈ {1, . . . , n}. The connection between the evolution of an enhanced mobile membrane and the evolution of its corresponding coloured Petri net CPN can be expressed formally by the following result.
Theorem 40 [17]: This implies that the reachability problem can be decided in membrane systems. Proposition 9 [17]: If the reachability problem is decidable for CPN , then the reachability problem is also decidable for .
A similar approach was presented in [18] where coloured Petri nets are used to simulate the mutual mobile membranes with objects on surface. A different approach is in [12], where the temporal logic of actions is used to study the mutual mobile membranes with objects on their surfaces.

VII. CONCLUSION
This paper presented the results obtained for mobile membranes in which the movement is inspired from the biological mobility of living membranes by exocytosis and endocytosis. Several classes of mobile membranes (simple, enhanced, mutual and mutual with objects on surface) are defined, and their efficiency and computational power are studied. Finally, we presented the connections of mobile membranes with brane calculus, with mobile ambients and with coloured Petri nets. The practical application of mobile membrane remains still an open problem. GABRIEL CIOBANU is currently a Researcher at the Romanian Academy of Sciences (Iaşi Branch) and at Alexandru Ioan Cuza University of Iaşi. His research interests include distributed systems (process calculi), formal methods (semantics, logics), and natural computing (membrane systems). For his scientific contributions, he received awards from the Romanian Academy, in 2000, 2004, and 2013, Ad Astra Association, in 2018, and International Membrane Computer Society, in 2019. He is the Editor-in-Chief of the Scientific Annals of Computer Science. He is a member of Academia Europaea (the Academy of Europe).