On the number of useful objects in P systems with active membranes
aa r X i v : . [ c s . F L ] A ug On the number of useful objects in P systemswith active membranes ⋆ Zsolt Gazdag, K´aroly Hajagos, and Szabolcs Iv´an
Institute of InformaticsUniversity of Szeged, Szeged, Hungary { gazdag,hajagos,szabivan } @inf.u-szeged.hu Abstract.
In this paper we investigate the number of objects actuallyused in the terminating computations of a certain variant of polarization-less P systems with active membranes. The P systems we consider herehave no in-communication rules and have no different rules triggered bythe same object to manipulate the same membrane. We show that if weconsider such a P system Π and its terminating computation C , then wecan compute the result of C by setting a polynomial upper bound on thecontent of each region in C . The computational power of P systems with active membranes [14] is widelyinvestigated, mainly due to the fact that they can provide efficient solutionsto computationally hard problems. The first such solutions include [9,14,18,24],where NP -complete problems were solved using only elementary membrane di-vision. Non-elementary membrane division was also investigated and it turnedout that using this type of rules, these P systems can already solve PSPACE -complete problems [1,20]. Since then, many variants of P systems with activemembranes were used to solve computationally hard problems, see, for example,[2,3,5,6,12,13,17,19,22]. A recent survey can be found in [21].In [3] the authors considered such P systems where non-elementary mem-brane division was allowed but the use of polarizations was not, and it was shownthat this variant is still powerful enough to solve the
PSPACE -complete QSATproblem. On the other hand, it is still open, whether NP -complete problemscan be solved without polarization using only elementary membrane division.Due to the famous conjecture of Gh. P˘aun [15], the answer to this question isexpected to be negative. To prove this conjecture however is challenging sinceeven a polarizationless P system can produce exponentially many regions andexponentially many objects in the regions in linear time.Nevertheless, there are several partial solutions to P˘aun’s conjecture (whichis often called the P conjecture in the literature), see, for example, [7,8,10,11,23]. ⋆ Ministry of Human Capacities, Hungary grant 20391-3/2018/FEKUSTRAT is ac-knowledged. Szabolcs Iv´an was supported by the J´anos Bolyai Scholarship of theHungarian Academy of Sciences. n these partial solutions, the authors investigate a certain class of P systems,called recognizer P systems [18]. Recognizer P systems are common tools whenP systems are used to decide problems. They have many useful properties, forexample, all of their computations halt and they have two designated objects yes and no which are sent to the environment exactly in the last step of thesystem. These objects are the outputs of the system and are used to indicate theacceptance or rejection of the input. If we use such a system to decide a problem,we require the system to be confluent meaning that all of the computations withthe same input should give the same output. More precisely, given an instance I of a decision problem L , a recognizer P system Π deciding L should halt on I with the correct output yes or no , according to whether I is a positive instanceof L or not. That is, even if Π can have many different computations on I , all ofthese computations should halt with the same output. Therefore, if we want totell the output of Π on I , it is enough to simulate an arbitrary computation of Π .This implies that we can assume that Π has no different rules involving the samemembrane and triggered by the same object. To see this, assume that Π has thefollowing two rules r : [ a → u ] and r : [ a ] → [ b ] [ c ] (that is, r and r areusual evolution and division rules, respectively, with the usual specification of a, b, c , and u ). Clearly, whenever Π can apply r , it can apply r , too. Therefore,if we drop r from Π , then Π still has at least one computation which gives thecorrect output, for every possible input. According to this, to give a polynomialupper bound on the running time of recognizer P systems, it is enough to consideronly certain computations of them, or even we can assume that these systemshave the syntactic restrictions described above. Similar observations are usedin some of the partial solutions of the P conjecture in order to simplify theconsidered P systems.Another concept that is frequently arises in this research line is that of the dependency graph [4] and its variants. Roughly, these graphs describe how anobject (or in same cases a configuration) evolves when certain rules are applied onit. For example, in [7], object division trees, a restricted variant of dependencygraphs, are used to follow the evolution of objects under the application ofdivision rules. These graphs are usually used to find such computations of aP system that can be simulated efficiently. For example, in [10] and [23], thesimulated computations are such that only a reasonable small part of them hasto be represented in order to determine the result of these computations. In[7], the object division trees are used to find such computations which can besimulated by polynomial multiplication.In the above mentioned partial solutions of the P conjecture, P systems withrestricted initial membrane structure were investigated. In this paper we considerP systems with arbitrary initial membrane structure. Our P systems have no in-communication rules and have no different rules involving the same membraneand triggered by the same object. We show that for such a P system, we cancompute the result of the terminating computations by setting a polynomialupper bound (depending of the length of the computation) on the content ofeach region. With our result, even if a P system Π can produce exponentiallyany objects in the regions, it is possible to simulate Π by keeping only apolynomial number of objects in each region (there can be exponentially manyregions of Π , though).In the proof of our result, we will need to use precisely such well knownnotions of Membrane Computing as maximal parallelism and the computationstep of a P system. Thus we will give their formal definitions in the first partof the paper. Moreover, it will be convenient for us to treat a P system suchthat the rules of the system and the configurations the rules are working on areseparated. Thus we will define membrane grammars , which consist of similarrules as P systems except that evolution rules have the form [ a ] ℓ → [ u ] ℓ ( ℓ , a ,and u are specified as usual). Moreover, we will define membrane configurations which are nonempty, finite, rooted, directed, edge-labeled trees. The nodes ofsuch a tree will represent the regions of a membrane configuration, while thelabeled edges are the membranes between the regions.The paper is structured as follows. First, we give the necessary notions andnotations in the next section. Then, in Section 3, we give the main results ofthe paper. We discuss the possible extensions and applications of our results inSection 4. In this section, we introduce the necessary notions and notations. In particular,we define the notion of membrane grammars which is a novel representation ofpolarizationless P systems with active membranes. Nevertheless, we assume thatthe reader is familiar with the basic concepts of membrane computing techniques(for a comprehensive guide see e.g. [16]). N stands for the set of natural numbers including zero, and for arbitrary i, j ∈ N , i ≤ j, [ i, j ] denotes the set { i, . . . , j } . Furthermore, if i = 1, then [ i, j ] isdenoted by [ j ].Let O be an alphabet of objects and H be a set of membrane labels. Weassume that H always contains the special label skin . A membrane structure is a triple ( V, E, L ) where (
V, E ) is a nonempty, finite, rooted, directed tree,having exactly one node in depth one, with edges directed towards the root, and L : E → H assigns labels to each edge, such that only the unique edge leadingto the root can be labeled by the symbol skin , and it has to be labeled by skin .Edges are called membranes of the structure, the nodes are its regions . The rootis also called the environment . For each non-environment region x ∈ V , theoutgoing edge ( x, y ) ∈ E towards the parent of x is called the outer membraneof the region, the edges directed into x are called the inner membranes of x .We can assume that initially, each membrane has its unique label. The regionsthat have only an outgoing edge are the leaves of the structure, and a membranebetween a leaf and its parent is called an elementary membrane .A membrane configuration is a tuple ( V, E, L, ω ) where (
V, E, L ) is a mem-brane structure and ω : V → O ∗ is a function which assigns a finite word ofobjects to each region. We view these words as multisets of O , and also employhe functional notation: if w ∈ O ∗ and o ∈ O , then w ( o ) denotes the number ofoccurrences of o in w . The empty word is denoted by ε . The difference u − v oftwo words u and v is defined if for each o ∈ O , u ( o ) ≥ v ( o ), in which case u − v is a word w with w ( o ) + v ( o ) = u ( o ) for each o ∈ O . Sum of u and v is definedas their concatenation u + v = uv . If t ∈ N , then the multiplication t · u , for aword u , is a word v with v ( o ) = t · u ( o ) for each i ∈ O .A membrane grammar G over ( O, H ) is a finite set of rules of the followingform: [ a ] ℓ → [ u ] ℓ for some a ∈ O, u ∈ O ∗ and ℓ ∈ H (evolution)[ a ] ℓ → b for some a, b ∈ O and ℓ ∈ H − { skin } (dissolution)[ a ] ℓ → [ b ] ℓ [ c ] ℓ for some a, b, c ∈ O and ℓ ∈ H − { skin } (division)[ a ] ℓ → [] ℓ b for some a, b ∈ O and ℓ ∈ H (out)[] ℓ a → [ b ] ℓ for some a, b ∈ O and ℓ ∈ H − { skin } (in)The semantics of these rules will be described later, when we formally define acomputation step of a membrane grammar. We just note here that it will bedefined in the same way as in the case of P systems with active membranes [14].However, we impose some restrictions on the rules of a membrane grammar aswe have discussed it in the Introduction. We require that for each a ∈ O and ℓ ∈ H , there is at most one rule with left-hand side [ a ] ℓ , and there is at mostone rule with left-hand side [] ℓ a . We say that the pair ( o, ℓ ) is an evolution- /dissolution- / division- / out-pair, if there is an evolution / dissolution / division/ out rule with left-hand side [ o ] ℓ , or an in-pair, if there is an in-rule with [] ℓ o on its left-hand side. Division for non-elementary membranes is not allowed. Example 1.
Let G be a membrane grammar over ( O, H ) and let (
V, E, L, ω ) bea membrane configuration, where – O = { o , o } , – H = { ℓ , . . . , ℓ , skin } , – V = { env, s, x , . . . , x } , – E = { ( s, env ) , ( x , s ) , ( x , s ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) } , – and the rules of G are:[ o ] ℓ → [] ℓ o (out)[ o ] ℓ → o (dissolution)[ o ] ℓ → [] ℓ o (out)[ o ] ℓ → o (dissolution)[] ℓ o → [ o ] ℓ (in)[] ℓ o → [ o ] ℓ (in)[ o ] ℓ → [ o ] ℓ [ o ] ℓ (division)[ o ] ℓ → [ o o ] ℓ (evolution)urthermore, let L : E → H be defined as follows: L ( s, env ) = skin L ( x , s ) = ℓ L ( x , s ) = ℓ L ( x , x ) = ℓ L ( x , x ) = ℓ L ( x , x ) = ℓ L ( x , x ) = ℓ . Finally, we define ω : V → O ∗ as follows: ω ( env ) = ω ( s ) = ε ω ( x ) = o o o ω ( x ) = o o o ω ( x ) = o o ω ( x ) = o o ω ( x ) = o o ω ( x ) = o o . Figure 1 shows the membrane configuration (
V, E, L, ω ). The regions x , x , x and x are the leaves, therefore ℓ , ℓ , ℓ , and ℓ are elementary membranes. env so o o x o o o x o o x o o x o o x o o x skin ℓ ℓ ℓ ℓ ℓ ℓ Fig. 1.
A membrane configuration
Next, we define a function on the edges (that is, on the membranes) of aconfiguration and call it viable transition . It is used to assign objects to themembranes of a configuration according to the well-known notion of maximalparallelism.
Definition 1.
Let G be a membrane grammar over ( O, H ) and C = ( V, E, L, ω ) be a membrane configuration. The partial function f : E → O × {↑ , ↓} is calleda viable transition , if it satisfies the following conditions:0. If f ( x, y ) = ( o, ↑ ) for some x, y ∈ V and o ∈ O , then there has to be anon-evolution rule with the left-hand side [ o ] L ( x,y ) . Moreover, if x is not aleaf, then this rule cannot be a division rule. If f ( x, y ) = ( o, ↓ ) , then therehas to be an in-rule with the left-hand side [] L ( x,y ) o .. For each region, there are at least as many objects inside the region who leaveit. Formally, for each x ∈ V and o ∈ O we have |{ ( y, x ) ∈ E : f ( y, x ) = ( o, ↓ ) }| + |{ ( x, y ) ∈ E : f ( x, y ) = ( o, ↑ ) }| is at most ω ( x )( o ) . Note that the secondset contains at most one edge, the one leading from x to its parent.2. For each region, if some object could use the outer membrane (and thuscannot evolve inside the membrane since, by assumption, G has no evolutionrules to do so), but the outer membrane is not used, then all those objectsare occupied with in-rules. (Maximal parallelism.) Formally, for each edge ( x, y ) with f ( x, y ) being undefined, and for each o ∈ O for which ( o, L ( x, y )) is a dissolution-, division- or out-pair, it has to be the case that |{ ( z, x ) ∈ E : f ( z, x ) = ( o, ↓ ) }| = ω ( x )( o ) .3. Similarly, for each region, if some object could use one of the inner mem-branes but it is not used, then all of those objects are occupied with evolution,or using the outer membrane or one of the inner membranes. Formally, foreach edge ( y, x ) with f ( y, x ) being undefined, and for each o ∈ O such that ( o, L ( y, x )) is an in-rule, one of the following cases has to hold, where ( x, z ) is the outer membrane of the region x : – ( o, L ( x, z )) is an evolution-pair; – or |{ ( y ′ , x ) ∈ E : f ( y ′ , x ) = ( o, ↓ ) }| + |{ ( x, z ) : f ( x, z ) = ( o, ↑ ) }| = ω ( x )( o ) . Again, observe that the second term of the sum is either zero or one.
Notice that viable transitions do not say explicitly which rules are attachedto the membranes but, since G has no different rules triggered by the same objectfor a membrane, we can tell which rules are about to use. A viable transition ofthe membrane grammar occurring in Example 1 can be seen in Figure 2. Definition 2.
Let C = ( V, E, L, ω ) be a membrane configuration and f : E → O × {↑ , ↓} be a viable transition for C . Then the computation step C f ⊢ C ′ forthe membrane configuration C ′ is defined via several sub-steps discussed below.During the definition, we demonstrate the given steps by an example. We usethe membrane grammar given in Example 1 and show the application of its ruleson the configuration shown in Figure 1 according to the viable transition givenin Figure 2. The example is given via the corresponding figures below.1. (Membrane attachments.) First, we remove those objects from the regionsthat use membranes. The new configuration C = ( V, E, L, ω ) is defined asfollows: for each region x ∈ V and object o ∈ O , let ω ( x )( o ) = ω ( x )( o ) − |{ ( y, x ) ∈ E : f ( y, x ) = ( o, ↓ ) }|− |{ ( x, y ) ∈ E : f ( y, x ) = ( o, ↑ ) }| . Since f is viable for C , these values are nonnegative for each x and o . nv so o o x o o o x o o x o o x o o x o o x skin ℓ , o , ↑ ℓ , o , ↑ ℓ , o , ↓ ℓ ℓ , o , ↑ ℓ Fig. 2.
A viable transition for the configuration of Example 1. Here, the applicablerules are [ o ] ℓ → [] ℓ o , [ o ] ℓ → o , [] ℓ o → [ o ] ℓ , [ o ] ℓ → [ o ] ℓ [ o ] ℓ and [ o ] ℓ → [ o o ] ℓ . env so x o o x o o x o o x o x o o x skin ℓ , o , ↑ ℓ , o , ↑ ℓ , o , ↓ ℓ ℓ , o , ↑ ℓ Fig. 3.
After membrane attachment. Object o of x is attached to ℓ and one other o is attached to ℓ ; o of x is attached to ℓ and o of x is attached to ℓ . . (Evolutions.) Next, we apply the evolution rules. The new configuration C =( V, E, L, ω ) is defined as follows: for each region x ∈ V , let ω ( x ) = o o . . . o m and ℓ = L ( x, y ) be the label of the outer membrane of x . Then define ω ( x ) as u u . . . u m where u i = ( u if there is an evolution rule [ o i ] ℓ → [ u ] ℓ o i otherwise . env so x o o x o o x o o x o x o o o x skin ℓ , o , ↑ ℓ , o , ↑ ℓ , o , ↓ ℓ ℓ , o , ↑ ℓ Fig. 4.
After evolutions. Inside ℓ the object o got rewritten to o o .
3. (Movements.) Applying the in- and out-rules we get C = ( V, E, L, ω ) whichis defined as follows: for each region x ∈ V and o ∈ O , let ω ( x )( o ) = ω ( x )( o )+ |{ ( y, x ) ∈ E : f ( y, x ) = ( o, ↑ ) , ( o, L ( y, x )) is an out-rule }| + |{ ( x, y ) ∈ E : f ( x, y ) = ( o, ↓ ) , ( o, L ( x, y )) is an in-rule }| .
4. (Dissolutions.) Applying the dissolution rules, the current configuration is C = ( V , E , L , ω ) which we define as follows. Initially let C = C .Iterating from the leaves towards the root, we dissolve membranes step by stepas follows: if for an edge ( x, y ) ∈ E , f ( x, y ) = ( o , ↑ ) so that ( o , L ( x, y )) is a dissolution-pair with a rule [ o ] L ( x,y ) → o for some o , o ∈ O , then weset ω ( y ) = ω ( y ) + ω ( x ) + o . Moreover, for each z ∈ V with ( z, x ) ∈ E ,we add a new edge ( z, y ) to E , set L ( z, y ) = L ( z, x ) , E = E − { ( z, x ) } .Furthermore, we set E = E − { ( x, y ) } and V = V − { x } . We repeat thisprocess till we handled all the dissolution-marked membranes. nv o so x o o x o o o x o o x o x o o o x skin ℓ ℓ , o , ↑ ℓ ℓ ℓ , o , ↑ ℓ Fig. 5.
After movements. The membrane ℓ released o as o upwards, ℓ released o as o downwards. env o o o o so x o o o x o o x o x o o o x skin ℓ ℓ ℓ ℓ , o , ↑ ℓ Fig. 6.
After dissolutions. The membrane ℓ got dissolved under the rule [ o ] ℓ → o . . (Divisions.) Finally, C ′ = ( V ′ , E ′ , L ′ , ω ′ ) is defined as follows. Initially let C ′ = C . For each membrane ( x, y ) ∈ E ′ such that f ( x, y ) = ( o, ↑ ) and ( o, L ′ ( x, y )) is a division-pair with a rule [ o ] L ′ ( x,y ) → [ o ] L ′ ( x,y ) [ o ] L ′ ( x,y ) forsome o, o , o ∈ O , let V ′ = V ′ ∪ { x ′ } , where x ′ is a new child of y with ω ′ ( x ′ ) = ω ′ ( x ) + o and L ′ ( x ′ , y ) = L ′ ( x, y ) . Then we set ω ′ ( x ) = ω ′ ( x ) + o . env o o o o so x o o o x o o x o o x o o x ′ o o o x skin ℓ ℓ ℓ ℓ ℓ ℓ Fig. 7.
After divisions. The membrane ℓ became divided under the rule [ o ] ℓ → [ o ] ℓ [ o ] ℓ . The membrane configuration we end in up after this last step is C ′ with C f ⊢ C ′ . If there is such a viable f , then we write C ⊢ C ′ . Consider a membrane grammar G and a terminating computation C ⊢ . . . ⊢ C t of G . In this section, we show that if G does not have in-rules, then G actuallyuses at most t copies of each object of each membrane of C . In other words, wecan apply a threshold t on the content of the membranes in C without affectingthe result of the computation. First, we give a formal definition of applying athreshold on a multiset or a configuration.When w is a multiset over O and t ∈ N is a threshold, then let w | t be themultiset with w | t ( o ) = ( w ( o ) if w ( o ) < tt otherwise.or a configuration C = ( V, E, L, ω ) and a number t ∈ N , let C | t denote theconfiguration we get by applying the threshold t on the content of each region of C , hence C | t = ( V, E, L, ω ′ ), where ω ′ ( x ) = ω ( x ) | t for each x ∈ V (an examplecan be seen in Figure 8). We say that the multisets w , w over O are t -equivalent( w ≈ t w ), if w | t = w | t . Similarly, two membrane configurations C and C are t -equivalent, denoted C ≈ t C , if they have the same membrane structure( V, E, L ) and for each region x ∈ V , we have ω ( x ) ≈ t ω ( x ) | t . Clearly, C ≈ t C | t since we got C | t by applying the threshold t on the content of each region of C . env o o o so o x o o x o o o x env o so o x o x o o x skin ℓ ℓ ℓ skin ℓ ℓ ℓ Fig. 8.
Membrane configurations C and C | Theorem 1.
Let G be a membrane grammar over ( O, H ) . Assume there are noin-rules in G , C and C ′ are t -equivalent configurations for some t > , and C ⊢ C . Then there exists a configuration C ′ with C ′ ⊢ C ′ and C ≈ t − C ′ .Proof. Let us write in more detail C f ⊢ C and let C = ( V, E, L, µ ) and C ′ =( V, E, L, µ ′ ) . We show that the same function f is viable for C ′ as well, and forthe configuration C ′ with C ′ f ⊢ C ′ we have C ≈ t − C ′ .As there are no in-rules, the conditions for viability become simpler. We checkthem one by one:0. Since this condition depends only on the rules, it is satisfied for C ′ as well.1. Since f is viable for C , we have that for each x ∈ V and o ∈ O , |{ ( x, y ) ∈ E : f ( x, y ) = ( o, ↑ ) }| is at most µ ( x )( o ) . Since the cardinality of this set isither one or zero and t > , we have that whenever µ ( x )( o ) is at least one,then so is µ ′ ( x )( o ) , so this condition is satisfied.2. As there are no in-rules, the set in the second condition of viability is empty,meaning µ ( x )( o ) = 0 for those edges and objects. Hence, by t -equivalencewe get that µ ′ ( x )( o ) = 0 as well for these pairs ( x, o ) , thus this condition isalso satisfied.3. As there are no in-rules, this condition is also vacuously satisfied.Thus, f is viable for C ′ as well. Now let C ′ be the configuration with C ′ f ⊢ C ′ .We show that for each intermediate step in the definition of a computation step,the configurations are ( t − -equivalent.1. After the membrane attachment step, since there is no in-rule, ω ( x )( o ) ei-ther remains the same or decreases by one for each region x and object o ,depending on f . Thus, if µ ( x )( o ) = µ ′ ( x )( o ) , then they will be the sameafter attachment; if both of them are at least t , then both of them will be atleast t − after the attachment step.2. Let x be a region whose outer membrane is labeled by ℓ . For each o ∈ O , let u o denote u ∈ O ∗ if there is an evolution rule [ o ] ℓ → [ u ] ℓ , and o otherwise.After evolution, ω ( x ) will become P o ∈ O (cid:0) ω ( x )( o ) · u o (cid:1) . Since if t , t ≥ t ,then t · u ≈ t t · u for any multiset u , we get that since before evolutionthe (intermediate) configurations were ( t − -equivalent, so are they afterevolution.3. Movements can only increase the contents of a region (in fact, the membraneattachment step is the only one decreasing the content), and by the sameamount as they depend only on f . Clearly, u ≈ t − v implies ( u + o ) ≈ t − ( v + o ) for any object o and multisets u, v over O (actually, if u ′ ≈ t − v ′ for any other pair of multisets, then ( u + u ′ ) ≈ t − ( v + v ′ ) ). Thus, aftermovements the configurations are still ( t − -equivalent.4. Dissolving a membrane ( x, y ) with a rule [ o ] ℓ → o increases ω ( y ) by ω ( x ) + o . Since, for arbitrary ( x, y ) ∈ E , µ ( y ) ≈ t − µ ′ ( y ) and µ ( x ) ≈ t − µ ′ ( x ) by the assumption, moreover f dissolved x according to the same dis-solution rule in C and C ′ (thus the object on the right-hand side of this ruleis the same), ( t − -equivalence is retained after each dissolution.5. After performing a division by applying a rule [ o ] ℓ → [ o ] ℓ [ o ] ℓ , the membranestructure will be the same in both configurations. Moreover, if before thedivision the configurations are ( t − -equivalent, then so they are after thedivision, and when we insert the two objects o and o , the correspondingregions remain ( t − -equivalent.Hence, for C ′ we indeed have C ≈ t − C ′ . Using Theorem 1, we can show that for a computation C which terminates in t steps, we can give another computation C ′ such that the following holds. Thenumber of each object in the first configuration of C ′ is bounded by t and theenvironment at the end of both computations contains the same objects. orollary 1. Let C = C ⊢ . . . ⊢ C t be a terminating computation for some t > and let C ′ = C | t . Then there exists another terminating computation C ′ = C ′ ⊢ . . . ⊢ C ′ t with C t ≈ C ′ t .Proof. We prove more: there exists a terminating computation C ′ = C ′ ⊢ . . . ⊢ C ′ t such that C i ≈ t − i +1 C ′ i for each i ∈ [ t ]. We prove this by induction on i .If i = 1, then we have C ≈ t C ′ . If i >
1, then by induction we have C i − ≈ t − ( i − C ′ i − . Since C i − ⊢ C i , using Theorem 1 we get that thereexists a configuration C ′ i with C ′ i − ⊢ C ′ i and C i ≈ t − i +1 C ′ i .Thus C t ≈ C ′ t , which concludes the proof of the statement. Consider a membrane grammar G over ( O, H ) such that G has no in-rules. Bythe iterated application of Corollary 1, we can simulate a terminating compu-tation C ⊢ . . . ⊢ C t of G as follows. We compute a configuration sequence D , D , . . . , D t such that, for each i ∈ [ t ], D i ≈ t − i +1 C i and each membrane in D i contains only a polynomial number of objects in | O | + t . On the other hand,these configurations can contain exponentially many membranes. Nevertheless,we believe that our result can be used to give polynomial-time simulations ofcertain variants of polarizationless P systems with active membranes. For ex-ample, in [7], a novel method was given to simulate simple polarizationless Psystems efficiently. The simulated P systems are such that they have only onemembrane in the skin membrane at the beginning of the computation. To ex-tend the simulation given in [7] to P systems having an arbitrary membranestructure, we need that those membranes that become elementary during thecomputation contain polynomially many objects. By the results of this paper,we have a chance to achieve this property. The elaboration of the details is atopic for future work.Moreover, we think that the proof of Theorem 1 can be extended to mem-brane grammars having more general rules, such as rules dividing non-elementarymembranes and rules having polarizations or the possibility of changing the la-bels of membranes. References
1. Alhazov, A., Mart´ın-Vide, C., Pan, L.: Solving a PSPACE-complete problem byP systems with restricted active membranes. Fundamenta Informaticae , 67–77(2003)2. Alhazov, A., Pan, L., P˘aun, Gh.: Trading polarizations for labels in P systems withactive membranes. Acta Informatica (2-3), 111–144 (2004)3. Alhazov, A., P´erez-Jim´enez, M.J.: Uniform solution of QSAT using polarization-less active membranes. International Conference on Machines, Computations andUniversality, 122-133 (2007). Cord´on-Franco, A., Guti´errez-Naranjo, M.A., P´erez-Jim´enez, M.J., Riscos-N´u˜nez,A.: Exploring computation trees associated with P systems. In: Mauri, G., Paun,Gh., P´erez-Jim´enez, M.J., Rozenberg, G., Salomaa, A. (eds.) Membrane Comput-ing, 5th International Workshop, WMC 2004, LNCS vol. 3365, 278-–286 (2005)5. Gazdag, Z.: Solving SAT by P systems with active membranes in linear time inthe number of variables. In: Alhazov, A., Cojocaru, S., Gheorghe, M., Rogozhin,Y., Rozenberg, G., Salomaa, A. (eds.) Membrane Computing: 14th InternationalConference, LNCS vol. 8340, 189–205 (2014)6. Gazdag, Z., Kolonits, G.: A new approach for solving SAT by P systems withactive membranes. In: Csuhaj-Varj´u, E., Gheorghe, M., Rozenberg, G., Salomaa,A., Vaszil, G. (eds.) Membrane Computing: 13th International Conference, LNCSvol. 7762, 195–207 (2013)7. Gazdag, Zs., Kolonits, G.: A new method to simulate restricted variants of polar-izationless P systems with active membranes. J. Membr. Comput. 1(4): 251–261(2019)8. Gutierrez-Naranjo, M.A., Perez-Jimenez, M.J., Riscos-N´u˜nez, A., Romero-Campero, F.J.: On the power of dissolution in P systems with active membranes.In: Freund, R., P˘aun, Gh., Rozenberg, G., Salomaa, A. (eds.) Membrane Comput-ing: 6th International Workshop, LNCS vol. 3850, 224–240 (2006)9. Krishna, S.N., Rama, R.: A variant of P systems with active membranes: SolvingNP-complete problems. Romanian Journal of Information Science and Technology,2, 4, 357–367 (1999)10. Leporati, A., Manzoni, L., Mauri, G., Porreca, A.E., Zandron, C.: Solving a specialcase of the P conjecture using dependency graphs with dissolution. In: Gheorghe,M., Rozenberg, G., Salomaa, A., Zandron, C. (eds.) Membrane Computing: 18thInternational Conference, LNCS vol. 10725, 196-213 (2017)11. Murphy, N., Woods, D.: Active membrane systems without charges and using onlysymmetric elementary division characterise P. In: Eleftherakis, G., Kefalas, P.,P˘aun, Gh., Rozenberg, G., Salomaa, A. (eds.) Membrane Computing: 8th Inter-national Workshop, LNCS vol. 4860, 367–384 (2007)12. Pan, L., Alhazov, A.: Solving HPP and SAT by P Systems with Active Membranesand Separation Rules. Acta Informatica (2), 131–145 (2006)13. Pan, L., Alhazov, A., Ishdorj, T.-O.: Further remarks on P systems with activemembranes, separation, merging, and release rules. Soft Computing (9), 686–690(2004)14. P˘aun, Gh.: P systems with active membranes: attacking NP-complete problems.Journal of Automata, Languages and Combinatorics (1), 75–90 (2001)15. P˘aun, Gh.: Further twenty six open problems in membrane computing. In: ThirdBrainstorming Week on Membrane Computing. F´enix Editora, Sevilla, 249–262(2005)16. P˘aun, Gh., Rozenberg, G., Salomaa, A. (eds.): The Oxford Handbook of MembraneComputing. Oxford University Press, Oxford, England (2010)17. P´erez-Jim´enez, M.J., Romero-Campero, F.J.: Trading polarization for bi-stablecatalysts in P systems with active membranes. In: Mauri, G., P˘aun, Gh., P´erez-Jim´enez, M.J., Rozenberg, G., Salomaa, A. (eds.) Membrane Computing: 5th In-ternational Workshop, LNCS vol. 3365, 373–388 (2005)18. P´erez-Jim´enez, M.J., Romero-Jim´enez, ´A., Sancho-Caparrini, F.: Complexityclasses in models of cellular computing with membranes. Natural Computing (3),265–285 (2003)9. P´erez-Jim´enez, M.J., Romero-Jim´enez, ´A., Sancho-Caparrini, F.: A polynomialcomplexity class in P systems using membrane division. Journal of Automata,Languages and Combinatorics (4), 423–434 (2006)20. Sos´ık, P.: The computational power of cell division in P systems. Natural Comput-ing (3), 287–298 (2003)21. Sos´ık, P.: P systems attacking hard problems beyond NP: a survey. J. Membr.Comput. 1, 198–208 (2019)22. Sos´ık, P., Rodr´ıguez-Pat´on, A.: Membrane computing and complexity theory: Acharacterization of PSPACE. Journal of Computer and System Sciences73