Antiport Tissue P Systems
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Minimal Cooperation in Symport/Antiport Tissue P Systems
Artiom Alhazov∗ Yurii Rogozhin Institute of Mathematics and Computer Science Academy of Sciences of Moldova Academiei 5, Chi¸sin˘ au, MD-2028, Moldova {alhazov,rogozhin}@math.md Sergey Verlan† LACL, D´ epartement Informatique, Universit´ e Paris 12, 61, av. G´ en´ eral de Gaulle, 94010 Cr´ eteil, France [email protected]
We investigate tissue P systems with symport/antiport with minimal cooperation, i.e., when only 2 objects may interact. We show that 2 cells are enough in order to generate all recursively enumerable sets of numbers. Moreover, constructed systems simulate register machines and have purely deterministic behavior. We also investigate systems with one cell and we show that they may generate only finite sets of numbers.
1. Introduction P systems were introduced by Gh. P˘ aun in [13] as distributed parallel computing devices of biochemical inspiration. These systems are inspired from the structure and the functioning of a living cell. The cell is considered as a set of compartments (membranes) nested one in another and which contain objects and evolution rules. The basic framework specifies neither the nature of these objects, nor the nature of rules. Numerous variants specify these two parameters by obtaining a lot of different models of computing, see [19] for a comprehensive bibliography. One of these variants, P systems with symport/antiport, was introduced in [12]. This variant uses one of the most important properties of P systems: the communication. This property is so powerful, that it suffices by itself to reach the computational power of Turing machines. These systems have two types of rules: symport rules, when several objects go together from one membrane to another, and antiport rules, when several objects from two membranes are exchanged. In spite of a simple definition, they may compute all Turing computable sets of numbers [12]. This result was ∗ Also Research Group on Mathematical Linguistics, Rovira i Virgili University, Tarragona, Spain, [email protected] † Corresponding
author 1
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improved with respect to the number of used membranes and/or the weight of symport/antiport rules ([6], [8], [10], [14], [5]). Rather unexpectedly, minimal symport/antiport P systems, i.e., when one uses only one object in symport rules and a pair of objects in antiport rules are universal. The proof of this result may be found in [4] and the corresponding system has 9 membranes. This result was improved first by reducing the number of membranes to six [9], after that to five [5] and four [7]. In [17] it is shown that three membranes are sufficient to generate all recursively enumerable sets of numbers using 5 superfluous objects in the output membrane. In [3], a stronger result is reported where the output membrane did not contain superfluous objects. Similarly, when minimal symport is used, i.e., when only symport rules dealing with at most two objects are permitted, three membranes suffice to generate all recursively enumerable sets of numbers [3]. A recent result shows that this number can be decreased down to two for both minimal symport and minimal antiport cases [1], yet only modulo a terminal alphabet. At last in [2] it was shown that the number of additional objects can be bounded by 3 in minimal antiport case and by 6 in minimal symport case. The inspiration for tissue P systems comes from two sides. From one hand, P systems previously introduced may be viewed as transformations of labels associated to nodes of a tree. Therefore, it is natural to consider same transformations on a graph. On the other hand, they may be obtained by following the same reflections as for P systems, but starting from a tissue of cells and no more from a single cell. Tissue P systems were initially investigated in [15] and [16]. They have richer possibilities and the advantages of new topology have to be investigated. Tissue P systems with symport/antiport were first considered in [14] where several results having different values of parameters (graph size, maximal size of connected component, weight of symport and antiport rules) are presented. In this paper we consider symport and antiport rules with minimal cooperation, i.e., we consider minimal antiport rules (of weight 1) that permit to exchange two objects, as well as minimal symport rules (of weight 2) that permit to send two objects together. Systems with minimal antiport were first investigated in [18] where it was shown that 3 cells are sufficient to generate any recursively enumerable set of numbers. In this article, we investigate both variants and we show that in both cases we can construct systems defined on a graph with 2 cells that simulate any (non-)deterministic register machine. Moreover, in the deterministic case, obtained systems are also deterministic and only one evolution is possible at any time. Therefore, if the computation stops, then we are sure that the corresponding register machine stops on the provided input. Another difference from previous proofs is that we use a very small amount of symbols present in an infinite number of copies in the environment. We also show that if only one cell is considered, then corresponding systems may
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generate no more than finite sets of numbers. The minimal symport case was already discussed in [8], so we present only the antiport one. Therefore, we found a boundary between the decidability and the undecidability for corresponding systems. 2. Definitions We denote by N the set of all non-negative natural numbers, {0, 1, . . . }, and by N′ the set {1, 2, . . . }. A multiset S over O is a mapping fS : O −→ N. The mapping fS specifies the number of occurrences of each element of S. The size of the multiset S is P |S| = x∈O fS (x). We use the ordinary set notation in order to specify a multiset. In this case we either indicate the number of occurrences of each element as its power, or we give the mapping function fS . For example the multiset containing 3 occurrences of element a, one occurrence of element b and zero occurrences of element c will be specified as {a3 , b} or a → 3, b → 1, c → 0. We shall also use a string notation in order to specify a multiset. In this case we write all elements of the multiset having a positive multiplicity in a string. For example, the previous multiset will be written as aaab or a3 b. The sum (union) of two multisets P and Q over O, denoted P ∪ Q, is a multiset S such that fS (a) = fP (a) + fQ (a) for all a in O. Similarly, the difference of two multisets P and Q is a multiset S having fS (a) = fP (a) ⊖ fQ (a) where ⊖ is the positive subtraction. A deterministic register machine is the following construction: M = (Q, R, q0 , qf , P ), where Q is a set of states, R = {r1 , . . . , rk } is the set of registers, q0 ∈ Q is the initial state, qf ∈ Q is the final state and P is a set of instructions (called also rules) of the following form: (1) (p, A+, q) ∈ P , p, q ∈ Q, p 6= q, A ∈ R (being in state p, increase register A and go to state q). (2) (p, A−, q, s) ∈ P , p, q, s ∈ Q, A ∈ R (being in state p, decrease register A and go to q if successful or to s if A is zero). (3) (qf , ST OP ) (may be associated only to the final state qf ). We note that for each state p there is only one instruction of the type above. A configuration of a register machine is given by the (k +1)-tuple (q, n1 , . . . , nk ), where q ∈ Q and ni ∈ N, 1 ≤ i ≤ k, describing the current state of the machine as well as the contents of all registers. A transition of the register machine consists in updating/checking the value of a register according to an instruction of one of types above and by changing the current state to another one. We say that the machine stops, if it reaches the state qf .
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We say that M computes a value y ∈ N on the input x ∈ N if starting from the initial configuration (q0 , x, 0, . . . , 0) it reaches the final configuration (qf , y, 0, . . . , 0). We say that M recognizes the set S ⊆ N if for any input x ∈ S the machine stops and for any y 6∈ S the machine does not stop (performs an infinite computation). It is known that (deterministic) register machines recognize all recursively enumerable sets of numbers [11]. We may also consider non-deterministic register machines where the first type of instruction is of the form (p, A+, q, s) and with the following meaning: if the machine is in state p, then the register A is increased and the current state is changed to q or s non-deterministically. A non-deterministic register machine generates the set of all values of the first register resulting from computations that stop on an empty input. We assume that the machine empties all registers except the first register before stopping. It is known that non-deterministic register machines generate all recursively enumerable sets of non-negative integers. A tissue P system with symport/antiport of degree m ≥ 1 is a construct Π = (O, G, w1 , . . . , wm , E, R, i0 ), where O is the alphabet of objects and G is the underlying directed labelled graph of the system. The graph G has m + 1 nodes and the nodes are numbered from 0 to m. We shall also call nodes from 1 to m cells and node 0 the environment. There is an edge between each cell i, 1 ≤ i ≤ m, and the environment. Each cell contains a multiset of objects, initially cell i, 1 ≤ i ≤ m, contains multiset wi . The environment is a special node which contains symbols from E in infinite multiplicity as well as a finite multiset over O\E, but initially this multiset is empty. The symbol i0 ∈ (1 . . . m) indicates the output cell, and R is a finite set of rules (associated to edges) of the following forms: (1) (i, x, j), 0 ≤ i ≤ m, 0 ≤ j ≤ m, i 6= j, x ∈ O+ and not i = 0 & x ∈ E + (symport rules for the communication). (2) (i, x/y, j), 0 ≤ i, j ≤ m, i 6= j, x, y ∈ O+ (antiport rules for the communication). We remark that G may be deduced from relations of R. More exactly, G contains m + 1 vertices and there is an oriented edge between vertex i and j if and only if there is a rule (i, x, j) in R and edges between i and j and j and i if and only if there is a rule (i, x/y, j) in R. However, we prefer to indicate both G and R because it simplifies the presentation. The rule (i, x, j) sends a multiset of objects x from node i to node j. The rule (i, x/y, j) exchanges multisets x and y situated in nodes i and j respectively. The weight of symport rule (i, x, j) is equal to |x|, while the weight of an antiport rule is equal to max{|x|, |y|}. A computational step is made by applying all applicable rules from R in a nondeterministic maximal parallel way. A configuration of the system is an (m + 1)tuple (z0 , z1 , . . . , zm ) where each zi , 1 ≤ i ≤ m, represents the contents of cell i and
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z0 represents the multiset of objects that appear with a finite multiplicity in the environment (initially z0 is the empty multiset). The computation stops when no rule may be applied. The result of a computation is given by the number of objects situated in cell i0 , i.e., by the size of the multiset from cell i0 . We denote by N OtPm,n (symp , antiq ) the family of all sets of numbers computed by tissue P systems with symport/antiport of degree at most m with the maximal size of the connected component of the restriction of the graph to cells equal to n and which have symport rules of weight at most p and antiport rules of weight at most q. Example 1. Let us consider system Π = (O, G, w1 , . . . , w6 , E, R, 2), with O = {A, s, s′ , s′′ , B, C}, E = {B, C} and which has the following underlying graph G (for commodity, we labelled its edges). We note that the environment (node 0) is connected to all other cells, but we indicate on the graph below only connections that are used. '&%$ !"# jj 0 HHH 3 HH HH jjjj j j $$ !"# j '&%$ !"# '&%$ 4 2 4 Hpp H 3 HH 5 vvv OO HH v v H 7 vv 8 // '&%$ !"# '&%$ !"# 6 10 2 ddHH 6 HH9 HH H 11 '&%$ !"# !"# // '&%$ 5 1 1 jjjj
Now we shall give rules and objects of Π. Node Object(s) Edge 0. B ∞ , C ∞ 1. 1.1 : (4, C/B, 0) 1. ss′ An 2. 2.1 : (2, sB/CC, 0) 2. B 3. 3.1 : (0, s, 3) 3. 4. 4.1 : (2, ss′ , 4) 4. 5. 5.1 : (3, s/s′ , 2) ′′ 5. s 6. 6.1 : (4, s′ /s′′ , 5) 6. 7. 7.1 : (4, sB, 6) 8. 8.1 : (6, B, 2) 9. 9.1 : (1, sA, 2) 10. 10.1 : (1, s′ , 3) 11. 11.1 : (5, ss′ , 1)
Rules 2.2 : (2, A, 0) 4.2 : (2, C, 4) 5.2 : (3, s′ , 2) 6.2 : (4, ss′′ , 5) 7.2 : (4, s′′ /s, 6) 7.3 : (6, s′′ , 4)
Now, let us see what the system Π computes. In the initial configuration rules 9.1 and 10.1 are applicable. This brings s and A in the second cell and s′ in cell 3. Now rules 2.1, 3.1, 5.1 and 5.2 permit to replace each B situated in cell 2 by two copies of C. When there are no more copies of B, rule 4.1 can be applied, bringing s and s′ to cell 4. In the meanwhile, all C from cell 2 go to cell 4 by rule 4.2. In cell 4 all C are replaced by B. Rules 7.1, 7.2, 7.3 and 8.1 permit to move these B
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to cell 6 and after that to cell 2. When there are no more B in cell 4, rules 6.2 and 11.1 are used bringing symbols s, s′ and s′′ to their original locations. This last configuration has one A less in cell 1 and the double number of B in cell 2 comparing to the initial configuration. It is easy to observe that after n such cycles the computation will stop because there will be no more A in cell 1. The result of the computation may be read in cell 2 which contains 2n elements B. Therefore, given a number n, system Π computes 2n . 3. The power of symport/antiport of weight 1 In this section we investigate the power of systems having symport/antiport rules of weight 1. Firstly, we show that systems having only one cell are not very powerful. Theorem 1. N OtP1,1 (sym1 , anti1 ) ⊆ N F IN . Proof. Consider an arbitrary P system Π with 1 cell and symport/antiport rules of weight 1, Π = (O, G, w1 , E, R, 1). Consider also an arbitrary halting computation, ending in some configuration C = (z0 , z1 ∪ ze ) with z1 ∈ (O \ E)∗ and ze ∈ E ∗ and w0 ∈ (O \ E)∗ . We claim that |z1 | + |ze | ≤ |w1 |. We shall prove this assertion by contradiction. Let us assume the contrary. Since the number of objects in the membrane can only increase by symport rules, some rule p0 : (0, s0 , 1) had to be applied at some step (by definition s0 ∈ O \ E). This implies that s0 has been brought to the environment. We can assume that rules pi : (1, si /si−1 , 0), 1 ≤ i < n, have been applied (n ≥ 0), si ∈ O \ E, 1 ≤ i ≤ n. Suppose also that n is maximal (sn was not brought to the environment by antiport with another object from O \ E). Thus R either contains rule p : (1, sn , 0), or p′ : (1, sn /a, 0), a ∈ O. Now let us examine the final (halting) configuration. If s0 is in z1 , then p0 can be applied, hence the configuration is not halting. Therefore s0 is in w0 . For all 1 ≤ i ≤ n, given si−1 in z0 , if si is in z1 , then pi can be applied, hence the configuration is not halting. Consequently, si is in z0 as well. By induction, we obtain that sn is in z0 . However, this implies that either p ∈ R and p can be applied, or some p′ ∈ R and p′ can be applied, therefore the configuration is not halting. This implies that any computation where the number of objects inside the membrane is increased cannot halt. Therefore, Π can only generate numbers not exceeding |w|. The statement of the theorem follows directly from here. The following lemma shows that two cells are already sufficient to simulate a register machine. Lemma 1. For any deterministic register machine M and for any input In there is a tissue P system with symport/antiport of degree two having symport and antiport rules of weight 1, which simulates M on this input and produces the same result, i.e., such that:
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(1) If M computes On on input In , then L(Π) will contain a unique element On . (2) If M does not halt on In , then L(Π) will be the empty set. Proof. We consider an arbitrary deterministic register machine M = (Q, R, q0 , qf , P ) with k registers (|R| = k), and we construct a tissue P system with symport/antiport Π that will simulate this machine on the input In . We consider a more general problem: we shall simulate M on any initial configuration (q0 , N1 , . . . , Nk ). We organize the proof as follows. First, we give the formal definition of Π. After that, we present in an informal way how instructions of M are simulated. Finally, a more detailed proof is given. We define the system as follows. Π = (O, G, w1 , w2 , E, RΠ , 2). 0 ′ ′′ − O = Q ∪ R ∪ {A+ pq | ∃(p, A+, q) ∈ P } ∪ {p , p , Qpqs , Qpqs | ∃(p, A−, q, s) ∈ P }. E = R. We consider the following underlying graph G:
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!"# w2 '&%$ 2
w0 '&%$ !"# 0 >> >>1 >> > '&%$ !"# 1 w1 2
w1 = (Q \ {q0 }) ∪ {Apq | (p, A+, q) ∈ P } ∪ {p′ , p′′ , Q′pqs , Q0pqs | (p, A−, q, s) ∈ P }. N w2 = {q0 } ∪ {rj j | 1 ≤ j ≤ k}.
Below we give in tables rules of the system. In fact, RΠ is the union of all cells in all tables. Rules are numbered according to the following convention: the second number is the number of the edge where the rule is located, the first number indicates which instruction is simulated using this rule: 1 for incrementing, 2 for decrementing, 3 for stop and 0 for rules not related to these categories. The third is the number of corresponding rule in group. Rules of RΠ which do not fall in any category (q ∈ Q): Edge Rules 1. 2. 3. 0.3.1 : (0, q, 2) For any rule (p, A+, q) ∈ P there are the following rules in RΠ : Edge Rules 1. 1.1.1 : (0, A+ pq /q, 1) 2. 1.2.1 : (2, p/A+ pq , 1) 3. 1.3.1 : (2, A+ /A, 0) pq
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For any rule (p, A−, q, s) ∈ P there are the following rules in RΠ : Edge Rules 1. 2.1.1 : (0, p′ /Q′pqs , 1) 2.1.2 : (1, p′′ , 0) 2.1.3 : (0, Q′pqs /Q0pqs , 1) 2.1.4 : (0, Q0pqs /s, 1) 2. 2.2.1 : (2, p/p′ , 1) 2.2.2 : (2, p′′ /Q0pqs , 1) ′ 2.2.3 : (2, Qpqs /q, 1) 2.2.4 : (2, Q0pqs , 1) 3. 2.3.1 : (2, p′ /p′′ , 0) 2.3.2 : (2, A/Q′pqs , 0) Rules associated to the STOP instruction: Edge Rules 1. 2. 3.2.1 : (2, qf , 1) 3. We organize system Π as follows. Cell 2 contains the current configuration of machine M . Cell 1 contains one copy of objects that correspond to each state. In the same cell, there are additional symbols used for the simulation of the decrementing operation and which are present in one copy. The environment (node 0) contains symbols rj , 1 ≤ j ≤ k, which are used to increment registers. These symbols are present in an infinite number of copies. Each configuration (p, n1 , . . . , nk ) of machine M is encoded as follows. Cell 2 contains objects rj of the multiplicity nj , 1 ≤ j ≤ k, as well as the object p. It is easy to observe that the initial configuration of Π corresponds to an encoding of the initial configuration of M . Now we shall discuss the simulation of instructions of M . Incrementing Suppose that M is in configuration (p, n1 , . . . , nk ) and that there is a rule (p, rj +, q) in P . Let us denote rj by A and suppose that the value of counter A is equal to n (nj = n). This corresponds to the following configuration of Π where we indicate only symbols that we effectively use.
An '&%$ !"# p 2
A∞ '&%$ !"# 0 >> >> >> > A+ '&%$ !"# 1 pq q
It is clear that in order to simulate the above instruction p and q should be exchanged and one A should come from the environment to cell 2. This is done by symbol A+ pq , which travels from cell 1 to cell 2, the environment and back exchanging with p, A and q.
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Now we present the evolution of the system in more details. The symbol p in the second cell that encodes the current state of M triggers the application of rule 1.2.1 and it exchanges with A+ pq which comes to the second cell. After that, the last symbol brings an object A to cell 2, which corresponds to an incrementing of register A. Further, symbol A+ pq goes to cell 1 and brings from there to the environment the new state q. After that, symbol q moves to cell 2.
An+1'&%$ !"# 2 q
A∞ '&%$ !"# 0 >> >> >> > A+ '&%$ !"# 1 pq p
The last configuration differs from the first one by the following. In cell 2, there is one more copy of object A and the object p was replaced by the object q. All other symbols remained on their places and the symbol p was moved to cell 1 which contains as before one copy of each state of the register machine that is different from the current. This corresponds to the following configuration of M : (q, n1 , . . . , nj + 1, . . . , nk ), i.e., the corresponding instruction of M was simulated. Decrementing Suppose that M is in configuration (p, n1 , . . . , nk ) and that there is a rule (p, rj −, q, s) in P . Let us denote rj by A and suppose that the value of A is n (nj = n). This corresponds to the following configuration of Π where we indicate only symbols that we effectively use.
An '&%$ !"# p 2
p′′ ,A∞ '&%$ !"# 0 >> >> >> p′ > '&%$ !"# q, s 1 Q0pqs , Q′pqs
The idea of the decrementing is very simple. First we duplicate the information that the current state is p by using objects p′ and p′′ . After that, these objects are moved in opposite directions: p′ clockwise and p′′ anticlockwise. If possible, the first object will take off one A from cell 2. The zero test is made based on the reciprocal position of these objects after some steps of computation. If they are in cells 2 and 1 respectively, then this means that one A was subtracted. Otherwise, if p′ is in the environment and p′′ is in cell 1, which means that the value of register A is zero. Based on this information the corresponding branch of the computation is selected. Now we give more details on the evolution of the system. The symbol p in the second cell that encodes the current state of M triggers the application of rule 2.2.1 and it exchanges with p′ which comes to the second cell. After that, the last symbol goes to the environment bringing at the same time the symbol p′′ in cell 2. During
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next step, symbol p′ exchanges with Q′pqs and symbol p′′ exchanges with Q0pqs . The obtained configuration is shown below. Q′pqs , A∞ '&%$ !"# 0 >> >> >> > p, q, s n A '&%$ '&%$ !"# !"# 1 p′ , p′′ 0 2 Qpqs Now there are two cases, n > 0 and n = 0, and the system behaves differently in each case.
CASE A: n > 0 In this case, Q′pqs goes to cell 2 and brings a symbol A to the environment, hence decrementing the register. At the same time symbol p′′ returns to the environment. Finally, the symbol Q′pqs brings the symbol q to cell 2:
An−1'&%$ !"# 2 q
p′′ , A∞ '&%$ !"# 0 >> >> >> p, s > ′ '&%$ !"# 1 p ′ Qpqs , Q0pqs
We can see that the obtained configuration differs from the first one by the following. In cell 2, there is one copy of object A less and the object p was replaced by the object q. All other symbols remained on their places and the symbol p was moved to cell 1 which contains as before one copy of each state of the register machine that is different from current. This corresponds to the following configuration of M : (q, n1 , . . . , nj − 1, . . . , nk ), i.e., the corresponding instruction of M was simulated.
CASE B: n = 0 In this case, Q′pqs remains in the environment for one more step. After that, it exchanges with Q0pqs which returns to cell 1. After that Q0pqs brings to the environment the symbol s which moves after that to cell 2 (see configuration at the right). We remark that the first exchange takes place only if the value of register A is equal to zero, otherwise rule 2.3.2 is applied and the exchange above cannot happen.
!"# s '&%$ 2
p′′ , A∞ '&%$ !"# 0 >> >> >> p, q > ′ '&%$ !"# 1 p ′ Qpqs ,Q0pqs
We can see that the obtained configuration differs from the first one by the following. In cell 2, the object p was replaced by the object s. All other symbols remained on their places and the symbol p was moved to cell 1 which contains as before one copy of each state of the register machine that is not current. This
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corresponds to the following configuration of M : (s, n1 , . . . , nj−1 , 0, nj+1 , . . . , nk ), i.e., the corresponding instruction of M was simulated. Stop If the system is in halting state qf , then rule 3.2.1 moves the symbol qf to cell 1. Since cell 2 contain only symbols corresponding to the value of the output register, the system cannot evolve any more and the computation stops. It is clear that we simulate the behavior of M . Indeed, we simulate an instruction of M and all additional symbols return to their places which permits to simulate the next instruction of M . Moreover, this permits to reconstruct easily a computation in M from a successful computation in Π. For this it is enough to look for configurations which have a state symbol p in cell 2. We stop the computation when rule 3.2.1 is used and symbol qf goes to cell 1. In this case, cell 2 contains the result of the computation. Theorem 2. N OtP2,2 (sym1 , anti1 ) = N RE. Proof. It is easy to observe that the system of previous lemma may simulate a non-deterministic register machine. Indeed, in order to simulate a rule (p, A+, q, s) of such machine, we use the same rules and objects as for rule (p, A+, q). We only need to add a rule 1.1.1′ : (0, A+ pq /s, 1) to edge 1. Descriptional Complexity Let M be a register machine having n states, k registers and n1 incrementing instructions, i.e., M has n − n1 − 1 decrementing instructions. In this case the system constructed as in Lemma 1 need at most n1 + 4(n − n1 − 1) + n = 5n − 3n1 − 4 symbols present in one copy, k symbols present in infinite number of copies, 3n1 +8(n−n1 −1) = 8n−5n1 −8 antiport rules and n+2(n−n1 −1)+1 = 3n−2n1 −1 symport rules. 4. The power of symport of weight 2 In this section we consider systems with symport rules of weight 2 and 1. Similarly to the case discussed in the previous section, when only one cell is used then corresponding systems are not very powerful and may generate only finite sets of numbers [8]. With two cells the computational power increases and becomes equal to the power of register machines. Lemma 2. For any deterministic register machine M and for any input In there is a tissue P system with symport/antiport of degree 2 having only symport rules of weight at most 2, which simulates M on this input and produces the same result, i.e., such that: (1) If M computes On on input In , then L(Π) will contain a unique element On . (2) If M does not halt on In , then L(Π) will be the empty set.
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Proof. As in previous case we consider an arbitrary deterministic register machine M = (Q, R, q0 , qf , P ) with k registers (k = |R|), and we construct a tissue P system with symport/antiport that will simulate this machine on any initial configuration (q0 , N1 , . . . , Nk ). We organize the proof as follows. First, we give the formal definition of Π. After that, since the simulation is very similar to the one from Lemma 1, we present only things that are different from the previous proof. We assume, by commodity, that we may have objects initially present in a finite number of copies in the environment and we denote this multiset by w0 . We shall show later that this assumption is not necessary. Let m be the number of instructions of M and m1 the number of incrementing instructions. Let l = m + 5(m − m1 − 1) + m1 = 6m − 4m1 − 5. We define the system as follows. Π = (O, G, w1 , w2 , E, RΠ , 2). ′ O = Q ∪ R ∪ {Xq | q ∈ Q} ∪ {A+ pq | ∃(p, A+, q) ∈ P } ∪ {Y, E, EY , E , S, V } ′ ′ ∪ {p′ , p′′ , p0 , Zpqs , Zpqs , Dpqs , Dpqs , Xpqs , Xp0 | ∃(p, A−, q, s) ∈ P }
∪ {Ej , Ej′ | 1 ≤ j ≤ l + 1}. E = R. Consider the following multisets over O. V1 = {Xq | q ∈ Q}, V2 = {Xpqs | (p, A−, q, s) ∈ P }, V3 = {Xp0 | (p, A−, q, s) ∈ P }, V4 = {p′ , A+ st | (p, A−, q, l) ∈ P, (s, B+, t) ∈ P } and V = V1 ∪ V2 ∪ V2 ∪ V3 ∪ V3 ∪ V4 , Let n1 = |V1 |, n2 = |V2 |, n3 = |V3 |, n4 = |V4 | and n = |V |, i.e., n = n1 + 2n2 + 2n3 + n4 (it is clear that n = l). We number elements of V = {a1 , . . . , an } as follows. Elements of V1 will receive numbers from 1 to n1 . Elements of V2 will receive numbers from n1 + 1 to n1 + 2n2 . Elements of V3 will receive numbers from n1 + 2n2 + 1 to n1 + 2n2 + 2n3 . Elements of V4 will receive numbers from n1 + 2n2 + 2n3 to n. Now we define initial multisets (1 ≤ j ≤ n + 1, 1 ≤ t ≤ k). We consider that initially the environment may contain a non-empty multiset of objects w0 . We shall show later that this assumption is not necessary. ′′ ′ w0 = {p′′ , Dpqs , Dpqs , Zpqs | (p, A−, q, s) ∈ P } ∪ {Ej′ , E ′ }. ′ w1 = (Q \ {q0 }) ∪ {p0 , Dpqs , Zpqs | (p, A−, q, s) ∈ P } ∪ {Ej , EY }. ′ w2 = {Y, q0 } ∪ {rtNt } ∪ {A+ pq | (p, A+, q) ∈ P } ∪ {p | (p, A−, q, s) ∈ P }
∪ {E, S, Xpqs , Xpqs , Xp0 , Xp0 , Xq }.
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We consider the following underlying graph G:
3
!"# w2 '&%$ 2
w0 '&%$ !"# 0 >> >>1 >> > '&%$ !"# 1 w1 2
Below we give in tables rules of the system. In fact, RΠ is the union of all cells in all tables. Rules are numbered according to the following convention: the second number is the number of the edge where the rule is located, the first number indicates which instruction is simulated using this rule: 1 for incrementing, 2 for decrementing, 3 for stop and 0 for rules not related to these categories. The third is the number of corresponding rule in group. Rules of RΠ which do not fall in any category (q ∈ Q): Edge Rules 1. 0.1.1 : (1, A, 0) 2. 3. 0.3.1 : (0, qY, 2) 0.3.2 : (2, Y, 0) For any rule (p, A+, q) ∈ P there are the following rules in RΠ : Edge Rules 1. 1.1.1 : (0, A+ pq q, 1) 2. 1.2.1 : (2, pA+ pq , 1) 3. 1.3.1 : (2, A+ pq A, 0) For any rule (p, A−, q, s) ∈ P there are the following rules in RΠ :
Edge Rules 1. 2.1.1 : (1, p′ p0 , 0) 2.1.2 : (0, p′′ p0 , 1) ′ ′′ 2.1.3 : (1, Dpqs Dpqs , 0) 2.1.4 : (1, Dpqs q, 0) ′ ′ 2.1.5 : (0, Zpqs Zpqs , 1) 2.1.6 : (1, Zpqs s, 0) 2. 2.2.1 : (2, pp′ , 1) 2.2.2 : (2, Dpqs A, 1) ′ 2.2.3 : (1, p′′ Zpqs , 2) 2.2.4 : (2, Dpqs Zpqs , 1) ′′ 2.2.5 : (2, Dpqs V, 1) 2.2.6 : (1, V, 2) ′′ ′ , 2) Dpqs 3. 2.3.1 : (0, p′ Dpqs , 2) 2.3.2 : (0, Dpqs ′′ 2.3.3 : (2, Dpqs Zpqs , 0) 2.3.4 : (2, p , 0) Below we give rules and objects associated to the STOP instruction (1 ≤ i ≤ n, 1 ≤ j ≤ n + 1):
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Edge Rules 1. 3.1.1 : (1, EEY , 0) 3.1.2 : (0, EY Y, 1) 3.1.3 : (1, SE1 , 0) 3.1.4 : (1, Ei′ Ei+1 , 0) 3.1.5 : (1, Xq q, 0) 3.1.6 : (1, Xpqs Zpqs , 0) 3.1.7 : (1, Xp0 p0 , 0) 2. 3.2.1 : (2, qf E, 1) 3.2.2 : (2, E ′ S, 1) 3.2.3 : (2, Ei ai , 1) 3.2.4 : (2, Ei′ V, 1) 3.2.5 : (2, En+1 E, 1) 3. 3.3.1 : (0, EE ′ , 2) 3.3.2 : (0, Ej Ej′ , 2) 3.3.3 : (2, V En+1 , 0) We organize system Π as in Lemma 1. Cell 2 contains the current configuration of machine M . All cells contain additional symbols used for the simulation of M . The simulation of M is similar to Lemma 1. The incrementing of a register is done directly using the symbol A+ pq . The decrementing operation based on same ideas as for Lemma 1. More exactly, the signal that the system has to perform the decrementing operation is duplicated (p′ and p′′ ) and one symbol tries to decrement the corresponding register, while the other one is delayed in order to make the zero check. The key point consists in the fact that if the register is non-empty, then rules 2.2.2 and 2.2.3 are applied simultaneously and symbols Dpqs and Zpqs cannot meet in cell 2. Contrarily, if the corresponding register is empty, then these symbols meet in cell 2 permitting the simulation of the corresponding branch. The key difference from the previous lemma consists in the termination procedure. Since the system contains a lot of additional symbols, in particular in cell 2, this cell must be cleaned at the end of the computation avoiding at the same time possible conflicts with other rules. This is done in several stages. More precisely, the main problem is to move symbols p′ from cell 2 to cell 1. In order to solve this, it is sufficient to disable the group of rules 2.1.1. This might be done in several stages. (1) (2) (3) (4) (5) (6) (7)
Move symbol Y to cell 1 (thus disabling rule 0.3.1). Move all state symbols (q) to the environment (disabling rule 2.1.6). ′ Move symbols Zpqs to cell 1 (disabling rule 2.1.5). Move symbols Zpqs to the environment (disabling rule 2.2.3). Move symbols p′′ to cell 1 (disabling rule 2.1.2). Move symbols p0 to the environment (disabling rule 2.1.1). Finally move symbols p′ to cell 1.
Rules 3.2.1, 3.1.1 and 3.1.2 permit to realize the first stage. Rules 3.3.2, 3.2.3, 3.2.4 and 3.1.4 permit to move symbols ai from cell 2 to cell 1 one after another. Now the main idea is to use existing rules from the decrementing phase and a special numeration of symbols in a way that will permit to accomplish all stages. First symbols Xq are processed. After being moved to cell 1 these symbols move together with corresponding states q to the environment. Since the symbol Y is no
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more present, symbols q will remain there. Next, symbols Xpqs are processed. These symbols are present in two copies. When the first copy arrives in cell 1, it moves the corresponding symbol Zpqs to the ′ environment. After that, symbol Zpqs brings symbol Zpqs to cell 1 (rule 2.1.5). Now, the second copy of Xpqs will definitively move symbols Zpqs to the environment, hence realizing stages 3 and 4. In a similar way, symbols Xp0 permit to move symbols p′′ to cell 1 and symbols 0 p to the environment. Finally, all remaining symbols from cell 2 are transferred to cell 1 or to the environment. Following same arguments like in Lemma 1 it is clear that we simulate the behavior of M . Now we will show that the assumption that w0 is not empty is not necessary. Indeed, it is enough to place initially all symbols si that are in w0 , except E ′ , in cell 2 as well as |w0 | − 1 copies of symbol U . After that, rules (2, si U, 0) must be added. It is clear that during the first step all symbols si will move to the environment. The remaining symbol E ′ must be placed in cell 1, as well as a copy of the symbol U . A similar rule (1, E ′ U, 0) will move E ′ to the environment at the first step of the computation. Finally, in order to avoid a possible application of rules 2.2.5 and 3.2.4, the symbol V must be initially placed in cell 1. We remark that if we permit a simple encoding of the result, then there is no need for the group of rules associated to the STOP instruction. In this case, if the result of computation of M over input In is s, then system Π will compute number s + m (recall that m is the number of instructions of Π). Theorem 3. N OtP2,2 (sym2 , anti0 ) = N RE. Proof. It is easy to observe that the system of previous lemma may simulate a non-deterministic register machine. Indeed, in order to simulate a rule (p, A+, q, s) of such machine, we use the same rules and objects as for rule (p, A+, q). We only need to add a rule 1.1.1′ : (0, A+ pq s, 1) to the edge 1. Descriptional Complexity Let M be a register machine having n states, k registers and n1 incrementing instructions, i.e., M has n − n1 − 1 decrementing instructions. In this case the system constructed as in Lemma 2 needs at most 14n − 16n1 − 13 symbols present in one copy, k symbols present in infinite number of copies 42n − 29n1 − 28 symport rules of weight 2 and n − n1 + k + 1 symport rules of weight 1. 5. Conclusions In this article we studied tissue P systems with symport/antiport. We considered variants using minimal cooperation (symport of weight two or antiport of weight one) and we showed that they are able to generate any recursively enumerable set of
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numbers with only two cells. We remark that contrarily to other proofs concerning P systems with symport/antiport, the system that we constructed is deterministic. Consequently, if we have a deterministic register machine M and an initial configuration (q0 , n1 , . . . , nk ) of this machine, then the corresponding system constructed as described in Lemma 1 and Lemma 2 will have a deterministic behavior, i.e., there will be only one possible evolution at each step and it will halt if and only if M halts on the above configuration. We highlight this deterministic evolution because it makes the behavior of the system more predictable and makes such systems good candidates for possible implementations. Another advantage of our proof technique with respect to other proofs is that we need only k symbols present in an infinite number of copies, where k is the number of registers of M . All other symbols are present in one or two copies. We also showed that if only one cell is considered, then corresponding systems generate at most finite sets of numbers. Therefore, we completely solved the open question about the minimal antiport from [18], as well as about the minimal symport. We remark that the proof of Theorem 1 holds also in the case of ordinary P systems. However, the difference between two models in this case is minimal. An open problem concerns the number of symport rules used for minimal antiport systems. In our system we used 3n − 2n1 − 1 symport rules. Using ideas from [18] it is possible to decrease the number of symport rules down to 4. The question if we may obtain the same result using a smaller number of symport rules remains open. We remark that because we deal with antiport rules of size one, we need at least one symport rule, otherwise we cannot change the number of objects in the system. Acknowledgements The authors acknowledge the project 06.411.03.04P from the Supreme Council for Science and Technological Development of the Academy of Sciences of Moldova. References [1] A. Alhazov, R. Freund, and Yu. Rogozhin. Some optimal results on symport/antiport P systems with minimal cooperation. In Proceedings of the ESF Exploratory Workshop on Cellular Computing(Complexity Aspects), Sevilla (Spain), January 31st February 2nd, p. 23–36, 2005. [2] A. Alhazov, R. Freund, and Yu. Rogozhin. Computational power of symport/antiport: History, advances, and open problems. In R. Freund, Gh. P˘ aun, G. Rozenberg, and A. Salomaa, eds., Membrane Computing: 6th International Workshop, WMC 2005, Vienna, Austria, July 18-21, 2005, Revised Selected and Invited Papers, vol. 3850 of Lecture Notes in Computer Science, p. 1–30. Springer-Verlag, 2006. [3] A. Alhazov, M. Margenstern, V. Rogozhin, Yu. Rogozhin, and S. Verlan. Communicative P systems with minimal cooperation. In G. Mauri, Gh. P˘ aun, M. Perez-Jimenez, G. Rozenberg, and A. Salomaa, eds., Membrane Computing, 5th International Workshop, WMC5, Milano, Italy, June, 14-16, 2004, Revised Papers, vol. 3365 of Lecture Notes in Computer Science, p. 161–177, 2005. [4] F. Bernardini and M. Gheorghe. On the power of minimal symport/antiport. In
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[6]
[7]
[8] [9]
[10] [11] [12] [13] [14] [15] [16]
[17]
[18]
[19]
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A. Alhazov, C. Mart´ın-Vide, and G. P˘ aun, eds., Preproceedings of the Workshop on Membrane Computing, p. 72–83, Tarragona, July 17-22 2003. F. Bernardini and A. P˘ aun. Universality of minimal symport/antiport: Five membranes suffice. In C. Mart´ın-Vide, G. Mauri, Gh. P˘ aun, G. Rozenberg, and A. Salomaa, eds., Membrane Computing, International Workshop, WMC 2003, Tarragona, Spain, July, 17-22, 2003, Revised Papers, vol. 2933 of Lecture Notes in Computer Science, p. 43–54. Springer-Verlag, 2003. R. Freund and A. P˘ aun. Membrane systems with symport/antiport: Universality results. In Gh. P˘ aun, G. Rozenberg, A. Salomaa, and C. Zandron, eds., Membrane Computing: International Workshop, WMC-CdeA 2002, Curtea de Arges, Romania, August 19-23, 2002. Revised Papers., vol. 2597 of Lecture Notes in Computer Science, p. 270–287. Springer-Verlag, 2003. P. Frisco. About P systems with symport/antiport. In G. Pıaun, A. Riscos-N´ un ˜ez, A. Romero-Jim´enez, and F. Sancho-Caparrini, eds., Second Brainstorming Week on Membrane Computing, number TR 01/2004, p. 224–236. University of Seville, 2004. P. Frisco and H. Hoogeboom. P systems with symport/antiport simulating counter automata. Acta Informatica, 41(2-3):145–170, 2004. L. Kari, C. Mart´ın-Vide, and A. P˘ aun. On the universality of P systems with minimal symport/antiport rules. In N. Jonoska, Gh. P˘ aun, and G. Rozenberg, eds., Aspects of Molecular Computing: Essays Dedicated to Tom Head, on the Occasion of His 70th Birthday, vol. 2950 of Lecture Notes in Computer Science, p. 254–265. SpringerVerlag, 2004. C. Mart´ın-Vide, A. P˘ aun, and Gh. P˘ aun. On the power of P systems with symport rules. Journal of Universal Computer Science, 8(2):317–331, 2002. M. Minsky. Computations: Finite and Infinite Machines. Prentice Hall, Englewood Cliffts, NJ, 1967. A. P˘ aun and Gh. P˘ aun. The power of communication: P systems with symport/antiport. New Generation Computing, 20(3):295–305, 2002. Gh. P˘ aun. Computing with membranes. Journal of Computer and System Sciences, 1(61):108–143, 2000. Also TUCS Report No. 208, 1998. Gh. P˘ aun. Membrane Computing. An Introduction. Springer-Verlag, 2002. Gh. P˘ aun, Y. Sakakibara, and T. Yokomori. P systems on graphs of restricted forms. Publicationes Mathematicae Debrecen, 60:635–660, 2002. Gh. P˘ aun and T. Yokomori. Membrane computing based on splicing. vol. 54 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, p. 217–232. American Mathematical Society, 1999. G. Vaszil. On the size of P systems with minimal symport/antiport. In Preliminary proceedings of WMC 2004, Workshop on Membrane Computing, Milano, Italy, June, 14-16, 2004, p. 422–431, 2004. S. Verlan. Tissue P systems with minimal symport/antiport. In C. S. Calude, E. Calude, and M. J. Dinneen, eds., Developments in Language Theory: 8th International Conference, DLT 2004. Auckland, New Zealand, December 13-17, 2004. Proceedings, vol. 2240 of Lecture Notes in Computer Science, p. 418–430. Springer-Verlag, 2004. C. Zandron. The P systems web page. http://psystems.disco.unimib.it/.
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Artiom Alhazov∗ Yurii Rogozhin Institute of Mathematics and Computer Science Academy of Sciences of Moldova Academiei 5, Chi¸sin˘ au, MD-2028, Moldova {alhazov,rogozhin}@math.md Sergey Verlan† LACL, D´ epartement Informatique, Universit´ e Paris 12, 61, av. G´ en´ eral de Gaulle, 94010 Cr´ eteil, France [email protected]
We investigate tissue P systems with symport/antiport with minimal cooperation, i.e., when only 2 objects may interact. We show that 2 cells are enough in order to generate all recursively enumerable sets of numbers. Moreover, constructed systems simulate register machines and have purely deterministic behavior. We also investigate systems with one cell and we show that they may generate only finite sets of numbers.
1. Introduction P systems were introduced by Gh. P˘ aun in [13] as distributed parallel computing devices of biochemical inspiration. These systems are inspired from the structure and the functioning of a living cell. The cell is considered as a set of compartments (membranes) nested one in another and which contain objects and evolution rules. The basic framework specifies neither the nature of these objects, nor the nature of rules. Numerous variants specify these two parameters by obtaining a lot of different models of computing, see [19] for a comprehensive bibliography. One of these variants, P systems with symport/antiport, was introduced in [12]. This variant uses one of the most important properties of P systems: the communication. This property is so powerful, that it suffices by itself to reach the computational power of Turing machines. These systems have two types of rules: symport rules, when several objects go together from one membrane to another, and antiport rules, when several objects from two membranes are exchanged. In spite of a simple definition, they may compute all Turing computable sets of numbers [12]. This result was ∗ Also Research Group on Mathematical Linguistics, Rovira i Virgili University, Tarragona, Spain, [email protected] † Corresponding
author 1
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improved with respect to the number of used membranes and/or the weight of symport/antiport rules ([6], [8], [10], [14], [5]). Rather unexpectedly, minimal symport/antiport P systems, i.e., when one uses only one object in symport rules and a pair of objects in antiport rules are universal. The proof of this result may be found in [4] and the corresponding system has 9 membranes. This result was improved first by reducing the number of membranes to six [9], after that to five [5] and four [7]. In [17] it is shown that three membranes are sufficient to generate all recursively enumerable sets of numbers using 5 superfluous objects in the output membrane. In [3], a stronger result is reported where the output membrane did not contain superfluous objects. Similarly, when minimal symport is used, i.e., when only symport rules dealing with at most two objects are permitted, three membranes suffice to generate all recursively enumerable sets of numbers [3]. A recent result shows that this number can be decreased down to two for both minimal symport and minimal antiport cases [1], yet only modulo a terminal alphabet. At last in [2] it was shown that the number of additional objects can be bounded by 3 in minimal antiport case and by 6 in minimal symport case. The inspiration for tissue P systems comes from two sides. From one hand, P systems previously introduced may be viewed as transformations of labels associated to nodes of a tree. Therefore, it is natural to consider same transformations on a graph. On the other hand, they may be obtained by following the same reflections as for P systems, but starting from a tissue of cells and no more from a single cell. Tissue P systems were initially investigated in [15] and [16]. They have richer possibilities and the advantages of new topology have to be investigated. Tissue P systems with symport/antiport were first considered in [14] where several results having different values of parameters (graph size, maximal size of connected component, weight of symport and antiport rules) are presented. In this paper we consider symport and antiport rules with minimal cooperation, i.e., we consider minimal antiport rules (of weight 1) that permit to exchange two objects, as well as minimal symport rules (of weight 2) that permit to send two objects together. Systems with minimal antiport were first investigated in [18] where it was shown that 3 cells are sufficient to generate any recursively enumerable set of numbers. In this article, we investigate both variants and we show that in both cases we can construct systems defined on a graph with 2 cells that simulate any (non-)deterministic register machine. Moreover, in the deterministic case, obtained systems are also deterministic and only one evolution is possible at any time. Therefore, if the computation stops, then we are sure that the corresponding register machine stops on the provided input. Another difference from previous proofs is that we use a very small amount of symbols present in an infinite number of copies in the environment. We also show that if only one cell is considered, then corresponding systems may
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generate no more than finite sets of numbers. The minimal symport case was already discussed in [8], so we present only the antiport one. Therefore, we found a boundary between the decidability and the undecidability for corresponding systems. 2. Definitions We denote by N the set of all non-negative natural numbers, {0, 1, . . . }, and by N′ the set {1, 2, . . . }. A multiset S over O is a mapping fS : O −→ N. The mapping fS specifies the number of occurrences of each element of S. The size of the multiset S is P |S| = x∈O fS (x). We use the ordinary set notation in order to specify a multiset. In this case we either indicate the number of occurrences of each element as its power, or we give the mapping function fS . For example the multiset containing 3 occurrences of element a, one occurrence of element b and zero occurrences of element c will be specified as {a3 , b} or a → 3, b → 1, c → 0. We shall also use a string notation in order to specify a multiset. In this case we write all elements of the multiset having a positive multiplicity in a string. For example, the previous multiset will be written as aaab or a3 b. The sum (union) of two multisets P and Q over O, denoted P ∪ Q, is a multiset S such that fS (a) = fP (a) + fQ (a) for all a in O. Similarly, the difference of two multisets P and Q is a multiset S having fS (a) = fP (a) ⊖ fQ (a) where ⊖ is the positive subtraction. A deterministic register machine is the following construction: M = (Q, R, q0 , qf , P ), where Q is a set of states, R = {r1 , . . . , rk } is the set of registers, q0 ∈ Q is the initial state, qf ∈ Q is the final state and P is a set of instructions (called also rules) of the following form: (1) (p, A+, q) ∈ P , p, q ∈ Q, p 6= q, A ∈ R (being in state p, increase register A and go to state q). (2) (p, A−, q, s) ∈ P , p, q, s ∈ Q, A ∈ R (being in state p, decrease register A and go to q if successful or to s if A is zero). (3) (qf , ST OP ) (may be associated only to the final state qf ). We note that for each state p there is only one instruction of the type above. A configuration of a register machine is given by the (k +1)-tuple (q, n1 , . . . , nk ), where q ∈ Q and ni ∈ N, 1 ≤ i ≤ k, describing the current state of the machine as well as the contents of all registers. A transition of the register machine consists in updating/checking the value of a register according to an instruction of one of types above and by changing the current state to another one. We say that the machine stops, if it reaches the state qf .
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We say that M computes a value y ∈ N on the input x ∈ N if starting from the initial configuration (q0 , x, 0, . . . , 0) it reaches the final configuration (qf , y, 0, . . . , 0). We say that M recognizes the set S ⊆ N if for any input x ∈ S the machine stops and for any y 6∈ S the machine does not stop (performs an infinite computation). It is known that (deterministic) register machines recognize all recursively enumerable sets of numbers [11]. We may also consider non-deterministic register machines where the first type of instruction is of the form (p, A+, q, s) and with the following meaning: if the machine is in state p, then the register A is increased and the current state is changed to q or s non-deterministically. A non-deterministic register machine generates the set of all values of the first register resulting from computations that stop on an empty input. We assume that the machine empties all registers except the first register before stopping. It is known that non-deterministic register machines generate all recursively enumerable sets of non-negative integers. A tissue P system with symport/antiport of degree m ≥ 1 is a construct Π = (O, G, w1 , . . . , wm , E, R, i0 ), where O is the alphabet of objects and G is the underlying directed labelled graph of the system. The graph G has m + 1 nodes and the nodes are numbered from 0 to m. We shall also call nodes from 1 to m cells and node 0 the environment. There is an edge between each cell i, 1 ≤ i ≤ m, and the environment. Each cell contains a multiset of objects, initially cell i, 1 ≤ i ≤ m, contains multiset wi . The environment is a special node which contains symbols from E in infinite multiplicity as well as a finite multiset over O\E, but initially this multiset is empty. The symbol i0 ∈ (1 . . . m) indicates the output cell, and R is a finite set of rules (associated to edges) of the following forms: (1) (i, x, j), 0 ≤ i ≤ m, 0 ≤ j ≤ m, i 6= j, x ∈ O+ and not i = 0 & x ∈ E + (symport rules for the communication). (2) (i, x/y, j), 0 ≤ i, j ≤ m, i 6= j, x, y ∈ O+ (antiport rules for the communication). We remark that G may be deduced from relations of R. More exactly, G contains m + 1 vertices and there is an oriented edge between vertex i and j if and only if there is a rule (i, x, j) in R and edges between i and j and j and i if and only if there is a rule (i, x/y, j) in R. However, we prefer to indicate both G and R because it simplifies the presentation. The rule (i, x, j) sends a multiset of objects x from node i to node j. The rule (i, x/y, j) exchanges multisets x and y situated in nodes i and j respectively. The weight of symport rule (i, x, j) is equal to |x|, while the weight of an antiport rule is equal to max{|x|, |y|}. A computational step is made by applying all applicable rules from R in a nondeterministic maximal parallel way. A configuration of the system is an (m + 1)tuple (z0 , z1 , . . . , zm ) where each zi , 1 ≤ i ≤ m, represents the contents of cell i and
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z0 represents the multiset of objects that appear with a finite multiplicity in the environment (initially z0 is the empty multiset). The computation stops when no rule may be applied. The result of a computation is given by the number of objects situated in cell i0 , i.e., by the size of the multiset from cell i0 . We denote by N OtPm,n (symp , antiq ) the family of all sets of numbers computed by tissue P systems with symport/antiport of degree at most m with the maximal size of the connected component of the restriction of the graph to cells equal to n and which have symport rules of weight at most p and antiport rules of weight at most q. Example 1. Let us consider system Π = (O, G, w1 , . . . , w6 , E, R, 2), with O = {A, s, s′ , s′′ , B, C}, E = {B, C} and which has the following underlying graph G (for commodity, we labelled its edges). We note that the environment (node 0) is connected to all other cells, but we indicate on the graph below only connections that are used. '&%$ !"# jj 0 HHH 3 HH HH jjjj j j $$ !"# j '&%$ !"# '&%$ 4 2 4 Hpp H 3 HH 5 vvv OO HH v v H 7 vv 8 // '&%$ !"# '&%$ !"# 6 10 2 ddHH 6 HH9 HH H 11 '&%$ !"# !"# // '&%$ 5 1 1 jjjj
Now we shall give rules and objects of Π. Node Object(s) Edge 0. B ∞ , C ∞ 1. 1.1 : (4, C/B, 0) 1. ss′ An 2. 2.1 : (2, sB/CC, 0) 2. B 3. 3.1 : (0, s, 3) 3. 4. 4.1 : (2, ss′ , 4) 4. 5. 5.1 : (3, s/s′ , 2) ′′ 5. s 6. 6.1 : (4, s′ /s′′ , 5) 6. 7. 7.1 : (4, sB, 6) 8. 8.1 : (6, B, 2) 9. 9.1 : (1, sA, 2) 10. 10.1 : (1, s′ , 3) 11. 11.1 : (5, ss′ , 1)
Rules 2.2 : (2, A, 0) 4.2 : (2, C, 4) 5.2 : (3, s′ , 2) 6.2 : (4, ss′′ , 5) 7.2 : (4, s′′ /s, 6) 7.3 : (6, s′′ , 4)
Now, let us see what the system Π computes. In the initial configuration rules 9.1 and 10.1 are applicable. This brings s and A in the second cell and s′ in cell 3. Now rules 2.1, 3.1, 5.1 and 5.2 permit to replace each B situated in cell 2 by two copies of C. When there are no more copies of B, rule 4.1 can be applied, bringing s and s′ to cell 4. In the meanwhile, all C from cell 2 go to cell 4 by rule 4.2. In cell 4 all C are replaced by B. Rules 7.1, 7.2, 7.3 and 8.1 permit to move these B
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to cell 6 and after that to cell 2. When there are no more B in cell 4, rules 6.2 and 11.1 are used bringing symbols s, s′ and s′′ to their original locations. This last configuration has one A less in cell 1 and the double number of B in cell 2 comparing to the initial configuration. It is easy to observe that after n such cycles the computation will stop because there will be no more A in cell 1. The result of the computation may be read in cell 2 which contains 2n elements B. Therefore, given a number n, system Π computes 2n . 3. The power of symport/antiport of weight 1 In this section we investigate the power of systems having symport/antiport rules of weight 1. Firstly, we show that systems having only one cell are not very powerful. Theorem 1. N OtP1,1 (sym1 , anti1 ) ⊆ N F IN . Proof. Consider an arbitrary P system Π with 1 cell and symport/antiport rules of weight 1, Π = (O, G, w1 , E, R, 1). Consider also an arbitrary halting computation, ending in some configuration C = (z0 , z1 ∪ ze ) with z1 ∈ (O \ E)∗ and ze ∈ E ∗ and w0 ∈ (O \ E)∗ . We claim that |z1 | + |ze | ≤ |w1 |. We shall prove this assertion by contradiction. Let us assume the contrary. Since the number of objects in the membrane can only increase by symport rules, some rule p0 : (0, s0 , 1) had to be applied at some step (by definition s0 ∈ O \ E). This implies that s0 has been brought to the environment. We can assume that rules pi : (1, si /si−1 , 0), 1 ≤ i < n, have been applied (n ≥ 0), si ∈ O \ E, 1 ≤ i ≤ n. Suppose also that n is maximal (sn was not brought to the environment by antiport with another object from O \ E). Thus R either contains rule p : (1, sn , 0), or p′ : (1, sn /a, 0), a ∈ O. Now let us examine the final (halting) configuration. If s0 is in z1 , then p0 can be applied, hence the configuration is not halting. Therefore s0 is in w0 . For all 1 ≤ i ≤ n, given si−1 in z0 , if si is in z1 , then pi can be applied, hence the configuration is not halting. Consequently, si is in z0 as well. By induction, we obtain that sn is in z0 . However, this implies that either p ∈ R and p can be applied, or some p′ ∈ R and p′ can be applied, therefore the configuration is not halting. This implies that any computation where the number of objects inside the membrane is increased cannot halt. Therefore, Π can only generate numbers not exceeding |w|. The statement of the theorem follows directly from here. The following lemma shows that two cells are already sufficient to simulate a register machine. Lemma 1. For any deterministic register machine M and for any input In there is a tissue P system with symport/antiport of degree two having symport and antiport rules of weight 1, which simulates M on this input and produces the same result, i.e., such that:
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(1) If M computes On on input In , then L(Π) will contain a unique element On . (2) If M does not halt on In , then L(Π) will be the empty set. Proof. We consider an arbitrary deterministic register machine M = (Q, R, q0 , qf , P ) with k registers (|R| = k), and we construct a tissue P system with symport/antiport Π that will simulate this machine on the input In . We consider a more general problem: we shall simulate M on any initial configuration (q0 , N1 , . . . , Nk ). We organize the proof as follows. First, we give the formal definition of Π. After that, we present in an informal way how instructions of M are simulated. Finally, a more detailed proof is given. We define the system as follows. Π = (O, G, w1 , w2 , E, RΠ , 2). 0 ′ ′′ − O = Q ∪ R ∪ {A+ pq | ∃(p, A+, q) ∈ P } ∪ {p , p , Qpqs , Qpqs | ∃(p, A−, q, s) ∈ P }. E = R. We consider the following underlying graph G:
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!"# w2 '&%$ 2
w0 '&%$ !"# 0 >> >>1 >> > '&%$ !"# 1 w1 2
w1 = (Q \ {q0 }) ∪ {Apq | (p, A+, q) ∈ P } ∪ {p′ , p′′ , Q′pqs , Q0pqs | (p, A−, q, s) ∈ P }. N w2 = {q0 } ∪ {rj j | 1 ≤ j ≤ k}.
Below we give in tables rules of the system. In fact, RΠ is the union of all cells in all tables. Rules are numbered according to the following convention: the second number is the number of the edge where the rule is located, the first number indicates which instruction is simulated using this rule: 1 for incrementing, 2 for decrementing, 3 for stop and 0 for rules not related to these categories. The third is the number of corresponding rule in group. Rules of RΠ which do not fall in any category (q ∈ Q): Edge Rules 1. 2. 3. 0.3.1 : (0, q, 2) For any rule (p, A+, q) ∈ P there are the following rules in RΠ : Edge Rules 1. 1.1.1 : (0, A+ pq /q, 1) 2. 1.2.1 : (2, p/A+ pq , 1) 3. 1.3.1 : (2, A+ /A, 0) pq
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For any rule (p, A−, q, s) ∈ P there are the following rules in RΠ : Edge Rules 1. 2.1.1 : (0, p′ /Q′pqs , 1) 2.1.2 : (1, p′′ , 0) 2.1.3 : (0, Q′pqs /Q0pqs , 1) 2.1.4 : (0, Q0pqs /s, 1) 2. 2.2.1 : (2, p/p′ , 1) 2.2.2 : (2, p′′ /Q0pqs , 1) ′ 2.2.3 : (2, Qpqs /q, 1) 2.2.4 : (2, Q0pqs , 1) 3. 2.3.1 : (2, p′ /p′′ , 0) 2.3.2 : (2, A/Q′pqs , 0) Rules associated to the STOP instruction: Edge Rules 1. 2. 3.2.1 : (2, qf , 1) 3. We organize system Π as follows. Cell 2 contains the current configuration of machine M . Cell 1 contains one copy of objects that correspond to each state. In the same cell, there are additional symbols used for the simulation of the decrementing operation and which are present in one copy. The environment (node 0) contains symbols rj , 1 ≤ j ≤ k, which are used to increment registers. These symbols are present in an infinite number of copies. Each configuration (p, n1 , . . . , nk ) of machine M is encoded as follows. Cell 2 contains objects rj of the multiplicity nj , 1 ≤ j ≤ k, as well as the object p. It is easy to observe that the initial configuration of Π corresponds to an encoding of the initial configuration of M . Now we shall discuss the simulation of instructions of M . Incrementing Suppose that M is in configuration (p, n1 , . . . , nk ) and that there is a rule (p, rj +, q) in P . Let us denote rj by A and suppose that the value of counter A is equal to n (nj = n). This corresponds to the following configuration of Π where we indicate only symbols that we effectively use.
An '&%$ !"# p 2
A∞ '&%$ !"# 0 >> >> >> > A+ '&%$ !"# 1 pq q
It is clear that in order to simulate the above instruction p and q should be exchanged and one A should come from the environment to cell 2. This is done by symbol A+ pq , which travels from cell 1 to cell 2, the environment and back exchanging with p, A and q.
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Now we present the evolution of the system in more details. The symbol p in the second cell that encodes the current state of M triggers the application of rule 1.2.1 and it exchanges with A+ pq which comes to the second cell. After that, the last symbol brings an object A to cell 2, which corresponds to an incrementing of register A. Further, symbol A+ pq goes to cell 1 and brings from there to the environment the new state q. After that, symbol q moves to cell 2.
An+1'&%$ !"# 2 q
A∞ '&%$ !"# 0 >> >> >> > A+ '&%$ !"# 1 pq p
The last configuration differs from the first one by the following. In cell 2, there is one more copy of object A and the object p was replaced by the object q. All other symbols remained on their places and the symbol p was moved to cell 1 which contains as before one copy of each state of the register machine that is different from the current. This corresponds to the following configuration of M : (q, n1 , . . . , nj + 1, . . . , nk ), i.e., the corresponding instruction of M was simulated. Decrementing Suppose that M is in configuration (p, n1 , . . . , nk ) and that there is a rule (p, rj −, q, s) in P . Let us denote rj by A and suppose that the value of A is n (nj = n). This corresponds to the following configuration of Π where we indicate only symbols that we effectively use.
An '&%$ !"# p 2
p′′ ,A∞ '&%$ !"# 0 >> >> >> p′ > '&%$ !"# q, s 1 Q0pqs , Q′pqs
The idea of the decrementing is very simple. First we duplicate the information that the current state is p by using objects p′ and p′′ . After that, these objects are moved in opposite directions: p′ clockwise and p′′ anticlockwise. If possible, the first object will take off one A from cell 2. The zero test is made based on the reciprocal position of these objects after some steps of computation. If they are in cells 2 and 1 respectively, then this means that one A was subtracted. Otherwise, if p′ is in the environment and p′′ is in cell 1, which means that the value of register A is zero. Based on this information the corresponding branch of the computation is selected. Now we give more details on the evolution of the system. The symbol p in the second cell that encodes the current state of M triggers the application of rule 2.2.1 and it exchanges with p′ which comes to the second cell. After that, the last symbol goes to the environment bringing at the same time the symbol p′′ in cell 2. During
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next step, symbol p′ exchanges with Q′pqs and symbol p′′ exchanges with Q0pqs . The obtained configuration is shown below. Q′pqs , A∞ '&%$ !"# 0 >> >> >> > p, q, s n A '&%$ '&%$ !"# !"# 1 p′ , p′′ 0 2 Qpqs Now there are two cases, n > 0 and n = 0, and the system behaves differently in each case.
CASE A: n > 0 In this case, Q′pqs goes to cell 2 and brings a symbol A to the environment, hence decrementing the register. At the same time symbol p′′ returns to the environment. Finally, the symbol Q′pqs brings the symbol q to cell 2:
An−1'&%$ !"# 2 q
p′′ , A∞ '&%$ !"# 0 >> >> >> p, s > ′ '&%$ !"# 1 p ′ Qpqs , Q0pqs
We can see that the obtained configuration differs from the first one by the following. In cell 2, there is one copy of object A less and the object p was replaced by the object q. All other symbols remained on their places and the symbol p was moved to cell 1 which contains as before one copy of each state of the register machine that is different from current. This corresponds to the following configuration of M : (q, n1 , . . . , nj − 1, . . . , nk ), i.e., the corresponding instruction of M was simulated.
CASE B: n = 0 In this case, Q′pqs remains in the environment for one more step. After that, it exchanges with Q0pqs which returns to cell 1. After that Q0pqs brings to the environment the symbol s which moves after that to cell 2 (see configuration at the right). We remark that the first exchange takes place only if the value of register A is equal to zero, otherwise rule 2.3.2 is applied and the exchange above cannot happen.
!"# s '&%$ 2
p′′ , A∞ '&%$ !"# 0 >> >> >> p, q > ′ '&%$ !"# 1 p ′ Qpqs ,Q0pqs
We can see that the obtained configuration differs from the first one by the following. In cell 2, the object p was replaced by the object s. All other symbols remained on their places and the symbol p was moved to cell 1 which contains as before one copy of each state of the register machine that is not current. This
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corresponds to the following configuration of M : (s, n1 , . . . , nj−1 , 0, nj+1 , . . . , nk ), i.e., the corresponding instruction of M was simulated. Stop If the system is in halting state qf , then rule 3.2.1 moves the symbol qf to cell 1. Since cell 2 contain only symbols corresponding to the value of the output register, the system cannot evolve any more and the computation stops. It is clear that we simulate the behavior of M . Indeed, we simulate an instruction of M and all additional symbols return to their places which permits to simulate the next instruction of M . Moreover, this permits to reconstruct easily a computation in M from a successful computation in Π. For this it is enough to look for configurations which have a state symbol p in cell 2. We stop the computation when rule 3.2.1 is used and symbol qf goes to cell 1. In this case, cell 2 contains the result of the computation. Theorem 2. N OtP2,2 (sym1 , anti1 ) = N RE. Proof. It is easy to observe that the system of previous lemma may simulate a non-deterministic register machine. Indeed, in order to simulate a rule (p, A+, q, s) of such machine, we use the same rules and objects as for rule (p, A+, q). We only need to add a rule 1.1.1′ : (0, A+ pq /s, 1) to edge 1. Descriptional Complexity Let M be a register machine having n states, k registers and n1 incrementing instructions, i.e., M has n − n1 − 1 decrementing instructions. In this case the system constructed as in Lemma 1 need at most n1 + 4(n − n1 − 1) + n = 5n − 3n1 − 4 symbols present in one copy, k symbols present in infinite number of copies, 3n1 +8(n−n1 −1) = 8n−5n1 −8 antiport rules and n+2(n−n1 −1)+1 = 3n−2n1 −1 symport rules. 4. The power of symport of weight 2 In this section we consider systems with symport rules of weight 2 and 1. Similarly to the case discussed in the previous section, when only one cell is used then corresponding systems are not very powerful and may generate only finite sets of numbers [8]. With two cells the computational power increases and becomes equal to the power of register machines. Lemma 2. For any deterministic register machine M and for any input In there is a tissue P system with symport/antiport of degree 2 having only symport rules of weight at most 2, which simulates M on this input and produces the same result, i.e., such that: (1) If M computes On on input In , then L(Π) will contain a unique element On . (2) If M does not halt on In , then L(Π) will be the empty set.
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Proof. As in previous case we consider an arbitrary deterministic register machine M = (Q, R, q0 , qf , P ) with k registers (k = |R|), and we construct a tissue P system with symport/antiport that will simulate this machine on any initial configuration (q0 , N1 , . . . , Nk ). We organize the proof as follows. First, we give the formal definition of Π. After that, since the simulation is very similar to the one from Lemma 1, we present only things that are different from the previous proof. We assume, by commodity, that we may have objects initially present in a finite number of copies in the environment and we denote this multiset by w0 . We shall show later that this assumption is not necessary. Let m be the number of instructions of M and m1 the number of incrementing instructions. Let l = m + 5(m − m1 − 1) + m1 = 6m − 4m1 − 5. We define the system as follows. Π = (O, G, w1 , w2 , E, RΠ , 2). ′ O = Q ∪ R ∪ {Xq | q ∈ Q} ∪ {A+ pq | ∃(p, A+, q) ∈ P } ∪ {Y, E, EY , E , S, V } ′ ′ ∪ {p′ , p′′ , p0 , Zpqs , Zpqs , Dpqs , Dpqs , Xpqs , Xp0 | ∃(p, A−, q, s) ∈ P }
∪ {Ej , Ej′ | 1 ≤ j ≤ l + 1}. E = R. Consider the following multisets over O. V1 = {Xq | q ∈ Q}, V2 = {Xpqs | (p, A−, q, s) ∈ P }, V3 = {Xp0 | (p, A−, q, s) ∈ P }, V4 = {p′ , A+ st | (p, A−, q, l) ∈ P, (s, B+, t) ∈ P } and V = V1 ∪ V2 ∪ V2 ∪ V3 ∪ V3 ∪ V4 , Let n1 = |V1 |, n2 = |V2 |, n3 = |V3 |, n4 = |V4 | and n = |V |, i.e., n = n1 + 2n2 + 2n3 + n4 (it is clear that n = l). We number elements of V = {a1 , . . . , an } as follows. Elements of V1 will receive numbers from 1 to n1 . Elements of V2 will receive numbers from n1 + 1 to n1 + 2n2 . Elements of V3 will receive numbers from n1 + 2n2 + 1 to n1 + 2n2 + 2n3 . Elements of V4 will receive numbers from n1 + 2n2 + 2n3 to n. Now we define initial multisets (1 ≤ j ≤ n + 1, 1 ≤ t ≤ k). We consider that initially the environment may contain a non-empty multiset of objects w0 . We shall show later that this assumption is not necessary. ′′ ′ w0 = {p′′ , Dpqs , Dpqs , Zpqs | (p, A−, q, s) ∈ P } ∪ {Ej′ , E ′ }. ′ w1 = (Q \ {q0 }) ∪ {p0 , Dpqs , Zpqs | (p, A−, q, s) ∈ P } ∪ {Ej , EY }. ′ w2 = {Y, q0 } ∪ {rtNt } ∪ {A+ pq | (p, A+, q) ∈ P } ∪ {p | (p, A−, q, s) ∈ P }
∪ {E, S, Xpqs , Xpqs , Xp0 , Xp0 , Xq }.
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We consider the following underlying graph G:
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!"# w2 '&%$ 2
w0 '&%$ !"# 0 >> >>1 >> > '&%$ !"# 1 w1 2
Below we give in tables rules of the system. In fact, RΠ is the union of all cells in all tables. Rules are numbered according to the following convention: the second number is the number of the edge where the rule is located, the first number indicates which instruction is simulated using this rule: 1 for incrementing, 2 for decrementing, 3 for stop and 0 for rules not related to these categories. The third is the number of corresponding rule in group. Rules of RΠ which do not fall in any category (q ∈ Q): Edge Rules 1. 0.1.1 : (1, A, 0) 2. 3. 0.3.1 : (0, qY, 2) 0.3.2 : (2, Y, 0) For any rule (p, A+, q) ∈ P there are the following rules in RΠ : Edge Rules 1. 1.1.1 : (0, A+ pq q, 1) 2. 1.2.1 : (2, pA+ pq , 1) 3. 1.3.1 : (2, A+ pq A, 0) For any rule (p, A−, q, s) ∈ P there are the following rules in RΠ :
Edge Rules 1. 2.1.1 : (1, p′ p0 , 0) 2.1.2 : (0, p′′ p0 , 1) ′ ′′ 2.1.3 : (1, Dpqs Dpqs , 0) 2.1.4 : (1, Dpqs q, 0) ′ ′ 2.1.5 : (0, Zpqs Zpqs , 1) 2.1.6 : (1, Zpqs s, 0) 2. 2.2.1 : (2, pp′ , 1) 2.2.2 : (2, Dpqs A, 1) ′ 2.2.3 : (1, p′′ Zpqs , 2) 2.2.4 : (2, Dpqs Zpqs , 1) ′′ 2.2.5 : (2, Dpqs V, 1) 2.2.6 : (1, V, 2) ′′ ′ , 2) Dpqs 3. 2.3.1 : (0, p′ Dpqs , 2) 2.3.2 : (0, Dpqs ′′ 2.3.3 : (2, Dpqs Zpqs , 0) 2.3.4 : (2, p , 0) Below we give rules and objects associated to the STOP instruction (1 ≤ i ≤ n, 1 ≤ j ≤ n + 1):
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Edge Rules 1. 3.1.1 : (1, EEY , 0) 3.1.2 : (0, EY Y, 1) 3.1.3 : (1, SE1 , 0) 3.1.4 : (1, Ei′ Ei+1 , 0) 3.1.5 : (1, Xq q, 0) 3.1.6 : (1, Xpqs Zpqs , 0) 3.1.7 : (1, Xp0 p0 , 0) 2. 3.2.1 : (2, qf E, 1) 3.2.2 : (2, E ′ S, 1) 3.2.3 : (2, Ei ai , 1) 3.2.4 : (2, Ei′ V, 1) 3.2.5 : (2, En+1 E, 1) 3. 3.3.1 : (0, EE ′ , 2) 3.3.2 : (0, Ej Ej′ , 2) 3.3.3 : (2, V En+1 , 0) We organize system Π as in Lemma 1. Cell 2 contains the current configuration of machine M . All cells contain additional symbols used for the simulation of M . The simulation of M is similar to Lemma 1. The incrementing of a register is done directly using the symbol A+ pq . The decrementing operation based on same ideas as for Lemma 1. More exactly, the signal that the system has to perform the decrementing operation is duplicated (p′ and p′′ ) and one symbol tries to decrement the corresponding register, while the other one is delayed in order to make the zero check. The key point consists in the fact that if the register is non-empty, then rules 2.2.2 and 2.2.3 are applied simultaneously and symbols Dpqs and Zpqs cannot meet in cell 2. Contrarily, if the corresponding register is empty, then these symbols meet in cell 2 permitting the simulation of the corresponding branch. The key difference from the previous lemma consists in the termination procedure. Since the system contains a lot of additional symbols, in particular in cell 2, this cell must be cleaned at the end of the computation avoiding at the same time possible conflicts with other rules. This is done in several stages. More precisely, the main problem is to move symbols p′ from cell 2 to cell 1. In order to solve this, it is sufficient to disable the group of rules 2.1.1. This might be done in several stages. (1) (2) (3) (4) (5) (6) (7)
Move symbol Y to cell 1 (thus disabling rule 0.3.1). Move all state symbols (q) to the environment (disabling rule 2.1.6). ′ Move symbols Zpqs to cell 1 (disabling rule 2.1.5). Move symbols Zpqs to the environment (disabling rule 2.2.3). Move symbols p′′ to cell 1 (disabling rule 2.1.2). Move symbols p0 to the environment (disabling rule 2.1.1). Finally move symbols p′ to cell 1.
Rules 3.2.1, 3.1.1 and 3.1.2 permit to realize the first stage. Rules 3.3.2, 3.2.3, 3.2.4 and 3.1.4 permit to move symbols ai from cell 2 to cell 1 one after another. Now the main idea is to use existing rules from the decrementing phase and a special numeration of symbols in a way that will permit to accomplish all stages. First symbols Xq are processed. After being moved to cell 1 these symbols move together with corresponding states q to the environment. Since the symbol Y is no
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more present, symbols q will remain there. Next, symbols Xpqs are processed. These symbols are present in two copies. When the first copy arrives in cell 1, it moves the corresponding symbol Zpqs to the ′ environment. After that, symbol Zpqs brings symbol Zpqs to cell 1 (rule 2.1.5). Now, the second copy of Xpqs will definitively move symbols Zpqs to the environment, hence realizing stages 3 and 4. In a similar way, symbols Xp0 permit to move symbols p′′ to cell 1 and symbols 0 p to the environment. Finally, all remaining symbols from cell 2 are transferred to cell 1 or to the environment. Following same arguments like in Lemma 1 it is clear that we simulate the behavior of M . Now we will show that the assumption that w0 is not empty is not necessary. Indeed, it is enough to place initially all symbols si that are in w0 , except E ′ , in cell 2 as well as |w0 | − 1 copies of symbol U . After that, rules (2, si U, 0) must be added. It is clear that during the first step all symbols si will move to the environment. The remaining symbol E ′ must be placed in cell 1, as well as a copy of the symbol U . A similar rule (1, E ′ U, 0) will move E ′ to the environment at the first step of the computation. Finally, in order to avoid a possible application of rules 2.2.5 and 3.2.4, the symbol V must be initially placed in cell 1. We remark that if we permit a simple encoding of the result, then there is no need for the group of rules associated to the STOP instruction. In this case, if the result of computation of M over input In is s, then system Π will compute number s + m (recall that m is the number of instructions of Π). Theorem 3. N OtP2,2 (sym2 , anti0 ) = N RE. Proof. It is easy to observe that the system of previous lemma may simulate a non-deterministic register machine. Indeed, in order to simulate a rule (p, A+, q, s) of such machine, we use the same rules and objects as for rule (p, A+, q). We only need to add a rule 1.1.1′ : (0, A+ pq s, 1) to the edge 1. Descriptional Complexity Let M be a register machine having n states, k registers and n1 incrementing instructions, i.e., M has n − n1 − 1 decrementing instructions. In this case the system constructed as in Lemma 2 needs at most 14n − 16n1 − 13 symbols present in one copy, k symbols present in infinite number of copies 42n − 29n1 − 28 symport rules of weight 2 and n − n1 + k + 1 symport rules of weight 1. 5. Conclusions In this article we studied tissue P systems with symport/antiport. We considered variants using minimal cooperation (symport of weight two or antiport of weight one) and we showed that they are able to generate any recursively enumerable set of
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numbers with only two cells. We remark that contrarily to other proofs concerning P systems with symport/antiport, the system that we constructed is deterministic. Consequently, if we have a deterministic register machine M and an initial configuration (q0 , n1 , . . . , nk ) of this machine, then the corresponding system constructed as described in Lemma 1 and Lemma 2 will have a deterministic behavior, i.e., there will be only one possible evolution at each step and it will halt if and only if M halts on the above configuration. We highlight this deterministic evolution because it makes the behavior of the system more predictable and makes such systems good candidates for possible implementations. Another advantage of our proof technique with respect to other proofs is that we need only k symbols present in an infinite number of copies, where k is the number of registers of M . All other symbols are present in one or two copies. We also showed that if only one cell is considered, then corresponding systems generate at most finite sets of numbers. Therefore, we completely solved the open question about the minimal antiport from [18], as well as about the minimal symport. We remark that the proof of Theorem 1 holds also in the case of ordinary P systems. However, the difference between two models in this case is minimal. An open problem concerns the number of symport rules used for minimal antiport systems. In our system we used 3n − 2n1 − 1 symport rules. Using ideas from [18] it is possible to decrease the number of symport rules down to 4. The question if we may obtain the same result using a smaller number of symport rules remains open. We remark that because we deal with antiport rules of size one, we need at least one symport rule, otherwise we cannot change the number of objects in the system. Acknowledgements The authors acknowledge the project 06.411.03.04P from the Supreme Council for Science and Technological Development of the Academy of Sciences of Moldova. References [1] A. Alhazov, R. Freund, and Yu. Rogozhin. Some optimal results on symport/antiport P systems with minimal cooperation. In Proceedings of the ESF Exploratory Workshop on Cellular Computing(Complexity Aspects), Sevilla (Spain), January 31st February 2nd, p. 23–36, 2005. [2] A. Alhazov, R. Freund, and Yu. Rogozhin. Computational power of symport/antiport: History, advances, and open problems. In R. Freund, Gh. P˘ aun, G. Rozenberg, and A. Salomaa, eds., Membrane Computing: 6th International Workshop, WMC 2005, Vienna, Austria, July 18-21, 2005, Revised Selected and Invited Papers, vol. 3850 of Lecture Notes in Computer Science, p. 1–30. Springer-Verlag, 2006. [3] A. Alhazov, M. Margenstern, V. Rogozhin, Yu. Rogozhin, and S. Verlan. Communicative P systems with minimal cooperation. In G. Mauri, Gh. P˘ aun, M. Perez-Jimenez, G. Rozenberg, and A. Salomaa, eds., Membrane Computing, 5th International Workshop, WMC5, Milano, Italy, June, 14-16, 2004, Revised Papers, vol. 3365 of Lecture Notes in Computer Science, p. 161–177, 2005. [4] F. Bernardini and M. Gheorghe. On the power of minimal symport/antiport. In
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A. Alhazov, C. Mart´ın-Vide, and G. P˘ aun, eds., Preproceedings of the Workshop on Membrane Computing, p. 72–83, Tarragona, July 17-22 2003. F. Bernardini and A. P˘ aun. Universality of minimal symport/antiport: Five membranes suffice. In C. Mart´ın-Vide, G. Mauri, Gh. P˘ aun, G. Rozenberg, and A. Salomaa, eds., Membrane Computing, International Workshop, WMC 2003, Tarragona, Spain, July, 17-22, 2003, Revised Papers, vol. 2933 of Lecture Notes in Computer Science, p. 43–54. Springer-Verlag, 2003. R. Freund and A. P˘ aun. Membrane systems with symport/antiport: Universality results. In Gh. P˘ aun, G. Rozenberg, A. Salomaa, and C. Zandron, eds., Membrane Computing: International Workshop, WMC-CdeA 2002, Curtea de Arges, Romania, August 19-23, 2002. Revised Papers., vol. 2597 of Lecture Notes in Computer Science, p. 270–287. Springer-Verlag, 2003. P. Frisco. About P systems with symport/antiport. In G. Pıaun, A. Riscos-N´ un ˜ez, A. Romero-Jim´enez, and F. Sancho-Caparrini, eds., Second Brainstorming Week on Membrane Computing, number TR 01/2004, p. 224–236. University of Seville, 2004. P. Frisco and H. Hoogeboom. P systems with symport/antiport simulating counter automata. Acta Informatica, 41(2-3):145–170, 2004. L. Kari, C. Mart´ın-Vide, and A. P˘ aun. On the universality of P systems with minimal symport/antiport rules. In N. Jonoska, Gh. P˘ aun, and G. Rozenberg, eds., Aspects of Molecular Computing: Essays Dedicated to Tom Head, on the Occasion of His 70th Birthday, vol. 2950 of Lecture Notes in Computer Science, p. 254–265. SpringerVerlag, 2004. C. Mart´ın-Vide, A. P˘ aun, and Gh. P˘ aun. On the power of P systems with symport rules. Journal of Universal Computer Science, 8(2):317–331, 2002. M. Minsky. Computations: Finite and Infinite Machines. Prentice Hall, Englewood Cliffts, NJ, 1967. A. P˘ aun and Gh. P˘ aun. The power of communication: P systems with symport/antiport. New Generation Computing, 20(3):295–305, 2002. Gh. P˘ aun. Computing with membranes. Journal of Computer and System Sciences, 1(61):108–143, 2000. Also TUCS Report No. 208, 1998. Gh. P˘ aun. Membrane Computing. An Introduction. Springer-Verlag, 2002. Gh. P˘ aun, Y. Sakakibara, and T. Yokomori. P systems on graphs of restricted forms. Publicationes Mathematicae Debrecen, 60:635–660, 2002. Gh. P˘ aun and T. Yokomori. Membrane computing based on splicing. vol. 54 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, p. 217–232. American Mathematical Society, 1999. G. Vaszil. On the size of P systems with minimal symport/antiport. In Preliminary proceedings of WMC 2004, Workshop on Membrane Computing, Milano, Italy, June, 14-16, 2004, p. 422–431, 2004. S. Verlan. Tissue P systems with minimal symport/antiport. In C. S. Calude, E. Calude, and M. J. Dinneen, eds., Developments in Language Theory: 8th International Conference, DLT 2004. Auckland, New Zealand, December 13-17, 2004. Proceedings, vol. 2240 of Lecture Notes in Computer Science, p. 418–430. Springer-Verlag, 2004. C. Zandron. The P systems web page. http://psystems.disco.unimib.it/.
