New Choice for Small Universal Devices: Symport/Antiport P Systems
aa r X i v : . [ c s . CC ] J un T. Neary, D. Woods, A.K. Seda and N. Murphy (Eds.):The Complexity of Simple Programs 2008.EPTCS 1, 2009, pp. 235–242, doi:10.4204/EPTCS.1.23 c (cid:13)
S. Verlan and Y. Rogozhin
New Choice for Small Universal Devices: Symport/Antiport PSystems
Sergey Verlan
LACL, D´epartement Informatique, Universit´e Paris Est,61 av. G´en´eral de Gaulle, 94010 Cr´eteil, FranceInstitute of Mathematics and Computer ScienceAcademy of Sciences of Moldova, Academiei, 5, MD-2028, Moldova [email protected]
Yurii Rogozhin
Institute of Mathematics and Computer ScienceAcademy of Sciences of Moldova, Academiei, 5, MD-2028, MoldovaRovira i Virgili University,Research Group on Mathematical Linguistics,Pl. Imperial T`arraco 1, 43005 Tarragona, Spain [email protected]
Symport/antiport P systems provide a very simple machinery inspired by corresponding operationsin the living cell. It turns out that systems of small descriptional complexity are needed to achievethe universality by these systems. This makes them a good candidate for small universal devices re-placing register machines for different simulations, especially when a simulating parallel machineryis involved. This article contains survey of these systems and presents different trade-offs betweenparameters.
The idea of symport/antiport P systems comes from simple observations in cell biology. In a living cell,there is a permanent chemical exchange with the environment. Water, ions and other chemicals enteror exit the cell depending on its necessity. Some of these exchanges use a passive transport where noenergy is consumed and the chemicals are moved along the chemical gradient, while others use an activetransport , which consumes energy in order to move chemicals against the gradient. Very often the activetransport uses co-transporters , i.e. molecules that facilitate the penetration of the transported substancethrough the cell membrane. The most common co-transporters either travel together with the transportedsubstance, in this case we speak about symport , or they are exchanged with the transported substance, inthis case we speak about antiport .This transport mechanism is formalized by symport/antiport P systems [11], [12] which abstract thecell by a set of nested compartments enclosed by membranes and chemicals by a multiset of objects. Thesymport transport is then represented by a rule ( y , out ) or ( y , in ) that specifies that objects present in themultiset y travel together outside or inside the current compartment. The antiport is formalized by therule ( x , out ; y , in ) which indicates that objects given by x and present in the compartment will exchangewith objects given by y situated outside the compartment.The evolution of a symport/antiport P system is done in a maximally parallel way (other evolutionstrategies are discussed in [8]), starting from an initial distribution of objects in membranes and the resultis obtained by counting objects in some membrane when the system cannot evolve anymore.36 NewChoice for SmallUniversal Devices: Symport/Antiport PSystemsFurther generalization of the model leads to symport/antiport tissue P systems where the underlyingmembrane structure is no more represented by a tree as in the case of P systems but by an arbitrary graphcorresponding to a tissue of cells. More generalizations and a presentation of P systems (not necessarilyusing symport and antiport operations) can be found in [12] and [16].The computational model given by symport/antiport (tissue) P systems is very simple, however it wasshown that if a cooperation of three objects is permitted, then one membrane is sufficient to generate allrecursively enumerable sets of numbers [7] and [9]. After that other descriptional complexity parametersstared to be investigated, in particular, systems with minimal symport or antiport, where only two objectscan cooperate. Such systems are of great interest because the biological variants of symport and antiportinvolve only two objects in most of the cases. These systems first were investigated in [5], where ninemembranes were used to achieve computational completeness. This number was progressively decreasedand finally established to two membranes in [3].Other complexity parameters like the number of used objects or the number of rules were investigatedand trade-offs between different parameters were established. In this article we present a survey ofdifferent complexity measures and best known results.
We recall here some basic notions of formal language theory we need in the rest of the paper. We referto [14] for further details.We denote by N the set of all non-negative integers. Let O = { a , . . . , a k } be an alphabet. A finite mul-tiset M over O is a mapping M : O −→ N , i.e., for each a ∈ O , M ( a ) specifies the number of occurrencesof a in M . The size of the multiset M is | M | = (cid:229) a ∈ O M ( a ) . A multiset M over O can also be repre-sented by any string x that contains exactly M ( a i ) symbols a i for all 1 ≤ i ≤ k , e.g., by a M ( a ) . . . a M ( a k ) k ,or else by the set { a M ( a i ) i | ≤ i ≤ k } . For example, the multiset over { a , b , c } defined by the mapping a → , b → , c → a b or { a , b } . An empty multiset is represented by l .We may also consider mappings M of form M : O −→ N ∪ { ¥ } , i.e., elements of M may have aninfinite multiplicity; we shall call them infinite multisets .In the following we briefly recall the basic notions concerning P systems with symport/antiport rules.For more details on these systems and on P systems in general, we refer to [12].A P system with symport/antiport of degree n is a construct P = ( O , m , w , . . . , w n , E , R , . . . , R n , i ) , where:1. O is a finite alphabet of symbols called objects,2. m is a membrane structure consisting of n membranes that are labeled in a one-to-one manner by1 , , . . . , n .3. w i ∈ O ∗ , for each 1 ≤ i ≤ n is a multiset of objects associated with the region i (delimited bymembrane i ),4. E ⊆ O is the set of objects that appear in the environment in infinite numbers of copies,5. R i , for each 1 ≤ i ≤ n , is a finite set of symport/antiport rules associated with the region i and whichhave the following form ( x , in ) , ( y , out ) , ( y , out ; x , in ) , where x , y ∈ O ∗ ,.Verlan and Y.Rogozhin 2376. i is the label of an elementary membrane of m that identifies the corresponding output region.A symport/antiport P system is defined as a computational device consisting of a set of k hierarchi-cally nested membranes that identify k distinct regions (the membrane structure m ), where to each region i there are assigned a multiset of objects w i and a finite set of symport/antiport rules R i , 1 ≤ i ≤ n . Asymport rule ( x , in ) ∈ R i permits to move x into region i from the immediately outer region. Notice thatrules of the form ( x , in ) , where x ∈ E ∗ are forbidden in the skin (the outermost) membrane. A symportrule ( x , out ) ∈ R i permits to move the multiset x from region i to the outer region. An antiport rule ( y , out ; x , in ) exchanges two multisets y and x , which are situated in region i and the outer region of i respectively.A computation in a symport/antiport P system is obtained by applying the rules in a non-deterministicmaximally parallel manner, i.e. all rules that can be applied together should be applied. Other possibil-ities not using the maximal parallelism are discussed in [8]. The computation is restricted to movingobjects through membranes, since symport/antiport rules do not allow the system to modify the objectsplaced inside the regions. Initially, each region i contains the corresponding finite multiset w i ; whereasthe environment contains only objects from E that appear in infinitely many copies.A computation is successful if starting from the initial configuration it reaches a configuration whereno rule can be applied. The result of a successful computation is the natural number that is obtainedby counting the objects that are presented in region i . Given a P system P , the set of natural numberscomputed in this way by P is denoted by N ( P ) .We denote by NOP n ( sym r , anti t ) the family of sets of natural numbers that are generated by a Psystem with symport/antiport of degree at most n >
0, symport rules of size at most r ≥
0, and antiportrules of size at most t ≥
0. The size of a symport rule ( x , in ) or ( x , out ) is given by | x | , while the sizeof an antiport rule ( y , out ; x , in ) is given by max {| x | , | y |} . We denote by NRE the family of recursivelyenumerable sets of natural numbers.P systems as defined above have an underlying tree-like membrane structure. It is possible to applya similar reasoning to an arbitrary graph. This leads us to the idea of tissue P systems.A tissue P system with symport/antiport of degree n ≥ P = ( O , G , w , . . . , w n , E , R , i ) , where O is the alphabet of objects and G is the underlying directed labeled graph of the system. Thegraph G has n + n . We shall also call nodes from 1 to n cells and node 0 the environment. There is an edge between each cell i , 1 ≤ i ≤ n , and the environment.Each cell contains a multiset of objects, initially cell i , 1 ≤ i ≤ n , contains multiset w i . The environmentis 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 i ∈ ( . . . n ) indicates the output cell, and R is afinite set of rules (associated to edges) of the following forms:1. ( i , x , j ) , 0 ≤ i ≤ n , ≤ j ≤ n , i = j , x ∈ O + and not i = x ∈ E + (symport rules for the commu-nication).2. ( i , x / y , j ) , 0 ≤ i , j ≤ n , i = j , x , y ∈ O + (antiport rules for the communication).We remark that G may be deduced from relations of R . More exactly, G contains n + i and j if and only if there is a rule ( i , x , j ) in R and edgesbetween 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 indicateboth G and R because it simplifies the presentation.38 NewChoice for SmallUniversal Devices: Symport/Antiport PSystemsThe rule ( i , x , j ) sends a multiset of objects x from node i to node j . The rule ( i , x / y , j ) exchangesmultisets x and y situated in nodes i and j respectively. The size of symport rule ( i , x , j ) is equal to | x | ,while the size of an antiport rule is equal to | x | + | y | .As in the case of P systems a computational step is made by applying all applicable rules from R in anon-deterministic maximal parallel way. A configuration of the system is an ( n + ) -tuple ( z , z , . . . , z n ) where each z i , ≤ i ≤ n , represents the contents of cell i and z represents the multiset of objects thatappear with a finite multiplicity in the environment (initially z is the empty multiset). The computationstops when no rule may be applied. The result of a computation is given by the number of objects situatedin cell i , i.e., by the size of the multiset from cell i .We denote by NOtP n ( sym p , anti q ) the family of all sets of numbers computed by tissue P systemswith symport/antiport of degree at most n and which have symport rules of size at most p and antiportrules of size at most q .The following theorem shows the basic results for symport/antiport [tissue] P systems: Theorem 1 NO [ t ] P ( sym ) = NO [ t ] P ( anti ) = RE . We can also consider accepting (tissue) P systems where an input multiset is placed in some fixedcell/membrane and it is accepted if and only if the corresponding system halts. Theorem 1 holds as wellin the accepting case, however it is possible to use a deterministic construction for the proof.
Theorems from the previous section show that using symport or antiport rules of size three the compu-tational completeness is achieved with only one membrane. The situation changes completely if rules ofsize two, called minimal antiport or minimal symport rules, are considered – in one membrane or cell,we only get finite sets: Theorem 2 NO [ t ] P ( sym , anti ) ∪ NO [ t ] P ( sym ) ⊆ NFIN.
The theorem follows from the fact that the number of symbols inside the membrane cannot be in-creased using minimal symport or antiport rules. Hence at least two membranes are needed for thecomputational completeness. This number is sufficient, as the following result holds.
Theorem 3 NO [ t ] P ( sym , anti ) = NO [ t ] P ( sym ) = NRE.
The proof significantly differs if tissue or tree-like P systems are considered. In the tissue case, theproof is based on the possibility to reach a membrane from another one by two roads, directly or viathe environment, which have a different length. In this way, a temporal de-synchronization of pairs ofobjects is obtained and it can be used to simulate the instructions of a register machine.Moreover, in the tissue case, we have a deterministic construction for the acceptance of recursivelyenumerable sets. In the tree-like case it is not possible to use a similar technique, because only the rootis connected to the environment, which considerably restricts the accepting power of deterministic Psystems:
Theorem 4
For any deterministic P system with minimal symport and minimal antiport rules (of typesym and anti ), the number of objects present in the initial configuration of the system cannot be in-creased during halting computations. .Verlan and Y.Rogozhin 239Hence, deterministic P systems with minimal symport and antiport rules with any number of mem-branes can generate only finite languages.However, if non-deterministic systems are considered, then it is possible to reach computational com-pleteness for the accepting case with two membranes: an initial pumping phase is performed to introducea sufficient number of working objects needed to carry out the computation (a non-deterministic guessfor the number of working objects is done). After that, the system simulates a register machine therebyconsuming the number of working objects. We can generalize the idea of minimal antiport and symport and introduce the concept of minimal in-teraction tissue P systems . These are tissue P systems where at most two objects may interact, i.e.,one object is moved with respect to another one. Such interactions can be described by rules of the form ( a , i )( b , j ) → ( a , k )( b , l ) , which indicate that if symbol a is present in membrane i and symbol b is presentin membrane j , then a will move to membrane k and b will move to membrane l . We may impose severalrestrictions on these interaction rules, namely by superposing several cells. Some of these restrictionsdirectly correspond to antiport or symport rules of size 2.Below we define all possible restrictions (modulo symmetry): let O be an alphabet and let ( a , i )( b , j ) → ( a , k )( b , l ) be an interaction rule with a , b ∈ O , i , j , k , l ≥
0. Then we distinguish the following cases:1. i = j = k = l : the conditional-uniport-out rule sends b to membrane l provided that a and b are inmembrane i .2. i = k = l = j : the conditional-uniport-in rule brings b to membrane i provided that a is in thatmembrane.3. i = j = k = l : the symport2 rule corresponds to the minimal symport rule, i.e., a and b movetogether from membrane i to k .4. i = l = j = k : the antiport1 rule corresponds to the minimal antiport rule, i.e., a and b are ex-changed in membranes i and k .5. i = k = j = l : the presence-move rule moves the symbol b from membrane j to l , provided thatthere is a symbol a in membrane i .6. i = j = k = l : the separation rule sends a and b from membrane i to membranes k and l , respec-tively.7. k = l = i = j : the joining rule brings a and b together to membrane i .8. i = l = j = k or i = j = k = l : the chain rule moves a from membrane i to membrane k while b ismoved from membrane j to membrane i , i.e., where a previously has been.9. i = j = k = l : the parallel-shift rule moves a and b in independent membranes.A minimal interaction tissue P system may have rules of several types as defined above. With respectto the computational power of such systems we immediately see that when only antiport1 rules or onlysymport2 rules are used, the number of objects in the system cannot be increased, hence, such systemscan generate only finite sets of natural numbers. However, if we allow uniport rules (i.e., rules of theform ( a , i ) → ( a , k ) specifying that, whenever an object a is present in cell i , this may be moved to cell k ),then minimal interaction tissue P systems with symport2 and uniport rules or with antiport1 and uniportrules become tissue P systems with minimal symport or minimal symport and antiport, respectively.By combining conditional-uniport-in rules and conditional-uniport-out rules, computational com-pleteness can be achieved by simulating a register machine. The best known construction from [2] is40 NewChoice for SmallUniversal Devices: Symport/Antiport PSystemsusing 14 cells, but it is very probably that this number can be decreased. A register machine may be alsosimulated by using only the parallel-shift rule with 19 cells [15]. In all other cases, when only one of thetypes of rules defined above is considered, it is not even clear whether infinite sets of natural numberscan be generated.Another interesting problem is to investigate how an interaction rule may be simulated by somerestricted variants. Such a study may lead to a formulation of sufficient conditions on how combinationsof variants of rules ( a , i )( b , j ) → ( a , k )( b , l ) may guarantee that the system can be realized by using onlyspecific restricted variants of rules in an equivalent minimal interaction tissue P system. After that, asystem satisfying sufficient conditions of several restrictions may be automatically rewritten in terms ofany corresponding restricted variants. A list of such results can be found in [15]. Another complexity parameter that can be investigated is the number of objects that can be used. Themain results for P systems with antiport (and symport) rules can be summarized in the following table: objects5 NRE4 2 NRE3 1 2 NRE2 C 1 2 NRE1 A B B B B1 2 3 4 m ( ≥ ) membranes In the above table, the class of P systems indicated by A generates exactly
NFIN , the class indicatedby B generates at least
NREG , in the case of C at least
NREG can be generated and at least
NFIN can be accepted, while a class indicated by a number d can simulate any d -register machine. The mostinteresting questions still remaining open are to characterize the families generated or accepted by Psystems with only one symbol.In the tissue case the situation changes as the additional links between every cell and the environmentpermit to easily simulate a register machine [1]. However, the definition used by the authors is slightlydifferent and it imposes a sequentiality for the communication between two cells, i.e. if two rules thatinvolve same two cells may be applied at the same time, then only one of them will be chosen. The tablebelow shows the obtained results. In the table A indicates that the corresponding family includes at least NREG , and B indicates that the corresponding family can generate more than NFIN .objects4
NREG NRE NREG A NREG A NRE NRE NRE NRE NRE NFIN B A A A A NRE
In this section we consider universal symport/antiport P systems having a small number of rules. Sucha bound can be obtained if we simulate a universal device for which a bound on the number of rules isalready known. Since P systems with antiport and symport rules can easily simulate register machines,it is natural to consider simulations of register machines having a small number of instructions. An.Verlan and Y.Rogozhin 241example of such a machine is the register machine U described in [10], which has 22 instructions (9increment and 13 decrement instructions). The table below summarizes the best results known on thistopic, showing the trade-off between the number of antiport rules and their size:number of rules 73 56 47 43 30 23size of rules 3 5 6 7 11 19The results for columns 1, 4 and 5 were established in [6], while other results are taken from [4].The last column in this table is particularly interesting, because the register machine U which was thestarting point of the construction uses 25 computational branches. Symport/antiport P systems were heavily investigated (there are more than 60 articles on this topic) anda lot of results about them are known, in particular, about systems having low complexity parameters.This information combined with their simple construction makes them an ideal object to be used inuniversality proofs where they can replace register machines, in particular for parallel computing devices.They are particularly well suited as a simulated device for different classes of P systems which permitsto obtain different descriptional complexity improvements.Even if there are a lot of results on P systems with symport/antiport, there remain a lot of openquestions; we would like to highlight the importance of the investigation of generalized minimal com-munication models as this can show new communication strategies that can be further used in othervariants of P systems. Another important topic is the number of rules of universal antiport P systemswith one membrane. This is especially interesting because such systems directly correspond to maxi-mally parallel multiset rewriting systems (MPMRS), see [4] for a formal definition of MPMRS. Sincealmost all types of object-based P systems can be represented in terms of MPMRS, this will give a lowerbound on the number of rules needed for an universal P system.
Acknowledgments
The authors gratefully acknowledge support by the Science and Technology Centerin Ukraine, project 4032.
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