On acceptance conditions for membrane systems: characterisations of L and NL
TT. Neary, D. Woods, A.K. Seda and N. Murphy (Eds.):The Complexity of Simple Programs 2008.EPTCS 1, 2009, pp. 172–184, doi:10.4204/EPTCS.1.17 c (cid:13)
N. Murphy and D. Woods
On acceptance conditions for membrane systems:characterisations of L and NL
Niall Murphy ∗ [email protected] Department of Computer Science, National University of Ireland Maynooth, Ireland
Damien Woods †Deptartment of Computer Science & Artificial Intelligence, University of Seville, Spain [email protected]
In this paper we investigate the affect of various acceptance conditions on recogniser membranesystems without dissolution. We demonstrate that two particular acceptance conditions (one easier toprogram, the other easier to prove correctness) both characterise the same complexity class, NL . Wealso find that by restricting the acceptance conditions we obtain a characterisation of L . We obtainthese results by investigating the connectivity properties of dependency graphs that model membranesystem computations. In the membrane systems (also known as P-systems [11]) computational complexity community it iscommon practice to explore the power of systems by allowing and prohibiting different developmentalrules. This technique has yielded several interesting results such as the role of membrane dissolution inrecognising
PSPACE -complete problems [5] and the role of membrane division in recognising problemsoutside of P [15].In this paper we do not vary the rules permitted in membrane systems but instead we vary the accep-tance conditions and observe the change (or lack of change) this makes to the computing power of the sys-tem. Our main technique is to analyse the structure, and connectivity, of dependency graphs [5] that areinduced by acceptance conditions. Our approach builds on previous work on dependency graphs [5, 4]to give a number of new techniques and results. Our techniques and results should be of interest tothose who wish to characterise complexity classes, those studying acceptance conditions for membranesystems, and those characterising the power of membrane systems.This research was motivated by the realisation that in prior work [9] we were using a seemingly moregeneral halting condition than is used by the membrane community. Previously, we showed that AC -uniform families of active membrane systems without dissolution, and using the acceptance conditionsspecified in Section 3.1, characterise NL [9]. However, most researchers use a more restricted acceptancecondition (see Section 3.2). We show here that this more restricted definition also characterises NL .This means that the two definitions are equivalent in terms of computing power for ( AC ) – PMC ∗ A M − d systems. The choice of which definition to use is now mostly a matter of personal taste as we have shown ∗ Funded by the Irish Research Council for Science, Engineering and Technology † Funded by Junta de Andaluc´ıa grant TIC-581 All membrane systems in this paper are AC -uniform and run for polynomial time. . Murphy and D. Woods AC reductions, i.e. there is a (very efficient) compiler to translate onedefinition to another.In Section 3.3 we show that active membrane systems without dissolution, and using a restriction onthe standard acceptance definition, characterise L . This demonstrates that not all (minor) restrictions onhalting definitions yield systems that characterise NL .We note here that the three definitions that we consider in Section 3 all characterise P if they aregeneralised to use P -uniformity. The P lower bound of this characterisation is a trivial corollary of thefact that such membrane systems can easily embed polynomial time deterministic Turing machines, andis not related to the differences in their definitions. In this section we define membrane systems and some complexity classes. These definitions are basedon those from P˘aun [11, 10], Sos´ık and Rodr´ıguez-Pat´on [13], Guti´errez-Naranjo et al. [5], and P´erez-Jim´enez et al. [12]. Previous works on complexity and membrane systems spoke of solving a problemin a “uniform way”, that is, in a manner reminiscent of how families of circuits solve a problem. Sos´ıkand Rodr´ıguez-Pat´on defined uniformity for membrane systems in a similar manner to circuit uniformity,this allows us to refer to uniform families of membrane systems.
Active membrane systems are a class of membrane systems with membrane division rules. Divisionrules can either only act on elementary membranes, or else on both elementary and non-elementarymembranes. An elementary membrane is one which does not contain other membranes (a leaf node, intree terminology).
Definition 1.
An active membrane system without charges is a tuple Π = ( O , H , µ , w , . . . , w m , R ) where,1. m ≥ is the initial number of membranes;2. O is the alphabet of objects;3. H is the finite set of labels for the membranes;4. µ is a membrane structure in the form of a tree, consisting of m membranes (nodes), labelled withelements of H. The parent of all membranes (the root node) is called the “environment” and haslabel env ∈ H;5. w , . . . , w m are strings over O, describing the multisets of objects placed in the m regions of µ .6. R is a finite set of developmental rules, of the following forms:(a) [ a → u ] h , for h ∈ H , a ∈ O , u ∈ O ∗ (b) a [ ] h → [ b ] h , for h ∈ H , a , b ∈ O(c) [ a ] h → [ ] h b, for h ∈ H , a , b ∈ O(d) [ a ] h → b, for h ∈ H , a , b ∈ O(e) [ a ] h → [ b ] h [ c ] h , for h ∈ H , a , b , c ∈ O.( f ) [ a [ ] h [ ] h [ ] h ] h → [ b [ ] h [ ] h ] h [ c [ ] h [ ] h ] h ,for h , h , h , h ∈ H , a , b , c ∈ O. These rules are applied according to the following principles:74
On acceptance conditions for membrane systems: characterisations of L and NL • All the rules are applied in a maximally parallel manner. That is, in one step, one object of amembrane is used by at most one rule (chosen in a non-deterministic way), but any object whichcan evolve by one rule of any form, must evolve. • If at the same time a membrane labelled with h is divided by a rule of type ( e ) or ( f ) and there areobjects in this membrane which evolve by means of rules of type ( a ), then we suppose that firstthe evolution rules of type ( a ) are used, and then the division is produced. This process takes onlyone step. • The rules associated with membranes labelled with h are used for membranes with that label. Atone step, a membrane can be the subject of only one rule of types ( b )–( f ). • Rules of type ( f ) are division rules for non-elementary membranes. These rules allow us duplicatean entire branch of the membrane structure in the following manner. If the membrane (label h )to which the non-elementary division rule is applied contains objects and child membranes thencopies of those membranes and all of their contents (including their own child membranes) arefound in both resulting copies of h . In this paper one of our goals is to unify and clarify definitions for language recognising variants ofmembrane systems. To achieve this, we consider three different notions of acceptance for recognisersystems, one in each of Sections 3.1 to 3.3. Each of these three definitions is a restriction on the general(and purposely vague) Definition 2 below.We recall from [5] that a computation of the system is a sequence of configurations such that eachconfiguration (except the initial one) is obtained from the previous one by a transition. A computationthat reaches a configuration where no more rules can be applied to the existing objects and membranesis called a halting computation.
Definition 2. A recognizer membrane system is a membrane system with external output (that is, theresults of halting computations are encoded in the environment) such that:1. the working alphabet contains two distinguished elements yes and no ;2. if C is a computation of the system, then it is either an accepting or a rejecting computation. This definition is vague since we have not defined accepting and rejecting computations. In Section 3we show the set of problems that a membrane system accepts when using various notions of accepting(or rejecting) computations.
Consider a decision problem X , i.e. a set of instances X = { x , x , . . . } over some finite alphabet suchthat to each x i there is an unique answer “yes” or “no”. We say that a family of membrane systemssolves a decision problem if each instance of the problem is solved by some family member. We denoteby | x | = n the length of any instance x ∈ X . Throughout this paper, AC circuits are DLOGTIME -uniform, polynomial sized (in input length n ), constant depth, circuits with AND, OR and NOT gates, andunbounded fanin [2]. The complexity class L ( NL ) is the set of problems solved by (non-)deterministicTuring machines using only O ( log n ) space, where n is the length of the input instance. Definition 3.
Let D be a class of membrane systems and let f : N → N be a total function. Theclass of problems solved by AC -uniform families of membrane systems of type D in time f , denoted ( AC ) – MC D ( f ) , contains all problems X such that: . Murphy and D. Woods • There exists an AC - uniform family of membrane systems, ΠΠΠ X = ( Π X ( ) , Π X ( ) , . . . ) of type D :that is, there exists an AC circuit family such that on unary input n the n th member of the circuitfamily constructs Π X ( n ) . • There exists an AC circuit family such that on input x ∈ X , of length | x | = n, the n th member ofthe family encodes x as a multiset of input objects placed in the distinct input membrane of Π X ( n ) . • ΠΠΠ X is sound and complete with respect to problem X : Π X ( n ) starting with an encoding of inputx ∈ X of length n accepts iff the answer to x is “yes”. • ΠΠΠ X is f -efficient: Π X ( n ) always halts in at most f ( n ) steps. Definition 3 describes AC -uniform families and we generalise this to define AC -semi-uniformfamilies of membrane systems ΠΠΠ X = ( Π X ( x ) ; Π X ( x ) ; . . . ) where there exists an AC circuit familywhich, on an input x ∈ X , constructs membrane system Π X ( x ) . Here a single circuit family (rather thantwo) is used to construct the semi-uniform membrane family, and so the problem instance is encodedusing objects, membranes, and rules. In this case, for each instance of x ∈ X we have a special membranesystem which does not need a separately constructed input. The resulting class of problems is denoted by ( AC ) – MC ∗ D ( f ) . Obviously, ( AC ) – MC D ( f ) ⊆ ( AC ) – MC ∗ D ( f ) for any given class D and a valid [1]complexity function f .We define ( AC ) – PMC D and ( AC ) – PMC ∗ D as ( AC ) – PMC D = (cid:91) k ∈ N ( AC ) – MC D ( n k ) , and ( AC ) – PMC ∗ D = (cid:91) k ∈ N ( AC ) – MC ∗ D ( n k ) . In other words, ( AC ) – PMC D (and ( AC ) – PMC ∗ D ) is the class of problems solvable by uniform (re-spectively semi-uniform) families of membrane systems in polynomial time. We let A M denote theclass of membrane systems with active membranes and no charges. We let ( AC ) – PMC ∗ A M − d denotethe class of problems solvable by AC -semi-uniform families of membrane systems in polynomial timewith no dissolution rules. In an abuse of notation, we often let ( AC ) – PMC ∗ A M − d refer to the classof such membrane systems (rather than problems). For brevity we often write Π X instead of Π X ( n ) or Π X ( x ) . Remark 4.
A membrane system is confluent if it is both sound and complete. That is a Π X is confluent ifall computations of Π X with the same input give the same result; either always accepting or else alwaysrejecting. In a confluent membrane system, given a fixed initial configuration, the system non-deterministicallychooses one from a number of valid configuration sequences, but all of the reachable configurationsequences must lead to the same result, either all accepting or all rejecting.
The dependency graph (first introduced by Guti´errez-Naranjo et al. [5]) is an indispensable tool forcharacterising the computational complexity of membrane systems without dissolution. This technique76
On acceptance conditions for membrane systems: characterisations of L and NL is reminiscent of configuration graphs for Turing Machines. Similarly to a configuration graph, a de-pendency graph helps visualise a computation. However, it differs in its approach by representing amembrane system configuration as a set of nodes rather than as a single node in configuration space.Looking at membrane systems without dissolution as dependency graphs allows us to employ theexisting, mature corpse of techniques and complexity results for graph problems. As we show in thispaper, this greatly simplifies the process of proving upper and lower bounds for such systems. A keytechnique we use in this paper is to transfer from a dependency graph to a new membrane system, Π → G Π → Π G Π . This new system accepts iff the original membrane system accepts, since their dependencygraphs are isomorphic. Also, the new system is considerably simplified as it uses only one membrane(the environment) and all rules are of type ( a ). This is used as a normal form for membrane systemswithout dissolution.In Sections 3.1 to 3.3 we define reachability problems for dependency graphs such that if the answerto the graph reachability problem is yes, then the membrane system it represents is an accepting system.This is because the nodes of a dependency graph represent an object being in a certain membrane,and an edge between two nodes represents a developmental rule that causes that object to be in thatmembrane. Thus if the object yes arrives in the environment (the acceptance signal) of the membranesystem, then there is a directed path leading from one special node ( in ) to another special node ( yes ) inthe dependency graph. For more details about how a dependency graph is constructed and its proof ofcorrectness see Guti´errez-Naranjo et al. [5, 4].The dependency graph for a membrane system Π is a directed graph G = ( V G , E G , in , yes , no ) where in ⊆ V G represents the input multiset, and yes , no ∈ V G , represent the accepting and rejecting signalsrespectively. Each vertex a ∈ V G is a pair a = ( o , h ) ∈ O × H , where O is the set of objects in Π and H is the set of membrane labels in Π . An edge ( a , b ) exists iff there is a developmental rule in Π suchthat the left hand side of the rule has the same object-membrane pair as a and the right hand side hasan object-membrane pair matching b . Since there is no membrane dissolution allowed, the parent/childrelationships of membranes does not change during the computation. This allows us to determine thecorrect parent and child membranes for type ( b ) and type ( c ) rules.Previously [5], the graph G was constructed from Π in polynomial time. We make the observationthat the graph G can be constructed in AC . We use a common circuit technique known as “masking”whereby using AND gates and a desired pattern we filter out the bits of the input string that we are inter-ested in. We take as input a binary string x that encodes a membrane system, Π . To make a dependencygraph from a membrane system requires a constant number of parallel steps that are as follows. First,a row of circuits identifies all type ( b ) and ( c ) rules and uses the membrane structure to determine thecorrect parent membranes, then writes out (a binary encoding of) edges representing these rules. Next,a row of circuits writes out all edges representing type ( e ) and ( f ) rules (see [5] for more details aboutthe representation of these rules in dependency graphs). For ( a ) rules it is possible to have polynomiallymany copies of polynomially many distinct objects on the right hand side of a rule. To write out edgesfor these rules in constant time we take advantage of the fact that we require at most one edge for eachobject-membrane pair in O × H . We have a circuit for each element of { o h | o ∈ O , h ∈ H } . The circuit for o h takes as input (an encoding of) all rules in R whose left hand side is of the form [ o ] h . The circuit then,in a parallel manner, masks (an encoding of) the right hand side of the rule (for example [ bbcdc ] h ) withthe encoding of each object in O , (in the example, masking for (encoded) b would produce (encoded) bb [ o ] h now writes out an encoding of the edge ( o h , b h ) and an encoding of all other edges forobjects that existed on the right hand side of this rule in parallel. . Murphy and D. Woods Remark 5.
Of course one can take the opposite view. We observe that to convert a dependency graph G = ( V G , E G , in , yes , no ) into a new membrane system, Π G , we simply convert the edges of the graphinto object evolution rules. The set of objects of Π G is O G = V G . The rules of Π G are { [ v → S ( v )] env | ∀ v ∈ V G } where S ( v ) = { s ∈ V G | ( v , s ) ∈ E G } . The nodes in , yes , no become the inputmultiset, yes object, and no object respectively. We compute this in AC . This new membrane system, Π G , highlights some points about active membrane systems withoutdissolution. These give rise to significant simplifications and normal forms. Lemma 6.
Any ( AC ) – PMC ∗ A M − d , Π , with m membranes can be simulated by a ( AC ) – PMC ∗ A M − d system, Π (cid:48) , that (1) has no membranes other than the environment and (2) uses only rules of type (a). By simulate we mean that the latter system accepts on input in iff the former does. To see thatLemma 6 holds, first notice how the dependency graph represents an (object, label) pair as a single node.Also if we convert the dependency graph G into a membrane system Π G , (1) it uses a single membranewith label env , and each node is modelled by a single object. (2) Each edge in G becomes a rule oftype ( a ). Notice that the dependency graphs of Π and Π G are isomorphic. Lemma 7.
Any ( AC ) – PMC ∗ A M − d system, Π , which has, as usual, multisets of objects in each mem-brane can be simulated by another ( AC ) – PMC ∗ A M − d system, Π (cid:48) , which has sets of objects in eachmembrane. We verify Lemma 7 by observing that in a dependency graph, G , the multiset of objects is encodedas a set of vertices, no information is kept regarding object multiplicities. Thus when G is converted intoa new membrane system, Π G , there are no rules with a right hand side with more than one instance ofeach object. The resulting system Π G accepts iff Π accepts since the dependency graphs of both systemsare isomorphic. Thus object multiplicities do not affect whether the system accepts or rejects. Here we present three different acceptance conditions for membrane systems with active membranesand show what complexity class they characterise. We define each acceptance condition; define a graphreachability problem that models the computation of such a system; then prove both upper and lowerbounds on the computational power of the system. Each of Definitions 8, 13, 19, is a more concrete re-placement for Definition 2. Most results in this section use reductions to and from reachability problemson membrane dependency graphs. Solving these reachability problems is equivalent to simulating such amembrane system since we translate (via AC reductions) from a membrane system to a correspondingreachability problem, and vice-versa. In previous works [9, 8] we used a definition of recogniser membrane systems that is more general thanis typical of other work in the area (i.e. Section 3.2). In this more general definition it is possible forthe membrane system to output both yes and no symbols. However, when the first of these symbols isproduced we call it the accepting/rejecting step of the computation. (Note that it is forbidden for both yes and no to be produced in the same timestep.) We now define this acceptance condition and then goon to show that ( AC ) – PMC ∗ A M − d systems with this acceptance condition characterise NL .78 On acceptance conditions for membrane systems: characterisations of L and NL
Definition 8. A general recognizer membrane system , Π , is a membrane system with external output(that is, the results of halting computations are encoded in the environment) such that:1. the working alphabet contains two distinguished elements yes and no ;2. if C is a computation of the system, (i) then a yes or no object is released into the environment,(ii) but not in the same timestep. If yes is released before no then the computation is accepting,otherwise the computation is rejecting. in yesno Figure 1: An example dependency graph G for some unspecified general recogniser membrane system (Definition 8). Note that this represents a rejecting computation since the minimum directed path from in to no is of length 6, while the minimum directed path from in to yes is of length 7.We now define the reachability problem for ( AC ) – PMC ∗ A M − d systems whose acceptance condi-tions are as in Definition 8. Solving this problem is equivalent (via a reduction) to simulating such asystem. Problem 9 ( GENREC ) . Instance:
A dependency graph G = ( V G , E G , in , yes , no ) where { in , yes , no } ⊆ V G , representing therules of a general recogniser membrane system Π as defined in Definition 8. Problem:
Is the shortest directed path from in to yes of length less than the shortest directed path from in to no ? We also define the problem
STCON , the canonical NL -complete problem [7]. This problem is alsoknown as PATH , REACHABILITY , and
GAP . Problem 10 ( STCON ) . Instance:
A directed acyclic graph G = ( V , E , s , t ) where { s , t } ⊆ V .
Problem:
Is there a directed path in G from s to t?
We now provide a result which is used to show that ( AC ) – PMC ∗ A M − d systems whose acceptanceconditions are as in Definition 8 characterise NL (this characterisation has been published elsewhere [9],we present a shorter proof here). Theorem 11.
GENREC is NL -completeProof. First we show
STCON ≤ AC GENREC . Given an instance G = ( V , E , s , t ) of STCON , we con-struct a dependency graph G = ( V G , E G , in , yes , no ) such that V G = V ∪ { no } and E G = E . We replaceall instances of s with in , and t with yes , in G . Clearly there is a path from in to yes iff there is apath from s to t in G . We also add a directed path of length | V | + in to no in G . This ensures . Murphy and D. Woods s to t in G , than no is reached after all other paths have terminated. Thisreduction is computed in AC .We now prove the correctness of the above reduction. Since GENREC is defined in terms of thegeneral recogniser membrane systems (Definition 8), we often appeal to Definition 8 in the proof. Recallthat, via Remark 5, we can translate G to a membrane system Π G in AC . • By adding a path of length | V | + in to no we are guaranteeing that object no is not producedby the membrane system Π G at the same time as any other object, this satisfies point 2(ii) ofDefinition 8. • If there is a path from s to t in G (and yes is evolved in Π G ) the reduction ensures that a path from in to yes exists in G . Also in either case a path from in to no is created by the reduction thatensures the correct output from Π G . Thus we satisfy point 2(i) of Definition 8.We now show that GENREC ∈ NL . Let M be a non-deterministic Turing machine with two variables x and y . Finding the shortest path between two nodes is well known to be computable in NL via ≤ n iterations of a STCON algorithm. Set x to be the shortest path from in to yes . Set y to be the shortestpath from in to no . If x < y , M accepts, otherwise M rejects. Thus M uses a non-deterministic algorithmand two binary counters to solve GENREC and so the problem is in NL . Theorem 12. NL is characterised by ( AC ) – PMC ∗ A M − d using the general acceptance conditions fromDefinition 8. The proof is omitted, but can be obtained by using standard techniques along with Remark 5, Theo-rem 11, and Definition 8.
In this section we discuss the “standard” definition for recogniser membrane systems, i.e. the definitionthat most researchers use when proving results about recogniser membrane systems. On a given input,these systems produce either a yes object or a no object, but not both. Also it is assumed that this occursin the last timestep of the computation where no other rules are applicable.By showing an NL characterisation for such systems, we are showing that this definition has equalpower to the more general definition discussed above in Section 3.1. Furthermore, we have provided a“compiler,” via reductions, to translate a system that uses the general definition into a system that usesthe standard definition. This is significant since the general definition is often easier to program, while itis often easier to prove certain properties (such as correctness) for the standard definition. We begin witha definition of standard recogniser membrane systems from Guti´errez-Naranjo et al. [5]. Definition 13 ([5]) . A recognizer membrane system , Π , is a membrane system with external output (thatis, the results of halting computations are encoded in the environment) such that:1. the working alphabet contains two distinguished elements yes and no ;2. all computations halt; and3. if C is a computation of the system, then (i) either object yes or object no (but not both) musthave been released into the environment, and (ii) only in the last step of the computation. If yes isreleased then the computation is accepting, otherwise the computation is rejecting. Remark 14.
Definition 13 affects the dependency graph of such systems so that we can define the fol-lowing subsets of the objects O.O yes = { o | o ∈ O and o eventually evolves yes } ,O no = { o | o ∈ O and o eventually evolves no } , and O other = O \ ( O yes ∪ O no ) . On acceptance conditions for membrane systems: characterisations of L and NL
Lemma 15. O yes ∩ O no = /0 .Proof. Assume that object o ∈ O yes ∩ O no , this implies that both a yes and a no object are produced bythe confluent system on a given input which contradicts point 3(i) of Definition 13.These observations are illustrated in Figure 2. in yesno Figure 2: An example dependency graph G for some unspecified standard recogniser membrane system(Definition 13). Note that via Lemma 15 there are no directed paths from O yes to O no , they are weaklyconnected .We now define the reachability problem for ( AC ) – PMC ∗ A M − d systems whose acceptance condi-tions are as in Definition 13. We remind the reader that these systems are confluent via Definition 3 andRemark 4. Problem 16 ( STDREC ) . Instance:
A dependency graph G = ( V G , E G , in , yes , no ) where { in , yes , no } ⊆ V G , representing therules of a ( AC ) – PMC ∗ A M − d recogniser membrane system Π as defined in Definition 13. Problem:
Is there a directed path from in to yes ? We now provide the main result needed to show that standard ( AC ) – PMC ∗ A M − d characterises NL . Theorem 17.
STDREC is NL -complete.Proof. First we show
STCON ≤ AC STDREC . Given an instance G = ( V , E , s , t ) of STCON , we con-struct a dependency graph G = ( V G , E G , in , yes , no ) such that V G = V ∪ { yes , no } and E G = E . Wereplace s with in in G . We add a directed path of | V | + t to yes to ensure that allother computations have halted before yes is evolved. Clearly there is a path from in to yes in G iffthere is a path from s to t in graph G .So far, G we have shown that ( AC ) – PMC ∗ A M − d recogniser membrane systems, as in Definition 13, accept words in STCON . However, the construction does not explicitly say how to reject words that arenot in the language, which is a requirement of Definition 13. We extend the proof as follows. Let
STCON be the complementary problem to
STCON , i.e. given an acyclic graph G (cid:48) is there no directed path from s (cid:48) to t (cid:48) ? STCON is coNL -complete (via the same reduction that is used to show the NL -completeness of STCON ), and so is also NL -complete (since NL = coNL [6, 14]). Now we define a third NL -completeproblem STCON – STCON ; the set of graphs with two disjoint components G , G (cid:48) that are related in thefollowing sense: s eventually yields t in G iff s (cid:48) does not eventually yield t (cid:48) in G (cid:48) . Now we reduce thisgraph to a dependency graph G in a similar manner as the above reduction. That is, we place an edge . Murphy and D. Woods in to s and from in to s (cid:48) . We add a directed path of | V | + t to yes , and anotherdirected path of | V | + t (cid:48) to no . Then the induced membrane system Π G correctlydecides STCON – STCON since it answers yes iff s leads to t , otherwise it answers no . This reduction iscomputed in AC .We now prove the correctness of the above reduction. Recall that, via Remark 5, we translate G to amembrane system Π G in AC . • Since an instance of
STCON – STCON is an acyclic graph we trivially satisfy point 2 of Defini-tion 13. • In the induced membrane system Π G the node in can only lead to one of yes or no , but not both,since the embedded STCON and
STCON problems are complementary. This satisfies point 3(i)of Definition 13. • Π G outputs (either yes or no ) in the last step because we add | V | + t and t (cid:48) sothat the accepting or rejecting path is the longest in the dependency graph, satisfying point 3(ii) ofDefinition 13.Now we show that ( AC ) – PMC ∗ A M − d , as in Definition 13, can recognise no more than NL byshowing that STDREC ≤ AC STCON . We observe that an instance of
STDREC is a directed acyclicgraph (via point 2 of Definition 13). Given an instance G = ( V G , E G , in , yes , no ) of STDREC , weconstruct G = ( V , E , s , t ) such that V = V G and E = E G and replace all instances of in with s and yes with t in G . Clearly there is a path from s to t in G iff there is a path from in to yes in the dependencygraph G . This reduction is computed in AC . Theorem 18. NL is characterised by ( AC ) – PMC ∗ A M − d using the standard acceptance conditions fromDefinition 13. The proof is omitted, but can be obtained by using standard techniques along with Remark 5, Theo-rem 17, and Definition 13.
We now consider a restriction on the standard definition of recogniser membrane systems. Above inSection 3.2, we forbid an object that eventually yielded a yes from also yielding a no (and vice versa).Now we further restrict the system and require that all descendent nodes of in must eventually yield yes , or all must eventually yield no . Notice that this restriction forbids objects that do not contribute tothe final answer (accept or reject) and forbids rules of the form [ a → λ ] where λ is the empty word. Definition 19. A restricted recogniser membrane system , Π , is a membrane system with external output(that is, the results of halting computations are encoded in the environment) such that:1. the working alphabet contains two distinguished elements yes and no ;2. all computations halt;3. if C is a computation of the system, then (i) either object yes or object no (but not both) musthave been released into the environment, and (ii) only in the last step of the computation. If yes isreleased then the computation is accepting, otherwise the computation is rejecting.4. each object o ∈ O must, via a sequence of zero or more developmental rules, lead to yes , or elselead to no , but not both. On acceptance conditions for membrane systems: characterisations of L and NL in yesno
Figure 3: An example dependency graph G for some unspecified restricted recogniser membrane system(Definition 19).The definition has the following effect on the dependency graph. Remark 20.
Since every object eventually yields exactly one yes , or exactly one no , the graph G consistsof exactly two disjoint components. We now define a graph reachability problem for ( AC ) – PMC ∗ A M − d systems whose acceptance con-ditions are as in Definition 19. Problem 21 ( RSTREC ) . Instance:
A dependency graph G = ( V G , E G , in , yes , no ) where { in , yes , no } ⊆ V G , representing therules of an ( AC ) – PMC ∗ A M − d recogniser membrane system Π as defined in Definition 19. Problem:
Is there a directed path from in to yes ? We define the L -complete problem D IRECTED F OREST A CCESSIBILITY ( DFA ) [3].
Problem 22 ( DFA [3]) . Instance:
An acyclic directed graph G = ( V , E , s , t ) where { s , t } ⊆ V and each node is of out-degree or . Property:
Is there a directed path from s to t?
Theorem 23.
RSTREC is L -completeProof. First we show
DFA ≤ AC RSTREC . Given an instance G = ( V , E , s , t ) of DFA , we construct adependency graph G = ( V G , E G , in , yes , no ) such that V G = V ∪ { no } and E G = E \{ ( t , v ) | v ∈ V } . Wealso replace s with in , and add a directed path of length | V | + t to yes in G . Clearly there is a pathfrom in to yes in G iff there is a path from s to t in graph G . Note that since we removed the edge (if itexists) leaving t , every computation halts (in the induced membrane system Π G ) upon evolving yes . Wealso add an edge from all nodes, except yes , of out-degree 0 to no . There is now a path from in to no iffthere is no path from s to t in G because all paths that do not lead to yes now lead to no . This reductionis computed in AC .We now prove the correctness of the above reduction. Recall that, via Remark 5, we translate G to amembrane system Π G in AC . • Since G (as a forest) is acyclic, our reduction ensures G , and hence any computation of Π G , isacyclic also, satisfying point 2 of Definition 19. • Our reduction ensures that exactly 2 nodes in G have out-degree 0, the (sink) nodes yes and no ,this implies that the only objects that have no applicable rules in Π G are yes and no . This satisfiespoints 1 and 3(ii) of Definition 19. . Murphy and D. Woods • Since every node in G has out-degree 0 or 1, then every node in G has out-degree 0 or 1 (and everyobject in Π G has 0 or 1 applicable developmental rules). Combined with the previous point, thisimplies that all nodes in G are on a path to either yes or no , and that all objects in Π G eventuallyyield either yes or no , satisfying points 4 and 3(i) of Definition 19.Now we show RSTREC is contained in L by outlining a deterministic logspace Turing machine M that decides RSTREC . The input tape of M encodes an instance G = ( V G , E G , in , yes , no ) of RSTREC .Starting with the input node in , M stores this node in a variable called x on its work tape. If x is neither yes nor no then M searches the set of edges E G on its input tape, upon finding an edge ( x , v ) , the machinesets x to be v (overwriting the previous value). The computation carries on in this fashion until either x equals no causing M to reject, or yes , in which case M accepts.The algorithm correctly decides RSTREC because each node in the data-structure has out-degree 0or 1 and we simply trace along a path until we reach a sink. If the sink is yes , we accept, otherwise wereject. Since only one node is stored on M ’s work tape at any time, M uses O ( log n ) space (where n isthe input length). Thus RSTREC ∈ L . Theorem 24. L is characterised by ( AC ) – PMC ∗ A M − d using the restricted acceptance conditions fromDefinition 19. The proof is omitted, but can be obtained by using standard techniques along with Remark 5, Theo-rem 23, and Definition 19.
In this paper we have shown how the acceptance conditions of membrane systems affect the computa-tional complexity of the system. We have presented an analysis of three different acceptance conditionsand proved that they each characterise one of two logspace complexity classes, NL or L .In our previous work [9] we used Definition 8 as our acceptance condition. Systems using thisdefinition are relatively easy to program (construct a membrane system to solve a problem) becauseone is not concerned with ensuring the system halts or that only yes or only no is output. HoweverDefinition 13 is the more common definition that is used when discussing active membrane systems asit is easier to prove correctness for these systems. The results in Sections 3.1 and 3.2 reveal that whenworking with ( AC ) – PMC ∗ A M − d systems, both of Definitions 8 and 13 characterise NL . Our resultgives an AC computable compiler to turn a system obeying one definition into a system obeying theother definition. This makes the choice of either definition a matter of taste and convenience.We also have given the first complexity class defined by membrane systems that characterises L .It is interesting to note that the rules of ( AC ) – PMC ∗ A M − d systems allow for the generation of anexponential amount of objects and membranes. However these systems decide only those problems thata (non-)deterministic Turing machine uses logarithmic space to decide.Here we looked at a number of acceptance conditions for active membrane systems and then char-acterised the computational complexity classes of the systems. However, it is also possible to go in theother direction, that is, to choose a complexity class and then try to engineer an acceptance condition inorder to characterise the class. This technique may give rise to interesting new characterisations. Fur-thermore, we would hope that it may even be useful to help solve some open questions on the power ofcertain classes of membrane systems.84 On acceptance conditions for membrane systems: characterisations of L and NL
We intend to extend this research to see what effect, if any, acceptance conditions have on the com-plexity of uniform active membrane systems. The techniques may also prove useful for exploring otherclasses of membrane systems such as tissue P-systems.
Acknowledgements
We would like to thank Mario J. P´erez-Jim´enez and Agust´ın Riscos-N´u˜nez for interesting discussion andclarification on the standard definition of recogniser membrane systems. We would also like to thankPetr Sos´ık for his comments on an earlier draft of this paper.
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