scispace - formally typeset

Book ChapterDOI

A Characterisation of NL Using Membrane Systems without Charges and Dissolution

25 Aug 2008-pp 164-176

TL;DR: It turns out that the computational power of some systems is lowered from P to NL, so it seems that the tighter uniformities are more reasonable for these systems.

AbstractWe apply techniques from complexity theory to a model of biological cellular membranes known as membrane systems or P-systems. Like circuits, membrane systems are defined as uniform families. To date, polynomial time uniformity has been the accepted uniformity notion for membrane systems. Here, we introduce the idea of using AC 0and L -uniformities and investigate the computational power of membrane systems under these tighter conditions. It turns out that the computational power of some systems is lowered from P to NL , so it seems that our tighter uniformities are more reasonable for these systems. Interestingly, other systems that are known to be lower bounded by P are shown to retain their computational power under the new uniformity conditions. Similarly, a number of membrane systems that are lower bounded by PSPACE retain their power under the new uniformity conditions.

Topics: PSPACE (53%), Bounded function (52%)

Summary (2 min read)

1 Introduction

  • Membrane systems [12] are a model of computation inspired by living cells.
  • In the active membrane model it is also possible for a membrane to completely dissolve, and for a membrane to divide into two child membranes.
  • The authors also show that the PSPACE lower and upper bounds mentioned above still hold under these restricted uniformity conditions.
  • So far the authors have shown that four models, that characterise P when polynomial time uniformity is used, actually only characterise NL when restricted to be AC0 uniform.
  • Interestingly, the authors also show that two other polynomial time uniform membrane system that are known [9] to be lower bounded by P actually retain this P lower bound when restricted to be AC0 uniform.

2 Membrane Systems

  • In this section the authors define membrane systems and complexity classes.
  • The authors also introduce the notion of AC0 uniformity for membrane systems.

2.1 Recogniser membrane systems

  • Active membranes systems are membrane systems with membrane division rules.
  • Division rules can either only act on elementary membranes, or else on both elementary and non-elementary membranes.
  • These rules are applied according to the following principles: – All the rules are applied in maximally parallel manner.
  • The result of the computation (a solution to the instance) is “yes” if a distinguished object yes is expelled during the computation, otherwise the result is “no”.
  • Therefore, the following interpretation holds: given a fixed initial configuration, a confluent membrane system nondeterministically chooses one from a number of valid configuration sequences, but all of them must lead to the same result.

2.2 Complexity classes

  • Here the authors introduce the notion of AC0 uniformity to membrane systems.
  • Previous work on the computational complexity of membrane systems used (Turing machine) polynomial time uniformity [14].
  • – Each ΠX(n) is confluent: all computations of ΠX(n) with the same input x of size n give the same result; either always “yes” or else always “no”.
  • The authors denote by AM0+ne the classes of membrane systems with active membranes, and both non-elementary and elementary membrane division and no charges.
  • The authors now show that the use of AC0 uniformity does not change this lower bound.

3 NL Upper bound on active membranes without dissolution rules

  • Previously the upper bound on all active membrane systems without dissolution was P [5].
  • The authors give an overview rather than the full details.
  • The authors make the observation that the graph GΠ can be constructed in deterministic logspace, and even in AC0.
  • Since the authors have shown that the problem of simulating a membrane system without charges and without dissolution can be encoded as an NL-complete problem they have proved Theorem 1.

4 NL lower bound for semi-uniform active membranes without dissolution

  • The algorithm works by having each edge in the problem instance graph represented as a membrane.
  • The initial multisets are all empty except Mcount = {c2n+1}.
  • The authors also have a counter that counts down in parallel with the above steps.
  • Note that the authors encode the edges of the graph as membranes, rather than objects.
  • In the membrane computing framework, for uniform membrane systems, inputs must be specified as objects.

4.1 PARITY lower bound for uniform active membranes without dissolution

  • The previous proof gave a lower bound for a semi-uniform membrane system.
  • The authors show that PARITY ∈ PMCAM0−d,+u by providing an AC0 uniform membrane system that can solve instances of the problem.
  • PARITYis the problem of telling whether the number of 1 symbols in the input word is odd.
  • A type (a) rule is created mapping every even object with i “1” symbols to the odd object with i− 1 “1” symbols in it.
  • The AC0 uniformity machine (a CRAM) rearranges the input word w by moving all 1 symbols to the left and all 0 symbols to the right, to give w′.

5 P lower bound on uniform families of active membrane systems with dissolving rules

  • In this section the authors show that does not happen for all models with at least P power.
  • Naturally this result also holds for the semi-uniform case.
  • The resulting membrane system directly solves the instance of CVP in polynomial time.
  • The authors simulate multiple fanouts by outputting multiple copies of the resulting truth value of each gate.
  • The output of a gate moves up through the layers of the membrane system until it reaches the correct gate according to its tag.

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report

A characterisation of NL using membrane
systems without charges and dissolution
Niall Murphy, Damien Woods
Technical Report
NUIM-CS-TR-2008-01
Department of Computer Science
National University of Ireland, Maynooth
Ireland

A characterisation of NL using membrane
systems without charges and dissolution
Niall Murphy
1
and Damien Woods
2
1
Department of Computer Science, National University of Ireland, Maynooth,
Ireland
nmurphy@cs.nuim.ie
2
Department of Computer Science, University College Cork, Ireland
d.woods@cs.ucc.ie
Abstract. We apply techniques from complexity theory to a model of
biological cellular membranes known as membrane systems or P-systems.
Like circuits, membrane systems are defined as uniform families. To
date, polynomial time uniformity was the accepted uniformity notion
for membrane systems. Here, we introduce the idea of using AC
0
and L
uniformities and investigate the computational power of membrane sys-
tems under these tighter conditions. It turns out that the computational
power of some systems is lowered from P to NL, so it seems that our
tighter uniformities are more reasonable for these systems. Interestingly,
other systems that are known to be lower bounded by P are shown to
retain their computational power under the new uniformity conditions.
Similarly, a number of membrane systems that are lower bounded by
PSPACE retain their power under the new uniformity conditions.
1 Introduction
Membrane systems [12] are a model of computation inspired by living cells.
In this paper we explore the computational power of cell division (mitosis) and
dissolution (apoptosis) by investigating a variant of the model called active mem-
branes [11]. An instance of the model consists of a number of (possibly nested)
membranes, or compartments, which themselves contain objects. During a com-
putation, the objects, depending on the compartment they are in, become other
objects or pass through membranes. In the active membrane model it is also
possible for a membrane to completely dissolve, and for a membrane to divide
into two child membranes.
This membrane model can be regarded as a model of parallel computation
but it has a number of features that make it somewhat unusual when compared
to other parallel models. For example, object interactions are nondeterministic
so confluence plays an important role, membranes contain multisets of objects,
there are many parameters to the model, etc. In order to clearly see the power
of the model we analyse it from the computational complexity point of view, the
goal being to characterise the model in terms of the set of problems that it can
solve in reasonable time.

Another, more specific, motivation is the so-called P-conjecture [13] which
states that recogniser membranes systems with division rules (active membranes),
but without charges, characterise P. On the one hand, it was shown that this
conjecture does not hold for systems with non-elementary division as PSPACE
upper [16] and lower [1] bounds were found for this variant (non-elementary di-
vision is where a membrane containing multiple membranes and objects may be
copied in a single timestep). On the other hand, the P-conjecture was shown to
hold for all active membrane systems without dissolution rules, when Guti´errez-
Naranjo et al. [5] gave a P upper bound. The corresponding P lower bound
(trivially) came from the fact that the model is defined to be P-uniform.
However, here we argue that the aforementioned P lower bound highlights a
problem with using P uniformity, as it does not tell us whether this membrane
model itself has (in some sense) the ability to solve all of P in polynomial time,
or if the uniformity condition is providing the power. In this paper we show that
in fact when we use weaker, and more reasonable, uniformity conditions the
model does not have the ability to solve all problems in P (assuming P 6= NL).
We find that with either AC
0
or L uniformity the model characterises NL in
the semi-uniform case, and we give an NL upper bound for the uniform case.
We also show that the PSPACE lower and upper bounds mentioned above still
hold under these restricted uniformity conditions.
Using the notation of membrane systems (to be defined later) our upper
bound on L-uniform and L-semi-uniform membrane systems can be stated as
follows.
Theorem 1. PMC
AM
0
d
NL
Essentially this theorem states that polynomial time active membrane systems,
without dissolution rules, solve no more than those problems in NL. Despite
the fact that these systems run for polynomial time (and can even create expo-
nentially many objects), they can not solve all of P (assuming NL 6= P). This
result is illustrated by the bottom four nodes in Figure 1.
The upper bound in Theorem 1 is found by showing that the construction
in [5] can be reduced to an instance of the NL-complete problem s-t-connectivity
(STCON). The full proof appears in Section 3. Next we give a corresponding
lower bound.
Theorem 2. NL PMC
AM
0
d,u
To show this lower bound we provide an AC
0
-semi-uniform membrane family
that solves STCON. The full proof is in Section 4 and the result is illustrated
by the bottom left two nodes in Figure 1. Therefore, in the semi-uniform case
we have a characterisation of NL.
Corollary 1. NL = PMC
AM
0
d,u
We have not yet shown an analogous lower bound result for uniform families. To
date our best lower bound is PARITY, which is known not to be in AC
0
[4].
We describe this in Section 4.1.

NL
PSPACE
P
PSPACE
P
NL
PARITY
NL
NL
PARITY
PSPACE PSPACE
-d, -ne, -u
+d, -ne, -u +d, -ne, +u
-d, -ne, +u
-d, +ne, -u -d, +ne, +u
+d, +ne, -u +d, +ne, +u
Fig. 1. An inclusion diagram showing the currently known upper and lower bounds
on the variations of the model. The top part of a node represents the best known
upper bounds, and the lower part the best known lower bounds. An undivided node
represents a characterisation.
So far we have shown that four models, that characterise P when polynomial
time uniformity is used, actually only characterise NL when restricted to be AC
0
uniform. Interestingly, we also show that two other polynomial time uniform
membrane system that are known [9] to be lower bounded by P actually retain
this P lower bound when restricted to be AC
0
uniform. This result is stated as
a P lower bound on membrane systems with dissolution:
Theorem 3. P PMC
AM
0
+d,+u
The proof appears in Section 5 and is illustrated by the top front two nodes in
Figure 1.
In Section 2.3 we observe that the known PSPACE upper and lower bounds
(top four nodes in Figure 1) remain unchanged under AC
0
uniformity conditions.
2 Membrane Systems
In this section we define membrane systems and complexity classes. These def-
initions are from aun [11, 12], and Sos´ık and Rodr´ıguez-Pat´on [16]. We also
introduce the notion of AC
0
uniformity for membrane systems.
2.1 Recogniser membrane systems
Active membranes systems are membrane systems with membrane division rules.
Division rules can either only act on elementary membranes, or else on both
elementary and non-elementary membranes. An elementary membrane is one
which does not contain other membranes (a leaf node, in tree terminology).

Definition 1. An active membrane system without charges is a tuple Π =
(O, H, µ, w
1
, . . . , w
m
, R) where,
1. m > 1 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, consisting of m membranes, labelled with ele-
ments of H;
5. w
1
, . . . , 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 v ]
h
,
for h H, a O, v O
(b) a[
h
]
h
[
h
b ]
h
,
for h H, a, b O
(c) [
h
a ]
h
[
h
]
h
b,
for h H, a, b O
(d) [
h
a ]
h
b,
for h H, a, b O
(e) [
h
a ]
h
[
h
b ]
h
[
h
c ]
h
,
for h H, a, b, c O.
(f) [
h
o
[
h
1
]
h
1
[
h
2
]
h
2
[
h
3
]
h
3
]
h
0
[
h
0
[
h
1
]
h
1
[
h
3
]
h
3
]
h
0
[
h
0
[
h
2
]
h
2
[
h
3
]
h
3
]
h
0
,
for h
0
, h
1
, h
2
, h
3
H.
These rules are applied according to the following principles:
All the rules are applied in maximally parallel manner. That is, in one step,
one object of a membrane is used by at most one rule (chosen in a non-
deterministic way), but any object which can 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 are objects in this membrane which evolve by means
of rules of type (a), then we suppose that first the evolution rules of type
(a) are used, and then the division is produced. This process takes only one
step.
The rules associated with membranes labelled with h are used for membranes
with that label. At one step, a membrane can be the subject of only one rule
of types (b)-(f).
In this paper we study the language recognising variant of membrane systems
that solves decision problems. A distinguished region contains, at the beginning
of the computation, an input a description of an instance of a problem. The
result of the computation (a solution to the instance) is “yes” if a distinguished
object yes is expelled during the computation, otherwise the result is “no”. Such
a membrane system is called deterministic if for each input a unique sequence of
configurations exists. A membrane system is called confluent if it always halts
and, starting from the same initial configuration, it always gives the same re-
sult, either always “yes” or always “no”. Therefore, the following interpretation

Citations
More filters

Book ChapterDOI
03 Sep 2009
TL;DR: Even though systems with dissolution, elementary division and where each membrane initially has at most one child membrane may create exponentially many membranes, it is shown that their power is upperbounded by P.
Abstract: Membrane systems with dividing and dissolving membranes are known to solve PSPACE problems in polynomial time. However, we give a P upperbound on an important restriction of such systems. In particular we examine systems with dissolution, elementary division and where each membrane initially has at most one child membrane. Even though such systems may create exponentially many membranes, each with different contents, we show that their power is upperbounded by P.

13 citations


Cites background from "A Characterisation of NL Using Memb..."

  • ...AC or L), then we conjecture that a P lowerbound can be found by improving a result in [6]....

    [...]

  • ...What is the lowerbound on the power of the systems that we consider? If P uniformity is used, then we get a trivial P lowerbound [6]....

    [...]

  • ...Given a (properly encoded) set of rules for a membrane system Π, the dependency graph GΠ is created in logspace [6]....

    [...]


Journal ArticleDOI
TL;DR: This work gives analogous results for membrane systems by showing that certain classes of uniform membrane systems are strictly weaker than the analogous semi-uniform classes, which solves a known open problem in the theory of membrane systems.
Abstract: We investigate computing models that are presented as families of finite computing devices with a uniformity condition on the entire family Examples of such models include Boolean circuits, membrane systems, DNA computers, chemical reaction networks and tile assembly systems, and there are many others However, in such models there are actually two distinct kinds of uniformity condition The first is the most common and well-understood, where each input length is mapped to a single computing device (eg a Boolean circuit) that computes on the finite set of inputs of that length The second, called semi-uniformity, is where each input is mapped to a computing device for that input (eg a circuit with the input encoded as constants) The former notion is well-known and used in Boolean circuit complexity, while the latter notion is frequently found in literature on nature-inspired computation from the past 20 years or so Are these two notions distinct? For many models it has been found that these notions are in fact the same, in the sense that the choice of uniformity or semi-uniformity leads to characterisations of the same complexity classes In other related work, we showed that these notions are actually distinct for certain classes of Boolean circuits Here, we give analogous results for membrane systems by showing that certain classes of uniform membrane systems are strictly weaker than the analogous semi-uniform classes This solves a known open problem in the theory of membrane systems We then go on to present results towards characterising the power of these semi-uniform and uniform membrane models in terms of NL and languages reducible to the unary languages in NL, respectively

9 citations


Cites background from "A Characterisation of NL Using Memb..."

  • ...In this paper and others [33, 34, 35, 36, 37], we have put forward the idea of exploring the power of membrane systems under tight uniformity conditions....

    [...]


Journal ArticleDOI
01 Dec 2019
TL;DR: A new approach is given based on the concept of object division polynomials introduced in this paper to simulate certain computations of polarizationless P systems with active membranes and how to compute efficiently the result of these computations using these polynmials.
Abstract: According to the P conjecture by Gh. Paun, polarizationless P systems with active membranes cannot solve $${\mathbf {NP}}$$-complete problems in polynomial time. The conjecture is proved only in special cases yet. In this paper we consider the case where only elementary membrane division and dissolution rules are used and the initial membrane structure consists of one elementary membrane besides the skin membrane. We give a new approach based on the concept of object division polynomials introduced in this paper to simulate certain computations of these P systems. Moreover, we show how to compute efficiently the result of these computations using these polynomials.

9 citations


Cites background from "A Characterisation of NL Using Memb..."

  • ...It is also widely investigated how certain restrictions on P systems with active membrane affect the computation power of these systems (see for example [6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 25])....

    [...]


Journal ArticleDOI
TL;DR: It is demonstrated that two particular acceptance conditions (one easier to program, the other easier to prove correctness) both characterise the same complexity class, NL, and by restricting the acceptance conditions, by obtaining a characterisation of L.
Abstract: In this paper we investigate the affect of various acceptance conditions on recogniser membrane systems without dissolution We demonstrate that two particular acceptance conditions (one easier to program, the other easier to prove correctness) both characterise the same complexity class, NL We also find that by restricting the acceptance conditions we obtain a characterisation of L We obtain these results by investigating the connectivity properties of dependency graphs that model membrane system computations

6 citations


Book ChapterDOI
20 Aug 2014
TL;DR: This work presents new results on the weight of promoters and inhibitors of non-cooperative P systems with either promoters or inhibitors, as well as characterizing the systems with priorities only.
Abstract: Membrane systems (with symbol objects) are distributed controlled multiset processing systems. Non-cooperative P systems with either promoters or inhibitors (of weight not restricted to one) are known to be computationally complete. Since recently, it is known that the power of the deterministic subclass of such systems is subregular. We present new results on the weight (one and two) of promoters and inhibitors, as well as characterizing the systems with priorities only.

6 citations


Cites background or methods from "A Characterisation of NL Using Memb..."

  • ...Membrane systems have been introduced as a computational model inspired by cellular biology [9], and have been later applied to the description of biological systems [6]....

    [...]

  • ...Various possibilities how to “go beyond Turing” to be already found in the literature are discussed in [9]; most of the definitions and results for red-green Turing machines are taken from this paper....

    [...]

  • ...To get the reader familiar with the basic idea of red-green automata, we give a short sketch of the proofs for some well-known results (see [9]):...

    [...]

  • ...Various classes of membrane systems (also called P systems) have been defined in [9], while several applications of these systems are described in [3]....

    [...]

  • ...We assume that the reader is familiar with the basics of formal language theory and membrane computing; for more information, we refer to the monograph [10], and the handbooks [9] and [8]....

    [...]


References
More filters

Book
19 Dec 1990
TL;DR: The Handbook of Theoretical Computer Science provides professionals and students with a comprehensive overview of the main results and developments in this rapidly evolving field.
Abstract: "Of all the books I have covered in the Forum to date, this set is the most unique and possibly the most useful to the SIGACT community, in support both of teaching and research.... The books can be used by anyone wanting simply to gain an understanding of one of these areas, or by someone desiring to be in research in a topic, or by instructors wishing to find timely information on a subject they are teaching outside their major areas of expertise." -- Rocky Ross, "SIGACT News" "This is a reference which has a place in every computer science library." -- Raymond Lauzzana, "Languages of Design" The Handbook of Theoretical Computer Science provides professionals and students with a comprehensive overview of the main results and developments in this rapidly evolving field. Volume A covers models of computation, complexity theory, data structures, and efficient computation in many recognized subdisciplines of theoretical computer science. Volume B takes up the theory of automata and rewriting systems, the foundations of modern programming languages, and logics for program specification and verification, and presents several studies on the theoretic modeling of advanced information processing. The two volumes contain thirty-seven chapters, with extensive chapter references and individual tables of contents for each chapter. There are 5,387 entry subject indexes that include notational symbols, and a list of contributors and affiliations in each volume.

3,089 citations


Book
01 Jan 1997
TL;DR: Throughout the book, Sipser builds students' knowledge of conceptual tools used in computer science, the aesthetic sense they need to create elegant systems, and the ability to think through problems on their own.
Abstract: From the Publisher: Michael Sipser's philosophy in writing this book is simple: make the subject interesting and relevant, and the students will learn. His emphasis on unifying computer science theory - rather than offering a collection of low-level details - sets the book apart, as do his intuitive explanations. Throughout the book, Sipser - a noted authority on the theory of computation - builds students' knowledge of conceptual tools used in computer science, the aesthetic sense they need to create elegant systems, and the ability to think through problems on their own. INTRODUCTION TO THE THEORY OF COMPUTATION provides a mathematical treatment of computation theory grounded in theorems and proofs. Proofs are presented with a "proof idea" component to reveal the concepts underpinning the formalism. Algorithms are presented using prose instead of pseudocode to focus attention on the algorithms themselves, rather than on specific computational models. Topic coverage, terminology, and order of presentation are traditional for an upper-level course in computer science theory. Users of the Preliminary Edition (now out of print) will be interested to note several new chapters on complexity theory: Chapter 8 on space complexity; Chapter 9 on provable intractability, and Chapter 10 on advanced topics, including approximation algorithms, alternation, interactive proof systems, cryptography, and parallel computing.

2,842 citations


"A Characterisation of NL Using Memb..." refers methods in this paper

  • ...STCON is also known as PATH [17] and REACHABILITY [12]....

    [...]


Book
01 Jan 2002
TL;DR: This chapter discusses Membrane Computing, What It Is and What It is Not, and attempts to get back to reality with open problems and Universality results.
Abstract: Preface.- 1. Introduction: Membrane Computing, What It Is and What It Is Not.- 2. Prerequisites.- 3. Membrane Systems with Symbol-Objects.- 4. Trading Evolution for Communication.- 5. Structuring Objects.- 6. Networks of Membranes.- 7. Trading Space for Time.- 8. Further Technical Results.- 9. (Attempts to Get) Back to Reality.- Open Problems.- Universality Results. Bibliography.- Index.

1,744 citations


Book
30 Nov 1993
TL;DR: Computational complexity is the realm of mathematical models and techniques for establishing impossibility proofs for proving formally that there can be no algorithm for the given problem which runs faster than the current one.
Abstract: Once we have developed an algorithm (q.v.) for solving a computational problem and analyzed its worst-case time requirements as a function of the size of its input (most usefully, in terms of the O-notation; see ALGORITHMS, ANALYSIS OF), it is inevitable to ask the question: "Can we do better?" In a typical problem, we may be able to devise new algorithms for the problem that are more and more efficient. But eventually, this line of research often seems to hit an invisible barrier, a level beyond whch improvements are very difficult, seemingly impossible, to come by. After many unsuccessful attempts, algorithm designers inevitably start to wonder if there is something inherent in the problem that makes it impossible to devise algorithms that are faster than the current one. They may try to develop mathematical techniques for proving formally that there can be no algorithm for the given problem which runs faster than the current one. Such a proof would be valuable, as it would suggest that it is futile to keep working on improved algorithms for this problem, that further improvements are certainly impossible. The realm of mathematical models and techniques for establishing such impossibility proofs is called computational complexity.

965 citations


Additional excerpts

  • ...STCON is also known as PATH [17] and REACHABILITY [12]....

    [...]


Journal ArticleDOI
TL;DR: A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function and connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.
Abstract: A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function. Introducing the notion of polynomial-size, constant-depth reduction, similar results are shown for the majority, multiplication, and transitive closure functions. Connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.

857 citations


Frequently Asked Questions (1)
Q1. What are the contributions in "A characterisation of nl using membrane systems without charges and dissolution" ?

Here, the authors introduce the idea of using AC and L uniformities and investigate the computational power of membrane systems under these tighter conditions.