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 P˘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
