Membrane dissolution and division in P
Damien Woods
1
, Niall Murphy
2
, Mario J. P´erez–Jim´enez
1
, Agust´ın
Riscos–N´u˜nez
1
1
Research Group on Natural Computing, Department of Computer Science and
Artificial Intelligence, University of Seville, Spain
2
Department of Computer Science, National University of Ireland Maynooth, Ireland
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.
1 Introduction
Bacteria and other cells provide an interesting avenue to consider in Natural
Computing. Their native ability to make copies of themselves inspires us with
visions of exponentially multiplying processors churning away, solving intractable
problems. But can cells actually solve such hard problems? To answer this we
should abstract away from the mind-boggling intricacy of a living cell and create
a model that strips away everything but the features that we want to analyse.
The model we investigate is called active membrane systems [9], a variant
of P systems [10]. Active membrane systems were introduced to analyse the
computational complexity of nested membranes that have the ability to divide
and dissolve. It was shown by Sos´ık [11] that the model was extremely powerful
and could solve problems as quickly as parallel machines that satisfy the parallel
computation thesis. Later Sos´ık and Rodr´ıguez-Pat´on showed [12] that the model
was no more powerful than such parallel machines. This result holds for a very
general definition of active membrane systems and so can be interpreted as an
upperbound on systems with dividing and dissolving membranes.
These results used non-elementary division, that is, when a membrane di-
vides, all sub-membranes are copied. For several years a tighter upperbound
(or lowerbound) has been sought for the model where only membranes with
no sub-membranes may divide. This would close an open problem known as
the P conjecture which states that active membrane systems without charges
characterise P [8].
However, the proof has been elusive (see Mauri et al. [3] for a more detailed
survey). While attempting to answer this conjecture it was discovered that mem-
brane dissolution was a powerful aspect of the system when it was found that
without dissolution the system was upperbounded by P [1]. It is also known
that systems without division rules are upperbounded by P [13]. This highlights
that membrane dissolution when combined with division are the most difficult
aspects of the model.
A restriction of the model that uses symmetric cell division, where the two
resulting membranes are identical, was shown to solve no more than P [5]. In this
paper we analyse the more general division rule where two resulting membranes
of a division may be different (in a manner similar to the mechanism of stem
cells) combined with dissolution rules.
Our result gives a P upperbound on systems with dissolution and elementary
division, where at the initial timestep, the depth of membrane nesting is equal
to the total number of membranes. In the notation of membrane systems, this
may be stated as follows.
Theorem 1. PMC
∗
D
⊆ P, where D is the class of systems in AM
0
+d,−ne,−e,−c
having an initial membrane structure that is a single path.
While all the machinery is in place for a proof for a membrane structure that
(initially) is a tree, we leave the full proof to a more complete version of the
paper and only consider here the case where the structure is initially a single
path.
Our simulation algorithm does not explicitly simulate all (of the exponential
number of) membranes. Even though these systems may create exponentially
many membranes, each with different contents, we observe that the elementary
membranes are only important to simulate if they release a dissolution object to
a parent membrane, or some other ancestor.
To prove our main result we use a number of simulation techniques. The main
novelties in our approach are the following. We carefully choose a computation
to simulate: the membrane systems we simulate are confluent (nondeterminis-
tic, but all computations produce the same answer), and our technique finds a
sufficiently simple computation to simulate out of a large set of more compli-
cated computations. Also, we model important aspects of the computation using
special division graphs that are inspired by, and a generalisation of, dependency
graphs. Finally, our simulation algorithm carefully selects a small number of
important membranes to explicitly simulate, while ignoring up to exponentially
many other membranes. This technique is nicer than a brute force approach as it
highlights which aspects of the computation actually influence the answer, and
which do not.
Although our main motivation is the P conjecture, our work has applications
for membrane system simulators. For previous P upperbounds, this point could
perhaps be more difficult to argue since such upperbounds on membrane sys-
tems often have large hidden constants. However given a membrane system that
creates an exponential number of membranes, our algorithm actually explicitly
simulates very few membranes (less than double the number of membranes that
are in the initial configuration).
Unlike some previous techniques, we feel that the present approach will be
fruitful for proving the P conjecture for systems that have dissolution, elementary
2
division, evolution and communication rules. Our technique seems suitable for
generalisation, and does not suffer from a combinatorial explosion of cases.
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]. However for uniformity
that is tighter than P (e.g. AC
0
or L), then we conjecture that a P lowerbound
can be found by improving a result in [6].
2 Membrane Systems
In this section we define membrane systems and some complexity classes. These
definitions are based on those from P´erez-Jim´enez et al. [2], P˘aun [9,10], Sos´ık
and Rodr´ıguez-Pat´on [12], and Murphy and Woods [4].
2.1 Active membrane systems
Active membrane systems are a class of membrane systems with membrane di-
vision rules. In this paper, division rules can act only on elementary membranes,
which are membranes that do not contain other membranes (and are represented
as leaves in the membrane structure tree).
Definition 2. 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 tree that represents the membrane structure, consisting of m nodes,
labelled with elements of H. The root node (representing the parent of all
membranes) is called the “skin” and has label 1 ∈ 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 → u ]
h
, for h ∈ H, a ∈ O, u ∈ O
∗
(object evolution)
(b) a[ ]
h
→ [ b ]
h
, for h ∈ H, a, b ∈ O (communication in)
(c) [ a ]
h
→ [ ]
h
b, for h ∈ H, a, b ∈ O (communication out)
(d) [ a ]
h
→ b, for h ∈ H, a, b ∈ O (membrane dissolution)
(e) [ a ]
h
→ [ b ]
h
[ c ]
h
, for h ∈ H, a, b, c ∈ O (elementary membrane division)
The semantics of these rules are described elsewhere [10], however we note that
they are applied according to the following principles:
– All the rules are applied in a 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.
– 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)–(e), although in this paper we use only rules of type (d) and (e).
3
2.2 Recogniser membrane systems
We recall [1] that a computation of a membrane system is a sequence of config-
urations such that each configuration (except the initial one) is obtained from
the previous one by a transition (one-step maximally parallel application of the
rules). Systems are nondeterministic, therefore on a given input there are mul-
tiple possible computations.
Definition 3 ([1]). A recogniser membrane system is a membrane system where
each computation outputs either object yes or no (to the membrane with label 1).
When this output occurs, no other rules are applicable.
2.3 Complexity classes
Consider a decision problem X, i.e. a set of instances X = {x
1
, x
2
, . . .} over
some finite alphabet such that to each x
i
there is a unique answer “yes” or “no”.
We say that a family of membrane systems solves a decision problem if each
instance of the problem is solved by some family member. We let |x| denote the
length of x. In this paper, the functions classes E, F, H are each contained in the
class of functions that are polynomial time computable on deterministic Turing
machines.
Definition 4. Let D be a class of membrane systems, let E and F be classes of
functions, and let t : N → N be a total function. The class of problems solved
by (E, F )-uniform families of membrane systems of type D in time t, denoted
(E, F)–MC
D
(t), contains all problems X such that:
– There exists an f-uniform family of membrane systems,
Π
X
= {Π
X
(1), Π
X
(2), . . .} of type D: that is, there exists a function f ∈ F ,
f : {1}
∗
→ Π
X
such that f (1
n
) = Π
X
(n).
– There exists an input encoding function e ∈ E, e : X → Σ
∗
where e(x) rep-
resents the input multiset (as a word), which is placed in the input membrane
of Π
X
(n).
– The pair Π
X
, e is sound and complete with respect to X: Π
X
(n) given the
encoded input e(x), where |x| = n, accepts iff the answer to x is “yes”.
– Π
X
is t-efficient: Π
X
(n) always halts in at most t(n) steps.
Definition 4 describes (e, f)-uniform family (i.e. with input) and we
generalise this to define (h)-semi-uniform family of membrane systems Π
X
=
{Π
X
(x
1
), Π
X
(x
2
), . . .} where there exists a function h ∈ H, h : X → Π
X
such
that h(x) = Π
X
(x). Here a single function (rather than two) is used to construct
the semi-uniform membrane family, and so the problem instance is encoded us-
ing objects, membranes, and rules. In this case, for each instance of x ∈ X we
have a (possibly unique) membrane system. The resulting class of problems is
denoted by (H)–MC
∗
D
(t). We define
(E, F)–PMC
D
=
[
k∈N
(E, F)–MC
D
(n
k
),
4
(H)–PMC
∗
D
=
[
k∈N
(H)–MC
∗
D
(n
k
).
We let AM
0
+d,−ne,−e,−c
denote the class of active membrane systems without
charges and using only dissolution and elementary division rules. Theorem 1 is
shown for such (polynomial time) systems. For consistency with the membrane
systems literature (which often uses P uniformity and semi-uniformity), we sim-
ply write PMC
D
and PMC
∗
D
when E, F, H are classes of functions that are
polynomial time computable on deterministic Turing machines. The results in
this paper hold for P, or tighter, uniformity and semi-uniformity.
Remark 5. A recogniser membrane system is confluent if it is both sound and
complete. That is a Π
X
is confluent if all computations of Π
X
, with the same
encoded input, give the same result.
Therefore the following interpretation holds: in a confluent membrane system,
given a fixed initial configuration, the system non-deterministically chooses one
from a number of valid computations, but all computations must lead to the
same result, either all accepting or all rejecting.
2.4 Notation
For a membrane system Π we define ∆
i
= {δ | [δ]
i
→ o is a rule in Π, o ∈ O, i ∈
H}. In other words, ∆
i
is the set of objects that are on the left hand side of a rule
that dissolves a membrane with label i. Also, ∆
1
= {yes, no}, and ∆ = ∪
i∈H
∆
i
.
The parent of a membrane with label i is denoted p(i). We often explicitly write
the contents of a multiset as a vector C of multiplicities, as shown in Section 4.4.
3 Object division graphs
We introduce object division graphs which represent all divisions that result
from a given object and membrane label. We use the notion of dependency
graph [1], a directed acyclic graph previously used to represent non-dissolution
rules [i.e. types (a), (b), (c), and (e)]. Given a (properly encoded) set of rules for
a membrane system Π, the dependency graph G
Π
is created in logspace [6]. An
object division graph is a directed acyclic graph G
div(o,h)
that is the dependency
graph for division rules only (ignoring all other rules), and that contains only
those nodes and edges that are on paths leading from (o, h). We can also define
an object division graph as follows. For each division rule [ o
i
]
h
→ [ o
j
]
h
[ o
k
]
h
we define the three nodes {o
i,h
, o
j,h
, o
k,h
} and the two edges (o
i,h
, o
j,h
) and
(o
i,h
, o
k,h
). Then, the object division graph G
div(o
`
,h)
is the subgraph that is
reachable from the node o
`,h
.
For example, Figure 1 shows four object division graphs (each is contained
in a ‘cloud’). The three rules that gave rise to the leftmost object division graph
(rooted at a) are as follows, [ a ] → [ b ][ c ], [ b ] → [ δ
p(h
max
)
][ e ], [ c ] → [ e ][ δ
3
].
(All membranes are labeled h
max
and this label is omitted for brevity.)
5
