Priorities, Promoters and Inhibitors in
Deterministic Non-Cooperative P Systems
Artiom Alhazov
1
, Rudolf Freund
2
1
Institute of Mathematics and Computer Science
Academy of Sciences of Moldova
Academiei 5, Chi¸sin˘au MD-2028 Moldova
artiom@math.md
2
Faculty of Informatics, Vienna University of Technology
Favoritenstr. 9, 1040 Vienna, Austria
rudi@emcc.at
Summary. Membrane systems (with symbol objects) are distributed controlled multiset
pro cessing 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 of promoters and inhibitors, as well as for characterizing
the systems with priorities only.
1 Introduction
The most famous membrane computing model where determinism is a criterion of
universality versus decidability is the model of catalytic P systems, see [3] and [6].
It is also known that non-cooperative rewriting P systems with either promoters
or inhibitors are computationally complete, [2]. Moreover, the pro of satisfies some
additional properties:
• Either promoters of weight 2 or inhibitors of weight 2 are enough.
• The system is non-deterministic, but it restores the previous configuration if
the guess is wrong, which leads to correct simulations with probability 1.
Recently, in [1] it was shown that the computational completeness cannot
be achieved by deterministic non-cooperative systems with promoters, inhibitors
and priorities (in maximally parallel or asynchronous mode, unlike the sequential
mode), and characterizations of the corresponding classes were obtained:
28 A. Alhazov, R. Freund
NF IN ∪ coNF IN = N
deta
OP
asyn
1
(ncoo, pro
1,∗
, inh
1,∗
)
= N
deta
OP
maxpar
1
(ncoo, pro
1,∗
)
= N
deta
OP
maxpar
1
(ncoo, inh
1,∗
)
= N
deta
OP
asyn
1
(
ncoo, (pro
∗,∗
, inh
∗,∗
)
∗
, pri
)
= N
deta
OP
maxpar
1
(
ncoo, (pro
∗,∗
, inh
∗,∗
)
∗
, pri
)
, but
NRE = N
deta
OP
sequ
1
(ncoo, pro
1,1
, inh
1,1
) .
A few interesting questions have been left open. For instance, what is the power
of P systems, e.g., in the maximally parallel mode, when we only use priorities, or
when we restrict the weight of the promoting/inhibiting multisets. These are the
questions we address in this paper.
2 Definitions
An alphabet is a finite non-empty set V of abstract symbols. The free monoid
generated by V under the operation of concatenation is denoted by V
∗
; the empty
string is denoted by λ, and V
∗
\ {λ} is denoted by V
+
. The set of non-negative
integers is denoted by N; a set S of non-negative integers is called co-finite if N \ S
is finite. The family of all finite (co-finite) sets of non-negative integers is denoted
by NF IN (coNF IN , respectively). The family of all recursively enumerable sets
of non-negative integers is denoted by N RE. In the following, we will use ⊆ both
for the subset as well as the submultiset relation.
Since flattening the membrane structure of a membrane system preserves b oth
determinism and the model, in the following we restrict ourselves to consider mem-
brane systems as one-region multiset rewriting systems.
A (one-region) membrane system (P system) is a tuple
Π = (O, Σ, w, R
′
) ,
where O is a finite alphabet, Σ ⊆ O is the input sub-alphabet, w ∈ O
∗
is a string
representing the initial multiset, and R
′
is a set of rules of the form r : u → v,
u ∈ O
+
, v ∈ O
∗
.
A configuration of the system Π is represented by a multiset of objects from O
contained in the region, the set of all configurations over O is denoted by C (O).
A rule r : u → v is applicable if the current configuration contains the multiset
specified by u. Furthermore, applicability may be controlled by context conditions,
specified by pairs of sets of multisets.
Definition 1. Let P
i
, Q
i
be (finite) sets of multisets over O, 1 ≤ i ≤ m. A rule
with context conditions (r, (P
1
, Q
1
) , · · · , (P
m
, Q
m
)) is applicable to a configuration
C if r is applicable, and there exists some j ∈ {1, · · · , m} for which
• there exists some p ∈ P
j
such that p ⊆ C and
• q ⊆ C for all q ∈ Q
j
.
Controls in Deterministic Non-Coop erative P Systems 29
In words, context conditions are satisfied if there exists a pair of sets of multisets
(called promoter set and inhibitor set, respectively) such that at least one multiset
in the promoter set is a submultiset of the current configuration, and no multiset
in the inhibitor set is a submultiset of the current configuration.
Definition 2. A P system with context conditions and priorities on the rules is a
construct
Π = (O, Σ, w, R
′
, R, >) ,
where (O, Σ, w, R
′
) is a (one-region) P system as defined above, R is a set of rules
with context conditions and > is a priority relation on the rules in R; if rule r
′
has
priority over rule r, denoted by r
′
> r, then r cannot be applied if r
′
is applicable.
Throughout the paper, we will use the word control to mean that at least one of
these features is allowed (context conditions or promoters or inhibitors only and
eventually priorities).
In the sequential mode (seq u), a computation step consists in the non-
deterministic application of one applicable rule r, replacing its left-hand side
(lhs (r)) with its right-hand side (rhs (r)). In the maximally parallel mode
(maxpar), a multiset of applicable rules may be chosen non-deterministically to
be applied in parallel to the underlying configuration to disjoint submultisets, pos-
sibly leaving some objects idle, under the condition that no further applicable rule
can be added to that multiset (i.e., no supermultiset of the chosen multiset is
applicable to the same configuration). Maximal parallelism is the most common
computation mode in membrane computing, see also Definition 4.8 in [5]. In the
asynchronuous mode (asyn), any positive number of applicable rules may be cho-
sen non-deterministically to be applied in parallel to the underlying configuration,
to disjoint submultisets. The computation step between two configurations C and
C
′
is denoted by C → C
′
, thus yielding the binary relation ⇒: C (O) × C (O). A
computation halts when there are no rules applicable to the current configuration
(halting configuration) in the corresponding mode.
The computation of a generating P system starts with w, and its result is |x|
if it halts, an accepting system starts with wx, x ∈ Σ
∗
, and we say that |x| is
its results – is accepted – if it halts. The set of numbers generated/accepted by a
P system working in the mode α is the set of results of its computations for all
x ∈ Σ
∗
and denoted by N
α
g
(Π) and N
α
a
(Π), respectively. The family of sets of
numbers generated/accepted by a family of (one-region) P systems with context
conditions and priorities on the rules with rules of type β working in the mode
α is denoted by N
δ
OP
α
1
(
β, (pro
k,l
, inh
k
′
,l
′
)
d
, pri
)
with δ = g for the generating
and δ = a for the accepting case; d denotes the maximal number m in the rules
with context conditions (r, (P
1
, Q
1
) , · · · , (P
m
, Q
m
)); k and k
′
denote the maximum
number of promoters/inhibitors in the P
i
and Q
i
, respectively; l and l
′
indicate
the maximum of weights of promotors and inhibitors, respectively. If any of these
numbers k, k
′
, l, l
′
is not bounded, we replace it by ∗. As types of rules we are
going to distinguish between cooperative (β = coo) and non-cooperative (i.e., the
left-hand side of each rule is a single object; β = ncoo) ones.
30 A. Alhazov, R. Freund
In the case of accepting systems, we also consider the idea of determinism,
which means that in each step of any computation at most one (multiset of)
rule(s) is applicable; in this case, we write deta for δ.
In the literature, we find a lot of restricted variants of P systems with con-
text conditions and priorities on the rules, e.g., we may omit the priorities or
the context conditions completely. If in a rule (r, (P
1
, Q
1
) , · · · , (P
m
, Q
m
)) we have
m = 1, we say that (r, (P
1
, Q
1
)) is a rule with a simple context condition, and
we omit the inner parentheses in the notation. Moreover, context conditions only
using promoters are denoted by r|
p
1
,··· ,p
n
, meaning (r, {p
1
, · · · , p
n
} , ∅), or, equiva-
lently, (r, (p
1
, ∅) , · · · , (p
n
, ∅)); context conditions only using inhibitors are denoted
by r|
¬q
1
,··· ,¬q
n
, meaning (r, λ, {q
1
, · · · , q
n
}), or r|
¬{q
1
,··· ,q
n
}
. Likewise, a rule with
both promoters and inhibitors can be specified as a rule with a simple context con-
dition, i.e., r|
p
1
,··· ,p
n
,¬q
1
,··· ,¬q
n
stands for (r, {p
1
, · · · , p
n
} , {q
1
, · · · , q
n
}). Finally,
promoters and inhibitors of weight one are called atomic.
Remark 1. If we do not consider determinism, then (the effect of) the rule
(r, (P
1
, Q
1
) , · · · , (P
m
, Q
m
)) is equivalent to (the effect of) the collection of rules
{(r, P
j
, Q
j
) | 1 ≤ j ≤ m}, no matter in which mode the P system is working (ob-
viously, the priority relation has to be adapted accordingly, too).
Remark 2. Let (r, {p
1
, · · · , p
n
} , Q) be a rule with a simple context condition; then
we claim that (the effect of) this rule is equivalent to (the effect of) the collection
of rules
{(r, {p
j
} , Q ∪ {p
k
| 1 ≤ k < j}) | 1 ≤ j ≤ m}
even in the the case of a deterministic P system: If the first promoter is chosen
to make the rule r applicable, we do not care about the other promoters; if the
second promoter is chosen to make the rule r applicable, we do not allow p
1
to
appear in the configuration, but do not care about the other promoters p
3
to p
m
;
in general, when promoter p
j
is chosen to make the rule r applicable, we do not
allow p
1
to p
j−1
to appear in the configuration, but do not care about the other
promoters p
j+1
to p
m
; finally, we have the rule {(r, {p
m
} , Q ∪ {p
k
| 1 ≤ k < m})}.
If adding {p
k
| 1 ≤ k < j} to Q has the effect of prohibiting the promotor p
j
from
enabling the rule r to be applied, this makes no harm as in this case one of the
promoters p
k
, 1 ≤ k < j, must have the possibility for enabling r to be applied.
By construction, the domains of the new context conditions now are disjoint, so
this transformation does not create (new) non-determinism. In a similar way, this
transformation may be performed on context conditions which are not simple.
Therefore, without restricting generality, the set of promoters may be assumed to
be a singleton. In this case, we may omit the braces of the multiset notation for
the promoter multiset and write (r, p, Q).
Remark 3. As in a P system (O, Σ, w, R
′
, R, >) the set of rules R
′
can easily be
deduced from the set of rules with context conditions R, we omit R
′
in the de-
scription of the P system. Moreover, for systems having only rules with a simple
Controls in Deterministic Non-Coop erative P Systems 31
context condition, we omit d in the description of the families of sets of numbers
and simply write
N
δ
OP
α
1
(β, pro
k,l
, inh
k
′
,l
′
, pri) .
Moreover, each control mechanism not used can be omitted, e.g., if no priorities
and only promoters are used, we only write N
δ
OP
α
1
(β, pro
k,l
).
3 Results
3.1 Recent results
We first recall from [1] the bounding operation over multisets, with a parameter
k ∈ N as follows:
for u ∈ O
∗
, b
k
(u) = v with |v|
a
= min(|u|
a
, k) for all a ∈ O.
The mapping b
k
“crops” the multisets by removing copies of every object a
present in more than k copies until exactly k remain. For two multisets u, u
′
,
b
k
(u) = b
k
(u
′
) if for every a ∈ O, either |u|
a
= |u
′
|
a
< k, or |u|
a
≥ k and
|u
′
|
a
≥ k. Mapping b
k
induces an equivalence relation, mapping O
∗
into (k + 1)
|O|
equivalence classes. Each equivalence class corresponds to specifying, for each a ∈
O
∗
, whether no copy, one copy, or · · · k − 1 copies, or “k copies or more” are
present. We denote the range of b
k
by {0, · · · , k}
O
.
Lemma 1. [1] Context conditions are equivalent to predicates defined on bound-
ings.
Theorem 1. [1] Priorities are subsumed by conditional contexts.
Remark 4. It is worth to note, see also [4], that if no other control is used, the
priorities can be mapped to sets of atomic inhibitors. Indeed, a rule is inhibited
precisely by the left side of each higher priority rule. This is straightforward in
case when the priority relation is assumed to be a partial order.
If it is not, then both the semantics of computation in P systems and the
reduction of priorities to inhibitors is a bit more complicated, but the claim still
holds.
Fix an arbitrary deterministic controlled non-cooperative P system. Take k as
the maximum of size of all multisets in all context conditions. Then, the bounding
does not influence applicability of rules, and b
k
(u) is halting if and only if u is
halting. We recall that bounding induces equivalence classes preserved by any
computation.
Lemma 2. [1] Assume u → x and v → y. Then b
k
(u) = b
k
(v) implies b
k
(x) =
b
k
(y).
Corollary 1. [1] If b
k
(u) = b
k
(v), then u is accepted if and only if v is accepted.