In this work, we propose a variant of P system based on the rewriting of string-objects by means of evolutionary rules. The membrane structure of such a P system seems to be a very natural tool for simulating the filters in accepting networks of evolutionary processors with filtered connections. We discuss an informal construction supporting this simulation. A detailed proof is to be considered in an extended version of this work

A. Păun

C. Drăgoi

F. Manea

G. Păun

J. Castellanos

M. Madhu

S.R. Krishna

W.D. Hillis

Juan Castellanos

Victor Mitrana

Eugenio Santos

José M. Sempere

Crossref

Simulating Accepting Networks of Evolutionary Processors with Filtered Connections by Accepting Evolutionary P Systems

Castellanos Peñuela, Juan Bautista

Mitrana, Victor

Santos Menéndez, Eugenio

Sempere, José M.

Archivo Digital UPM

Simulating accepting networks of evolutionary processors with filtered connections by accepting evolutionary P systems

In this work, we propose a variant of P system based on the rewriting of string-objects by means of evolutionary rules. The membrane structure of such a P system seems to be a very natural tool for simulating the filters in accepting networks of evolutionary processors with filtered connections. We discuss an informal construction supporting this simulation. A detailed proof is to be considered in an extended version of this work.Castellanos, J.; Mitrana, V.; Santos, E.; Sempere Luna, JM. (2011). Simulating accepting networks of evolutionary processors with filtered connections by accepting evolutionary P systems. En Foundations on Natural and Artificial Computation. Springer Verlag (Germany). 6686:295-302. doi:10.1007/978-3-642-21344-1_31S295302668

Castellanos, Juan

Mitrana, Víctor

Santos, Eugenio

Sempere Luna, José María

RiuNet

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Simulating Accepting Networks of Evolutionary
Processors with Filtered Connections by
Accepting Evolutionary P Systems
(extended abstract)
Juan Castellanos1 Victor Mitrana2 Eugenio Santos2
Jose´ M. Sempere3
1Department of Artificial Intelligence, Faculty of Informatics
Polytechnic University of Madrid, 28660 Boadilla del Monte, Madrid, Spain.
E-mail: jcastellanos@fi.upm.es
2Department of Organization and Structure of Information,
University School of Informatics,
Polytechnic University of Madrid, Crta. de Valencia km. 7 - 28031 Madrid, Spain.
E-mail: victor.mitrana@upm.es, esantos@eui.upm.es
3Department of Information Systems and Computation,
Polytechnic University of Valencia
Camino de Vera, s/n, 46022 Valencia, Spain.
E-mail: jsempere@dsic.upv.es
Abstract. In this work, we propose a variant of P system based on the
rewriting of string-objects by means of evolutionary rules. The mem-
brane structure of such a P system seems to be a very natural tool for
simulating the filters in accepting networks of evolutionary processors
with filtered connections. We discuss an informal construction support-
ing this simulation. A detailed proof is to be considered in an extended
version of this work.
Keywords: Network of Evolutionary Processors with Filtered Connections, Evo-
lutionary P system.
1 Introduction
A rather informal idea of what a network of evolutionary processor is consists of a
virtual (complete) graph in which each node hosts a very simple processor called
an evolutionary processor. This is a pretty common architecture for parallel and
distributed symbolic processing, related to the Connection Machine [5] as well as
the Logic Flow paradigm [3]. By an evolutionary processor we mean a processor
which is able to perform very simple operations, namely point mutations in a
DNA sequence (insertion, deletion or substitution of a pair of nucleotides). More
generally, each node may be viewed as a cell having genetic information encoded
in DNA sequences which may evolve by local evolutionary events, that is point
mutations.
Each node processor, which is specialized just for one of these evolutionary
operations, acts on the local data and then local data becomes a mobile agent
which can navigate in the network following a given protocol. Only that data
which is able to pass a filtering process can be communicated. This filtering pro-
cess may require to satisfy some conditions imposed by the sending processor,
by the receiving processor or by both of them. All the nodes send simultaneously
their data and the receiving nodes handle also simultaneously all the arriving
messages, according to some strategies, see [4, 5]. The filtering process that is
to be considered here is regulated by some context conditions associated to the
edges of the graph. This is the model introduced in [2] under the name of accept-
ing network of evolutionary processors with filtered connections (ANEPFC). The
reader interested in a survey of the main results regarding ANEPFCs is referred
to [9].
P systems [13] were introduced as a computational model inspired by the
information and biochemical product processing of living cells through the use
of membrane communication. In most of the works about P systems, informa-
tion is represented as multisets of symbol/objects which can interact and evolve
according to predefined rules. Nevertheless, the use of strings to represent the
information and the use of rules to transform strings instead of multiset objects
have always been present in the literature of this scientific area. So, in his mostly
referred book [13], Gh. Pa˘un overviews the use of string rules in P systems. Dif-
ferent variants of string-based P systems have been proposed along the time. We
can mention rewriting P systems [10], referred as membrane systems with worm
objects [1] in the case of genomic operations, insertion-deletion P systems [7] and
splicing P systems [12], among others. Observe that most of these models have
been used for language generation [11]. In [6, 8], the proposal of hybrid P systems
introduces the use of contextual rules and Chomsky rules to achieve universality
by generating all the recursively enumerable languages.
In this work, we propose the use of evolutionary rules of string rewriting
in all regions of a P system. The idea is not new (see for instance [7]) but
our approach has two different main goals. First, the P systems considered here
define languages by an accepting process in contrast with the generating variants
widely considered so far in the area of membrane computing. Second, we show
that the membrane structure is indispensable for simulating the filtering process
in an ANEPFC. This is also in contrast with very many constructions of P
systems in which the membrane structure plays actually a very minor role, most
of them being reduced to just one membrane. The main part of this note is the
informal construction of an evolutionary P system simulating an ANEPFC with
emphasis on the membrane hierarchies of each region associated with a filtered
connection.
The structure of this work is as follows: In section 2 and 3 we recall the
definition of ANEPFCs and evolutionary P systems, respectively. Then, we in-
formally describe a construction of an evolutionary P system simulating a given
ANEPFC. More precisely, we discuss the evolutionary rules in each region and
the membrane structure associated with each filtered connection in the simulated
network.
2 Accepting networks of evolutionary processors with
filtered connections
We start by summarizing the notions used throughout this work. An alphabet is
a finite and nonempty set of symbols. The cardinality of a finite set A is written
card(A). Any finite sequence of symbols from an alphabet V is called string
over V . The set of all strings over V is denoted by V ∗ and the empty string is
denoted by ε. The length of a string x is denoted by |x| while alph(x) denotes
the minimal alphabet W such that x ∈W ∗.
We say that a rule a→ b, with a, b ∈ V ∪{ε} and ab 6= ε is a substitution rule
if both a and b are not ε; it is a deletion rule if a 6= ε and b = ε; it is an insertion
rule if a = ε and b 6= ε. The set of all substitution, deletion, and insertion rules
over an alphabet V are denoted by SubV , DelV , and InsV , respectively.
Given a rule σ as above and a string w ∈ V ∗, we define the following actions of
σ on w:
• If σ ≡ a→ b ∈ SubV , then σ
∗(w) =
{
{ubv : ∃u, v ∈ V ∗ (w = uav)},
{w}, otherwise
• If σ ≡ a→ ε ∈ DelV , then σ
∗(w) =
{
{uv : ∃u, v ∈ V ∗ (w = uav)},
{w}, otherwise
σr(w) =
{
{u : w = ua},
{w}, otherwise
σl(w) =
{
{v : w = av},
{w}, otherwise
• If σ ≡ ε→ a ∈ InsV , then
σ∗(w) = {uav : ∃u, v ∈ V ∗ (w = uv)}, σr(w) = {wa}, σl(w) = {aw}.
α ∈ {∗, l, r} expresses the way of applying a deletion or insertion rule to a string,
namely at any position (α = ∗), in the left (α = l), or in the right (α = r) end
of the string, respectively. For every rule σ, action α ∈ {∗, l, r}, and L ⊆ V ∗, we
define the α-action of σ on L by σα(L) =
⋃
w∈L
σα(w). Given a finite set of rules
M , we define the α-action of M on the string w and the language L by:
Mα(w) =
⋃
σ∈M σ
α(w) and Mα(L) =
⋃
w∈LM
α(w),
respectively. In what follows, we shall refer to the rewriting operations defined
above as evolutionary operations since they may be viewed as linguistic formu-
lations of local DNA mutations.
For two disjoint subsets P and F of an alphabet V and a string w over V ,
we define the predicates:
ϕ(s)(w;P, F ) ≡ P ⊆ alph(w) ∧ F ∩ alph(w) = ∅
ϕ(w)(w;P, F ) ≡ alph(w) ∩ P 6= ∅ ∧ F ∩ alph(w) = ∅.
The construction of these predicates is based on random-context conditions de-
fined by the two sets P (permitting contexts/symbols) and F (forbidding con-
texts/symbols). Informally, the first condition requires that all permitting sym-
bols are present in w and no forbidding symbol is present in w, while the second
one is a weaker variant of the first, requiring that at least one permitting sym-
bol appears in w and no forbidding symbol is present in w. For every language
L ⊆ V ∗ and β ∈ {(s), (w)}, we define:
ϕβ(L,P, F ) = {w ∈ L | ϕβ(w;P, F )}.
An evolutionary processor over V is a tuple (M,PI, FI, PO, FO), where:
• M is a set of substitution, deletion or insertion rules over the alphabet V .
Formally: (M ⊆ SubV ) or (M ⊆ DelV ) or (M ⊆ InsV ). The set M represents
the set of evolutionary rules of the processor. As one can see, a processor is
“specialized” in one evolutionary operation, only.
• PI, FI ⊆ V are the input permitting/forbidding contexts of the processor,
while PO,FO ⊆ V are the output permitting/forbidding contexts of the pro-
cessor. Informally, the permitting input/output contexts are the set of symbols
that should be present in a string, when it enters/leaves the processor, while the
forbidding contexts are the set of symbols that should not be present in a string
in order to enter/leave the processor.
We denote the set of evolutionary processors over V by EPV .
An accepting network of evolutionary processors with filtered connections
(ANEPFC for short) is a 8-tuple Γ = (V, U,G,R,N , α, β, xI , xO), where:
⋄ V and U are the input and network alphabet, respectively, V ⊆ U .
⋄ G = (XG, EG) is an undirected graph without loops with the set of vertices
XG and the set of edges EG. G is called the underlying graph of the network.
⋄ R : XG −→ 2SubU ∪ 2DelU ∪ 2InsU is a mapping which associates with each
node the set of evolutionary rules that can be applied in that node. Note that
each node is associated only with one type of evolutionary rules, namely for
every x ∈ XG either R(x) ⊂ SubU or R(x) ⊂ DelU or R(x) ⊂ InsU holds.
⋄ N : EG −→ 2U × 2U is a mapping which associates with each edge e ∈ EG
the disjoint sets N (e) = (Pe, Fe), Pe, Fe ⊂ U .
⋄ β : EG −→ {s, w} defines the filter type of an edge.
We say that card(XG) is the size of Γ .
A configuration of an ANEPFC Γ as above is a mapping C : XG −→ 2V
∗
which associates a set of strings with every node of the graph. A configuration
may be understood as the sets of strings which are present in any node at a
given moment. A configuration can change either by an evolutionary step or by
a communication step.
When changing by an evolutionary step each component C(x) of the con-
figuration C is changed in accordance with the set of evolutionary rules Mx
associated with the node x and the way of applying these rules α(x). Formally,
we say that the configuration C′ is obtained in one evolutionary step from the
configuration C, written as C =⇒ C′, if and only if
C′(x) =M
α(x)
x (C(x)) for all x ∈ XG.
When changing by a communication step, each node processor x ∈ XG of
an ANEPFC sends one copy of each string it has to every node processor y
connected to x, provided they can pass the filter of the edge between x and y.
It keeps no copy of these strings but receives all the strings sent by any node
processor z connected with x providing that they can pass the filter of the edge
between x and z.
Formally, we say that the configuration C′ is obtained in one communication
step from configuration C, written as C ⊢ C′, iff
C′(x) = (C(x) \ (
⋃
{x,y}∈EG
ϕβ({x,y})(C(x),N ({x, y}))))
∪(
⋃
{x,y}∈EG
ϕβ({x,y})(C(y),N ({x, y})))
for all x ∈ XG. Note that a copy of a string remains in the sending node x only
if it not able to pass the filter of any edge connected to x.
Let Γ be an ANEPFC, the computation of Γ on the input string w ∈ V ∗
is a sequence of configurations C
(w)
0 , C
(w)
1 , C
(w)
2 , . . . , where C
(w)
0 is the initial
configuration of Γ defined by C
(w)
0 (xI) = {w} and C
(w)
0 (x) = ∅ for all x ∈ XG,
x 6= xI , C
(w)
2i =⇒ C
(w)
2i+1 and C
(w)
2i+1 ⊢ C
(w)
2i+2, for all i ≥ 0. By the previous
definitions, each configuration C
(w)
i is uniquely determined by the configuration
C
(w)
i−1. A computation as above is said to be an accepting computation if there
exists a configuration in which the set of strings existing in the output node xO
is non-empty. The language accepted by Γ is
L(Γ ) = {w ∈ V ∗ | the computation of Γ on w is an accepting one}.
We denote by L(ANEPFC) the class of languages accepted by ANEPFCs.
3 Accepting evolutionary P systems
We now informally describe the P system we are going to investigate. An Ac-
cepting evolutionary P system (AEPS for short) of degree m is a construct
Π = (V, U, µ, (R1, ρ1), · · · , (Rm, ρm)),
where:
- V is the input alphabet, U ⊇ V is the working alphabet,
- µ is a membrane structure consisting of m membranes,
- Ri, 1 ≤ i ≤ m is a finite set of evolutionary and/or dissolving rules over U
associated with the ith region and ρi is a partial order relation over Ri specifying
the priority among the rules. An evolutionary rule is a 4-tuple (a, b, α, β) (or
a→ bβα) where a, b ∈ U ∪ {λ}, α ∈ {here, out, in} and β ∈ {∗, l, r}. A dissolving
rule is 5-tuple (a, b, α, β, δ) (or a→ bβαδ), where a, b, α, β have the same meaning
as for evolutionary rules and δ ∈ U is the dissolving symbol.
The application of a rule a→ bβα in an arbitrary region of the system works
as follows: if there exists a string w in that region, such that w = u1au2, then
w is transformed into u1bu2 (observe that β establishes the way of applying
the evolutionary rule). Parameter α establishes where to send the new strings,
namely they are sent to the outer region, to all immediate inner regions (a copy
of each string is sent to all these regions), or remain in the same region, provided
that β is out, in, or here. If a string is to be sent to an inner region that does
not exist, then it remains where it is.
If the rule is a dissolving one (a → bβαδ, the membrane of the region is
dissolved after the rule application, provided that the membrane is different
from the skin one.
The input string is initially stored in the outmost region. Then, in a fully
parallel manner all the rules are applied to the strings existing in every region
according to their priorities. The system halts whenever: (1) No rule can be
applied, or (2) The system is reduced to only one region, namely the outmost
one.
The language accepted by Π is denoted by L(Π). A string is in L(Π) if and
only if it being initially stored in the outmost region reduces the system to only
one region.
4 A simulation of ANEPFCs by AEPSs
In this section we give just a very brief idea how an AEPS, say Π , can simulate
an ANEPFC, say Γ . The membrane structure of Π consists of the skin region
that includes and as many regions as connections between the processors of Γ .
For every connection between processors i and j we have the regions Rij and
Rji inside the skin membrane. Each region Rij has different structure depending
on the filter type. A symbol in the current string in Π indicates the fact that
the corresponding string in Γ has just arrived in the node i. We discuss the
simulation of one evolutionary step in the node i and the communication process
of the words from the node i to the node j in Γ .
Let us suppose that the set of permitting symbols for the filter on the con-
nection between processor i and j in Γ is defined by Pij = {b1, b2, · · · , bk}. If
this filter acts in the weak mode, then the membrane structure in the region
Rij is showed in the left part of the next figure, while if the filter acts in the
strong mode, then the membrane structure is showed in the right part of the
same figure.
Rij
· · ·
Rb1 Rbk
weak filters
Rij
Rb1
· · · Rbk
strong filters
Fig. 1. Membrane structures for the filters
Inside the inner regions in illustrated membrane structures we apply the
same evolutionary rules as those associated with the node i. In addition, the
new strings are moved through the membrane structure in order to check the
filter conditions. Thus evolutionary rules having the highest priority check the
presence of forbidden symbols. If such a symbol is present, then the string re-
mains blocked in an inner region. If these rules cannot be applied, then other
evolutionary rules check the presence of permitting symbols. As soon as one per-
mitting symbol is present, the string is sent to the outer region. Clearly, some
special symbols are used in order to manage the string movements. It is worth
noting the important role of the membrane structure in each region Rij by an
analogy with electronic circuits: the membranes checking in the weak mode the
presence of a permitting symbol form a sort of parallel circuit, while the mem-
branes checking in the strong mode the presence of a permitting symbol form a
sort of serial circuit.
When a string is going to enter a region of the form Rij where i is the output
node of the ANEPFC, then a new symbol is inserted; this symbol will dissolve
in turn all the membranes.
5 Conclusions and future work
In this paper we have proposed a string accepting P system based on a general-
ization of the evolutionary rules considered for ANEPFCs. We have intuitively
described a construction of an AEPS able to naturally simulate an ANEPFC.
We consider that the simulation process is natural as the membrane structure is
of a great importance in the simulation of the filtering process in ANEPFC. It
is worth mentioning that this is in contrast with very many constructions of P
systems in which the membrane structure plays actually a very minor role, most
of them being reduced to just one membrane.
A technical proof together with other computationally aspects of accepting
evolutionary P systems are to be considered for an extension of this note.
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In this work, we propose a variant of P system based on the rewriting of string-objects by means of evolutionary rules. The membrane structure of such a P system seems to be a very natural tool for simulating the filters in accepting networks of evolutionary processors with filtered connections. We discuss an informal construction supporting this simulation. A detailed proof is to be considered in an extended version of this work.Castellanos, J.; Mitrana, V.; Santos, E.; Sempere Luna, JM. (2011). Simulating accepting networks of evolutionary processors with filtered connections by accepting evolutionary P systems. En Foundations on Natural and Artificial Computation. Springer Verlag (Germany). 6686:295-302. doi:10.1007/978-3-642-21344-1_312953026686Castellanos, J., Păun, G., Rodríguez-Patón, A.: P systems with worm-objects. In: Proc. of the Seventh International Symposium on String Processing Information Retrieval (SPIRE 2000), pp. 64–74. IEEE Computer Society, Los Alamitos (2000)Drăgoi, C., Manea, F., Mitrana, V.: Accepting networks of evolutionary processors with filtered connections. Journal of Universal Computer Science 13, 1598–1614 (2007)Errico, L., Jesshope, C.: Towards a new architecture for symbolic processing. In: Artificial Intelligence and Information-Control Systems of Robots 1994, pp. 31–40 (1994)Fahlman, S.E., Hinton, G.E., Seijnowski, T.J.: Massively parallel architectures for AI: NETL, THISTLE and Boltzmann machines. In: Proc. of the National Conference on Artificial Intelligence, pp. 109–113 (1983)Hillis, W.D.: The Connection Machine. MIT Press, Cambridge (1979)Krishna, S.R., Lakshmanan, K., Rama, R.: Hybrid P systems. Romanian Journal of Information Science and Technology 4(1-2), 111–123 (2001)Krishna, S.R., Rama, R.: Insertion-deletion P systems. In: Jonoska, N., Seeman, N.C. (eds.) DNA 2001. LNCS, vol. 2340, pp. 360–370. Springer, Heidelberg (2002)Madhu, M., Krithivasan, K.: A note on hybrid P systems. Grammars 5(3), 239–244 (2002)Manea, F., Martín-Vide, C., Mitrana, V.: Accepting networks of evolutionary word and picture processors: A survey. In: Scientific Applications of Language Methods. Mathematics, Computing, Language, and Life: Frontiers in Mathematical Linguistics and Language Theory, vol. 2, pp. 523–560. World Scientific, Singapore (2010)Martín-Vide, C., Păun, G.: String-objects in P systems. In: Proc. of Algebraic Systems, Formal Languages and Computations Workshop, RIMS Kokyuroku, Kyoto Univ., pp. 161–169 (2000)Martín-Vide, C., Păun, G.: Language generating by means of membrane systems. Bulletin of the EATCS (75), 199–218 (October 2001)Păun, A., Păun, M.: On membrane computing based on splicing. In: Words, Sequences, Languages, pp. 409–422. Kluwer Academic Publishers, Dordrecht (2000)Păun, G.: Membrane Computing. An Introduction. Springer, Heidelberg (2002

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Simulating accepting networks of evolutionary processors with filtered connections by accepting evolutionary P systems

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