International Journal of Computers, Communications & Control
Vol. I (2006), No. 3, pp. 33-39

Descriptive Timed Membrane Petri Nets for Modelling of Parallel
Computing

Emilian Guţuleac

Abstract: In order to capture the compartmentation and behaviour of membrane
systems for modelling of parallel computing, we introduce the descriptive dynamic
rewriting Descriptive Membrane Timed Petri Nets (DM-nets) that can at in run-time
modify their own structure by rewriting some of their descriptive expression compo-
nents. Furthermore, this descriptive approach facilitates the understanding of com-
plex models and their component-based construction as well as the application of
modern computer engineering concepts.
Keywords: Descriptive Petri nets, membrane systems, modelling, parallel comput-
ing.

1 Introduction

Recent technological achievements require advances beyond the existing computational models in
order to be used effectively. Pragmatic aspects of current and future computer systems will be modelled
so that realistic estimates of efficiency can be given for algorithms in these new settings.

Petri nets (PN) are very popular formalism for the analysis and representation of parallel and dis-
tributed computing in concurrent systems that has draw much attention to modelling and verification of
this type of systems [1].

P systems, also referred to as membrane systems, are a class of parallel and distributed computing
models [6]. The interest of relating P systems with the PN model of computation lead to several important
results on simulation and decidability issues. Some efforts have been made to simulate P systems with
Petri nets [2, 5, 7] to verifying the many useful behavioral properties such as reachability, boundedness,
liveness, terminating, etc.

In this paper we propose a new approach to express the components of continuous-time P systems
[6] throughout components of escriptive Petri Nets (PN) using descriptive expressions (DE) [3] for mod-
elling of parallel computing. The DE are used for analytical representation and compositional con-
struction of PN models. To model specific rules of P-systems within the framework of the descriptive
Rewriting Timed PN (RTN) [4] we introduce a new extensions Ű the descriptive Membrane RTN, called
DM-nets, that can modify dynamically their own structures by rewriting rules some of their components.

2 Labeled Extended Petri Nets

In this section, we define a variant of PN called labeled extended PN. Let L be a set of labels L =
LP ]LT . Each place pi labeled l(pi) ∈ P a local state and transition t j has action labeled as l(t j) ∈ LT .

A labeled extended PN is structure as a Γ =< P, T, Pre, Post, Test, Inh, G, Pri, Kp, l >, where: P is the
finite set of places and T is a finite set of transitions that P ∩ T = /0. In the graphical representation, the
place is drawn as a circle and the transition is drawn as a black bar; The Pre, Test and Inh : P×T ×N|P| →
N+ respectively is a forward flow, test and inhibition functions and is a backward flow function in the
multi-sets of P, where defined the set of arcs A and describes the marking-dependent cardinality of arcs
connecting transitions and places. The set A is partitioned into tree subsets: Ad , Ah, and At . The subset
Ad contains the directed arcs which can be seen as Ad : ((P×T )∪(T ×P))×N|P| → N+ and are drawn
as single arrows. The inhibitory arcs Ah : (P × T ) × N|P| → N+are drawn with a small circle at the
end. The test arcs At : (P × T )×N|P| → N+are directed from a place to a transition, and are drawn as

Copyright c© 2006 by CCC Publications
Selected paper from ICCCC 2006



34 Emilian Guţuleac

dotted single arrows. It does not consume the content of the source place. The arc of net is drawn if
the cardinality is not identically zero and this is labeled next to the arc and by a default value being 1;
G : E × N|P| → {true, f alse} is the guard function transitions. For t ∈ T a guard function g(t, M) that
will be evaluated in each marking, and if it evaluates to true, the transition t may be enabled, otherwise
t is disabled (the default value is true); Pri : T → N+ defines the priority functions for the firing of each
transition that maps transitions onto natural numbers representing their priority level. The enabling of a
transition with higher priority disables all the lower priority transitions; Kp : P → N+ is the capacity of
places, and by default being infinite value; The l : T ∪ P → L, is a labeling function that assigns a label
to a transition and places. In this way that maps transition name into action names that l(t j) = l(tk) = α
but t j 6= tk and l(pi) = l(pn) = β but pi 6= pn.

A marked labeled extended PN net is a pair N =< Γ, M0 >, where Γ is a labeled PN structure and
M0 is the initial marking of the net. M : P → N+ is the current marking of net which is described by
a symbolic vector-column M = (mi pi), mi ≥ 0,∀pi ∈ P, where the (mi pi) is the number mi of tokens in
place pi. The M is the state of net that assigns to each place tokens, represented by black dots.

The details concerning on enabling and firing rules, and evolution for of N =< Γ, M0 > can be found
in [3] as they require a great deal of space.

3 Descriptive expressions of Petri nets

Due to the space restrictions we will only give a brief overview to this topic and refer the reader to
[3, 4] and the references therein. In following for abuse of notation, labels and name of transitions/places
are the same. We use the concept of a basic descriptive element (bDE) for a basic PN (bPN) introduced
in [2] as following: bDE = |α jt j m0i pi[W +i ,W −i ]|

αk
tk . The translation of this bPN is shown in figure 1a,

where respectively is input transition (action type α j ) and tk = p•i is the output transition (action type
αk ) of place pi ∈ P with initial marking m0i , and the flow type relation functions W +i = Pre(t j, pi)
and W −i = Post(t j, pi), respectively which return the multiplicity of input and output arcs of the place
pi ∈ P. The derivative elements of bDE are for p•i = /0,W −i = 0 is |

α j
t j m

0
i [Wi] with final place pi of t j and

• pi = /0,W +i = 0 is m
0
i pi[Wi]|αktk with entry place pi of tk. If the initial marking m0i of place is a zero tokens

we can omit m0i in bDE. By default, if the type of action α is not mentioned this to match the name of
a transition t. From a bDE we can build more complex DE of PN components by using composition
operations. Also by default,if W +i = W

−
i = 1, we present bDE and it derivatives as following:|

α j
t j m

0
i pi|αktk ,

|α jt j m0 pi or m0i pi|
αk
tk .

A descriptive expression (DE) of a labeled PN is either bDE or a composition of DE a N: DE ::=
bDE|DE ∗DE|◦DE, where ∗ represents any binary composition operation and ◦ any unary operation.

Descriptive Compositional Operations. In the following by default the labels of N are encoded in
the name of the transitions and places. The composition operations are reflected at the level of the DE
components of N models by fusion of places, fusion of transitions with same type and same name (label)
or sharing of as subnets.

Place-Sequential Operation. This binary operation, denoted by the “ | “ sequential operator, de-
termines the logic of a interaction between two local states pi (pre-condition) and pk (post-condition)
by t j action that are in precedence and succeeding (causality-consequence) relation relative of this ac-
tion. Sequential operator is the basic mechanism to build DE of N models. This operation is an
associative, reflexive and transitive property, but is not commutative operation. The means the fact
DE1 = m0i pi[Wi]|

α j
t j m

0
k pk[Wk] 6= m0k pk[Wk]|

α j
t j m

0
i pi[Wi] that the specified conditions (local state) associ-

ated with place-symbol pi are fulfilled always happens before then the occurrence of the conditions
associated with place-symbol pk by means of the action t j. Also, the PN modelling of the iteration
operation is obtained by the fusion of head (entry) place with the tail (final) place that are the same
name (closing operation) in DE which describes this net. The self-loop of N2 net described by an:



Descriptive Timed Membrane Petri Nets for Modelling of Parallel Computing 35

DE2 = m0i pi[Wi]|
α j
t j pi[Wi] = m

0
i p̃i[Wi]|

α j
t j , it is the test operator "p̃", i.e. represent the test arc. The trans-

lation of DE2 in N2 is shows in figure 2b.
Inhibition Operation. This unary operation is represented by inhibitory operator "- " (place-symbol

with overbar) and it DE3 = m0i p̄i[Wi]|
α j
t j describe the inhibitor arc with a weight Wi = Inh(pi,t j).

Synchronization Operation. This binary operation is represented by the " •" or " ∧”join operator
describe the rendez-vous synchronization (by transition tt j ) of a two or more conditions represented
respectively by symbol-place pi ∈• t j, i = 1, n, i.e. it indicate that all preceding conditions of occurrence
actions must have been completed. This operation is a commutative, associative and reflexive.

Split Operation . This binary operation represented by the " ♦" split operator and it describe the
causal relations between activity t jand its post-conditions: after completion of the preceding action of t j
concomitantly several other post-condition can take occurs in parallel ("message sending"). Property of
split operation is a commutative, associative and reflexive.

Competing Parallelism Operation. This compositional binary operation is represented by the " ∨"
competing parallelism operator, and it can be applied over two NA with DEA = A and NB with DEB = B
or internally into resulting NR with DER = R, between the places of a single NR which the symbol-places
with the same name are fused, respectively. We can represent the resulting DER = A ∨ B as a set of
ordered pairs of places with the same name to be fused, with the first element belonging to A the second
to B. The fused places will inherit the arcs of the place in A and B . Also, this compositional binary
operation is a commutative, associative and reflexive property.

Precedence Relations between the Operations. We introduce the following precedence relation be-
tween the compositional operations in the DE: a) the evaluation of operations in DE are applied left-to-
right; b) an unary operation binds stronger than a binary one; c) the "• "operation is superior to"|" and "
♦", in turn, its are superior the "∨ " operation. Further details on definitions, enabling and firing rules,
and evolution for of N can be found in [3] as they require a great deal of space.

4 Dynamic Rewriting Petri Nets

In this section we introduce the model of descriptive dynamic net rewriting PN system. Let X ρY is
a binary relation. The domain of is the Dom(ρ) = ρY and the codomain of ρ is the Cod(ρ) = X ρ . Let
A =< Pre, Post, Test, Inh > is a set of arcs belong to netΓ .

A descriptive dynamic rewriting PN system is a structure RN =< Γ, R, φ , Gtr, Gr, M >,
where: =< P, T, Pre, Post, Test, Inh, G, Pri, Kp, l >; R = r1, ..., rk is a finite set of rewriting rules about the
runtime structural modification of net that P ∩ T ∩ R = /0. In the graphical representation, the rewriting
rule is drawn as a two embedded empty rectangle. We let E = T ∪ R denote the set of events of the net;
φ : E → T, R is a function indicate for every rewriting rule the type of event can occur; Gtr : R×N|P| →
{true, f alse} and Gr : R × N|P| → {true, f alse} is the transition rule guard function associated with
r ∈ R and the rewriting rule guard function defined for each rule of r ∈ R , respectively. For ∀r ∈ R, the
gtr ∈ Gtr and gr ∈ Grwill be evaluated in each marking and if its are evaluates to true, the rewriting rule
r may be enabled, otherwise it is disabled. Default value of gtr ∈ Gtr is true and for gr ∈ Gr is false.
Let RN =< RΓ, M > and RΓ =< Γ, R, φ , Gtr, Gr > described with the descriptive expression DERΓ and
DERN , respectively. A dynamic rewriting structure modifying rule r ∈ R of RN is a map r : DEL ¤ DEW ,
where whose codomain of the rewriting operator ¤ is a fixed descriptive expression DEL of a subnet RNL
of current net RN, where RNL ⊆ RN,with PL ⊆ P, EL ⊆ Eand set of arcs AL ⊆ A and whose domain of the
¤ is a descriptive expression DEW of a new RNW subnet with PW ⊆ P, EW ⊆ E and set of arcs AW . The ¤
rewriting operator represent binary operation which produce a structure change in the DERN and the net
RN by replacing (rewriting) of the fixed current DEL of subnet RNL (DEL and RNL are dissolved) by the
new DEW of subnet RNW now belong to the new modified resulting DERN′ of net RN

′
= (RN\RNL)∪RNW

with P
′
= (P\PL) ∪ PW and E

′
= (E\EL) ∪ EW , where A

′
= (P\PL) ∪ AW the meaning of \ (and ∪) is



36 Emilian Guţuleac

operation to removing (adding) RNL from (RNW to) net RN. In this new net RN
′
, obtained by execution

(fires) of enabled rewriting rule r ∈ R , the places and events with the same attributes which belong RN′
are fused, respectively. By default the rewriting rules r : DEL ¤ /0 andr : /0 ¤ DEW describe the rewriting
rule which fooling holds RN

′
= (RN\RNL) and RN

′
= (RN ∪ RNW ), respectively. A state of a net RN is

a pair (RΓ, M), where RΓ is the configuration of net together with a current marking M. Also, the pair
(RΓ0, M0) with P0 ⊆ P, E0 ⊆ E and marking M0 is called the initial state of the net.

Enabling and Firing of Events. The enabling of events depends on the marking of all places. We
say that a transition t j of event e j is enabled in current marking M if the following enabling condition
ec(t j, M) is verified:

ec(t j, M) = (∧∀pi∈•t j (mi ≥ Pre(pi,t j))∧(∧∀pk∈◦t j (mk < Inh(pi,t j)))∧(∧∀pl∈∗t j (ml ≥ Test(pl ,t j)))∧
(∧∀pn∈t•j ((Kpn −mi) ≥ Post(pn,t j))))∧g(t j, M)).

Similarly, the rewriting rule r j ∈ R is enabled in current marking M if the following enabling condi-
tion ectr(r j, M)is verified:

ectr(r j, M) = (∧∀pi∈•r j (mi ≥ Pre(pi, r j))∧(∧∀pk∈◦r j (mk < Inh(pi, r j)))∧(∧∀pl∈∗r j (ml ≥ Test(pl , r j)))∧
(∧∀pn∈r•j ((Kpn −mi) ≥ Post(pn, r j))))∧g(r j, M)).

Let the T (M) and R(M) is respectively the set of enabled transitions and rewriting rule in current
marking M. Let the E(M) = T (M) ] R(M), is the set of enabled events in a current marking M. The
event e j ∈ E(M) fire if no other event ek ∈ E(M) with higher priority has enabled. Hence, for e j event
i f ((φ j = t j)∨(φ j = r j)∧(gtr(r j, M) = f alse)) then (the firing of transition t j ∈ T (M) or rewriting rule
r j ∈ R(M) change only the current marking:(RΓ, M)

e j→ (RΓ, M′) ⇔ (RΓ = RΓ′ and M[e j > M
′
)). Also,

for e j event i f ((φ j = r j)∧(gr(r j, M) = true) then (the event e j occur to firing of rewriting rule r j and it
occurrence change configuration and marking of current net:(RΓ, M)

r j→ (RΓ′, M′), M[r j > M
′
).

The accessible state graph of a netRN =< Γ, M > is the labeled directed graph whose nodes are the
states and whose arcs which is labeled with events of RN are of two kinds: a) firing of a enabled event
e j ∈ E(M): arcs from state (RΓ, M) to state(RΓ, M

′
) labeled with evente j then this event can fire in the

net configuration RΓ at marking M and leads to new marking M
′
: (RΓ, M)

e j→ (RΓ′, M′) ⇔ (RΓ = RΓ′
and M[e j > M

′
in RΓ); b) change configuration: arcs from state (RΓ, M) to state( RΓ

′
, M

′
) labeled with

rewriting rule r j :(RΓL, ML)¤ (RΓW , MW ) which represent the change configuration of current RN net:
(RΓ, M)

e j→ (RΓ′, M′) and M[r j > M
′
.

Figure 1: Translation of (a) DERΓ1in RN1 and (b) DERΓ2 in RN2

Let we consider the RN1 given by the following descriptive expression:DERΓ1 = p1|r1 p2 ∨ DE
′
RΓ1,

DE
′
RΓ1 = (p2· p5)|t1 p3|t2 p4|t3 (p1♦p5), M0 = (5p1, 1p5), gr(r1, M) = (m1 = 3)&(m5 = 0) and r1 : DERΓ1 ¤

DERΓ2. Also, for r j is required to identify if RNL belong the RΓ. Upon firing, the enabled events or
rewriting rule modify the current marking and/or and modify the structure and current marking of net
RN1 in RN2 given by: DERΓ2 = p1|t1 p2 ∨ DE

′
RΓ2, DE

′
RΓ2 = (p2 · p6)|t2 p3(|t3 p4|t4 p5 ∨|t5 p5|r2 (p1♦p6)),

M = (1p1, 3p2, 1p3), gr(r2, M) = (m1 = 4)&(m5 = 1), r2 = r
−1
1 : DERΓ2 ¤ DERΓ1.

Figure 1 show the translation of DERΓ1 in RN1 and DERΓ2 in RN2, respectively.



Descriptive Timed Membrane Petri Nets for Modelling of Parallel Computing 37

5 Dynamic Rewriting Timed Petri Nets

Systems are described in timed PN (TPN) as interactions of components that can performed a set of
activities associated with events. An event e = (α, θ ), where α ∈ E is the type of the activity (action
name), and θ is the firing delay.

A descriptive dynamic rewriting TPN as a RT N =< RN, θ >, where: RN =< Γ, R, φ , Gtr, Gr, M >,
Γ =< P, T, Pre, Post, Test, Inh, G, Pri, K p, l > (see Definition 2 and 3) with set of events E which can be
partitioned into a set E0 of immediate events and a set Eτ of timed events E = E0 ] Eτ . The immediate
event is drawn as a thin bar and timed event is drawn as a black rectangle for transition or a two embedded
empty rectangle for rewriting rules, and Pri(E0) > Pri(Eτ ) ; θ : E ×N|P| → R+ is the weight function that
maps events onto real numbers R+ (delays or weight speeds). Its can be marking dependent. The delays
θ (ek, M) = dk(M)defining the events firing parameters governing its duration for each timed events of
Eτ . If several timed events are enabled concurrently e j ∈ E(M) for e j ∈ • pi = ∀e j ∈ E : Pre(pi, e j) > 0 ,
either in competition or independently, we assume that a race race competition condition exists between
them. The evolution of the model will determine whether the other timed events have been aborted or
simply interrupted by the resulting state change. The θ (e j, M) = w j(M) is weight speeds of immediate
events e j∈E0 . If several enabled immediate events are scheduled to fire at the same time in vanishing
marking M with the weight speeds, and the probability to enabled immediate event e j can fire is: qi(M) =
w(e j, M)/ ∑el∈(E(M)&• pi) w(e j, M) ,where E(M) is the set of enabled events in M. An immediate events
e j ∈ T0 has a zero firing time.

6 P Systems and Descriptive Timed Membrane Petri Nets

Here we give a brief review of P systems and its encoding with DM-nets. The main components of P
systems are membrane structures consisting of membranes hierarchically embedded in the outermost skin
membrane. A full guide for P systems can be referred to [3]. In general, a basic evolution-communication
P system with active membranes (of degree n ≥ 0) is Π = (O, H, µ, Ω, (ρ, π)), where: O is the alphabets
of objects; H is a finite set of labels for membranes;µ is a membrane structure consisting of n membranes
labeled with elements h in H;Ω is the configuration, that is a mapping from membranes of Π(nodes in
µ ) to multisets of objects ωk ∈ Ω, k = 1, ,|Ω|, from O;ρ and π is respectively the set off developmental
rules ρh and πhits priorities , h = 0, 1, , n − 1. Thus the can be of two forms of rules: a) the object rules
(OR), i.e., evolving and communication rules concerning the objects; b) the membranes rules (MR), i.e.,
the rules about the structural modification of membranes.

Here we define DM-Nets for encoding of P systems mentioned above into descriptive dynamic rewrit-
ing TPN as a RT N. The basis for DM-Nets is a membrane RT N that is DE net structure comprise: places;
transitions; weighed directed arcs from places to transitions and vice-versa; a capacity for each place;
weighed inhibitory and test arcs; priority and guard function of transitions.

The DM − nets of degree n ≥ 0 is a construct DM = ∨n−1h=0[hDEh]h , where DEh is the descriptive
expression of RT Nh that represent the configuration of membrane [h ]h in a P system Π.

Consider the P system Π.The encoding of Π into RT NΠ is decomposed into two separate steps. First,
for every membrane [h ]h we associate: to each object ωi ∈ Ω one place ph,i = [hm0i pi]h labeled as ωi with
the initial marking m0i , and to each rule ρh, j ∈ ρ one event eh, j = [he j]h labeled as ρh, j that acts on the
this membrane. Second, for every membrane [h ]h we define the DEh of RT Nh that it correspond to the
initial configuration of the P system Π as [hDEh]h.

Let u, v , and u′, v′ , is a multiset of objects. The evolving object rule ρh′ , j : [h[h′ u → v]h′ ]h with
multiset of objects u, v , which will be kept in membrane [h]h is encoded as [h[h′ pu|t j pv]h′ ]h . The antiport
rule ρh′ , j : [hu[h′ v]h′ ]h → [hv

′
[h′ u

′
]h′ ]h , that realize a synchronized wich object c the exchange of objects,

is encoded as [h[h′ (pu · pv · p̃c)|t j (pu′♦pv′ )]h′ ]h. Also, the symport rule ρh′ ,k : [hu[h′ ]h′ ]h → [h[h′ u
′
]h′ ]h that



38 Emilian Guţuleac

move objects from inside to outside a membrane, or vice-versa is encoded as [h[h′ (pu · p̃c)|tk pu′ ]h′ ]h.
Because a configuration mean both a membrane structure and the associated multisets, we need rules

for processing membranes and multisets of objects as:
MR = Change, Dissolve,Create, Divide, Merge, Separate, Move.
The above membrane rewriting rules (realized by the rewriting events in DE) are defined as follows:
Changerewriting rule [h[h′ (DEh′ , Mh′ )]h′ ]h ¤ [h[h′ (DE

′
h
′ , M

′
h
′ )]h′ ]h that in runtime the current structure

and the multisets of objects to membrane h, encoded by descriptive expression DEh′ and marking Mh′ is
changed in a new structure DE

′
h′ with new marking M

′
h′ ;

Dissolve rewriting rule [h(DEh, Mh)[h′ (DEh′ , Mh′ )]h′ ]h ¤[h(DEh, M
′
h)]h that the objects and sub-membranes

of membrane h
′
now belong to its parent membrane h , the skin membrane cannot be dissolved;

Create rewriting rule [h(DEh, Mh)]h ¤ [h(DE
′
h, M

′
h)[h′ (DE

′′
h
′ , M

′′
h
′ )]h′ ]h with Mh = M

′
h + M

′′
h′ that the

new membrane h
′

is created and M
′′
h
′ are added into membrane h

′
, the rest remain in the parent mem-

brane h; Divide rewriting rule [h(DEh, Mh)]h ¤[h[h′ (DEh, Mh)]h′ [h′′ (DEh, Mh)]h′′ ]hthat the objects and sub-
membranes are reproduced and added into membrane h

′
and membrane h

′′
, respectively;

Merge rewriting rule that the objects of membrane h
′
and h

′′
are added to a new membrane h is:

[h[h′ (DE
′
h
′ , Mh′ )]h′ [h′′ (DE

′′
h
′′ , M

′′
h
′′ )]h′′ ]h ¤ [h(DE

′
h
′ ∨DE′′

h
′′ , Mh′ + M

′′
h
′′ )]h;

Separate rewriting rule is the counterpart of Merge is done by a rewriting rule of the form ¤[h(DE
′
h
′ ∨

DE
′′
h
′′ , Mh′ + M

′′
h
′′ )]h ¤ [h[h′ (DE

′
h
′ , Mh′ )]h′ [h′′ (DE

′′
h
′′ , M

′′
h
′′ )]h′′ ]h with the meaning that the content of mem-

brane h is split into two membranes, with labels h
′
and h

′′
.

Moverewriting rule where a membrane h
′′

can be moved out or moved into a membrane h
′
as a whole

is: [h[h′ (DEh′ , Mh′ )[h′′ (DE
′′
h
′′ , M

′′
h
′′ )]h′′ ]h′ ]h ¤ [h[h′ (DEh′ , Mh′ )]h′ [h′′ (DE

′′
h
′′ , M

′′
h
′′ )]h′′ ]h or

[h[h′ (DEh′ , Mh′ )]h′ [h′′ (DE
′′
h
′′ , M

′′
h
′′ )]h′′ ]h ¤ [h[h′ (DEh′ , Mh′ )[h′′ (DE

′′
h
′′ , M

′′
h
′′ )]h′′ ]h′ ]h.

Thus, using the DM − Nets facilitates a compact and flexible specification to visual simulate of P
systems with dynamic rewriting TPN nets that permit the verification of the its many useful behavioral
properties such as reachability, boundedness, liveness, terminating, etc., and the performance evaluation
of parallel computing models.

7 Summary and Conclusions

In this paper we have proposed an approach to the performance modeling of the behaviour of P-
systems through a class of Petri nets, called Descriptive Membrane Timed PN (DM-nets). Based upon
the introduction of a set of descriptive composition operation and rewriting rules attached with transitions
for the creation of dynamic rewriting TPN, the membrane structure can be successfully encoded as
a membrane descriptive rewriting timed Petri nets models which permit the description the behavioral
state based process run-time structure change of P systems. We are currently developing a software visual
simulator with a friendly interface for verifying and performance evaluation of descriptive rewriting TPN
models and DM-nets.

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Emilian Guţuleac,
Technical University of Moldova,

Computer Science Department,
Address: 168, Bd. Stefan cel Mare, MD-2004,

Chişinău, Republic of Moldova
E-mail: egutuleac@mail.utm.md