=Paper=
{{Paper
|id=Vol-2962/paper46
|storemode=property
|title= Properties of Communicating Reaction Systems
|pdfUrl=https://ceur-ws.org/Vol-2962/paper46.pdf
|volume=Vol-2962
|authors=Erzsébet Csuhaj-Varjú,Pramod Kumar Sethy
|dblpUrl=https://dblp.org/rec/conf/itat/Csuhaj-VarjuS21
}}
== Properties of Communicating Reaction Systems==
https://ceur-ws.org/Vol-2962/paper46.pdf
Properties of Communicating Reaction Systems
Erzsébet Csuhaj-Varjú and Pramod Kumar Sethy
Department of Algorithms and Their Applications
Faculty of Informatics, Eötvös Loránd University ELTE
Budapest, Hungary
{csuhaj,pksethy}@inf.elte.hu
Abstract: Communicating reaction systems are new vari- other feature of reaction systems which makes them differ-
ants of networks of reaction systems where the compo- ent from other bio-inspired computational models, as for
nents communicate with each other by sending products example, P systems, is the lack of permanency: the state of
or reactions. Reaction system, a mathematical formalism the system consists of the products of those reactions that
inspired by the biochemistry of the living cell, focuses were performed in the last step. Those reactants that were
on an abstract set-based representation of chemical reac- not involved in any reaction disappear from the system.
tions via facilitation and inhibition. In this paper we ex- This property is widely used in the theory of R systems.
amine bio-inspired properties of communicating reaction R systems have been studied in detail over the last six-
systems such as steady state and mass conservation. teen years. One interesting topic of their study is the
Keywords: reaction system, communicating reaction sys- theory of networks of reaction systems [4]. Such a con-
tems, steady state, conserved set struct is a virtual graph with a reaction system in each
node. These reaction systems are defined over the same
background set and work in a synchronized manner, gov-
1 Introduction erned by the same clock. After performing the reactions
enabled for the current set of reactants at a node, cer-
The concept of a reaction system (an R system) was in- tain products from other nodes can be added to the node’s
troduced by A. Ehrenfeucht and G. Rozenberg as a formal product set. The nodes, thus the reaction systems inter-
model of interactions between biochemical reactions. The act with each other using distribution and communication
interested reader is referred to [9] for the original moti- protocols. The set of products of each reaction system in
vation. The main idea of the authors was to model the the network forms a part of the environment of the net-
behavior of biological systems in which a large number of work. Important ideas and results on these constructs can
individual reactions interact with each other. be found in [5, 4]. A recent development in the area is
A reaction system consists of a finite set of objects that the concept of communicating reaction systems with direct
represent chemicals and a finite set of triplets that repre- communication (cdcR systems), introduced in [6]. A cdcR
sent chemical reactions. Each reaction consists of three system consists of a finite number of components. Each
nonempty finite sets: the set of reactants, the set of in- component consists of a finite number of extended reac-
hibitors, and the set of products. The set of reactants and tions, and these extended reactions are of the same type.
the set of inhibitors are disjoint. Let T be a set of reactants. Components of a cdcR system are defined over the same
A reaction is enabled for T and it can be performed if all of background set. The components, in addition to perform-
its reactants are present in T and none of its inhibitors are ing standard reactions, communicate products or reactions
in T . When the reaction is performed, then the set of its to certain predefined target components.
reactants is replaced by the set of its products. All enabled There are various research topics in the domain of reac-
reactions are applied in parallel. The set of products ob- tions systems. One type of investigations focuses on the
tained by the reactions performed in parallel is the union mathematical properties of reaction systems, for example
of the sets of products that were obtained by the reactions functions defined by reaction systems, state sequences, ef-
that were enabled for T . For further details on reaction fect of limited resources, cycles and connections to propo-
systems consult [7]. sitional logic. For details consult [8, 11, 12]. One other
Reaction systems are qualitative models, opposed to research direction focuses on the capabilities of reaction
membrane systems (P systems) that are quantitative ones. systems as a modeling framework. In [1, 2, 3] a series of
The model of reaction systems focuses only on the pres- such biologically inspired properties are defined and stud-
ence or absence of the chemical species, and does not con- ied.
sider their amounts. Multiple reactions that have common In this paper we examine some of these biologically
reactants do not interfere. All of the reactions that are en- inspired properties like steady state, conserved sets in
abled at a certain step are performed simultaneously. An- the frame of communicating reaction systems with direct
Copyright c 2021 for this paper by its authors. Use permitted un-
communication. A system is said to be in a steady state
der Creative Commons License Attribution 4.0 International (CC BY if it does not experience any changes over time. Studying
4.0). steady states is a relevant topic in many fields of science.
Similarly, also mass conservation plays important role in 3 Communicating Reaction Systems
many scientific areas.
The paper is organized as follows. In Section 2 we in- In this section we briefly recall the most important con-
troduce basic notions and notations concerning reactions cepts concerning a variant of communicating reaction sys-
and reaction systems. In Section 3 we recall communi- tems (cdcR(p) systems, for short), where the reaction sys-
cating reaction systems with direct communication and in tems directly communicate with each other. The concept
Section 4 we define different properties of cdcR systems was introduced in [6], and is related to the notion of a net-
which communicate products. Finally, we provide conclu- work of R systems [4]. A cdcR(p) system consists of a fi-
sions and few suggestions for further research in Section nite number of components, each component consists of a
5. finite number of extended reactions which are of the same
type. The components are defined over the same back-
ground set and in addition to performing standard reac-
2 Reaction Systems tions, communicate products to certain predefined target
components. The components of the cdcR(p) system work
in a synchronized manner, governed by the same clock.
For basic notions of formal languages and computation The products obtained as results of the reactions are asso-
theory the reader is encouraged to consult [10]. ciated with targets, i.e., with the label of the component
In this section we recall the basic notions concerning which the product is sent to. The target component need
reaction systems, following [9, 7]. For technical reasons, not to be different from the sender component. After per-
some notations are presented in a form that slightly devi- forming the reactions and the communication, the system
ates from the original one. performs a new transition, i.e. the procedure is repeated.
Let S be a finite nonempty set; S is called the back- Now we recall the notion of a cdcR(p) system from [6].
ground set. A reaction ρ over S is a triplet (R, I, P) where A cdcR system communicating by products (a cdcR(p)
R, I, P are nonempty subsets of S such that R ∩ I = 0. / Sets system, for short), of degree n, n ≥ 1, is an (n + 1)-tuple
R, I, P are called the set of reactants, the set of inhibitors, ∆ = (S, A1 , . . . , An ), where
and the set of products of ρ, respectively; they can also
be denoted by Rρ , Iρ , and Pρ . In this case the reaction is • S is a finite nonempty set, the background set of ∆;
denoted by ρ : (Rρ , Iρ , Pρ ).
• Ai , 1 ≤ i ≤ n, is the i-th component of ∆, where
A finite nonempty set of reactions over the same back-
ground set is a reaction system. Thus, a reaction system is – Ai is a finite nonempty set of extended reactions
an ordered pair A = (S, A), where S is a background set of type pc (pc-reactions, for short).
and A is a finite nonempty set of reactions over S.
– Each pc-reaction ρ of Ai is of the form ρ :
Now we recall how reaction systems operate over a set (Rρ , Iρ , Πρ ), where Rρ and Iρ are nonempty
of reactants. subsets of S and Rρ ∩ Iρ = 0, / and Πρ ⊆ Pρ ×
Let S be a background set, T ⊆ S, ρ : (Rρ , Iρ , Pρ ) be a {1, . . . , n}, Pρ is a nonempty subset of S. Rρ , Iρ ,
reaction over S, and let A be a finite set of reactions over Πρ are called the set of reactants, the set of in-
S. Then hibitors, and the set of products with targets. A
pair (b, j), 1 ≤ j ≤ n in Πρ means that product
1. ρ is enabled for T iff Rρ ⊆ T and Iρ ∩ T = 0;
/ b is communicated to component A j .
2. the result of applying ρ to T , denoted by resρ (T ), The name pc-reaction refers to reaction communicating
equals Pρ if ρ is enabled for T and is equal to the products.
emptyset, 0,
/ otherwise; The notions and notations concerning reaction systems
are extended to cdcR(p) systems, we recall them from [6].
the result of applying A to T , denoted by resA (T ), is
3. S If it is clear from the context, for singleton set {ρ} we use
ρ∈A resρ (T ). notation ρ.
A pc-reaction ρ : (Rρ , Iρ , Πρ ) is enabled for a set U ⊆ S
That is, a reaction ρ is enabled for a set of reactants T if T if Rρ ⊆ U and Iρ ∩ U = 0/ as in case of standard reaction
contains all reactants of ρ and none of its inhibitors. If ρ is systems; this fact is denoted by enρ (U).
enabled for T , then its products contribute to the successor Let ∆ = (S, A1 , . . . , An ) be a cdcR(p) system and let
state of the reaction system. For T ⊆ S, enA (T ) denotes U ⊆ S. Then, we define resAi (U) = {b | (b, j) ∈ Πρ , ρ ∈
the set of reactions in A that are enabled for T . It is easy Ai , enρ (U), 1 ≤ j ≤ n}.
to see that resA defines a function on 2S , called the result We consider result all of the products obtained by per-
function. forming the pc-reactions, including those ones that will
The state sequence of a reaction system A with initial leave the component by communication.
state T is given by successive iterations of the result func- cdcR(p) systems operate by transitions, i.e., by chang-
tion: (resnA (T ))n∈N = (T, resA (T ), res2A (T ), ...). ing their states. A state of a cdcR(p) system ∆ =
(S, A1 , . . . , An ) is an n-tuple (D1 , . . . , Dn ) where Di ⊆ S, A where Rρ 0 = {[x, i] | x ∈ Rρ }, Iρ 0 = {[y, i] | x ∈ Iρ },
1 ≤ i ≤ n; Di is called the state of component Ai , 1 ≤ i ≤ n. Pρ 0 = {[x, k] | (x, k) ∈ Πρ , 1 ≤ k ≤ n}. No other reaction is
Notice that Di can be empty set. in A0 . Then A is called the flattened reaction system of ∆.
A transition in ∆ means that every component of the
cdcR(p) system performs all of its enabled pc-reactions
on the current set of reactants and then communicates the 4 Properties of cdcR(p) Systems
obtained products to their target components, indicated in
the corresponding pc-reaction. Notice that the same prod- In [1] reaction systems as a modeling framework was ex-
uct from several components can be communicated to a amined and several formalizations of concepts in the fo-
component and by several pc-reactions. cus of interest in bio-modeling were introduced and then
The sequence of transitions starting with the initial state studied: mass conservation, invariants, steady states, sta-
forms the state sequence in ∆. Observe that for a given tionary processes, elementary fluxes, and periodicity. In
initial state there is only one state sequence in ∆, i.e. the this paper we extend some of these notions to networks of
sequence of transitions is deterministic. reaction systems, more precisely, to cdcR(p) systems.
Let ∆ = (S, A1 , . . . , An ), n ≥ 1, be a cdcR(p) sys- We first start with steady states from [1].
tem. The sequence D̄0 , . . . , D̄ j , . . . is called the state
sequence of ∆ starting with initial state D̄0 if the Definition 2. . Let A = (S, A) be a reaction system. We
following conditions are met: For every D̄ j , j ≥ 0 say that a nonempty set W ⊂ S is a steady state of A if
where D̄ j = (D1, j . . . , Di, j , . . . , Dn, j ), 1 ≤ i ≤ n it resA (W ) = W.
holds that D̄ j+1 = (D1, j+1 . . . , Di, j+1 , . . . , Dn, j+1 )
with Di, j+1 = ∪1≤k≤nComk→i (resAk (Dk, j )) where Notice that this property means that no change can be
Comk→i (resAk (Dk, j )) = {b | (b, i) ∈ Πρ , ρ : (Rρ , Iρ , Πρ ) ∈ experienced in this state in a process of evolution, i.e. if
Ak , enρ (Dk, j )}. Sequence Di,0 , Di,1 , . . . is said to be the A enters state W , then all elements following W in the
state sequence of component Ai of ∆, 1 ≤ i ≤ n. state sequence will be equal to W .
The state sequence does not end if resAi (Di, j ) is the Before defining the steady state for cdcR(p) systems,
empty set, since products can be communicated to the we make some remarks. As it was shown in [6], to each
component in the coming steps. cdcR(p) system ∆ a reaction system A can be given,
Let ∆ = (S, A1 , . . . , An ), n ≥ 1, be a cdcR(p) system and namely, its flattened version, which represents the compo-
let D̄0 , D̄1 . . . , D̄i , . . . be the state sequence of ∆ starting nents of ∆ and its operation corresponds to the operation
with D̄0 . Then every pair (D̄i , D̄i+1 ), i ≥ 0 is said to be of ∆. This implies that if the flattened reaction system A
a transition in ∆ and is denoted by D̄i =⇒ D̄i+1 . has a steady state W and ∆ has n components, n ≥ 1, then
In [6] it was shown that to every cdcR(p) system ∆ a W corresponds to a state D̄W = (W1 , . . . ,Wn ) of ∆ where
simulating R system A can be constructed. Namely, W = ∪ni=1W̃i , where W̃i = {[a, i] | a ∈ Wi }. Notice that W j
can be the empty set for some j, 1 ≤ j ≤ n. By Theorem 1,
Theorem 1. Let ∆ = (S, A1 , . . . , An ), n ≥ 1, be a cdcR(p) A is constructed in such way that resA (W ) corresponds to
system and let D̄0 = (D1,0 , . . . , Dn,0 ) be the initial state 0 = (W 0 , . . . ,W 0 ), a state of ∆, where D̄ =⇒ D̄0 holds.
D̄W 1 n W W
of ∆. We can construct an R system A = (S0 , A0 ), give If D̄W = D̄W0 , then we call D̄ a steady state of ∆. Notice
W
0
an initial state W0 of A and define mappings hi : 2S → that in case of cdcR(p) systems the reaction is extended,
2S , 1 ≤ i ≤ n such that for each i, 1 ≤ i ≤ n, the thus elements of S obtained by the extended reactions can
state sequence Di,0 , Di,1 , . . . , Di,k , . . . of component Ai of be communicated to a node from other nodes.
∆ is equal to the sequence hi (W0 ), hi (W1 ), . . . , hi (Wk ), . . . , Now we define the notion of a steady state of a cdcR(p)
where W0 ,W1 , . . . ,Wk , . . . , k ≥ 0 is the state sequence of A system.
starting from initial state W0 .
Definition 3. Let ∆ = (S, A1 , . . . , An ), n ≥ 1, be a cdcR(p)
The statement was proved by a so-called flattening tech- system and let D̄W = (W1 , . . . ,Wn ) be a state of ∆. Then D̄W
nique (frequently used in the theory of P systems) where is said to be a steady state of ∆ if for D̄W 0 = (W 0 , . . . ,W 0 )
1 n
the notation of the reactants at the nodes indicates the lo- 0 0
where DW =⇒ Dw it holds that Wi = Wi for i, 1 ≤ i ≤ n.
cation of the object (entity) as well. The reader interested
in the details is referred to [6]. Notice that for any Wi , 1 ≤ i ≤ n, resAi (Wi ) consists of all
The reaction system A obtained in this way is called products obtained by the performed pc-reactions, includ-
the flattened reaction system or a flattened version of ∆. ing those ones which will leave the component by commu-
We recall the definition from [6]. nication. Thus, Wi 6= resAi (Wi ) may hold.
Next we will present a statement concerning a connec-
Definition 1. Let ∆ = (S, A1 , . . . , An ), n ≥ 1, be a cdcR(p) tion between steady states of cdcR(P) systems and steady
system. Let reaction system A = (S0 , A0 ) be defined as states of their flattened reaction systems.
follows. Let S0 = {[x, i] | x ∈ S, 1 ≤ i ≤ n} be the back-
ground set of A . For any pc-reaction ρ : (Rρ , Iρ , Πρ ) Theorem 2. Let ∆ = (S, A1 , . . . , An ), n ≥ 1, be a cdcR(p)
of component Ai , we define reaction ρ 0 : (Rρ 0 , Iρ 0 , Pρ 0 ) of system and let A = (S0 , A0 ) be its flattened reaction system.
• Let W be a steady state of A . Then there exist map- system A = (A, S), deciding if there exists a nonempty
0
pings gi : 2S → 2S , 1 ≤ i ≤ n and a state D̄W = steady state W ⊂ S is an NP-complete problem.
(W1 , . . . ,Wn ) of ∆ such that D̄W = (W1 , . . . ,Wn ) is a In the following we deal with one other important prop-
steady state of ∆ and gi (W ) = Wi . erty, called mass-conservation. First, we recall some aux-
iliary notions.
• Let D̄W = (W1 , . . . ,Wn ) be a steady state of ∆. Then
0 For a reaction system A = (S, A), the support set of A
there exist mappings hi : 2S → 2S , 1 ≤ i ≤ n and W ⊆ is defined as supp(A ) = R ∪ P where R =
S
Rρ and
S such that W = ∪i=1 hi (Wi ) is a steady state of A .
0 n
ρ∈A
P=
S
Pρ .
Proof sketch. To prove the statement, we consider the ρ∈A
definition of the flattened reaction systems of ∆. It is given Next we define the notion of the support set for a com-
by A = (S0 , A0 ), where S0 = {[x, i] | x ∈ S, 1 ≤ i ≤ n} is ponent of a cdcR(p) system and then for the system itself.
the background set and for any pc-reaction ρ : (Rρ , Iρ , Πρ )
of component Ai of ∆, 1 ≤ i ≤ n, we define reaction Definition 4. The support set for a particular component
ρ 0 : (Rρ 0 , Iρ 0 , Pρ 0 ) of A where Rρ 0 = {[x, i] | x ∈ Rρ }, Iρ 0 = Ai of a cdcR(p) system ∆ = (S, A1 , . . . , An ), 1 ≤ i ≤ n, is
{[y, i] | x ∈ Iρ }, Pρ 0 = {[x, k] | (x, k) ∈ Πρ , 1 ≤ k ≤ n}. No defined as supp(Ai ) = Ri ∪ P̄i , where Ri = {a | a ∈ Rρ , ρ ∈
other reaction is in A0 . It is easy to see that if we define gi Ai , a ∈ S} and P̄i = {a | (a, j) ∈ Πρ , ρ ∈ Ai , a ∈ S, 1 ≤ j ≤
such way that it orders to each reactant [x, i] in A a reac- n}.
tant x at component Ai , and by hi we order to each reactant For a cdcR(p) system ∆ = (S, A1 , . . . , An ), n ≥ 1 the sup-
x of component Ai a reactant [x, i] of A , then we obtain port set of ∆ is defined as supp(∆) =
n
S
supp(Ai ).
from state W of A state D̄W = (W1 , . . . ,Wn ) of ∆ and re- i=1
versely. Furthermore if W is a steady state, then D̄W will
be a steady state as well, and reversely. We leave the de- We recall the notion of a conserved set of a reaction
tails to the reader. system [1].
Next we provide an example.
Definition 5. Let A = (S, A) be a reaction system, then a
Example 1. Let ∆ = (S, A1 , A2 , A3 ) be a cdcR(p) system set M ⊆ supp(A ) is conserved if for any W ⊆ supp(A ),
where S = {a, b, c} and components A1 , A2 and A3 are de- M ∩W 6= 0/ if and only if M ∩ resA (W ) 6= 0.
/
fined as follows:
In this notion it is crucial that supp(A ) ⊂ S.
A1 = {ρ1 : ({a, b}, {c}, {(a, 1), (b, 1)})}, M has a special property, namely if it has a joint subset
with a state W , then it has a joint subset with the state
A2 = {ρ2 : ({b, c}, {a}, {(b, 3), (c, 2)}), obtained after applying all enabled reactions to W as well.
This definition cannot be directly implemented for
ρ3 : ({a, c}, {b}, {(a, 3), (c, 2)})},
cdcR(p) systems. Instead, we define a notion to describe
conservation of sets.
A3 = {ρ4 : ({a, c}, {b}, {(a, 2), (c, 3)}),
Definition 6. Let ∆ = (S, A1 , . . . , An ), n ≥ 1 be a cdcR(p)
ρ5 : ({b, c}, {a}, {(b, 2), (c, 3)})}.
system and let Mi ⊆ S, 1 ≤ i ≤ n. We say that Mi ⊂
Let D̄0 = ({a, b}, {b, c}, {a, c}) be the initial state of supp(Ai ) is a conserved set for component Ai , i, 1 ≤ i ≤ n
∆. For component A1 , it is clear from the product if the following holds. For any two states D̄ = (D1 , . . . , Dn )
{(a, 1), (b, 1)} that after each transition the state does not and D̄0 = (D01 , . . . , D0n ) where D̄ =⇒ D̄0 , it holds that if
change, it always remains {a, b}. On the other hand, states there exists Wi ⊂ Mi such that Wi ⊂ Di , then there exists
of components A2 and A3 keep changing due to the product Wi0 ⊂ Mi such that Wi0 ⊂ D0i holds.
with in-built communication.
The above way of conservation concerns a particular
The above example inspires us to distinguish between component. Obviously, such conserved sets can appear
so-called "strong steady states" of a cdcR(p) system where at several components.
the states of the components do not change or so-called As in the case of steady states, we can find a connection
"weak steady states" where the support of the entire state between conserved sets of cdcR(p) systems and their flat-
remain unchanged but the states of the particular compo- tened reaction systems. Let ∆ = (S, A1 , . . . , An ), n ≥ 1 be a
nents may change. The support of the state of a cdcR(p) cdcR(p) system and let A = (S0 , A0 ) be its flattened reac-
system is the set of those elements of the background set tion system. By the construction of A it can easily be seen
that appear in some of the states of the particular compo- that if Wi ⊂ Di and Wi0 ⊂ D0i , then W̄i = {[a, i] | a ∈ Wi } and
nents either as reactant or elements of a product (or both). W̄i0 = {[b, i] | b ∈ Wi0 } are subsets of D̄i = {[c, i] | c ∈ Di }
The study of weak steady states is an interesting open and D̄0i = {[d, i] | d ∈ D0i }, respectively. It would be useful
problem. Interesting questions are decidability problems to develop such notion that describe a distributed manner
as well. For example, it is known that given a reaction of conservation in the entire system.
5 Conclusions and Further Research Developments in Language Theory, 8th International Con-
Directions ference, DLT, 2004, Auckland, New Zealand, December 13-
17, 2004, Proceedings, Lecture Notes in Computer Science,
3340, 27-29, Springer, 2004
In this paper we proposed steady states and mass conserva- [10] Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to
tion of communicating reaction systems by product com- automata theory, languages, and computation, 3rd Edition,
munication. Using the concepts of the corresponding flat- Pearson international edition, Addison-Wesley, 2007
tened reaction systems, we attempted to describe the ideas [11] Salomaa, A.: Functions and sequences generated by reac-
beyond the definitions. It will be a promising and use- tion systems. Theor. Comput. Sci.466 (2012) 87-96
ful research to study the concepts of invariants, stationary [12] Salomaa, A.: Functional constructions between reaction
processes, elementary fluxes and periodicity of cdcR(p) systems and propositional logic. Internat. J. Found. Comput.
systems. Another interesting research could be studying Sci. 24(1) (2013) 147-159
on all these bio-inspired properties for cdcR(r) (cdcR sys-
tems communicating reactions) and comparing all respec-
tive properties with cdcR(p).
6 Acknowledgment
The work of Erzsébet Csuhaj-Varjú was supported by the
National Research, Development, and Innovation Office
- NKFIH, Hungary, Grant no. K 120558. The work of
Pramod Kumar Sethy was supported by project ” Inte-
grált kutatói utánpótlás-képzési program az informatika és
számítástudomány diszciplináris területein”, EFOP 3.6.3-
VEKOP-16-2017-00002, a project supported by the Euro-
pean Union and co-funded by the European Social Fund.
References
[1] Azimi, S., Gratie, C., Ivanov, S., Manzoni, L., Petre, I., Por-
reca, A.E.: Complexity of model checking for reaction sys-
tems. Theor. Comput. Sci. 623 (2016) 103-113
[2] Azimi, S.: Steady states of constrained reaction systems.
Theor. Comput. Sci.701 (2017) 20-26
[3] Azimi, S., Gratie, C., Ivanov, S., Petre, I.: Dependency
graphs and mass conservation in reaction systems. Theor.
Comput. Sci.598 (2015) 23-39
[4] Bottoni, P., Labella, A., Rozenberg, G.: Networks of Reac-
tion Systems, Int. J. Found. Comput. Sci. 31(1) (2020) 53-71
[5] Bottoni, P., Labella, A., Rozenberg, G.: Reaction systems
with influence on environment, J. Membr. Comput., 1(1)
(2019) 3-19
[6] Csuhaj-Varjú, E., Sethy, P.K.: Communicating Reaction
Systems with Direct Communication. In: Freund, R., Ish-
dorj, T.O., Rozenberg, G., Salomaa, A., Zandron, C.,
(Eds) Membrane Computing -21st International Conference
(CMC) 2020, Vienna, Austria, September 14-18, 2020, Re-
vised Selected Papers, Lecture Notes in Computer Science,
vol 12687, Springer, Berlin, Heidelberg, 17-30, 2021
[7] Ehrenfeucht, A., Rozenberg, G.: Reaction Systems, Fun-
dam. Informaticae,75, (1-4)(2007) 263-280
[8] Ehrenfeucht, A., Main, M., Rozenberg, G.: Functions de-
fined by reaction systems. Internat. J. Found. Comput. Sci.
22 (01) (2011) 167-178
[9] Ehrenfeucht, A., Rozenberg, G.: Basic Notions of Reaction
Systems. In: Calude, C., Calude, E., Dinneen, M.J.,(Eds):