Reaction Systems: a Formal Framework for  Processes Based on Biochemical InteractionsThis paper is dedicated to Hans-Jörg Kreowski on the occasion of his 60th birthday.


Electronic Communications of the EASST
Volume 26 (2010)

Manipulation of Graphs, Algebras and Pictures

Essays Dedicated to Hans-Jörg Kreowski
on the Occasion of His 60th Birthday

Reaction Systems: a Formal Framework for
Processes Based on Biochemical Interactions

Andrzej Ehrenfeucht and Grzegorz Rozenberg

9 pages

Guest Editors: Frank Drewes, Annegret Habel, Berthold Hoffmann, Detlef Plump
Managing Editors: Tiziana Margaria, Julia Padberg, Gabriele Taentzer
ECEASST Home Page: http://www.easst.org/eceasst/ ISSN 1863-2122

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Reaction Systems: a Formal Framework for
Processes Based on Biochemical Interactions∗

Andrzej Ehrenfeucht1 and Grzegorz Rozenberg2

1 Department of Computer Science
University of Colorado at Boulder, USA

2 Department of Computer Science
University of Colorado at Boulder, USA

and
Leiden Institute of Advanced Computer Science

Leiden Center for Natural Computing,
Leiden University, The Netherlands

Abstract: This paper presents a formal framework for investigating processes driven
by interactions between biochemical reactions in living cells. These interactions are
based on the mechanisms of facilitation and inhibition, which underlie the defini-
tion of reaction systems – the central construct of our framework. We discuss in this
paper the basic setup for reaction systems, and its motivation. We also present an
important extension of reaction systems as well as some research topics and results.

Keywords: natural computing; biochemical interactions; reaction systems

1 Introduction

In this paper we investigate the interactions between biochemical reactions from the natural
computing point of view. Natural computing (see, e.g., [6, 7]) is concerned with human-designed
computing inspired by nature and also with computation taking place in nature (i.e., it also inves-
tigates processes taking place in nature in terms of information processing). The former strand
of research is quite well-established: some of the well-known examples are evolutionary com-
puting, neural computing, cellular automata, swarm intelligence, molecular computing, quantum
computing, artificial immune systems, and membrane computing. Examples of research themes
from the latter strand of research are: computational nature of self-assembly, computational
nature of developmental processes, computational nature of bacterial communication, compu-
tational nature of brain processes, computational nature of biochemical reactions, and system
biology approaches to bionetworks. A lot of research from this research strand underscores the
fact that computer science is also the fundamental science of information processing, and as such
a basic science for other scientific disciplines such as, e.g., biology.

This paper is concerned with the computational nature of processes driven by interactions
between biochemical reactions in living cells. It presents a formal framework for investigating
such processes, called the framework of reaction systems (see, e.g., [1, 2, 3, 4]). In particular,

∗ This paper is dedicated to Hans-Jörg Kreowski on the occasion of his 60th birthday.

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Processes Based on Biochemical Interactions

it provides basic definitions together with the intuition/motivation behind them as well as some
research themes, and results concerning the formation of structures (modules) during runs of
reaction systems.

2 Reactions

The functioning of a biochemical reaction is based on facilitation and inhibition: a reaction can
take place if all of its reactants are present and none of its inhibitors is present. If a reaction takes
place, then it creates its product. Therefore to specify a reaction one needs to specify its set of
reactants, its set of inhibitors, and its set of products – this leads to the following definition.

Definition 1 A reaction is a triplet a = (R,I,P), where R,I,P are finite sets. If S is a set such
that R,I,P ⊆ S, then a is a reaction in S.

The sets R,I,P are also denoted by Ra,Ia,Pa, and called the reactant set of a, the inhibitor set
of a, and the product set of a, respectively. Also, rac(S) denotes the set of all reactions in S.

For a finite set of reactions A,RA =
⋃

a∈A Ra, IA =
⋃

a∈A Ia, and PA =
⋃

a∈A Pa are called the
reactant set of A, the inhibitor set of A, and the product set of A, respectively.

The effect of a reaction a is conditional: if Ra is present and no element of Ia is present,
then Pa is produced; otherwise reaction does not take place and “nothing” is produced. This is
formalized as follows.

Definition 2 Let a be a reaction, A a finite set of reactions, and T a finite set.
(1) a is enabled by T , denoted by a en T , if Ra ⊆ T and Ia ∩ T = /0.
(2) The result of a on T , denoted by resa(T ), is defined by: resa(T ) = Pa if a en T , and resa(T ) =
/0 otherwise.
(3) The result of A on T , denoted by resA(T ), is defined by:
resA(T ) =

⋃
a∈A resa(T ).

Clearly, if Ra ∩ Ia 6= /0, then resa(T ) = /0 for every T . Therefore we assume that, for each
reaction a, Ra ∩ Ia = /0; in this paper we will also assume that Ra 6= /0,Ia 6= /0, and Pa 6= /0.

As an example consider the reaction a with Ra = {c,x1,x2}, Ia = {y1,y2}, and Pa = {c,z}.
We can interpret c as the catalyzer of a (it is needed for a to take place, but is not “consumed”
by a), x1,x2 as “real” reactants, y1,y2 as inhibitors (e.g., acids inhibiting the functioning of c
as the catalyzer), and z as the compound that is produced by this reaction. Then a en T for
T = {c,x1,x2,z}, and a is not enabled on neither {c,x1,x2,z,y1} nor on {x1,x2,z}.

An important notion is the activity of a set of reactions A on a finite set (state) T – it is denoted
by enA(T ), and defined by: enA(T ) = {a ∈ A : a en T }. Hence enA(T ) is the set of all reactions
from A that are enabled by (active on) T . Note that resA(T ) = resenA(T )(T ): only the reactions
from A which are enabled on T contribute to the result of A on T .

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3 Basic Assumptions and Intuition

We will discuss now in more detail the basic notions of enabling and application (result) of
reactions and sets of reactions, as they reflect our assumptions about biochemical reactions (mo-
tivated by organic chemistry of living organisms), which are very different from the underlying
assumptions of a majority of models (of human-designed systems) in theoretical computer sci-
ence.

A reaction a is enabled on a set T if T separates Ra from Ia (i.e., Ra ⊆ T and Ia ∩ T = /0). We
make no assumption about the relationship of Pa to either Ra or Ia. When a is enabled by a finite
set T , then resa(T ) = Pa. Thus the result of a on T is “locally determined” in the sense that it
uses only a subset of T , viz., the set of reactants Ra. However the result of the transformation
is global: in comparing T with Pa we note that all elements from T − Pa “vanished”. This is in
great contrast to classical models in theoretical computer science; e.g., in Petri nets (see, e.g.,
[5]) the firing of a single transition has only a local influence on the global marking which may
be changed only on places that are neighbouring the given transition. Our way of defining the
result of a reaction on a state T reflects our assumption that there is no permanency of elements:
an element (molecule) of a global state vanishes unless it is sustained by a reaction.

The result of applying a set of reactions A to a state T is cumulative: it is the union of results of
individual reactions from A. We do not set any conditions on the relationship between reactions
in A. In particular, we do not have the (standard) notion of conflict here: if a,b ∈ A with a en T
and b en T , then, even if Ra ∩ Rb 6= /0, still both a and b contribute to resA(T ), i.e., (resa(T ) ∪
resb(T )) ⊆ resA(T ). Such a conflict of resources (standard in classical models such as, e.g.,
Petri nets) does not exist here. There is no counting in reaction systems, and so we deal with
a qualitative rather than a quantitative model. This reflects our assumption about the “threshold
supply” of elements (molecules): either an element is present, and then there is “enough” of it,
or an element is not present.

We would like to mention here that there is a notion in reaction systems, viz., the notion of
consistency (see, e.g., [4]), that reflects an intuition of conflict. A set of reactions A is called
consistent if RA ∩ IA = /0, i.e., Ra ∩ Ib = /0 for any two reactions a,b ∈ A; clearly if Ra ∩ Ib 6= /0,
then a and b can never be together enabled.

4 Reaction Systems and Interactive Processes

We are ready now to define reaction systems.

Definition 3 A reaction system, abbreviated rs, is an ordered pair A = (S,A) such that S is a
finite set, and A ⊆ rac(S).

The set S is called the background set of A , and A is called the set of reactions of A . All the
notions and notations introduced for sets of reactions carry over to reaction systems through their
underlying sets of reactions. For example, for T ⊆ S,enA (T ) = enA(T ) and resA (T ) = resA(T )
– also, we say that T is active in A , if enA (T ) 6= /0.

It is important to note here that, in the setup of reaction systems, reactions are primary while
structures are secondary. Since we do not have permanency of elements – elements vanish unless

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Processes Based on Biochemical Interactions

they are sustained by reactions, (sets of) reactions create states rather than transform states, in
this sense reaction systems do not work in an environment but rather they create an environment.

The interactions of reaction systems is given by unions. For reaction systems A1 = (S1,A1)
and A2 = (S2,A2) their union, denoted A1 + A2, is defined by A1 + A2 = (S1 ∪ S2,A1 ∪ A2).
This way of combining reaction systems reflects the bottom-up modularity: local descriptions
(reaction systems A1,A2) are combined into the global picture (A1 + A2) in such a way that the
interactions of local descriptions is provided automatically. Thus a major difference with stan-
dard models in theoretical computer science is that no interface is given/needed for combining
reaction systems: the sheer fact that the sets of reactions A1,A2 operate in the same molecular
soup (tube) causes A1,A2 to interact (again through facilitation and inhibition). Thus union is the
basic mechanism for composing/decomposing reaction systems.

The dynamic behaviour of reaction systems is captured through the notion of an interactive
process which is formally defined as follows.

Definition 4 Let A = (S,A) be a rs. An interactive process in A is a pair π = (γ,δ ) of
finite sequences such that γ = C0,C1,...Cn, δ = D1,...,Dn,n ≥ 1, where C0,...,Cn , D1,...,Dn ⊆ S,
D1 = resA (C0), and Di = resA (Di−1 ∪Ci−1) for each 2 ≤ i ≤ n.

The sequence C0,...,Cn is the context sequence of π , and the sequence D1,...,Dn is the result
sequence of π . Let W0 = C0, and Wi = Di ∪Ci for all 1 ≤ i ≤ n. Then the sequence W0,...,Wn
is the state sequence of π , denoted sts(π), and W0 is the initial state of π . For each 0 ≤ j ≤ n,
C j is the context of W j . The sequence E0,...,En−1 of subsets of A such that Ei = enA(Wi), for
all 0 ≤ i ≤ n − 1, is the activity sequence of π , denoted act(π). If act(π) consists of nonempty
sets only, then sts(π) is active – in this case all states W1,...,Wn−1 are active. The set of all state
sequences of (all interactive processes in) A is denoted by ST S(A ).

The basic intuition behind the notion of an interactive process is rather straightforward. Con-
text C0 represents the initial state of π , i.e., the state in which π begins (is initiated), and the
contexts C1,...,Cn represent the influence of (the interaction with) the “rest of the world”. Then
D1 is the result of A on C0, i.e., the result of applying to C0 all the reactions from A enabled
on C0. Together with context C1,D1 forms the successor state W1 of the initial state. Then, iter-
atively, the result of applying A to state Wi−1 = Di−1 ∪Ci−1 yields the result Di which together
with the context Ci forms the successor state Wi . Note that even if Di = /0, Wi can still be an active
state (if enA (Ci) 6= /0). The definition of an interactive process is illustrated in Figure 1.

Note that the background set S provides the elements of all the sets (reactants, inhibitors,
products as well as contexts, results, states, . . . ) used in defining/analyzing a given reaction
system.

5 Extended Reaction Systems

Reaction systems form the basic construct of the broad “framework of reaction systems”. How-
ever, within this framework we use an “onion approach” meaning that additional levels/compo-
nents can be incrementally added (or removed) so that the resulting model is well fitted for the
research issue at hand. An example of such an (incremental) approach are extended reaction
systems which are suitable for investigating the issue of emergence of modules/structures in bio-

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Figure 1: An interactive process.

chemical systems, investigated in [3] and presented in the next section. We use the notation 2S

to denote the set of subsets of a set S.

Definition 5 An extended reaction system, abbreviated ers, is a triplet A = (S,A,R) such that
(S,A) is a reaction system, and R is a binary relation, R ⊆ 2S × 2S.

We refer to (S,A) as the underlying reaction system of A denoted by und(A ).
The role of the restriction relation is to restrict the set of interactive processes as follows. An

interactive process of A is an interactive process π = (γ,δ ) of und(A ) such that if sts(π) =
W0,W1,...,Wn , then, for each 0 ≤ i ≤ n − 1,(Wi,Wi+1) ∈ R. Thus interactive processes of A
are those interactive processes of und(A ), where each two consecutive states in the state se-
quence are related (allowed) by R. We also require that the restriction relation is not too re-
strictive, i.e., that for each state sequence W0,W1,...,Wn of A there exists Wn+1 ⊆ S such that
W0,W1,...,Wn,Wn+1 is also a state sequence of A . In other words, each interactive process of A
can be extended, as is the case in reaction systems.

A possible intuition for relation R is observability: the only state transitions that are observ-
able are those specified by R – therefore interactive processes of extended reaction systems are
observable processes. However, in general, R may express all kinds of restrictions on state tran-
sitions in reaction systems.

A distinct technical feature of extended reaction systems is the existence of periodic elements
– such elements cannot exist in reaction systems. An element t of an ers A is periodic (in A )
if there exists a positive integer n such that for each W0,W1,...,Wn ∈ ST S(A ),t ∈ W0 if and only
if t ∈ Wn; the smallest such n is called the period of t. Hence, if t is a periodic element with
period n,W0,W1,...,Wq ∈ ST S(A ), and 0 ≤ i ≤ q, then if t ∈ Wi then also t ∈ Wi−n (providing

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Processes Based on Biochemical Interactions

that i − n ≥ 0) and t ∈ Wi+n (providing that i + n ≤ q). The set of all periodic elements of A is
denoted by per(A ), and for each T ⊆ S, perA (T ) = T ∩ per(A ) is the set of periodic elements
of T .

The existence of periodic elements motivates the following definition of computing the images
of subsets of a given state (of an interactive process) in the successor state. Let A = (S,A,R)
be an ers, let τ = W0,W1,...,Wn ∈ ST S(A ), let i ∈ {0,...,n − 1}, and let Q ⊆ Wi. Then the
image of Q in Wi+1 (within τ ), denoted by imA ,τ,i(Q), is defined by imA ,τ,i(Q) = resEi(Q ∪
perA (Wi)) − perA (resEi(Wi)). The intuition behind this definition of an image is as follows:
since periodic elements are included in fixed states of state sequences “independently of the
applied reactions” (i.e., we can predict/compute the states of a state sequence where a periodic
element belongs without knowing reactions that are actually applied to states), they are added
to a “real argument” (i.e., Q) of the resEi when computing imA ,τ,i. For the same reason we
substract the periodic elements of resEi (Wi), because we want in the image of Q only the “real
results” (which excludes elements from perA (resEi (Wi)) which will be in Wi+1 anyhow because
of their periodicity).

6 Events and Modules

Among all the subsets of a state of an interactive process we will distinguish “material subsets” –
these are subsets that are the result of applying reactions of a system to subsets of the predecessor
state (in this interactive process). More formally, let τ = W0,W1,...,Wn be a state sequence of an
ers A , and let us consider state Wi for some 1 ≤ i ≤ n. A subset X ⊆ Wi is a “material subset”
of Wi if there exists a subset Y ⊆ Wi−1 such that X is the product of the set of reactions enabled
on Wi−1 applied to Y . Such products included in Wi are “modules” of Wi. If we now consider the
sequence of modules in consecutive states of τ initiated by some nonempty Y ⊆ Wi−1, beginning
with X in Wi and ending in some W j for j ≥ i, then we are tracing the fate of Y (as a sequence
of products) through ( j − i + 1) steps of (an interactive process π behind) τ . Such sequences of
modules are called events which are formally defined below. If we are interested in a module Q
in some Wk , for 1 ≤ k ≤ n, and follow backwards an event that produced Q in Wk , then we get a
possible history of Q, hence an explanation of why and how Q was created in Wk .

Definition 6 Let A be an ers, let τ = W0,W1,...,Wn ∈ ST S(A ), let i, j ∈ {1,...,n} be such
that i ≤ j, and let ω = Qi,...,Q j be such that Qi ⊆ Wi,...,Q j ⊆ Wj , and all Qi,...,Q j−1 are
nonempty. Then ω is an event in τ if there is a Qi−1 ⊆ Wi−1 such that, for each k ∈ {i,..., j},
Qk = imA ,τ,k−1(Qk−1).

We say that ω is passing through each of Wi,...,Wj ; if Q j = /0, then ω dies in W j . The sets
Qi,...,Q j are called the modules of ω in Wi,...,Wj , respectively. More specifically, each module
Ql , i ≤ l ≤ j, is called a l-module.

Thus, intuitively, an event (ω ) is tracing the fate of a subset (Qi−1) of a state (Wi−1 ) in a state
sequence τ within a segment (Wi,...,Wj ) of τ . More specifically, suppose that we are interested
in a state sequence τ (or in an interactive process π with sts(π) = τ ), and in particular we are
interested in the dynamic development of some Qi−1 ⊆ Wi−1 as τ evolves from Wi on until Wj is
reached. This dynamic development of Qi−1 in the segment Wi,...,Wj is the sequence Qi,...,Q j

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of material subsets (modules) of Wi,...,Wj , respectively. Note that both the notion of the result
of transforming Ql into Ql+1, l ∈ {i,..., j − 1}, and the notion of a material subset take into
account the existence of periodic elements in extended reaction systems.

When an event ω is passing through a state Wl then it leaves a “trace” there, viz., its module
Ql . The set of all such traces in Wl left there by all events passing through Wl is called the
snapshot of Wl in τ , denoted by sn pτ (l). Thus, for a given state sequence τ = W0,...,Wn we
get the corresponding sequence of snapshots sn p(τ) = S1,...,Sn, where Si = sn pτ (i) for each
1 ≤ i ≤ n, called the snapshot sequence of τ , and also called a snapshot sequence of A .

Given a snapshot sequence ρ = S1,...,Sn of a state sequence τ = W0,...,Wn there exists a
natural sequence of partial functions nextτ,1,nextτ,2,...,nextτ,n−1 transforming consecutive snap-
shots of ρ into their successor snapshots, where, for each 1 ≤ k ≤ n − 1, nextτ,k : Sk → Sk+1
is defined as follows. For Q ∈ Sk and Q′ ∈ Sk+1, nextτ,k(Q) = Q′ if and only if Q, Q′ are
nonempty and there exists an event ω in τ such that Q is the module of ω in Wk and Q′ is the
module of ω in Wk+1. If we extend the nextτ,k function also to pairs (Q,Q′) with Q′ possibly
empty, then the resulting function is denoted by sucτ,k . Thus, intuitively, the function nextτ,k
connects nonempty modules that are consecutive in an event passing through Wk and Wk+1 . In
this way the sequence of functions nextτ,1,...,nextτ,n−1 delineate all the events of τ as they are
passing through the states of τ , but it does not explicitly indicate the “moment of death” (if an
event dies). The sequence of functions sucτ,1,...,sucτ,n−1 does indicate also the death moments.
As a matter of fact the empty module has really no physical interpretation – it is clearly no ma-
terial subset, but rather its role is to signal the termination (the death) of an event. It is therefore
convenient to consider snapshots with the empty set removed. In this way, for a given snapshot
sequence ρ = S1,...,Sn we obtain its /0-free version ρ̄ = S̄1,...,S̄n, where for each 1 ≤ i ≤ n,
S̄i = Si − {/0}. Accordingly, each nextτ,k function is modified to the rnextτ,k function which is
nextτ,k restricted to S̄k .

We move now to present the structure of snapshots. First we need a couple of set-theoretical
notions.

Definition 7 Let L be a family of sets, and let F1,F2 ⊆ L be nonempty.
(1) We say that F1 is embedded in F2 if

⋃
F1 ⊆

⋂
F2.

(2) We say that F1 is separated from F2 in L if there exists U ∈ L such that
⋃

F1 ⊆ U ⊆
⋂

F2.

Theorem 1 Let A be an ers, let τ = W0,W1,...,Wn ∈ ST S(A ) where n ≥ 2, let sn p(τ) =
S1,...,Sn, and let 1 ≤ k ≤ n − 1. If F1,F2 ⊆ S̄k are nonempty families of sets such that F1 is
embedded in F2 and nextτ,k is defined on all modules in F1 ∪ F2, then nextτ,k(F1) is separated
from nextτ,k(F2) in S̄k+1.

This is a remarkable result as it allows us to view (extended) reaction systems as self-organizing
systems, where a possible self-organizing goal of interactive processes is to ensure (improve on)
separability!

An interactive process (hence a run) of an ers A produces a sequence ρ of snapshots S1,...,
Sk,...,Sn. In general such a sequence may be very “unstable” because there may be no “mathe-
matical similarity” between Sk and Sk+1: remember that the context of the state Wk (in the state
sequence τ = W0,...,Wn for which ρ = sn p(τ)) can “throw anything” into Wk and thus make

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Wk+1 and Sk+1 “arbitrarily different” from Wk and Sk respectively. So we can talk about local
stability (at Wk ) only if there is a strong mathematical similarity between Sk and Sk+1. Perhaps
the most natural choice for such a strong similarity is to require that rnextτ,k is an isomorphism
between partial orders (S̄k,⊆) and (S̄k+1,⊆). When this happens, we get a local stability – it
is local because, again, “anything can happen” to Sk+2 (through the context of Wk+1 ). Hence
we say that (Sk,Sk+1) is a locally stable situation if rnextk is an isomorphism between (S̄k,⊆)
and (S̄k+1,⊆). We want to point out that the situation is quite subtle here, e.g., the fact that
S̄k = S̄k+1 does not necessarily imply that rnextτ,k is an isomorphism of S̄k onto S̄k+1.

It turns out that under the local stability assumption snapshots posses an elegant mathematical
structure.

Theorem 2 Let A be an ers, let τ ∈ ST S(A ), and let S ,S ′ be two consecutive elements
of sn p(τ). If (S ,S ′) is a locally stable situation, then (S ′,⊆), and hence also (S ,⊆), is a
complete lattice.

7 Discussion

We have presented in this paper an introduction to the framework of reaction systems. It is
motivated by organic chemistry of living cells, and more specifically by interactions between
biochemical reactions. The basic notions here are reactions and their results, i.e., the way they
process states – this way of processing the states of a system is very different from the manner
that state processing happens in common models in theoretical computer science. The differ-
ences between (and motivation behind) them are discussed in detail in this paper. The basic
model of our framework are reaction systems, and the basic notion/tool to investigate their dy-
namics is an interactive process. Although reaction systems form the core of our framework, the
framework is constructed in an “incremental” way: depending on a research issue the notion of
reaction system can be modified so that the resulting model is well-suited for the investigation
of the given research issue. For example, reaction systems form a qualitative model where we
do not have counting (of elements), as is the case for models based on multisets rather than on
sets. However there are many situations where one needs to assign quantitative parameters to
states (e.g., when dealing with time issues). Our point of view is that a numerical value can be
assigned to a state T if there is a measurement of T yielding this value. This leads to the notion
of reaction systems with measurements, where a finite set of measurement functions is added as
a third component to reaction systems (see [4]).

Another example of research leading to an incremental modification of the notion of a reaction
system, is the investigation of the way that the products are formed and evolve within the runs of
biochemical systems. The resulting extended reaction systems and the formation of products (the
topics of [3]) are discussed in detail in this paper. The basic dynamic notion here is the notion of
an event which traces the formation of modules (products) within interactive processes of a sys-
tem. The rather surprising results that (extended) reaction systems can be seen as self-organizing
systems which in stable situations produce well-structured families of products/modules are also
presented.

Altogether this paper presents the basic setup of the framework of reaction systems and its

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motivation as well as some research themes and results.

Acknowledgements: The authors are indebted to the referees for useful comments, to Robert
Brijder and Hendrik Jan Hoogeboom for comments on this paper, and to Robert Brijder and
Marloes van der Nat for their help in producing this paper.

Bibliography

[1] Ehrenfeucht, A., Rozenberg, G., Basic notions of reaction systems, Lecture Notes in Com-
puter Science, v. 3340, 27–29, Springer, 2004.

[2] Ehrenfeucht, A., Rozenberg, G., Reaction systems, Fundamenta Informaticae, v. 75, 263–
280, 2007.

[3] Ehrenfeucht, A., Rozenberg, G., Events and modules in reaction systems, Theoretical Com-
puter Science, v. 376, 3–16, 2007.

[4] Ehrenfeucht, A., Rozenberg, G., Introducing time in reaction systems, Theoretical Com-
puter Science, v. 410, 310–322, 2009.

[5] Engelfriet, J., Rozenberg, G., Elementary net systems, Lecture Notes in Computer Science,
v. 1491, 12–121, Springer, 1998.

[6] Kari, L., Rozenberg, G., The many facets of natural computing, Communications of the
ACM, v. 51, 72–83, 2008.

[7] Rozenberg, G., Computer science, informatics, and natural computing – personal reflec-
tions, in S.B. Cooper, B. Löwe, A. Sorbi, eds., New Computational Paradigms – Changing
Conceptions of What is Computable, Springer, Berlin, Heidelberg, 2007.

9 / 9 Volume 26 (2010)


	Introduction
	Reactions
	Basic Assumptions and Intuition
	Reaction Systems and Interactive Processes
	Extended Reaction Systems
	Events and Modules
	Discussion