Reaction systems with influence on environment
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Abstract
Reaction systems, motivated by the functioning of the living cell, became a novel model of interactive computation. In this paper, we pursue this line of research. More specifically, we present a systematic investigation of possible interactions of a reaction system with its environment (context). While in the original definition this interaction is oneway, i.e., the behavior of a reaction system is influenced by its environment, we investigate now also the influence of the system on its environment, where a possible time delay of this influence is also considered. To understand the behavior of reaction systems when such a twoway interaction takes place, we establish its relationship to their contextindependent behavior (i.e., the behavior which is not influenced by the environment).
1 Introduction
 (1)
pre1, the set of reactants, required to be present in the current state,
 (2)
pre2, the set of inhibitors, required not to be present in the current state, and
 (3)
post1, the set of products of the reaction, to be present in the successor state.
While the original motivation for reaction systems came from biology (see, e.g., [1, 6, 9, 10, 12, 13]), they developed also as a novel model of interactive computation (see, e.g., [4, 8, 15, 20, 21, 22, 23, 24, 25]), where the behavior of a system (as defined by its processes) results from an interaction of the internal information processing performed by the system with the influence of the environment. The internal processing of information is determined by the cumulative effect of reactions on the current state, where each reaction represents the basic unit of internal information processing.
A standard manifestation of a process is its sequence of states. For reaction systems, three types of sequences are typically studied: the sequence of environmental context contributions, \(\langle {C_0},C_1,\ldots ,C_n\rangle\), the sequence of internal results of reactions, \(\langle {D_0},D_1,\ldots ,D_n\rangle\), and the overall state sequence, \(\langle {W_0},W_1,\ldots ,W_n\rangle\), with \(W_i\) depending on \(D_i\) and \(C_{i}\). Research concerning reaction systems includes investigations of their expressive power and their modeling abilities by studying their state sequences (see, e.g., [2, 4, 5, 7, 9, 11, 13, 14, 16, 17, 18, 19, 22, 23, 25]), as well as investigations of various extensions of the original model to include some typical mechanisms or to simulate typical phenomena observable in nature (see, e.g., [2, 5, 11, 12, 14, 18, 19]). In general, one would suppose that behaviors of extended models fall inbetween contextindependent behaviors, i.e., where the contributions by the environment are irrelevant, and general behaviors, i.e., with nonrestricted contributions by the environment.
Almost exclusively, in the studies of the extended models, it was assumed that the context could contribute in an arbitrary way to the determination of the successor state (or not contribute at all). This assumption was challenged in Bottoni and Labella [3], by considering processes where the context sequence correlates in some way with the result sequence. In particular, it was proved that if the context contribution is functionally dependent (through an environment function) on the current internal state of the system, then one obtains a family of processes which is genuinely intermediate between the contextindependent and the general behaviors.
In this paper, we proceed to investigate the interactive nature of processes in reaction systems and study in more detail a twoway communication between a system and its environment, and in particular we study possible influences of the reaction systems on the contexts provided by their environments. Such twoway interactions are very common in information processing systems, including biological systems.
If we symbolically express the immediate dependence of the system on the environment in the classical model of reaction systems by \(C_j\rightsquigarrow {W_j}\) at each step j, then we can express the opposite influence of the system on the environment as \(D_i\rightsquigarrow {C_j}\), for some \(j\ge {i}\). In this paper, we investigate the \(D_i\rightsquigarrow {C_j}\) influence through environment functions and study separately the cases of immediate (\(j=i\)) and delayed (\(j>i\)) influence.
The paper is organized as follows.
After recalling some relevant mathematical notions in Preliminaries, we provide basic notions and notations concerning reaction systems in Sect. 3. In Sect. 4, we reformulate dynamic processes in reaction systems (called interactive processes) in terms of configurations—this way of viewing processes is convenient for the considerations of our paper. In Sect. 5, we introduce reaction systems which influence the environment, called system–environment reaction systems, and prove some basic properties of their dynamic behavior in Sect. 6. In these systems, the contribution by the environment (the context set) at each stage of the process is determined by the current result of the reactions of the system (applied to the previous state of the system).
In Sect. 7, we consider the case of contextindependent system–environment reaction systems, while in Sect. 8 we consider reaction systems influencing the environment with some delay.
The properties of dynamic processes of such system–environment reaction systems with delay are investigated in Sect. 9 for delay \(d=1\) and in Sect. 10 for delay \(d>1\).
In Sect. 11, we compare the state sequences generative power of system–environment reaction systems with different delays.
The discussion in Sect. 12 concludes this paper.
2 Preliminaries
We recall here some basic mathematical notions and notations to be used in this paper.
We use \(\mathbb {N}\) and \(\mathbb {N}^+\) to denote the sets of natural numbers and positive integers, respectively.
Given sets X and Y, \(X\setminus {Y}\) denotes their difference, \(X\cup {Y}\) denotes their union, and \(X\cap {Y}\) denotes their intersection. The empty set is denoted by \(\emptyset\). Given a finite family of sets \({\mathcal{K}}=\{K_1,\dots ,K_n\}\), we denote its union by \(\bigcup {\mathcal{K}}\) and its intersection by \(\bigcap {\mathcal{K}}\). For a finite set X, X denotes its cardinality and \(\wp (X)\) denotes the set of all the subsets of X, also called the power set of X.
Also, \((\wp (X))^{k}\) denotes the set of all sequences of length k of subsets of X.
For a sequence \(\tau =\langle {Q_1},\ldots ,Q_n\rangle\), \(n\ge {1}\), of sets from \(\wp (X)\), and \(Y\subseteq {X}\), the projection of \(\tau\) on Y, denoted by \(proj_Y(\tau )\), is the sequence \(\langle {Q_1}\cap {Y},\ldots Q_n\cap {Y}\rangle\).
3 Reaction systems
In this section, we recall basic notions of the original setup of reaction systems, with most of the material taken from [4]. A finite set S such that \({S}\ge {2}\) is called a background set.
Definition 1
(Reaction) Let S be a background set. A triple \(b=(R,I,P)\), with R, I, P nonempty subsets of S such that \(R\cap {I}=\emptyset\), is a reactionoverS.
The sets R, I, and P are called the set of reactants, inhibitors, and products of b, respectively (and we use the notations \(R_b\), \(I_b\), and \(P_b\), respectively, to denote them).
We consider now the effect of a reaction (or of a set of reactions) on a specific state of a (biochemical) system. A state of a system is formalized as a finite set, namely the set of all different (e.g., biochemical) entities present in the state.
Definition 2
 1.
b is enabled in T if and only if \(R_b\subseteq {T}\) and \({I_b}\cap {T}=\emptyset\),
 2.
the result of applyingbinT, denoted by \(res_b(T)\), equals \(P_b\) if b is enabled in T and \(\emptyset\) otherwise,
 3.
the result of applyingAinT, denoted by \({res}_A(T)\), is defined as \(\bigcup \{{res}_b(T)\mid {b}\in {A}\}\).
Thus, for a given S, each set A of reactions over S induces the function \({res}_A:\wp (S)\rightarrow \wp (S)\), called the result function of A, such that, for each \(T\subseteq {S}\), \(res_A(T)\) is defined as above. In particular, if A is a singleton set, \(A=\{b\}\), then \({res}_{\{b\}}(T)={res}_b(T)\), defined as above.
Note that the definition of \({res}_A\) implies that the effect of reactions in A is cumulative. This implies that no conflict occurs between reactions using a common subset of reactants, i.e., reactions for which sets of reactants have a nonempty intersection.
We use \({en}_b(T)\) to denote the fact that b is enabled in T, and En(A, T) to denote the set of reactions from A enabled^{1} in T. Since \({res}_b(T)=\emptyset\) for each reaction b which is not enabled in T, only enabled reactions actually contribute to the result—therefore, \({res}_A(T)=\bigcup \{{res}_b(T)\mid {b}\in {{En}(A,T)}\}\).
Definition 3
(Reaction system) A reaction system is a pair \({\mathcal{A}}=(S,A)\) such that S is a background set and A is a set of reactions over S.
Thus, a reaction system is basically a finite set of reactions. In specifying a reaction system, we also specify its background set, hence \({\mathcal{A}}=(S,A)\). The elements of S, called entities, are used to define reactions in A, but S can contain additional entities, some of which may be used for specifying processes taking place in \({\mathcal{A}}\). Unless explicitly stated otherwise, we assume that A is nonempty.
The result function of\({\mathcal{A}}=(S,A)\), denoted by \({res}_{\mathcal{A}}\), is the function \({res}_{\mathcal{A}}:\wp (S)\rightarrow \wp (S)\) defined by: for each \(X\subseteq {S}\), \({res}_{\mathcal{A}}(X)={res}_{A}(X)\). Thus, transforming a state X by \(\mathcal{A}\) is in fact transforming it by the set of reactions of \(\mathcal{A}\).
In a usual setup for reaction systems, given a background set S, we will consider \(\wp (S)\) and sequences over \(\wp (S)\). Any element of \(\wp (S)\) is a state of \({\mathcal{A}}\). The behavior of a reaction system consists of all possible interactive processes taking place in it, where an interactive process is defined as follows.
Definition 4
(Interactive process) Let \({\mathcal{A}}=(S,A)\) be a reaction system. For \(n\in \mathbb {N}^+\), an (nstep) interactive process in \({\mathcal{A}}\) is a pair \(\pi =(\gamma ,\delta )\) of finite sequences over \(\wp (S)\), \(\gamma =\langle {C_0},{C_1,}\ldots ,C_n\rangle\) and \(\delta =\langle {D_0},{D_1,}\ldots ,D_n\rangle\), where^{2}, for all \(i\in \{1,\ldots ,n\}\), \(D_i={res}_A(D_{i1}\cup {C_{i1}})\).
The sequence \(\gamma\) is called the context sequence of \(\pi\) (denoted by \({cons}(\pi )\)), \(\delta\) is called the result sequence of \(\pi\) (denoted by \({ress}(\pi )\)) and \({\sigma =}\langle {W_0},{W_1,}\ldots ,W_n\rangle\), with \(W_i=D_i\cup {C_i}\) for all \(i\in \{0,\ldots ,n\}\), is called the state sequence of \(\pi\) (denoted by \({sts}(\pi )\)), with \(W_0=C_0\cup {D_0}\) called the initial state of \(\sigma\), denoted by \({init}(\sigma )\), also referred to as the initial state of\(\pi\). The suffix \(\langle {C_1},\ldots ,C_n\rangle\) of \({cons}(\pi )\) is called the proper context sequence of \(\pi\), denoted by \({pcs}(\pi )\). Similarly, \(\langle {W_1},\ldots ,W_n\rangle\) is called the proper state sequence of\(\pi\).
If \(C_i\subseteq {D_i}\) for all \(i\in \{{1},\ldots ,n\}\), then \(\pi\) is contextindependent. Thus, if \(\pi\) is contextindependent, \(W_i=C_i\cup {D_i}=D_i\) and consequently the context sequence of \(\pi\) does not influence the proper state sequence of \(\pi\). Obviously, if \(C_i=\emptyset\) for all \(i\in \{{1},\ldots ,n\}\), then \(\pi\) is contextindependent—we say then that \(\pi\) is an emptycontext interactive process.
The behavior of \({\mathcal{A}}\) is the set of all its interactive processes, denoted by \({PROC}({\mathcal{A}})\). The contextindependent behavior of\({\mathcal{A}}\) is the set of all contextindependent interactive processes in \({\mathcal{A}}\), denoted by \({{CIPROC}}({\mathcal{A}})\). Then \({STS}({PROC}({\mathcal{A}}))\) denotes the set of the state sequences of all interactive processes in \({\mathcal{A}}\) and \({STS}({CIPROC}({\mathcal{A}}))\) denotes the set of the state sequences of all contextindependent interactive processes in \({\mathcal{A}}\).
 1.
“No resurrection”: if \({\mathcal{A}}\) is in the empty state, then it remains in the empty state (because \({res}_{{{\mathcal{A}}}}({\emptyset })={\emptyset }\)),
 2.
“No saturation”: if the current state of \({\mathcal{A}}\) is S, then the successor state equals \({\emptyset }\) (because \({res}_A(S)={\emptyset }\)), and
 3.
“Once repeated, always repeated”: if a state sequence of \({\mathcal{A}}\) contains a subsequence \(\langle {W_i},W_{i+1},W_{i+2}\rangle\) such that \(W_i=W_{i+1}\), then \(W_{i+1}=W_{i+2}\) (because \(W_{i+2}={res}_{\mathcal{A}}(W_{i+1})={res}_{\mathcal{A}}(W_{i})=W_{i+1}\)).
4 Configurations of reaction systems
It is convenient, and standard in computation theory, to consider dynamic processes as sequences of configurations. This has been done also for reaction systems (see, e.g., [11], where configurations are referred to as instantaneous descriptions) and we will follow this approach.
Definition 5
(Sconfiguration) For a background set S, a Sconfiguration is a triple \(f=(C,D,W)\) with \(C,D,W\subseteq {S}\) and \(W=C\cup {D}\).
The sets C, D, and W are called the context of f, the result of f, and the state of f, respectively (and denoted by cn(f), re(f), and st(f), respectively). Also, we say that f is contextindependent if \({cn}(f)\subseteq {{re}(f)}\), and that f is emptycontext if \({cn}(f)=\emptyset\). When S is clear from the context of considerations, then we may use the term “configuration”, rather than “Sconfiguration”.
The notion of interactive process can now be reformulated as follows.
Definition 6
(Interactive process) Let \({\mathcal{A}}=(S,A)\) be a reaction system. For \(n\ge {1}\), an (nstep) interactive process in \({\mathcal{A}}\) is a sequence of Sconfigurations \(\sigma =\langle {f_0},f_1,\ldots ,f_n\rangle\) such that \({re}(f_{i+1})={res}_A({st}(f_i))\) for \(i\in \{0,1,\ldots ,n1\}\).
We call \(f_0\) the initial configuration of \(\sigma\) and denote it by \(init(\sigma )\). Also, we say that \(\sigma\) is contextindependent if for each \(i\in \{1,\dots ,n\}\)\(f_i\) is contextindependent and we say that \(\sigma\) is emptycontext if, for each \(i\in \{1,\dots ,n\}\), \(f_i\) is emptycontext.
Although traditionally the notion of context independence of an interactive process was defined by requiring that already the initial configuration \(f_0\) is contextindependent, we require now that context independence holds from the second configuration onwards, i.e., from \(f_1\) onwards. This change is well motivated from the point of view of composition of interactive processes.
An important obvious observation is that any subsequence of \(\sigma\) consisting of at least two configurations is also an interactive process in \({\mathcal{A}}\).
It is immediate to observe that an interactive process \(\pi\), in the sense of Definition 4, with \({cons}(\pi )={\langle {C_0},C_1,\ldots ,C_n\rangle }\), \({ress}(\pi )=\langle {D_0,D_1,\ldots ,D_n}\rangle\), and \({sts}(\pi )={\langle {W_0},W_1,\ldots ,W_n\rangle }\) translates into an interactive process \(\sigma =\langle {f_0},f_1,\ldots ,f_n\rangle\), in the sense of Definition 6, such that \({cn}(f_i)=C_i\), \({re}(f_i)=D_i\), and \({st}(f_i)=W_i\), for each \(i\in \{0,1,\ldots ,n\}\).
Sometimes, for a given reaction system \({\mathcal{A}}\), we are not interested in all interactive processes in \({\mathcal{A}}\) (respectively, contextindependent interactive processes in \({\mathcal{A}}\)), but rather only in those which begin in a specific set G of initial configurations. We use then \(PROC({\mathcal{A}},G)\) (respectively, \(CIPROC({\mathcal{A}},G)\)) to denote this subclass of interactive processes in \({\mathcal{A}}\).
The following notion is useful in analyzing interactive processes.
Definition 7
(Frame) For an interactive process \(\pi =\langle {f_0},f_1,\ldots ,f_n\rangle\), with \(f_i=(C_i,D_i,W_i)\) for \(i\in \{0,1,\ldots ,n\}\), the frame of \(\pi\), denoted by \({frm}(\pi )\), is the \((n+2)\)tuple \(\langle {C_0},D_0,C_1,\ldots ,C_n\rangle\).
Thus, the frame of \(\pi\) gives explicitly its initial configuration \(f_0\) (by specifying \(C_0\) and \(D_0\)) and the proper context sequence \(\langle {C_1},\ldots ,C_n\rangle\).
Theorem 1
Let\(\pi _1\)and\(\pi _2\)be two interactive processes of a reaction system\({\mathcal{A}}\). If\({frm}(\pi _1)= {frm}(\pi _2)\), then\(\pi _1=\pi _2\).
Proof
The proof follows directly from the definition of the \({res}_{\mathcal{A}}\) function.
Thus, for a given initial configuration \(f_0\) and a given proper context sequence \(\alpha ={\langle }C_1,\ldots ,C_n{\rangle }\), there exists exactly one interactive process \(\pi\) in \({\mathcal{A}}\) such that \({init}(\pi )=f_0\) and \({pcs}(\pi )=\alpha\). This generalizes the observation concerning the original formulation of reaction systems (where \(D_0={\emptyset }\), and consequently \(C_0=W_0\)) that the complete context sequence \(\langle {C_0},C_1,\ldots ,C_n\rangle\) determines \(\pi\).
5 Reaction systems influencing the environment
We will now study interactive processes where the context provided by the environment is influenced by the results of the reactions internal to the reaction systems.
More specifically, in this section we consider (nstep) interactive processes in which the context (which models the environment) is instantaneously influenced by the results (of applying the reactions of a reaction system), i.e., the dependency is of the form \(D_i\leadsto {C_i}\) for each \(i\in \{{0,}1,\dots ,n\}\). As a matter of fact, in this paper we consider the case that \(C_i\) is totally determined by \(D_i\) through an environment function\(\psi :\wp (S)\rightarrow \wp (S)\). Later in this paper, we will consider interactive processes where the influence of the reaction system on the environment happens with a delay.
Definition 8
(Environment function) Given a background set S, a function \(\psi :\wp (S)\rightarrow \wp (S)\) is called an environment function for S.
Definition 9
(System–environment reaction system) A system–environment reaction system, abbreviated se reaction system, is an ordered pair \({\mathcal{B}}=({\mathcal{A}},\psi )\), where \({\mathcal{A}}=(S,A)\) is a reaction system and \(\psi :\wp (S)\rightarrow \wp (S)\) is an environment function for S.
We call \({\mathcal{A}}\) the underlying reaction system of\({\mathcal{B}}\), denoted by \(und({\mathcal{B}})\), \(\psi\) the system–environment (se) function of\({\mathcal{B}}\), and S the background set of \({\mathcal{B}}\).
The “system–environment” prefix of the name indicates the fact that now the environment is not anymore independent of the system, but it is influenced (in fact it is determined) by the system. Clearly, this influence must be taken into account in the definition of the interactive processes for se reaction systems, which is done as follows.
Definition 10
((\(S,\psi\))configuration) Let S be a background set and let \(\psi\) be an environment function for S. A (\(S,\psi\))configuration is a Sconfiguration \(f=(X,Y,Z)\) such that \(\psi (Y)=X\).
The terminology and notation for Sconfigurations carries over to (\(S,\psi\))configurations.
Definition 11
(System–environment interactive process) Let \({\mathcal{B}}=({\mathcal{A}}\), \(\psi )\) be a se reaction system with \({\mathcal{A}}=(S,A)\). An (nstep) system–environment interactive process in\({\mathcal{B}}\), abbreviated seinteractive process in\({\mathcal{B}}\), is an (nstep) interactive process \(\pi =\langle {f_0},f_1,\dots ,f_n\rangle\) in \({\mathcal{A}}\), such that, for each \(i\in \{0,\dots ,n\}\), \(f_i=(C_i,D_i,W_i)\) is a (\(S,\psi\))configuration.
Intuitively, \(\pi\) is such that its context sequence is determined by the results of applying reactions from A. This influence is modeled through the se environment function \(\psi\), more specifically \(C_i=\psi (D_{i})\), for each \(i\in \{0,\ldots ,n\}\), as illustrated in Fig. 2.
6 Basic properties of state sequences of se reaction processes
 (1)
There is an important difference between se reaction systems and reaction systems. For a reaction system \({\mathcal{A}}=(S,A)\) every state of \({\mathcal{A}}\) (i.e., every subset of S) is relevant from the point of view of the behavior of \({\mathcal{A}}\), meaning that for each state W of \({\mathcal{A}}\), there exists an interactive process \(\pi\) in \({\mathcal{A}}\) such that W appears in \(\pi\). This does not hold for se reaction systems, because if \({\mathcal{B}}=({\mathcal{A}},\psi )\), with \({\mathcal{A}}=(S,A)\), is a se reaction system, then each \((S,\psi )\)configuration \(f=(C,D,W)\) is such that \(C=\psi (D)\) and thus \(W=D\ \cup \ \psi (D)\). Therefore, the only states which are relevant for \({\mathcal{B}}\), from the point of view of its behavior, are states of the form \(X\cup \psi (X)\), where \(X\subseteq S\). Only these states will appear in state sequences in \({STS}({SEPROC}({\mathcal{B}}))\). This issue is discussed in more detail in the next section of this paper.
 (2)
If \({\mathcal{B}}=({\mathcal{A}},\psi )\), with \({\mathcal{A}}=(S,A)\), is a se reaction system, and \(\pi =\langle {f_0},\ldots ,f_n\rangle\) is a se interactive process in \({\mathcal{B}}\), then \(\pi\) is uniquely determined by \(f_0\). In other words, for each \(n\ge {1}\) and each \((S,\psi )\)configuration f there exists exactly one nstep se interactive process \(\pi\) in \({\mathcal{B}}\) with \({init}(\pi )=f\). Therefore, for each \(n\ge {1}\), and each relevant state W of \({\mathcal{B}}\) (i.e., \(W\subseteq {S}\) such that \(W=Z\cup \psi (Z)\) for some \(Z\subseteq S\), see (1) above), there exists exactly one state sequence \(\langle {W_0},W_1,\ldots ,W_n\rangle\) in \({STS}({SEPROC}({\mathcal{B}}))\) such that \(W_0=W\). This is an important feature of se interactive processs—they share it with contextindependent processes of reaction systems.
 (3)
Let \({\mathcal{B}}=({\mathcal{A}},\psi )\) be a se reaction system with \({\mathcal{A}}=(S,A)\). Then \({STS}({{SE}}{PROC}({\mathcal{B}}))\subseteq {{STS}}({PROC}({\mathcal{A}}))\) because, if \(\pi =\langle {f_0},f_1,\ldots\), \(f_n\rangle\) is a se interactive process in \({\mathcal{B}}\), then, for each \(i\in \{0,1,\ldots ,n\)}, \(f_i\) is a (\(S,\psi\))configuration, and so also a Sconfiguration. Hence, \(\pi\) is also an interactive process in \({\mathcal{A}}\).
However, this inclusion is strict, because context sequences of interactive processes in \({\mathcal{A}}\) are not constrained by \(\psi\). More formally, it is seen as follows.
Let \(\pi =\langle {f_0,f_1,f_2}\rangle\), with \(f_i=(C_i,D_i,W_i)\) for all \(i\in \{0,1,2\}\), be a se interactive process in \({\mathcal{B}}\).It follows then from (i) and (ii) that \({STS}({SEPROC}({\mathcal{B}}))\subsetneq {{STS}({PROC}({\mathcal{A}}))}\). (i)
Assume that \(W_0\) is such that \(D_1={res}_{{\mathcal{A}}}(W_0)\ne {S}\)
Let then \(\pi ^\prime =\langle {f^\prime _0,f^\prime _1}\rangle\), with \(f^\prime _i=(C^\prime _i,D^\prime _i,W^\prime _i)\) for all \(i\in \{0,1\}\), be an interactive process in \({\mathcal{A}}\) such that \(f^\prime _0=f_0\) and \(C^\prime _1\) is defined as follows.Thus, \({sts}(\pi ^\prime )\ne {{sts}(\pi )}\) and since (see (2) above) \(W_0\) uniquely determines \(W_1\), \(\langle {W_0,W^\prime _1}\rangle ={sts}(\pi ^\prime )\not \in {{STS}}({SEPROC}({\mathcal{B}}))\). (i.1)
If \(W_1=D_1\cup {\psi (D_1)}=S\), then set \(C^\prime _1={\emptyset }\). Hence \(W^\prime _1=D_1\ne {S}\) and so \(W^\prime _1\ne {W_1}\).
 (i.2)
If \(W_1=D_1\cup {\psi (D_1)}\ne {S}\), then set \(C^\prime _1=S\). Hence \(W^\prime _1={S}\) and so \(W^\prime _1\ne {W_1}\).
 (i.1)
 (ii)
Assume that \(W_0\) is such that \(D_1={res}_{{\mathcal{A}}}(W_0)={S}\)
Thus \(W_1=D_1\cup \psi (D_1)=S\) and \(D_2={res}_{{\mathcal{A}}}(W_1)={\emptyset }\).
Let then \(\pi ^\prime =\langle {f^\prime _0,f^\prime _1,f^\prime _2}\rangle\), with \(f^\prime _i=(C^\prime _i,D^\prime _i,W^\prime _i)\) for all \(i\in \{0,1,2\}\), be an interactive process in \({\mathcal{A}}\) such that \(f^\prime _0=f_0\), \(f^\prime _1=f_1\) and \(C^\prime _2\) is defined as follows.Thus, \({sts}(\pi ^\prime )\ne {{sts}(\pi )}\) and since (see (2) above) \(W_0\) uniquely determines \(W_1\) and \(W_2\), \(\langle {W_0,W_1,W^\prime _2}\rangle ={sts}(\pi ^\prime )\not \in {{STS}}({SEPROC}({\mathcal{B}}))\). (ii.1)
If \({\psi (D_2)}=S\) (so that \(W_2=S\)), then set \(C^\prime _2=\emptyset\). Hence \(W^\prime _2=\emptyset\) and so \(W^\prime _2\ne {W_2}\).
 (ii.2)
If \({\psi (D_2)}\ne {S}\) (so that \(W_2=\emptyset \cup \psi (D_2)\ne {S}\)), then set \(C^\prime _2=S\). Hence \(W^\prime _2={S}\) and so \(W^\prime _2\ne {W_2}\).
 (ii.1)
 (i)
 (4)
Resurrection is possible. This happens if \(\psi\) is such that \(\psi (\emptyset )\ne \emptyset\).
 (5)
Saturation is possible. Again \(\psi (\emptyset )\ne \emptyset\) allows this.
In fact, one can get a “full saturation”, i.e., a sequence of consecutive states with all of them equal to S, as illustrated in Fig. 3. Here, we assume that \(\psi\) is such that \(\psi (\emptyset )=S\). Note that S is always a relevant state (see (1) above), because \(S\cup \psi (S)=S\), for each environment function \(\psi\) for S.
 (6)“Once repeated, always repeated”. This property of contextindependent interactive processes holds also for se interactive processes. Indeed, (see Fig. 4) assume that \(W_i=W_{i+1}\).
 (i)
Then \({D_{i+2}}={res}_{\mathcal{A}}(W_{i+1})=res_{\mathcal{A}}(W_{i})=D_{i+1}\).
 (ii)
Thus, \(C_{i+2}=\psi (D_{i+2})={\psi (D_{i+1})=C_{i+1}}\).
 (iii)
Consequently, \(W_{i+2}=D_{i+2}\cup {C_{i+2}}=D_{i+1}\cup {C_{i+1}}=W_{i+1}\).
 (i)
 (7)
In general, the se function \(\psi\) is arbitrary. One way of “taming” the power of \(\psi\) is to require that \(\psi\) satisfies the boundary conditions (which hold for any \({res}_A\) function), i.e., \(\psi\) is such that \(\psi (\emptyset )=\emptyset\) and \(\psi (S)=\emptyset\). In this case, the no resurrection and no saturation properties hold, even for not necessarily contextindependent interactive processes.
7 Context independence in se reaction systems
In general, a se reaction system \({\mathcal{B}}\) may be such that there are no contextindependent interactive processes in \({\mathcal{B}}\), or it can be such that allse interactive processes in \({\mathcal{B}}\) are contextindependent. We will discuss now the latter case.
Definition 12
(Contextindependent se reaction system) A se reaction system \({\mathcal{B}}\) is contextindependent if each se interactive process in \({\mathcal{B}}\) is contextindependent.
Thus, \({\mathcal{B}}=({\mathcal{A}},\psi )\), with \({\mathcal{A}} =(S,A)\), is contextindependent if \({SEPROC}({\mathcal{B}})\)\(\subseteq {{CIPROC}}({\mathcal{A}})\).
For a se reaction system \({\mathcal{B}}=({\mathcal{A}},\psi )\), its se interactive processes are formed by a “cooperation” between the result function \({res}_{{\mathcal{A}}}\) and the se environment function \(\psi\), where \({res}_{{\mathcal{A}}}\) determines the result component of the successor configuration and then, based on it, \(\psi\) determines the context. Then, various sorts of relationships between \({res}_{{\mathcal{A}}}\) and \(\psi\) may be used to define various classes of se interactive processes. We will demonstrate now how a specific kind of consistency between \({res}_{{\mathcal{A}}}\) and \(\psi\) characterizes contextindependent se reaction systems. First, we need some definitions.
Definition 13
 (1)
A set \(X\subseteq {S}\) is a \(\psi\)state of \({\mathcal{B}}\) if \(X=Z\cup \psi (Z)\) for some \(Z\subseteq {S}\). The set of all \(\psi\)states of \({\mathcal{B}}\) is the state domain of \({\mathcal{B}}\), denoted by \({std}({\mathcal{B}})\).
 (2)
The state range of \({\mathcal{B}}\), denoted by \({str}({\mathcal{B}})\), is defined by \({str}({\mathcal{B}})=\{Y\subseteq {S}\mid {Y}=res_{{\mathcal{A}}}(X), \text{ for } \text{ some } X\in {{std}}({{\mathcal{B}}})\}.\)
Definition 14
(\(\psi\)closure) Let S be a background set and let \(\psi :\wp (S)\rightarrow \wp (S)\). A set \(X\subseteq {S}\) is \(\psi\)closed if \(\psi (X)\subseteq {X}\).
The family of all \(\psi\)closed subsets of S is denoted by \(cls_\psi (S)\).
Definition 15
(ciconsistency) A se reaction system \({\mathcal{B}}=({\mathcal{A}},\psi )\), where \({\mathcal{A}}=(S,A)\), is ciconsistent if \({str}({\mathcal{B}})=cls_\psi (S)\).
Theorem 2
A se reaction system is contextindependent if and only if it is ciconsistent.
Proof
 (I)
Assume that \({\mathcal{B}}\) is contextindependent.
Let \(Y\in {{str}}({\mathcal{B}})\).
Hence, for some \(X\in {{std}}({\mathcal{B}})\), \(Y={res}_{{\mathcal{A}}}(X)\).
Since \(X\in {{std}}({\mathcal{B}})\), \(X=Z\cup \psi (Z)\) for some \(Z\subseteq {S}\). Therefore, \(f=(\psi (Z),Z,X)\) is a \((S,\psi )\)configuration. Consider then the contextindependent se interactive process \(\pi =\langle {f_0},f_1\rangle\) in \({\mathcal{B}}\), with \(f_i=(C_i,D_i,W_i)\) for \(i=\{0,1\}\), such that \(f_0=f\). Thus, \(C_0=\psi (Z)\), \(D_0=Z\), \(W=X\) and \(D_1={res}_{{\mathcal{A}}}(X)=Y\). Since \(\pi\) is contextindependent, \(C_1=\psi (D_1)\subseteq {D_1}\). Therefore \(\psi (Y)\subseteq {Y}\) and so \(Y\in {{cls}}_\psi (S)\).
Since Y was an arbitrary set from \({str}({\mathcal{B}})\), it follows that \({str}({\mathcal{B}})\subseteq {{cls}}_\psi (S)\).
Consequently, \({\mathcal{B}}\) is ciconsistent.
 (II)
Assume that \({\mathcal{B}}\) is ciconsistent.
Let \(\pi =\langle {f_0},f_1,\ldots ,f_n\rangle\), with \(f_i=(C_i,D_i,W_i)\) for \(i=\{0,1,\ldots ,n\}\), be a se interactive process in \({\mathcal{B}}\).
Let \(i\in \{1,\ldots ,n\}\) and consider \(f_i=(C_i,D_i,W_i)\). Hence, \(W_{i1}=D_{i1}\cup \psi (D_{i1})\) and so \(W_{i1}\in {{std}({\mathcal{B}})}\). Since \(D_i=\psi ({res}_{{\mathcal{A}}}(W_{i1}))\), \(D_i\in {{str}}({\mathcal{B}})\). Because \({\mathcal{B}}\) is ciconsistent, this implies that \(C_i=\psi (D_i)\subseteq {D_i}\). Since i was an arbitrary element of \(\{1,\ldots ,n\}\), this implies that for all \(i\in \{1,\ldots ,n\}\)\(C_i\subseteq {D_i}\) and so \(\pi\) is contextindependent.
Since \(\pi\) was an arbitrary se interactive process in \({\mathcal{B}}\), it follows then that \({\mathcal{B}}\) is contextindependent.
Adding environment functions to reaction systems (obtaining in this way se reaction systems) allows one to generate sets of state sequences which cannot be generated by reaction systems in a contextindependent fashion. As a matter of fact, the following relationship holds.
Theorem 3
 (I)
For each reaction system\({\mathcal{A}}\)there exists asereaction system\({\mathcal{B}}\)such that\({STS}({CIPROC}({\mathcal{A}}))= {STS}({SEPROC}({\mathcal{B}}))\).
 (II)
There existsereaction systems\({\mathcal{B}}\)such that for no reaction system\({\mathcal{A}}\), \({STS}({SEPROC}({\mathcal{B}}))= {STS}({CIPROC}({\mathcal{A}}))\)holds.
Proof
 ad (I)

Let \({\mathcal{A}}=(S,A)\) be a reaction system. Let then \({\mathcal{B}}=({\mathcal{A}},\psi )\) be a se reaction system such that \(\psi (X)=\emptyset\) for each \(X\subseteq {S}\). Since each \(X\subseteq {S}\) is \(\psi\)closed and hence a \(\psi\)–state of \({\mathcal{B}}\) (see Definitions 13 and 14), it is clear that \({STS}({CIPROC}({\mathcal{A}}))={STS}({SEPROC}({\mathcal{B}}))\).
 ad (II)

This follows because resurrection and saturation are possible for se interactive processes in se reaction systems (see properties (4) and (5) in Sect. 6) while they are not possible in contextindependent interactive processes in reaction systems.
However, contextindependent interactive processes in reaction systems can be used to represent se interactive processes in se reaction systems as follows. Recall that for a reaction system \({\mathcal{A}}\) and a set of configurations G, \(PROC({\mathcal{A}},G)\) (respectively, \({CIPROC}({\mathcal{A}},G)\)) denotes the set of interactive processes (respectively, contextindependent interactive processes) in \({\mathcal{A}}\) with initial configurations in G.
Theorem 4
Given asereaction system\({\mathcal{B}}=({\mathcal{A}},\psi )\), with\({\mathcal{A}}=(S,A)\), there exist an extension\({\mathcal{A}}^\prime =(S^\prime ,A^\prime )\)of\({\mathcal{A}}\)and a set of configurationsGover\(S^\prime\), such that\({STS}({{SE}}{PROC}({\mathcal{B}}))= {proj}_{{}_S}({STS}({CIPROC}({\mathcal{A}}^\prime ,G)))\).
Proof

\(S^\prime =S\cup \{\)¢,\({\$}\}\),

\(A^\prime =A\cup {A_{\psi }}\), with:
 \(A_{\psi }=A^1_\psi \cup {A^2_\psi }\cup {A^3_\psi }\) where:

\(A^1_\psi =\{(\{\)¢\(\},S^\prime {\setminus }\{\)¢\(\},\psi ({\emptyset })\cup \{\)¢\(\})\}\),

\(A^2_\psi =\{(S,{\{{\$}\}},\psi (S)\cup \{\)¢\(\})\}\), and

\(A^3_\psi =\{(Z,S{\setminus }{Z},\psi ({res}_A(Z))\cup \{\)¢\(\})\mid {Z}\subsetneq {S}\ \text{ and }\ Z\ne {\emptyset }\}\).

 (I)
The se interactive processes in \({\mathcal{B}}\) are modeled by contextindependent interactive processes in \({\mathcal{A}}^\prime\) as follows.
Let \(\pi =\langle {f_0},f_1,\dots ,f_n\rangle\) be a se interactive process in \({\mathcal{B}}\) such that \(f_i=(C_i,D_i,W_i)\) for each \(i\in \{0,1,\ldots ,n\}\). Then, \(\pi\) is modeled in \({\mathcal{A}}\) by a contextindependent interactive process \(\pi ^\prime =\langle {f^\prime _0},f^\prime _1,\ldots ,f^\prime _n\rangle\), with \(f^\prime _i=(C^\prime _i,D^\prime _i,W^\prime _i)\) for each \(i\in \{0,1,\ldots ,n\}\), in the way illustrated in Fig. 5, with \(\pi\) depicted in the top part and \(\pi ^\prime\) depicted in the bottom part of this figure.
The intuition behind the role of reactions in \(A^\prime\) is as follows.
Reactions in \(A_\psi\) produce in \(D^\prime _i\) the context set \(C_i\) so that \(D^\prime _{i}=C_i\cup \{\)¢\(\}\cup {D_i}\) (where \(D_i\) is produced by the reactions in A). This (the adding of \(C_i\)) is done by the reactions in \(A^3_\psi\) in the case that \(W_i\ne {S}\) and \(W_i\ne {\emptyset }\).
If \(W_i={\emptyset }\), then we still produce \(\psi ({res}_A({\emptyset }))=\psi ({\emptyset })\) by the reaction in \(A^1_\psi\), which also produces ¢, a dummy entity preventing that either the set of reactants or the set of products in a reaction are empty (note that it may be that \(\psi ({\emptyset })={\emptyset }\)).
If \(W_i=S\), then we still produce \(\psi ({res}_A(S))=\psi ({\emptyset })\) by the reaction in \(A^2_\psi\), where \({\$}\) is a dummy inhibitor entity, which is never produced (it just insures that the set of inhibitors is not empty).
It should be clear from the above description (and Fig. 5) that \({sts}(\pi )={proj}_{{}_S}({sts}(\pi ^\prime ))\).
Since \(\pi\) was an arbitrary se interactive process in \({\mathcal{B}}\) and \({init}(\pi ^\prime )\in {G}\), we conclude that \({STS}({SEPROC}({\mathcal{B}}))\subseteq {{proj}_{{}_S}}({STS}({CIPROC}({\mathcal{A}}^\prime ),G))\).
 (II)
Now, let \(\pi ^\prime =\langle {f^\prime _0},f^\prime _1,\ldots ,f^\prime _n\rangle\), with \(f^\prime _i=(C^\prime _i,D^\prime _i,W^\prime _i)\) for each \(i\in \{0,1,\ldots ,n\}\), be a contextindependent interactive process in \({\mathcal{A}}^\prime\) such that \(f^\prime _0\in {G}\) (hence \(W^\prime _0{\setminus }{S}=\{\)¢\(\}\)), \(C^\prime _0\subseteq {S}\) (hence ¢\(\in {D^\prime _0}\)), and \(\psi ({D}^\prime _0 {\setminus } \{\)¢\(\}) = {C}^\prime _0\).
It follows then from the discussion in (I) above that \(\pi ^\prime\) models the se interactive process \(\pi =\langle {f_0},f_1,\ldots ,f_n\rangle\) in \({\mathcal{B}}\), with \(f_i=(C_i,D_i,W_i)\) for each \(i\in \{0,1,\ldots ,n\}\), such that \(C_0=C^\prime _0\). Thus, \({proj}_{{}_S}({sts}(\pi ^\prime ))={sts}(\pi )\).
Since \(\pi ^\prime\) was an arbitrary contextindependent interactive process in \({\mathcal{A}}^\prime\) with \({init}(\pi ^\prime )\in {G}\), we conclude that \({{proj}_{{}_S}}({STS}({CIPROC}({\mathcal{A}}^\prime ,G))\subseteq {{STS}}({SEPROC}({\mathcal{B}}))\).
In the proof above, we have used the dummy entities ¢and \({\$}\) to ensure that the reactions we have constructed were “valid”, i.e., that for all of them their sets of reactants, inhibitors, and products are nonempty. We reserve the symbols ¢and \({\$}\) for denoting such dummy entities in the sequel of this paper.
8 Reaction systems with delayed system–environment influence
We will now consider the influence of reaction systems on their environments which takes place with some delay. Such delayed interactions between a system and its environment are abundant in information processing systems, including biological systems.
Definition 16
(System–environment reaction system with delay) A system–environment reaction system with delay, abbreviated se reaction system with delay, is a triplet \({\mathcal{B}}=({\mathcal{A}},\psi ,d)\), where \(({\mathcal{A}},\psi )\) is a se reaction system and \(d\in \mathbb {N}\).
We refer to \(({\mathcal{A}},\psi )\) as the underlying se reaction system of\({\mathcal{B}}\), to \({\mathcal{A}}\) as the underlying reaction system of\({\mathcal{B}}\), and to d as the delay of \({\mathcal{B}}\). Whenever specific \(\psi\) and d are relevant in the context of considerations, we refer to \({\mathcal{B}}\) as a \((\psi ,d)\)reaction system.
Definition 17
(System–environment interactive process with delay) Let \({\mathcal{B}}=({\mathcal{A}},\psi ,d)\) be a se reaction system with delay. An (nstep) system–environment interactive process with delay in\({\mathcal{B}}\), abbreviated seinteractive process with delay in\({\mathcal{B}}\), is a (nstep) interactive process \(\pi =\langle {f_0},f_1,\ldots ,f_n\rangle\) in \({\mathcal{A}}\), with \(f_i=(C_i,D_i,W_i)\) for each \(i\in \{0,1,\ldots ,n\}\), such that, for each \(j\in \{d,\ldots ,n\}\), \({C_j}=\psi (D_{jd})\).
We will use \({SE}{{D}}{PROC}({\mathcal{B}})\) to denote the set of all se interactive processes with delay in \({\mathcal{B}}\).
9 System–environment reaction systems with delay 1
In this section, we characterize behaviors of se reaction systems with delay 1 in terms of the contextindependent behaviors of reaction systems.
Theorem 5
Let\({\mathcal{B}}=({\mathcal{A}},\psi ,1)\)be asereaction system with delay\(d=1\), where\({\mathcal{A}}=(S,A)\). There exists an extension\({\mathcal{A}}^\prime =(S^\prime ,A^\prime )\)of\({\mathcal{A}}\)and a set of configurationsGover\(S^\prime\), such that\({STS}({{SE}}{{D}}{PROC}({\mathcal{B}}))={proj}_{{}_S}({STS}({{CI}}{PROC}({\mathcal{A}}^\prime\), G))).
Proof
 (i)
\(S^\prime =S\cup \{\)¢\(, {\$}\}\cup \{[\pounds _X,1]\mid {X}\subseteq {S}\}\cup \{\dagger _Y\mid {Y}\subseteq {S}\}\), and
 (ii)\(A^\prime =A\cup {A_{\psi ,1}}\cup {A_{\dagger ,1}}\), where

\(A_{\psi ,1}=\{(\{[\pounds _X,1]\},\{{\$}\},\psi (X)\cup \{\)¢\(\})\mid {X}\subseteq {S}\}\), and

\(A_{\dagger ,1}= \{(\{\dagger _Y,[\pounds _X,1]\}, \{{{\$}}\}, \{[\pounds _{{res}_{{}_A}(Y)},1],\dagger _{{res}_{{}_A}(Y)\cup \psi (X)}\})\mid {X,Y}\subseteq {S}\}\).

 (I)
The se interactive processes with delay in \({\mathcal{B}}\) are modelled by contextindependent interactive processes in \({\mathcal{A}}^\prime\) as follows.
Let \(\pi =\langle {f_0},f_1,\ldots ,f_n\rangle\) be a se interactive process in \({\mathcal{B}}\) such that \(f_i=(C_i,D_i\), \(W_i)\) for each \(i=\{0,1,\ldots ,n\}\). Then, \(\pi\) is modeled in \({\mathcal{A}}^\prime\) by a contextindependent interactive process \(\pi ^\prime =\langle {f^\prime _0},f^\prime _1,\ldots ,f^\prime _n\rangle\), with \(f^\prime _i=(C^\prime _i,D^\prime _i,W^\prime _i)\) for each \(i\in \{0,1,\ldots ,n\}\), in the way illustrated in Fig. 8, with \(\pi\) depicted in the top part and \(\pi ^\prime\) depicted in the bottom part of this figure.
The intuition behind the entities in \({S}^\prime {\setminus }{S}\) is as follows.From the details given in Fig. 8 (and the intuition behind the entities in \(S^\prime {\setminus }{S}\) given above) it should be clear that \({sts}(\pi )= {proj}_S({sts}(\pi ^\prime )).\)
\(\pounds\) remembers (through its subscript) the result sets—thus, \(\pounds _X\) remembers the result set X (result sets are sets produced by the \(res_A\) function).

\([\pounds _X,1]\) remembers the result set X and the fact that it will be “released” in the next step. In particular, the reactions of \(A_{\psi ,1}\) are such that \([\pounds _X,1]\) present in the current state X will contribute the set \(\psi (X)\) to the successor state.

\(\dagger\) remembers (through its subscript) the whole state—thus, \(\dagger _Y\) remembers the state Y. The reactions in \(A_{\dagger ,1}\) produce representations of the whole successor state of the current state Y (remembered in \(\dagger _{{res}_{{}_A}(Y)\cup \psi (X)})\), as well as the result of applying reactions of A to Y, remembered in \([\pounds _X,{1}]\).

¢and \({\$}\) are “dummy entities” preventing that either the product or the inhibitor set of production become empty sets.
Since \(\pi\) was an arbitrary se interactive process in \({\mathcal{B}}\) and \({init}(\pi ^\prime )\in {G}\), we obtain \({STS}({SE}{{D}}{PROC}({\mathcal{B}}))\subseteq {{proj}_S({STS}({CIPROC}({\mathcal{A}}^\prime ,G)))}\).

 (II)
Now, let \(\pi ^\prime =\langle {f_0^\prime },f^\prime _1,\ldots ,f^\prime _n\rangle\), with \(f^\prime _i=(C^\prime _i,D^\prime _i,W^\prime _i)\) for \(i\in \{0,1,\ldots ,{n}\}\), be a contextindependent interactive process in \({\mathcal{A}}^\prime\) such that \(f^\prime _0\in {G}\), hence \(C^\prime _0\subseteq {S}\), \(D^\prime _0=D_0\cup \{[\pounds _{D_{0}},1],\dagger _{W_0}\}\), for some \(D_0\subseteq {S}\) and \(W_0=C^\prime _0\cup \{D_0\}\), and \(\psi (D_0) = C^\prime _0\).
It follows then from the discussion in (I) above that \(\pi ^\prime\) models the se interactive process \(\pi =\langle {f_0},f_1,\ldots ,f_n\rangle\) in \({\mathcal{B}}\), with \(f_i=(C_i,D_i,W_i)\) for \(i\in \{0,1,\ldots ,{n}\}\), such that \(C_0=C^\prime _0\). Thus, \({proj}_S({sts}(\pi ^\prime ))={sts}(\pi )\).
Since \(\pi ^\prime\) was an arbitrary contextindependent interactive process in \({\mathcal{A}}^\prime\) with init\((\pi ^\prime )\in\)G, we obtain \({proj}_S({STS}({CIPROC}({\mathcal{A}}^\prime ,G)))\subseteq {{STS}}({SE{D}}\)\({PROC}({\mathcal{B}})).\)
10 System–environment reaction systems with delay bigger than 1
In this section, we characterize behaviors of se reaction systems with delay \(d\ge {2}\) in terms of contextindependent behaviors of reaction systems.
Theorem 6
Let\({\mathcal{B}}=({\mathcal{A}},\psi ,d)\)be asereaction system with delay such that\({\mathcal{A}}=(S,A)\)and\(d\ge {2}\). There exists an extension\({\mathcal{A}}^\prime =(S^\prime ,A^\prime )\)of\({\mathcal{A}}\)and a set of configurationsGover\(S^\prime\), such that\({STS}({{SE}}{{D}}{PROC}({\mathcal{B}}))={proj}_{{}_S}({STS}(\)\({{CI}}{PROC}({\mathcal{A}}^\prime ,G)))\).
Proof
 (1)
\(S^\prime = S\cup \{\)¢\(,{\$}\}\ \cup\)
\(\{[\pounds _X,k]\mid {X}\subseteq {S},k\in \{1,\ldots ,d\}\}\ \cup\)
\(\{\dagger _Y\mid {Y}\subseteq {S}\}\ \cup\)
\(\{[\Delta ,k,\mu ]\mid {k}\in \{2,\ldots ,d\},\mu \in (\wp (S))^{d+1}\}\), and
 (2)
\(A^\prime =A\cup {A_{\psi ,1}}\cup {A_{\dagger ,d}}\cup {A_{\pounds ,d}}\ \cup \bigcup _{\mu \in (\wp (S))^{d+1}}A_{\Delta ,d,\mu }\cup \bigcup _{\mu \in (\wp (S))^{d+1}}B_{\Delta ,2,\mu }\),
where:
\({A_{\psi ,1}}=\{(\{[\pounds _X,1]\},\{{\$}\},\psi (X)\cup \{\)¢\(\})\mid {X}\subseteq {S}\}\),

\({A_{\dagger ,d}}=\{(\{\dagger _Y,[\pounds _X,1]\},\{{\$}\},\{[\pounds _{{res}_A(Y)},d],\dagger _{{res}_A(Y)\cup \psi (X)}\})\mid {X,Y}\subseteq {S}\}\),

\({A_{\pounds ,d}}=\{(\{[\pounds _X,k]\},\{{\$}\},\{[\pounds _X,k1]\})\mid {X}\subseteq {S},k\in \{2,\ldots ,d\}\}\),

for each \(\mu =(C_0,D_0,C_1,\ldots ,C_{d1})\in (\wp (S))^{d+1}\), with \(W_{d1}=C_{d1}\cup {D_{d1}}\),
\(B_{\Delta ,2,\mu }=\{(\{[\Delta ,2,\mu ]\},\{{\$}\},C_{d1}\cup \{[\pounds _{D_0},1],[\pounds _{D_1},2],\ldots ,[\pounds _{D_{d1}},d],\dagger _{W_{d1}}\)\(\})\}\), and,

if \(d\ge {3}\), then, for each \(\mu =(C_0,D_0,C_1,\ldots ,C_{d1})\in (\wp (S))^{d+1}\),
\(A_{\Delta ,d,\mu }=\{(\{[\Delta ,k,\mu ]\},\{{\$}\},\{[\Delta ,k1,\mu ]\}\cup {C_{(dk)+1}})\mid {k\in \{3,\ldots ,d\}})\}\).

 (I)
The se interactive processes in \({\mathcal{B}}\) are modelled in \({\mathcal{A}}^\prime\) by contextindependent interactive processes as follows.
Let \(\pi =\langle {f_0},f_1,\dots ,f_n\rangle\) be a se interactive process in \({\mathcal{B}}\) such that \(f_i=(C_i,D_i,W_i)\) for each \(i\in \{0,1,\ldots ,n\}\). Then \(\pi\) is modelled in \({\mathcal{A}}^\prime\) by a contextindependent interactive process \(\pi ^\prime =\langle {f^\prime _0},f^\prime _1,\ldots ,f^\prime _n\rangle\), with \(f^\prime _i=(C^\prime _i,D^\prime _i,W^\prime _i)\), for each \(i\in \{0,1,\ldots ,n\}\), in two stages:The details of the modelling of \(\pi\) by \(\pi ^\prime\) are depicted in Figs. 9 and 10, corresponding to stages (1) and (2) respectively. (1)
the prefix of \(\pi\), up to configuration \(f_{d1}\), is modelled by \(\pi ^\prime\) so that all the “initial results”, \(D_0,D_1,\ldots ,D_{d1}\), are deposited in the configuration \(f^\prime _{d1}\) of \(\pi ^\prime\), and
 (2)
(if \(\pi\) is long enough, then) the determination by \(\psi\) of \(C_{d},\ldots ,C_{2d1}\) by \(D_0,\ldots ,D_{d1}\), of \(C_{2d},\ldots ,C_{3d1}\) by \(D_d,\ldots ,D_{2d1}\), of \(C_{3d},\ldots ,C_{4d1}\) by \(D_{2d},\ldots ,D_{3d1}\), \(\dots\) is modelled in \(\pi ^\prime\) in a routine “cyclic” way.
The intuition behind the entities in \(S^\prime \setminus {S}\) used in this modeling is as follows.In the first stage, illustrated by Fig. 9, \(\pi ^\prime\) collects, and carries on, all the information needed to proceed with simulating \(\pi\) from the configuration \(f_d\) on (when the result sets \(D_0,\dots ,D_{d1}\) begin to determine context sets \(C_d,\dots ,C_{2d1}\)). This information is deposited in the configuration \(f^\prime _{d1}\) of \(\pi ^\prime\) through the set \(\{[\pounds _{D_{0}},1],\ldots ,[\pounds _{D_{d1}},d],\dagger _{W_{d1}}\}\).
\(\pounds\) remembers (through its subscript) the result sets—thus, \(\pounds _X\) remembers the result set X (result sets are sets produced by the \(res_A\) function).

\([\pounds _X,k]\) remembers the result set X and the fact that it will be “released” in k steps from the current state. In particular, the reactions of \(A_{\psi ,1}\) are such that \([\pounds _X,1]\) present in the current state will contribute the set \(\psi (X)\) to the successor state, while the reactions of \({A_{\pounds ,d}}\) record the “ticking of the clock” towards the release of the set X.

\(\dagger\) remembers (through its subscript) the whole state, so that \(\dagger _{Y}\) remembers the state Y.

\(\Delta\) remembers \(C_0\) and \(D_0\) from the initial configuration \(f_0=(C_0,D_0\), \(W_0)\), the first d elements \(C_0,C_1,\ldots ,C_{d1}\) of the context sequence of \(\pi\) (these two items together form the frame \(\mu =(C_0,D_0,C_1,\ldots ,C_{d1})\)), and the instant k that this information will be released into the successor state (which happens when \(k=2\), by the production in \(B_{\Delta ,2,\mu }\)).

\([\Delta ,k,\mu ]\) remembers that, in the state produced in \(k2\) steps, \([\Delta ,2,\mu ]\) will be produced and this entity will release into the successor state the information contained in \(\mu\).

Recall that ¢and \({\$}\) are “dummy entities” preventing that either the product or the inhibitor set of a reaction become empty.
In the second stage, illustrated by Fig. 10, \(\pi ^\prime\) deposits the information collected in the previous d steps in consecutive configurations and, while doing this, it updates the information so that at each step it has the relevant information from the last d configurations. (Since Fig. 10 is very “dense”, we simplified the labels on the arrows so that we write A rather than \({res}_A\), \(A_{\psi ,1}\) rather than \({res}_{A_{\psi ,1}}\), etc.).
This information is deposited in the configuration \(f_{2d1}\) of \(\pi ^\prime\), through the set \(\{[\pounds _{D_{d}},1],\ldots ,[\pounds _{D_{2d1}},d],\dagger _{W_{2d1}}\}\), so that \(\pi ^\prime\) is “ready” now to simulate the next subsequence of \(sts(\pi )\), viz., \(\langle {W_{d}},\ldots ,W_{2d1}\rangle\). This simulation of fragments of \({sts}(\pi )\) of length d is iterated until the end of \({sts}(\pi )\).
From the discussion above, it follows that \({sts}(\pi )={proj}_S({sts}(\pi ^\prime ))\).
Since \(\pi\) was an arbitrary se interactive process in \({\mathcal{B}}\) and \(init(\pi ^\prime )\)\(\in {G}\), we obtain \({STS}({SE}{{D}}{PROC}({\mathcal{B}}))\subseteq {{proj}_{S}}({STS}({{CI}}{PROC}({\mathcal{A}}^\prime ,G)))\).
 (1)
 (II)
Let \(\pi ^\prime =\langle {f^\prime _0},f^\prime _1,\dots ,f^\prime _n\rangle\), with \(f^\prime _i=(C^\prime _i,D^\prime _i,W^\prime _i)\) for \(i\in \{0,1,\dots ,n\}\), be a contextindependent interactive process in \({\mathcal{A}}^\prime\), such that \(f^\prime _0\in {G}\), hence \(C^\prime _0\subseteq {S}\), \(D^\prime _0=D\cup \{[\Delta ,d,\mu ]\}\) for some \(D\subseteq {S}\) and \(\mu \in (\wp (S))^{d+1}\) such that the first component of \(\mu\) is \(C^\prime\) and the second component of \(\mu\) equals D, and \(\psi (D) = C^\prime\).
It follows then from the discussion in (I) above that \(\pi ^\prime\) models the se interactive process \(\pi =\langle {f_0},f_1,\ldots ,f_n\rangle\) in \({\mathcal{B}}\) (with \(f_i=(C_i,D_i,W_i)\) for \(i\in \{0,1,\ldots ,n\}\)), where \(\mu\) is the frame of \(\pi\), i.e., \(\mu =(C^\prime _0=C_0,D_0,C_1,\ldots ,C_n)\) and so \({proj}_S({sts}(\pi ^\prime ))={sts}(\pi )\).
Since \(\pi ^\prime\) was an arbitrary contextindependent process in \({\mathcal{A}}^\prime\) with \(init(\pi ^\prime )\)\(\in {G}\), we obtain \({{proj}_{S}}({STS}({{CI}}{PROC}({\mathcal{A}}^\prime ,G)))\subseteq {{STS}({SE}{{D}}{PROC}({\mathcal{B}}))}\).
The theorem follows now from (I) and (II). \(\square\)
11 Comparing delays
Each se reaction system with delay \({\mathcal{B}}\) is a generator of a set of state sequences, viz., \({STS}({SEDPROC}({\mathcal{B}}))\). In this section, we investigate the influence of delays on this generative power. More specifically, we compare the generative power of se reaction systems with different delays.
To begin with, we want to explicitly point out the following. Whenever the equality \({STS}({SEDPROC}({\mathcal{B}}))= {STS}({SEDPROC}({\mathcal{B}}^\prime ))\) holds for two se reaction systems with delay \({\mathcal{B}}\) and \({\mathcal{B}}^\prime\), then this implies that their background sets, S and \(S^\prime\), respectively, are equal. This holds because if \(S\ne {S^\prime }\), then there exists an entity, x, which is only in one of them, say \(x\in {S}{\setminus }{S^\prime }\). Consider then an Sconfiguration \(f=(C,D,W)\), such that \(x\in {D}\) (and hence \(x\in {W}\)), and a \(\pi \in {{SEDPROC}({\mathcal{B}})}\) such that f is its initial configuration and hence \(x\in {{init}({sts}(\pi ))}\). Since \(x\not \in {S^\prime }\), no process \(\pi ^\prime\) in \({\mathcal{B}}^\prime\) is such that \(x\in {{init}({sts}(\pi ^\prime ))}\). Consequently, \({sts}(\pi )\not \in {{STS}({SEDPROC}({\mathcal{B}}^\prime ))}\).
The following three auxiliary results will be useful in our considerations.
Lemma 1
Let \({\mathcal{B}}\) be a se reaction system and let \(\sigma \in {{STS}}({SEPROC}({\mathcal{B}}))\). If \(\sigma\) is such that \({init}(\sigma )={\emptyset }\), then each state in \(\sigma\) is the empty set.
Proof
Let \({\mathcal{B}}=({\mathcal{A}},\psi )\) be a se reaction system and let \(\sigma \in {{STS}}({SEDPROC}({\mathcal{B}}))\) be such that \({init}(\sigma )=\emptyset\). Let \(\pi =\langle {f_0},f_1,\ldots ,f_n\rangle\) be a se interactive process in \({\mathcal{B}}\), with \(f_i=(C_i,D_i,W_i)\) for \(i\in \{0,1,\ldots ,n\}\), such that \(\sigma ={sts}(\pi )\), hence \(\sigma =\langle {W_0},W_1,\ldots ,W_n\rangle\).
Since \(W_0={\emptyset }\), also \(C_0=D_0={\emptyset }\). But \(C_0=\psi (D_0)\) and so \(\psi (\emptyset )=\emptyset\). Since \(D_1={res}_{{\mathcal{A}}}(\emptyset )=\emptyset\), this implies that \(C_1=\psi (D_1)=\psi (\emptyset )=\emptyset\), and consequently \(W_1=\emptyset \cup \emptyset =\emptyset\). By iterating this reasoning, we obtain \(W_i=\emptyset\) for all \(i\in \{0,1,\ldots ,n\}\).
Lemma 2
Let\({\mathcal{B}}=({\mathcal{A}},\psi ,d)\)be asereaction system with delay\(d\ge {1}\)and let\(\sigma \in {{STS}}({SEDPROC}({\mathcal{B}}))\)be such that\(\sigma =\langle {W_0},W_1,\ldots\), \(W_d\rangle\). If\(W_i=\emptyset\)for all\(i\in \{0,1,\ldots ,d1\}\), then\(W_d=\psi (\emptyset )\).
Proof
Let \({\mathcal{B}}\) and \(\sigma\) be as in the assumption of the statement of the lemma. Assume that \(W_i=\emptyset\) for all \(i\in \{0,1,\ldots ,d1\}.\)
Since \(W_{d1}=\emptyset\) and \(res_{{\mathcal{A}}}(\emptyset )=\emptyset\), we get \(D_d=\emptyset\). Since \(W_0= \emptyset\), we get \(D_0=\emptyset\). But \(C_d=\psi (D_0)\) and so \(C_d=\psi (\emptyset )\). Thus, \(W_d=C_d\cup {D_d}=\psi (\emptyset )\).
Lemma 3
Let\({\mathcal{B}}=({\mathcal{A}},\psi ,d)\)be asereaction system with delay\(d\ge {1}\)and letSbe its background set. Then for every\(Z\subseteq {S}\), there exists a\(\sigma \in {{STS}}({SED}\)\({PROC}({\mathcal{B}}))\)such that\({init}(\sigma )=Z\).
Proof
The lemma holds, because for each Sconfiguration \(f=(C,D,W)\), the sequence \(\pi =\langle {f},f^\prime \rangle\), where \(f^\prime =(\psi ({res}_{{\mathcal{A}}}(W)),{res}_{{\mathcal{A}}}(W)\), \({res}_{{\mathcal{A}}}(W)\cup \psi ({res}_{{\mathcal{A}}}(\)W))), is such that \(\pi \in {{SEDPROC}}({\mathcal{B}})\).
So, for each \(Z\subseteq {S}\), if we set \(f=(\emptyset ,Z,Z)\), then \(\pi\) defined as above is such that \({{init}}({{sts}}(\pi ))=Z\).
We will demonstrate now that the generative powers (of sets of state sequences) of the classes of se reaction systems with different delays are incomparable.
Theorem 7
Let\(d,l\in \mathbb {N}\), with\(l>d\). There exists asereaction system\({\mathcal{B}}\)with delaylsuch that, for nosereaction system\({\mathcal{B}}^\prime\)with delayd, the equality\({STS}({SEDPROC}({\mathcal{B}}^\prime ))={STS}({SEDPROC}({\mathcal{B}}))\)holds.
Proof
 (1)
Let \(d=0\). Let \({\mathcal{B}}=({\mathcal{A}},\psi ,l)\) be a se reaction system with delay \(l\ge {1}\), such that \(\psi (\emptyset )\ne \emptyset\). Let then \(\pi =\langle {f_0},f_1,\ldots ,f_l\rangle\), with \(f_i=(C_i,D_i,W_i)\) for all \(i\in \{0,1,\ldots ,l\}\), be a se interactive process with delay in \({\mathcal{B}}\) such that \(W_i=\emptyset\) for all \(i\in \{0,1,\ldots ,l1\}\).
Since \({res}_{{\mathcal{A}}}(\emptyset )=\emptyset\), such a \(\pi\) is obtained by setting \(W_0=\emptyset\) and \(C_1=\ldots =C_{l1}=\emptyset\). Since \(W_0=\emptyset\), we get \(D_0=\emptyset\) and consequently \(C_l=\psi (D_0)=\psi (\emptyset )\ne \emptyset\). This implies that \(W_l\ne \emptyset\).
Thus, by Lemma 1, for no se reaction system \({\mathcal{B}}^\prime\) with delay \(d=0\), it occurs that the equality \({STS}({SEDPROC}({\mathcal{B}}^\prime ))={STS}({SEDPROC}({\mathcal{B}}))\) holds.
 (2)
Let \(d\ge {1}\). Let \({\mathcal{B}}=({\mathcal{A}},\psi ,l)\) be a se reaction system with delay \(l>{d}\) and the background set S. Let then \(\pi =\langle {f_0},f_1,\ldots ,f_d\rangle\) with \(f_i=(C_i,D_i,W_i)\), for all \(i\in \{0,1,\ldots ,d\}\), be a se interactive process with delay in \({\mathcal{B}}\), such that \(W_i=\emptyset\) for all \(i\in \{0,1,\ldots ,d1\}\). Since \(res_{{\mathcal{A}}}(\emptyset )=\emptyset\), such \(\pi\) is obtained by setting \(W_0=\emptyset\) and \(C_1=\dots =C_{d1}=\emptyset\). Moreover, since \(res_{{\mathcal{A}}}(\emptyset )=\emptyset\) and \(l>d\), for each \(Z\subseteq {S}\), the sequence of configurations \(\pi _Z\), which is obtained from \(\pi\) by setting \(C_d=Z\), so that \(f_d=(Z,\emptyset ,Z)\), is such that \(\pi _Z\in {{SEDPROC}({\mathcal{B}})}\).
Hence, for every \(Z\subseteq {S}\), the sequence of states \(\sigma _Z=\langle {T_0},T_1,\ldots ,T_d\rangle\) such that \(T_i=\emptyset\), for all \(i\in \{0,1,\ldots ,d1\}\), and \(T_d=Z\) is such that \(\sigma _Z\in {{STS}}({SEDPROC}({\mathcal{B}}))\).
Assume that there exists a se reaction system with delay \({\mathcal{B}}^\prime =({\mathcal{A}}^\prime ,\psi ^\prime ,d)\) such that \({STS}({SEDPROC}({\mathcal{B}}))={{STS}}({SEDPROC}({\mathcal{B}}^\prime ))\). This implies that for each \(Z\ne \psi ^\prime (\emptyset )\) the state sequence \(\sigma _Z\) defined as above is such that \(\sigma _Z\in {{STS}}({SEDPROC}({\mathcal{B}}^\prime ))\), which contradicts Lemma 2.
Consequently, for no se reaction system \({\mathcal{B}}^\prime\) with delay d it occurs that the equality \({STS}({SEDPROC}({\mathcal{B}}))\)\(={{STS}}({SEDPROC}({\mathcal{B}}^\prime ))\) holds.
Theorem 8
Let\(d,l\in \mathbb {N}\), with\(l>d\). There exists asereaction system\({\mathcal{B}}\)with delaydsuch that, for nosereaction system\({\mathcal{B}}^\prime\)with delayl, it occurs that\({STS}({SEDPROC}({\mathcal{B}}^\prime ))={STS}({SEDPROC}({\mathcal{B}}))\).
Proof
 (1)
If \(d=0\), then, for each \(\pi \in {{SEDPROC}}({\mathcal{B}})\), the initial configuration \((C_0,D_0,W_0)\) of \(\pi\) is such that \(W_0=S\), because \(\psi (D_0)=S\). Hence, each state sequence \(\sigma \in {STS}({SEDPROC}({\mathcal{B}}))\) is such that \(init(\sigma )=S\). Therefore, by Lemma 3, for no se reaction system \({\mathcal{B}}^\prime\) with delay \(l\ge {1}\) the equality \({STS}({SEDPROC}({\mathcal{B}}))\)\(={{STS}}({SEDPROC}({\mathcal{B}}^\prime ))\) holds.
 (2)
If \(d\ge {1}\), then consider the dstep se interactive process \(\pi\) with delay d in \({\mathcal{B}}\), \(\pi =\langle {f_0},f_1,\ldots ,f_d\rangle\) such that \(f_i=(\emptyset ,\emptyset ,\emptyset )\), for all \(i\in \{0,1,\ldots ,d1\}\). Since \({res}_{{\mathcal{A}}}(\emptyset )=\emptyset\) and \(\psi (\emptyset )=S\), \(C_d={\psi (D_0)}=\psi (\emptyset )=S\). Consequently, \(W_d=\emptyset \cup {S}=S\).
However, if \({\mathcal{B}}^\prime =({\mathcal{A}}^\prime ,\psi ^\prime ,l)\) is a se reaction system with delay \(l>d\) and \(\mu =\langle {f_0},f_1,\ldots ,f_{d1}\rangle \in {{SEDPROC}}({\mathcal{B}}^\prime )\) is such that \(f_i=(\emptyset ,\emptyset ,\emptyset )\) for all \(i\in \{0,1,\ldots ,d1\}\), then also for each \(Z\subseteq {S}\), \(\pi _Z=\langle {f_0},f_1,\ldots ,f_{d1},f_Z\rangle\), where \(f_Z=(Z,\emptyset ,Z)\) is such that \(\pi _Z\in {{SEDPROC}}({\mathcal{B}}^\prime )\). Thus, if we set \(Z\ne {S}\), then we obtain the state sequence \(\sigma _Z=\langle {W_0},W_1,\ldots ,W_d\rangle\) such that \(W_i=\emptyset\), for all \(i\in \{0,1,\ldots ,d1\}\) and \(W_d\ne {S}\). This contradicts the property of dstep se interactive processes with delay discussed in the previous paragraph.
Consequently, for no se reaction system \({\mathcal{B}}^\prime\) with delay \(l>d\) the equality \({STS}({SEDPROC}({\mathcal{B}}^\prime ))\)\(={{STS}}({SEDPROC}({\mathcal{B}}))\) holds.
12 Discussion
We have presented a novel perspective on the role of the environment in reaction systems, considering also the influence of the system on the environment.

In a se reaction system the environment function \(\psi :\wp (S)\rightarrow \wp (S)\) influences the enviroment by transforming the result \(D_i\) (of applying reactions to the previous state \(W_{i1}\)) into the context \(C_i=\psi (D_i)\). The resulting se interactive processes are sequences of configurations of the form (C, D, W) with \(C=\psi (D)\). One could also consider reaction systems influencing the environment through environment functions, but with the influence expressed in the form \(C_i=\psi (W_i)\) (rather than \(C_i=\psi (D_i)\)), i.e., the whole state\(W_i\) influences the context set \(C_i\) (through \(\psi )\). Here, the resulting interactive processes are sequences of configurations of the form (C, D, W) with \(C=\psi (W)\). It is certainly worthwhile to investigate such interactive processes. We note that such systems (processes) are more restrictive in the sense that one has less choice for environment functions—each such function must satisfy the subset restriction, i.e., \(\psi (X)\subseteq {X}\) for each \(X\in \wp (S)\). This follows from the fact that now all relevant configurations are of the form \((\psi (W),D,W)\), where \(\psi (W)\cup {D}=W\). Consequently, these environment functions automatically satisfy one of the boundary conditions, viz., \(\psi (\emptyset )=\emptyset\).

We have studied the case where the environment function \(\psi\)determines the context, i.e., \(C_j=\psi (D_i)\) for some \(i\le {j}\). A more general form of influence is when the system partially controls the context, i.e., when \(\psi (D_i)\subseteq {C_j}\), again for some \(i\le {j}\). In that case, we would define \(C_j{\setminus }\psi (D_i)\) as the genuine context \(C^\prime _j\).

We have investigated se reaction systems, where the system influences the context not directly, but through the environment function \(\psi\). It is natural then to consider the symmetric case where the context influences the system not directly through a proper context sequence \(\langle {C_1},\ldots ,C_n\rangle\), but rather indirectly through a system function \(\phi :\wp (S)\rightarrow \wp (S)\), so that the de facto proper context sequence influencing the states of the system becomes \(\langle \phi (C_1),\ldots ,\phi (C_n)\rangle\).
Thus, if \(\phi\) is the identity function, then we get the traditional (not transformed) influence by the environment through the proper context sequence \(\langle {C_1},\ldots ,C_n\rangle\).
Again, one can consider the instantaneous influence (where \(\phi (C_i)\) is a subset of \(W_i\)) or a delayed influence (where \(\phi (C_i)\) becomes a subset of \(W_i+d\) for some \(d\in \mathbb {N}^+\)).

One can consider delays associated with the complexity of the subsets influencing the environment. For example, one could assign weights to entities so that the environment will react in a faster way to “simple” subsets and more slowly to “complex” ones. This is somewhat analogous to reaction systems with duration (see [5]), where the duration function assigns life times to entities, so that entities with smaller duration decay (vanish) earlier than entities with bigger durations.
 We notice also the following duality between ordinary reaction systems and se reaction systems.
 1.
In ordinary reaction systems, the state of the system (\(W_i\)) is formed by adding to what the system did (viz., \(D_i\)) the influence of the environment (viz., \(C_i\)). If \(C_i \subseteq {D_i}\), then we get a contextindependent interactive process. (In systems with delay, one adds the delayed influence of the context, viz., \(C_{id}\), to what the system does, viz., \(D_i\)).
 2.
In se reaction systems, the state of the environment (\(C_i\)) is formed by adding to the genuine context produced by the environment (viz., \(C^\prime _i\)) the influence of the system (viz., \(\psi (D_i)\)). Now, if \(\psi (D_i)\subseteq {C'_i}\), then we get systemindependent interactive process (again, in se reaction systems with delay, this influence is delayed). Thus, when we consider reaction systems which partially control the environment, then their systemindependent interactive processes are in fact (represent) interactive processes of traditional reaction systems.
 1.
Footnotes
Notes
Acknowledgements
The authors are indebted to Robert Brijder and to three anonymous referees for their useful comments concerning this paper. Grzegorz Rozenberg was supported by the Visiting Professor Programme of Sapienza.
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