Evolving reaction systems

Accepted Manuscript Evolving reaction systems

Andrzej Ehrenfeucht, Jetty Kleijn, Maciej Koutny, Grzegorz Rozenberg

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S0304-3975(17)30035-X http://dx.doi.org/10.1016/j.tcs.2016.12.031 TCS 11035

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Theoretical Computer Science

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11 August 2016 13 November 2016 29 December 2016

Please cite this article in press as: A. Ehrenfeucht et al., Evolving reaction systems, Theoret. Comput. Sci. (2017), http://dx.doi.org/10.1016/j.tcs.2016.12.031

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Evolving Reaction Systems Andrzej Ehrenfeuchta , Jetty Kleijnb , Maciej Koutnyc , Grzegorz Rozenberga,b a

Department of Computer Science, University of Colorado at Boulder, 430 UCB Boulder, CO 80309-0430, U.S.A. b LIACS, Leiden University, P.O.Box 9512, NL-2300 RA Leiden, The Netherlands c School of Computing Science, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom

Abstract Reaction systems were introduced as a formal model of interactions between biochemical reactions. These interactions, which are based on two mechanisms: facilitation and inhibition, determine the functioning of the living cell. Processes taking place in a reaction system A are driven by the fixed set A of available reactions provided by A. In this paper we generalize this setup: as a process progresses from a state W to its successor W  , the set of available reactions may change from A in W to A in W  . This new framework of evolving reaction systems is introduced and studied in this paper. Also, the notion of enabling equivalence between sets of reactions and the notion of a transformation of a set of reactions are introduced and thoroughly studied. Keywords: functioning of the living cell, reaction, reaction system, interactive process, equivalence, evolving set of reactions, evolution, punctuated equilibrium 1. Introduction Reaction systems were introduced (see [11]) as a formal model of the functioning of the living cell. The underlying idea is that this functioning is determined by the interaction of biochemical reactions in the living cell, where these interactions are driven by two mechanism, facilitation and inhibition — the reactions (through their products) may facilitate or inhibit each other. This model takes into account the basic bioenergetics (flow of energy) of the living cell and the basic fact that the living cell is an open

Preprint submitted to Elsevier

January 6, 2017

system. Also (because of the level of abstraction it adopts) it is a qualitative rather than a quantitative model. A biochemical reaction is formalized as a 3-tuple of nonempty sets b = (R, I, P ), called a reaction, with R and I disjoint, where R is the set of reactants that b needs in order to take place, I is the set of inhibitors — if any of these is present in the current state of the system/cell, then b will not take place, and P is the product set — the set of entities contributed by b to the successor of the current state. Then a reaction system is basically a finite set of reactions, which reflects the point of view that the living cell is basically a reactor with a finite set of reactions taking place within it (where the reactor interacts with the environment). Formally, a reaction system is specified as an ordered pair A = (S, A), where A is a finite set of reactions and S is a finite (background) set containing all entities needed to define reactions in A and also interactions with the environment. The notion of reaction system is central for a broad framework of reaction systems, where one considers also various extensions of reaction systems motivated either by biological considerations or by considerations concerned with the need to understand the underlying computational nature of the models from this framework, see e.g., the tutorial and survey papers [7–9]. In fact, although the original motivation behind reaction systems came from biology, they became an interesting and novel model of computation, see, e.g., , [5, 6, 10, 12, 14, 16–18]. A central feature of models investigated in the framework of reaction systems is the invariance of the available set of reactions (the set of reactions of the considered reaction system). All processes are supported by this set of reactions, say A: in each state of each process the set of reactions enabled by this state is a subset of A. In this paper we abandon this “invariance point of view” and consider processes where a transition from a state to state may be accompanied by a change of the available set of reactions. We call this framework “Evolving Reaction Systems”. It is motivated by both biological considerations, in particular the evolution of biological systems (Section 7 of this paper deals with this theme), and computational considerations (considering systems where the available transformations change with time is quite traditional in the theory of computation, see, e.g., [4]). The paper is organized as follows. In Section 2 we introduce basic notions and notation concerning reactions and reaction systems. 2

The notion of enabling equivalence of sets of reactions, which is central for this paper, is introduced and analyzed in Section 3. As discussed above, in this paper we consider the framework of evolving reaction systems where the set of available reactions may change as the given state (say W ) is transformed to its successor (say W  ). If the set of available reactions at W is A and at W  it is A , then the change from A to A is governed by a transformation rule as introduced and analysed in Section 4. In Section 5 we introduce evolving interactive processes which differ from standard interactive processes considered in reaction systems by the fact that the set of available reactions may change as a process progresses from state to state. These processes are considered in evolving reaction systems. The main result of this paper, the Invisibility Theorem, is proved in Section 6. It provides conditions under which the changes of the sets of available reactions taking place in an evolving interactive process are not observable (hence invisible), meaning that they are not reflected in the state sequence of the process. In Section 7 we give an example illustrating the Invisibility Theorem which leads to an interpretation within the framework of evolving reaction systems of the notion of punctuated evolution, see, e.g., [11] and [18]. Finally in Section 8 we provide a brief discussion of the results of this paper. 2. Reactions and Reaction Systems Within this paper we will use standard mathematical terminology and notation. More specifically: The empty set is denoted by ∅. For sets X and Y , X \ Y , X ∪ Y , and X ∩ Y denote set difference, set union and set intersection, respectively. Also X ⊆ Y denotes set inclusion  and X ⊆ Y denotes the negation of set inclusion. For a family Z of sets, Z denotes the union of the sets from Z. The formal notion of a reaction captures the basic intuition behind a biochemical reaction: it can take place if all of its reactants and none of its inhibitors are present, and when it takes place it creates its products. Definition 2.1. A reaction is a triplet b = (R, I, P ), where R, I, P are finite nonempty sets with R ∩ I = ∅. If S is a set such that R, I, P ⊆ S, then b is a reaction over S. ♦

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The sets R, I, P are also written Rb , Ib , Pb , and called the reactant set of b, the inhibitor set of b, and the product set of b, respectively. Note that if b is a reaction over S, then |S| ≥ 2. Such finite sets (of cardinality at least 2) are called background sets. The set of all reactions over a background set S is denoted by rac(S). The dynamics of a single reaction and of a set of reactions is given by the following definition. Definition 2.2. Let S be a background set and let T ⊆ S. 1. Let b ∈ rac(S). Then b is enabled by T , denoted by en b (T ), if Rb ⊆ T and Ib ∩ T = ∅. The result of b on T , denoted by res b (T ), is defined by: res b (T ) = Pb if en b (T ), and res b (T ) = ∅ otherwise. 2. Let B ⊆ rac(S) be a finite set of reactions. The result of B on T , denoted by res B (T ), is defined by: res B (T ) = b∈B res b (T ). ♦ The above definition says how a reaction or a set of reactions behaves in a state of a biochemical system, where a state is formalized as a set T of biochemical entities (present in this state). Thus a reaction may happen (is enabled) if all of its reactants are present (Rb ⊆ T ) and none of its inhibitors are present (Ib ∩ T = ∅). If a reaction takes place in T , then it produces its product. Here res b (T ) = Pb means that b contributes Pb to the successor state of T and res b (T ) = ∅ means that b does not contribute to the successor of T . The result of a set of reactions B in T is cumulative, i.e., it is the union of the results of all reactions from B. Since res B (T ) is the union of res b (T ) for all reactions b from B which are enabled by T , an entity x ∈ S is sustained by B in T (i.e., x ∈ T and x ∈ res B (T )) if and only if x is produced by (at least) one reaction b from B. This is different from standard models of computation, where if an element from a current state is not “involved” in a transformation of this state, then it will be sustained (present in the successor state). This non-permanency property reflects the basic bioenergetics of the living cell (see e.g., [8] and [15]). The following notion of equivalence of single reactions and sets of reactions was introduced in [11]. Definition 2.3. Let S be a background set.

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1. Reactions b1 , b2 ∈ rac(S) are equivalent (over S), denoted by b1 eq S b2 , if and only if, for all T ⊆ S, res b1 (T ) = res b2 (T ). 2. Sets of reactions B1 , B2 ⊆ rac(S) are equivalent (over S), denoted by B1 eq S B2 , if and only if, for all T ⊆ S, res B1 (T ) = res B2 (T ). ♦ Whenever S is clear from the context of considerations we will simplify our terminology and notation and use the term “equivalence” and the notation eq. It turns out that the equivalence of single reactions can be characterized as follows. Theorem 2.4. [11] Let S be a background set and b1 , b2 ∈ rac(S). Then b1 eq S b2 if and only if Rb1 = Rb2 , Ib1 = Ib2 , and Pb1 = Pb2 . Thus single reactions are semantically equivalent if and only if they are syntactically equivalent (identical). The notion of covering of one reaction by another is useful when comparing results of reactions in a given state. Definition 2.5. Let S be a background set and b1 , b2 ∈ rac(S). We say that b1 covers b2 , denoted by b1 ≥S b2 , if and only if, for all T ⊆ S, res {b1 ,b2 } (T ) = res b1 (T ). ♦ Intuitively b1 ≥S b2 means that on its own b1 will accomplish as much as it will together with b2 . Also the notion of covering gets a syntactic characterization. Theorem 2.6. [11] Let S be a background set and b1 , b2 ∈ rac(S). Then b1 ≥S b2 if and only if Rb1 ⊆ Rb2 , Ib1 ⊆ Ib2 , and Pb2 ⊆ Pb1 . We are ready now to recall (see [11]) the formal notion of a reaction system. Definition 2.7. A reaction system is an ordered pair A = (S, A), where S is a background set and A ⊆ rac(S). ♦

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Thus a reaction system is essentially a finite set of reactions A. We also specify the background set S which includes all the entities needed to specify the reactions in A, but it also may include more entities which may be needed to reason about the behavior of A. Each subset T of S is called a state of A, and for each state T , the result of applying A to T , denoted by res A (T ), is defined by res A (T ) = res A (T ). Note that there is no counting in a reaction system — we deal with sets rather than multisets. Thus a reaction system is a qualitative (rather than quantitative) model which reflects the level of abstraction for modeling interactions between biochemical reactions. While A = (S, A) formalizes the static structure of a reaction system, its dynamic behavior is formalized through interactive processes which are defined as follows. Definition 2.8. Let A = (S, A) be a reaction system and let n be a positive integer. An (n-step) interactive process in A is an ordered pair π = (γ, δ) of finite sequences of finite sets such that γ = C0 , C1 , . . . , Cn and δ = D0 , D1 , . . . , Dn , where C0 , C1 , . . . , Cn , D0 , D1 , . . . , Dn ⊆ S, D0 = ∅, and Di = res A (Di−1 ∪ Ci−1 ) for all i ∈ {1, . . . , n}. ♦ The sequence γ is the context sequence of π and the sequence δ is the result sequence of π. Then the sequence τ = W0 , W1 , . . . , Wn such that Wi = Di ∪ Ci for all i ∈ {0, . . . , n} is the state sequence of π and W0 is the initial state of π. Note that since D0 = ∅, W0 = C0 . Hence the interactive process π runs as follows. It begins in the initial state W0 = C0 . The next state, W1 , consists of D1 , which is the result of applying to W0 the reactions from A enabled by W0 , and of context C1 , which formalizes the influence/effect of the environment. Then the consecutive states of τ are formed by iterating this procedure: for each i ∈ {1, . . . , n − 1}, Wi+1 is formed by the union of Di+1 = res A (Wi ) and Ci+1 . 3. Enabling equivalence When we consider the application of a single reaction b to a given state T , then we deal with a binary situation: either b is enabled by T or it is not. However, when we consider the application of a set of reactions B to T , then the situation is more involved: either none of the reactions in B is enabled by T or only some of the reactions in B are enabled by T or all reactions 6

in B are enabled by T . In this paper we will consider situations when a set of reactions B is acting as if it was a single reaction, which corresponds to considering only states T such that all reactions from B are enabled by T . This leads to the following definitions. Definition 3.1. Let S be a background set and let B ⊆ rac(S).  1. The reactant set of B, denoted by R B , is the set b∈B Rb ; the inhibitor set of B, denoted by IB , is  the set b∈B Ib ; and the product set of B, denoted by PB , is the set b∈B Pb . 2. For T ⊆ S, B is enabled by T , denoted by en B (T ), if RB ⊆ T and IB ∩ T = ∅. 3. B is consistent if RB ∩ IB = ∅.

♦

Note that if B is consistent by a state T , then all reactions from B are enabled by T . On the other hand if B is not consistent, then, for each T ⊆ S, B is not enabled by T (recall that if b is a reaction then Rb ∩ Ib = ∅, hence the notion of consistency is incorporated in the definition of a reaction). When we consider a set of reactions B as one “block” acting as a single reaction we get a different notion of equivalence for sets of reactions. Definition 3.2. Let S be a background set and let B1 , B2 ⊆ rac(S). We say that B1 is enabling equivalent to B2 (over S), denoted by B1 eeq S B2 , if and only if, for each T ⊆ S, (i) en B1 (T ) if and only if en B2 (T ), and (ii) if en B1 (T ), then res B1 (T ) = res B2 (T ).

♦

It is easily seen that eeq S is an equivalence relation. Whenever S is clear from the context of consideration, we will simplify our terminology and notations, using the term “enabling equivalence” and the notation eeq. Recall that B1 , B2 are equivalent if and only if for each T the results of applying B1 , B2 to T are equal independently of whether or not the whole sets B1 , B2 are enabled by T or only parts of them. On the other hand, B1 , B2 are enabling equivalent if and only if B1 and B2 are enabled by the same subsets T of S and on these subsets they give the same result. 7

First of all we notice that these two notions of equivalence are incomparable as demonstrated by the following examples. Example 3.3. Let S = {x, y, z, w, u} and let B1 , B2 ⊆ rac(S) be defined as follows: B1 = { ({x} , {z} , {w}) , ({y} , {z} , {u}) } B2 = { ({x, y} , {z} , {w, u}) } Since for T = {x}, res B1 (T ) = {w} and res B2 (T ) = ∅, B1 is not equivalent to B2 . Clearly, for each T ⊆ S, en B1 (T ) if and only if en B2 (T ). However, if en B1 (T ) (and hence also en B2 (T )) holds, then {x, y} ⊆ T and z ∈ / T ; but then res B1 (T ) = res B2 (T ) = {w, u}. Hence B1 is enabling equivalent to B2 . ♦ Example 3.4. Let S = {x, y, z} and let b1 , b2 ∈ rac(S) be defined as follows: b1 = ({x} , {z} , {y}) and b2 = ({x, y} , {z} , {y}). Let then B1 = {b1 } and B2 = {b1 , b2 }. Since, by Theorem 2.6, b1 ≥S b2 , for each T ⊆ S, res B1 (T ) = res B2 (T ) and so B1 is equivalent to B2 . However, B1 is enabled by T = {x} while B2 is not enabled by T = {x}. Thus B1 is not enabling equivalent to B2 . ♦ The relationship between the notion of consistency for a set of reactions and the notion of enabling equivalence is given by the following result. Lemma 3.5. Let S be a background set and let B1 , B2 ⊆ rac(S). 1. If B1 , B2 are not consistent, then B1 eeq S B2 . 2. If B1 eeq S B2 , then B1 is consistent if and only if B2 is consistent. Proof Follows directly from the definitions (of eeq S and consistency).

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The notion of enabling equivalence is a semantic (behavioral) notion which is global with respect to the space of all states (all subsets of S): to check whether or not two sets of reactions B1 and B2 are enabling equivalent, in general one has to test them in all states. We will provide now a syntactic characterization of enabling equivalence which allows one to test whether or not B1 eeqB2 by just inspecting the sets B1 and B2 . 8

Theorem 3.6. Let S be a background set and let B1 , B2 ⊆ rac(S) be consistent. Then B1 eeq S B2 if and only if RB1 = RB2 , IB1 = IB2 , and PB1 = PB2 . Proof (1) Assume that RB1 = RB2 , IB1 = IB2 , and PB1 = PB2 . (i) Since RB1 = RB2 and IB1 = IB2 , en B1 (T ) if and only if en B2 (T ), for each T ⊆ S. (ii) Since PB1 = PB2 , res B1 (T ) = res B2 (T ) whenever en B1 (T ) and en B2 (T ). It follows from (i) and (ii) that B1 eeq S B2 . (2) Assume that it is not true that (RB1 = RB2 , IB1 = IB2 , and PB1 = PB2 ). It follows than that one of the following three cases must hold: (i) RB1 = RB2 , (ii) IB1 = IB2 , and (iii) RB1 = RB2 , IB1 = IB2 , and PB1 = PB2 . We will consider each of the three cases separately. Assume that (i) holds. Without loss of generality we may assume that RB1 \ RB2 = ∅. Hence en B1 (RB2 ) does not hold. However, since B2 is consistent, en B2 (RB2 ). Consequently, it is not true that, for each T ⊆ S, en B1 (T ) if and only if en B2 (T ), and therefore B1 eeq S B2 does not hold. Assume that (ii) holds. Without loss of generality we may assume that IB1 \ IB2 = ∅. Let then y ∈ S be such that y ∈ IB1 \ IB2 . Hence it is not true that en B1 (RB2 ∪ {y}). However, since B2 is consistent, en B2 (RB2 ∪ {y}). Consequently, B1 eeq S B2 does not hold. Assume that (iii) holds. Since RB1 = RB2 and IB1 = IB2 (and B1 , B2 are consistent), en B1 (RB1 ) and en B2 (RB1 ). However, since PB1 = PB2 , res B1 (RB1 ) = res B2 (RB1 ). Consequently, B1 eeq S B2 does not hold. Since the cases (i), (ii) and (iii) are exhaustive, it follows that if it is not true that RB1 = RB2 , IB1 = IB2 , and PB1 = PB2 , then B1 eeq S B2 does not hold. The theorem follows now from (1) and (2). 2 Note that if B1 and B2 are singletons, B1 = {b1 } and B2 = {b2 }, then B1 eqB2 if and only if B1 eeqB2 . Hence, Theorem 3.6 generalizes the characterization of the equivalence of single reactions given in Theorem 2.4: reactions 9

b1 and b2 are equivalent if and only if Rb1 = Rb2 , Ib1 = Ib2 , and Pb1 = Pb2 . For single reactions b1 , b2 , this means that b1 and b2 are enabling equivalent if and only if they are identical, while two sets of reactions B1 , B2 may be enabling equivalent even if they are different provided that RB1 = RB2 , IB1 = IB2 , and PB1 = PB2 . The following technical corollary of Theorem 3.6 will be useful in the remainder of this paper. Corollary 3.7. Let S be a background set and let B1 , B2 , B3 ⊆ rac(S), where B2 is consistent. 1. If B1 eeq S B2 and B3 eeq S B2 , then (B1 ∪ B3 )eeq S B2 . 2. If B1 ⊆ B3 ⊆ B2 and B1 eeq S B2 , then B1 eeq S B3 and B3 eeq S B2 . Proof First we note that since B2 is consistent, the preconditions of (1) and (2) imply that also B1 and B3 are consistent (in (1) by Lemma 3.5(2) and in (2) because B1 ⊆ B2 and B3 ⊆ B2 ). (1) Follows directly from Theorem 3.6. (2) Since B1 eeq S B2 , it follows from Theorem 3.6 that RB1 = RB2 , IB1 = IB2 and PB1 = PB2 . Since B1 ⊆ B3 ⊆ B2 , RB1 ⊆ RB3 ⊆ RB2 , IB1 ⊆ IB3 ⊆ IB2 , and PB1 ⊆ PB3 ⊆ PB2 . Therefore, RB3 = RB2 , IB3 = IB2 and PB3 = PB2 , and so, by Theorem 3.6, B3 eeq S B2 . 2 4. Transformation rules In the standard setup an interactive process takes place within a given reaction system A = (S, A), where the set A of available reactions is invariant: at each stage of the process the set of available reactions is the same, viz., A, and if T is the state of A at this stage, then reactions from A that are enabled by T will transform T into res A (T ), which together with the context set available at this stage will form the successor state of T .

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In this paper we will consider “evolving situations” where the sets of available reactions may change as an interactive process progresses. Hence in an initial state W0 the set of available reactions is A0 , then at the successor state W1 it is A1 , at the following state W2 it is A2 , and so on. In order to investigate such evolving interactive processes we also need to define a mechanism which transforms A0 into A1 , A1 into A2 , and so on. The transformations that we consider are not arbitrary – they have to satisfy the “safety condition”. They have to preserve the enabling equivalence meaning that if a set of reactions A is transformed into a set of reactions B, then A and B must be enabling equivalent. A natural way to transform a set of reactions A into a set of reactions B is to remove some reactions (set of reactions A ) and add some new reactions (set of reactions A ), where the order of removing A and adding A is not important. Because of the transformation safety condition, the transformations we will consider preserve enabling equivalence not only between A and B but also between A and intermediate results (A \ A and A ∪ A ). At any moment of an implementation of transformation (or a sequence of transformations) beginning with a set of reactions A, the current set of reactions must be enabling equivalent to A. For example, for some reasons it may be important that, for a state T and an entity x, x ∈ / res A (T ). However, it may be the case that x ∈ / res B (T ) but x ∈ res A (T ), and so x will pop up in the intermediate state, while it may be important that x is not produced at all! Formally such transformations are defined as follows. Definition 4.1. A transformation rule is a 4-tuple q = (S, K, D, E), where S is a background set and K, D, E ⊆ rac(S) are such that: (i) K is consistent, (ii) D ⊆ K and E ∩ K = ∅, and (iii) Keeq S (K \ D) and Keeq S (K ∪ E). The outcome of q, denoted by out(q), is defined by out(q) = (K \ D) ∪ E. ♦ For a transformation rule q as above we say that S is the background set of q, and that q is a transformation rule over S — we use trr (S) to denote the set of transformation rules over S. Also, K is the kernel of q, D is the decrement of q, and E is the expansion of q, and we will use the notations Kq , 11

Dq , and Eq , to denote K, D, and E, respectively. To simplify the notation we may write simply q = (K, D, E) whenever S is understood from the context of considerations. Note that condition (iii) guarantees the transformation safety condition discussed above. A transformation rule q is trivial if Dq = Eq = ∅. Obviously, for a trivial transformation rule q, out(q) = Kq . For a transformation rule q over S and T ⊆ S, we say that q is enabled by T , denoted by en q (T ), if en Kq (T ), i.e., if the kernel Kq of q is enabled by T. The following lemma states a basic property of transformation rules. Lemma 4.2. Let q = (S, K, D, E) be a transformation rule. Then 1. K \ D is consistent, 2. K ∪ E is consistent, 3. Keeq S out(q), and 4. out(q) is consistent. Proof (1) This follows from Lemma 3.5(2), because K is consistent and Keeq S (K \ D). (2) This follows from Lemma 3.5(2), because K is consistent and Keeq S (K ∪ E). (3) Note that K \ D ⊆ (K \ D) ∪ E ⊆ K ∪ E. Moreover, since Keeq S (K \ D) and Keeq S (K ∪ E), we have (K \ D)eeq S (K ∪ E). Therefore, by Corollary 3.7(1), (K \ D) ∪ E eeq S (K \ D) and (since (K \ D)eeq S K) (K \ D) ∪ E = out(q)eeq S K. (4) This follows, by Lemma 3.5(2), from (3) and the consistency of K.

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Theorem 4.3. Let q = (S, K, D, E) be a transformation rule. If a 4-tuple q  = (S, K, D , E  ) is such that D ⊆ D and E  ⊆ E, then q  is also a transformation rule.

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Proof We will verify that all conditions for q  to be a transformation rule are satisfied. (i) Since q is a transformation rule, K is consistent. (ii) Since q is a transformation rule, D ⊆ K and E ∩ K = ∅. Hence, by D ⊆ D and E  ⊆ E, we get D ⊆ K and E  ∩ K = ∅. (iii) Since D ⊆ D, K \ D ⊆ K \ D . Since (K \ D)eeq S K, this implies (by Corollary 3.7(2), because K \ D ⊆ K \ D ⊆ K) that (K \ D )eeq S K. Since E  ⊆ E, K ⊆ K ∪ E  ⊆ K ∪ E. Since K ∪ Eeeq S K, this implies (by Lemma 4.2(2) and Corollary 3.7(2)) that (K ∪ E  )eeq S K. Consequently, q  is a transformation rule.

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We will define now how a transformation rule q transforms a set of reactions A — this depends on the state T in which the transformation of A takes place. Definition 4.4. Let q = (S, K, D, E) be a transformation rule and let T ⊆ S. 1. Let A ⊆ rac(S). The transformation of A by q in T , denoted by tr q,T (A), is defined by:  (A \ Kq ) ∪ out(q) if Kq ⊆ A and en q (T ) tr q,T (A) = A otherwise. 2. Let A = (S, A) be a reaction system. The transformation of A by q in T , denoted by tr q,T (A), is defined by tr q,T (A) = A , where A = (S, A ) with A = tr q,T (A). ♦ Note that Theorem 4.3 says that a transformation rule may be implemented piecewise and the order of “pieces” does not matter. For example, one can partition D into nonempty D1 , D2 , D3 and E into nonempty E1 , E2 . Then, whether one applies the sequence of transformation rules (S, K, D1 , ∅), (S, K \ D1 , D2 , ∅), (S, K \ D1 \ D2 , ∅, E1 ), (S, K \ D1 \ D2 ∪ E1 , D3 , E2 ), or the sequence of transformation rules 13

(S, K, D2 , E1 ), (S, K \D2 ∪E1 , D1 , E2 ), (S, K \D2 \D1 ∪E1 ∪E2 ), D3 , ∅) the final outcome of both sequences will be identical — it will be exactly the outcome of the original transformation rule q = (S, K, D, E). Moreover, by Lemma 4.2(3), after each of the sequential steps the outcome of the last transformation is enabling equivalent to the outcome of q ! Since (A \ Kq ) ∪ out(q) = (A \ Kq ) ∪ (Kq \ Dq ) ∪ Eq = (A \ Dq ) ∪ Eq , the definition of tr q,T (A) may be rewritten as  (A \ Dq ) ∪ Eq if Kq ⊆ A and en q (T ) tr q,T (A) = A otherwise. First we consider transformations by trivial transformation rules. Lemma 4.5. Let q be a trivial transformation rule over S. Then, for all T ⊆ S and A ⊆ rac(S), tr q,T (A) = A. Proof We consider separately three cases. (i) Kq ⊆ A. Then tr q,T (A) = A. (ii) Kq ⊆ A but en Kq (T ) does not hold. Then tr q,T (A) = A. (iii) Kq ⊆ A and en Kq (T ). Since q is trivial, out(q) = Kq , and consequently tr q,T (A) = (A \ Kq ) ∪ Kq = A. Then tr q,T (A) = A. It follows from (i), (ii), and (iii) that tr q,T (A) = A.

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The following result states the fundamental property of transformations of sets of reactions by transformation rules. Theorem 4.6. Let q, q  be transformation rules over S, for some background set S. If Kq = Kq , then for all T ⊆ S and all A ⊆ rac(S), res tr q,T (A) (T ) = res tr q ,T (A) (T ) .

14

Proof Let K = Kq = Kq . We consider separately three cases. Case 1: K ⊆ A. Then tr q,T (A) = tr q ,T (A) = A, and consequently res tr q,T (A) (T ) = res tr q ,T (A) (T ) = res A (T ). Case 2: K ⊆ A but en q (T ) does not hold (and hence also en q (T ) does not hold). Then tr q,T (A) = tr q ,T (A) = A, and consequently res tr q,T (A) (T ) = res tr q ,T (A) (T ) = res A (T ). Case 3: K ⊆ A and en q (T ) (and hence en q (T )). Then tr q,T (A) = (A \ K) ∪ out(q) and tr q ,T (A) = (A \ K) ∪ out(q  ). Consequently, res tr q,T (A) (T ) = res (A\K)∪out(q) (T ) res tr q ,T (A) (T ) = res (A\K)∪out(q ) (T )

(1)

Since Kq eeq S out(q) (by Lemma 4.2(3)), by Theorem 3.6, PKq = Pout(q) . Similarly, since Kq eeq S out(q  ), PKq = Pout(q ) . Consequently, since Kq = Kq , we get Pout(q) = Pout(q ) (2) By (1),

res tr q,T (A) (T ) = res A\K (T ) ∪ res out(q) (T ) res tr q ,T (A) (T ) = res A\K (T ) ∪ res out(q ) (T )

(3)

Since now (in Case 3) en q (T ) and en q (T ), by Lemma 4.2(3) we get en out(q) (T ) and en out(q ) (T ). Consequently, it follows from (3) that res tr q,T (A) (T ) = res A\K (T ) ∪ Pout(q) res tr q ,T (A) (T ) = res A\K (T ) ∪ Pout(q ) . Therefore, by (2), res tr q,T (A) (T ) = res tr q ,T (A) (T ), and the theorem holds. 2 The following corollary relates (for each state T ) the effect of a set of reactions A and the effect of the set of reactions resulting from transforming A at T . Corollary 4.7. Let q be a transformation rule over S, for some background set S. For all T ⊆ S and A ⊆ rac(S), res tr q,T (A) (T ) = res A (T ). 15

Proof Consider the trivial transformation rule q  = (Kq , ∅, ∅). By Theorem 4.6, for each T ⊆ S and each A ⊆ rac(S), res tr q,T (A) (T ) = res tr q ,T (A) (T ). Since, by Lemma 4.5, tr q ,T (A) = A, we get res tr q,T (A) (T ) = res A (T ). Thus the corollary holds. 2 Corollary 4.7 is quite remarkable and perhaps not very intuitive. It says that if a state T is converted by a set of reactions A into res A (T ) and q is a transformation rule, then also tr q,T (A) converts T into res A (T ). This result is an important technical tool for reasoning about chains of transformations and will be essential in the proof of the main result of this paper (Theorem 6.2 in Section 6). 5. Evolving Interactive Processes In this section we introduce evolving interactive processes where the underlying set of available reactions (hence the underlying reaction system) may change as the current state of an interactive process changes to the successor state. Definition 5.1. Let S be a background set. An evolving interactive process over S is a 5-tuple φ = (γ, δ, σ, ρ, λ) such that, for some n ≥ 1: • γ = C 0 , C 1 , . . . , Cn ,

where, for each i ∈ {0, . . . , n}, Ci ⊆ S,

• δ = D0 , D1 , . . . , Dn , where, for each i ∈ {0, . . . , n}, Di ⊆ S, • σ = W0 , W1 , . . . , Wn , where, for each i ∈ {0, . . . , n}, Wi ⊆ S • ρ = A0 , A1 , . . . , An , ∅,

where, for each i ∈ {0, . . . , n}, Ai ⊆ rac(S), Ai =

• λ = q0 , q1 , . . . , qn ,

where, for each i ∈ {0, . . . , n}, qi ∈ trr (S),

and the following relationships hold: 1. Wi = Ci ∪ Di ,

for each i ∈ {0, . . . , n},

2. Di = res Ai−1 (Wi−1 ),

for each i ∈ {1, . . . , n},

3. Kqi ⊆ Ai ,

for each i ∈ {0, . . . , n}, 16

γ δ σ ρ λ

0 C0 D0 W0 A0 q0

1 C1 D1 W1 A1 q1

2 C2 D2 W2 A2 q2

... ... ... ... ... ...

n−1 Cn−1 Dn−1 Wn−1 An−1 qn−1

n Cn Dn Wn An qn

Figure 1: An n-step evolving interactive process.

4. Ai = tr qi−1 ,Wi−1 (Ai−1 ), for each i ∈ {1, . . . , n}. The sequence γ is the context sequence of φ, denoted by con(φ); the sequence δ is the result sequence of φ, denoted by res(φ); the sequence σ is the state sequence of φ, denoted by st(φ); the sequence ρ is the sequence of sets of reactions of φ, denoted by sre(φ); and the sequence λ is the rule sequence of φ, denoted by rul (φ). The state W0 is the initial state of φ (and the initial state of st(φ)), denoted by in(φ) (and by in(st(φ))). We also say that φ is an n-step evolving interactive process over S. ♦ Note that it follows from Definition 4.4 (and the comment following it) that if i ∈ {0, . . . , n − 1} is such that en qi (Wi ) then, by Definition 5.1(points 3,4), Ai+1 = tr qi ,Wi (Ai ) = (Ai \ Kqi ) ∪ out(qi ) = (Ai \ Dq ) ∪ Eq . It is very convenient to represent an evolving n-step interactive process as a 5 × (n + 1) matrix, as shown in Figure 1. This representation shows that φ can be seen as a sequence of columns: column 0, column 1, . . . , column (n − 1), column n which portrays very well the intuition of an evolving interactive process. Each column represents the snapshot of a current situation — often called an instantaneous description in the theory of computation. An evolving interactive process is then a sequence f0 , f1 , . . . , fn of such instantaneous descriptions, as illustrated in Figure 2. We denote the context of fi by Ci , the transformation rule of fi by qi , etc. Here each successor instantaneous 17

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

C0 D0 W0 A0 q0 f0

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⇒ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

C1 D1 W1 A1 q1 f1

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⇒ ... ⇒ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

Cn−1 Dn−1 Wn−1 An−1 qn−1 fn−1

⎤

⎡

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⇒ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣

Cn Dn Wn An qn fn

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Figure 2: A sequence of instantaneous descriptions.

description fi+1 is obtained from its predecessor fi by the set of reactions Ai of fi , the transformation rule qi of fi , and the context Ci+1 of fi+1 . The intuition behind an evolving interactive process φ is that the initial snapshot of the situation is the instantaneous description f0 . Here the set of available reactions is A0 , the contribution (influence) of the environment (we deal with open systems) is C0 , which is also the initial state of the system, and q0 is the transformation rule which determines the set of available reactions in the following (successor) situation f1 . Then, inductively, for each instantaneous description fi , its set of reactions Ai applied to its state Wi determines the result Di+1 of the successor instantaneous description fi+1 which together with the context Ci+1 determines (by union) the state Wi+1 . The set Ai+1 of reactions available in fi+1 is determined by the rule qi of fi (applied to Ai in the state Wi ). Thus the context sequence con(φ) together with the rule sequence rul (φ) determine φ from the initial instantaneous description f0 . The sequence A0 , A1 , . . . , An of sets of reactions of φ induces the sequence A0 = (S, A0 ) A1 = (S, A1 )

...

An = (S, An )

of reaction systems over the same background set S. Thus one can see the pair (con(φ), res(φ)) as generalizing the notion of an interactive process of a reaction system by allowing the sequence of transformations (C0 , D0 ) → D1

(C1 , D1 ) → D2

...

(Cn−1 , Dn−1 ) → Dn

to be carried on by the sequence A0 , . . . , An−1 of reaction systems, where each reaction system Ai+1 is obtained from the reaction system Ai (and the state Wi ) by the transformation rule qi . 18

Definition 5.2. An evolving interactive process φ = f0 , f1 , . . . , fn is stationary, if, for each i ∈ {0, . . . , n}, rul (fi ) is trivial. ♦ To simplify our terminology we may use the term “stationary interactive process” rather than “stationary evolving interactive process”. Note that since, for each i ∈ {0, . . . , n} rul (fi ) is trivial, by Lemma 4.5, the sequence sre(φ) = A0 , . . . , An is such that A0 = A1 = . . . = An . Hence (referring to the above intuition of an evolving sequence A0 , A1 , . . . , An of reaction systems) φ is a process taking place within one reaction system A = A0 , i.e., π = (con(φ), res(φ)) is an interactive process in A0 . In this way the notion of an evolving interactive process generalizes the notion of an interactive process in reaction systems. 6. The invisibility theorem In this section we prove the main result of this paper. First we need the following definition. Definition 6.1. Let S be a background set and let B ⊆ rac(S). 1. A signature over S is a triplet (X, Y, Z) of subsets of S. It is called consistent if X, Y, Z = ∅ and X ∩ Y = ∅. 2. Let B ⊆ rac(S). A signature α = (X, Y, Z) over S is the signature of B, denoted by sig(B), if X = RB , Y = IB and Z = PB . 3. Let α = (X, Y, Z) be a consistent signature over S. A subset T of S is α-compatible if X ⊆ T and Y ∩ T = ∅. 4. Let q = (S, K, D, E) be a transformation rule. A signature α = (X, Y, Z) over S is the signature of q, denoted by sig(q), if α = sig(K). ♦ Note that if B is nonempty and consistent (Definition 3.1), then sig(B) is a reaction. Theorem 6.2. [Invisibility Theorem]. Let S be a background set and let α = (X, Y, Z) be a consistent signature over S. Let φ = f0 , f1 , . . . , fn be a stationary evolving interactive process, and and let ψ = f 0 , f 1 , . . . , f n be an evolving interactive process, where fi = (Ci , Di , Wi , Ai , qi ) and f i = (C i , Di , W i , Ai , q i ), 19

for each i ∈ {0, . . . , n}. Then res(φ) = res(ψ) and st(φ) = st(ψ) provided that: 1. Wi is α-compatible, for each i ∈ {0, . . . , n}, 2. sig(q i ) = α and Kqi ⊆ Ai , for each i ∈ {0, . . . , n}, 3. con(φ) = con(ψ), and 4. D0 = D0 and A0 = A0 .

♦

Note that in Condition 4 we do not require C 0 = C0 as this is guaranteed by Condition 3. Also, W 0 = W0 follows from D0 = D0 and Condition 3. Theorem 6.2 states that an evolving interactive process (ψ) can be such that the available sets of reactions (A0 , A1 , . . . , An ) change as the interactive process progresses from state to state (W 0 , W 1 , . . . , W n ) but these changes are not observable (hence invisible) in the consecutive states (W 0 , W 1 , . . . , W n ) of the process: a stationary evolving interactive process (φ) with the same context sequence (con(φ)) and the same initial situation (D0 = D0 , W0 = W 0 , and A0 = A0 ) will produce the same state sequence (and the same result sequence). Since (as usual in models of computation) processes are observable through their states, Theorem 6.2 is called the Invisibility Theorem. We precede the proof of the theorem by a technical lemma considering only one change of the available set of reactions. But first we introduce an auxiliary notion. Definition 6.3. Let S be a background set. Let φ = f0 , . . . , fn be a stationary evolving interactive process over S, where fi = (Ci , Di , Wi , Ai , qi ), for each i ∈ {0, . . . , n}. Let q be a transformation rule over S such that en q (W0 ) and Kq ⊆ A0 . Then an evolving interactive process ψ = f0 , f1 , . . . , fn , where fi = (Ci , Di , Wi , Ai , qi ), for each i ∈ {0, . . . , n}, is a q-change of φ if 1. f0 = (C0 , D0 , W0 , A0 , q), 2. con(φ) = con(ψ), and 3. qi is trivial, for each i ∈ {1, . . . , n}.

♦

The two processes, φ and ψ, appearing in the above definition are depicted in Figure 3. 20

con(φ)

C0

C1

C2

...

Cn

res(φ)

D0

D1

D2

...

Dn

st(φ)

W0

W1

W2

...

Wn

sre(φ)

A0

A1

A2

...

An

rul (φ)

q0

q1

q2

...

qn

φ

f0

f1

f2

...

fn

C0 = C0 D0 = D0 W0 = W0 A0 = A0 q = q0

C1 D1 W1 A1 q1

C2 D2 W2 A2 q2

...

Cn

...

Dn

...

Wn

...

An

...

qn

f0

f1

f2

...

fn

all equal to A0 all trivial

as in φ

ψ

con(ψ) = con(φ)

all equal to A1 all trivial

Figure 3: A stationary interactive process φ and a q-change ψ of φ as in Definition 6.3. Note that Kq ⊆ A0 „ A1 = tr q,W0 (A0 ) = (A0 \ Kq ) ∪ out(q), D1 = D1 , and W1 = W1 .

21

Thus ψ results from φ by replacing q0 by q in forming f0 and requiring that con(ψ) = con(φ) and all q1 , . . . , qn are trivial. Since each qi is trivial, for i ∈ {1, . . . , n − 1}, Ai+1 = Ai and, consequently, sre(ψ) = A0 , A1 , A1 , . . . , A1 . Hence there is only one initial change (determined by q) here from A0 to A1 = (A0 \ Kq ) ∪ out(q), after which the available set of reactions does not change anymore (it equals A1 ). Note that ψ is a a q-change of φ and not the q-change of φ because the choice of q1 , . . . , qn is “free” provided that all of them are trivial. We cannot require that q1 = q1 ,. . . ,qn = qn because the definition of an evolving interactive process requires that Kqi ⊆ Ai = A1 and Kqi ⊆ Ai = A0 , for each i ∈ {1, . . . , n}, and, in general, these conditions are not compatible as A0 may be different from A1 ! Lemma 6.4. [One-change Lemma]. Let S be a background set and let α = (X, Y, Z) be a consistent signature over S. Let φ = f0 , f1 , . . . , fn be a stationary evolving interactive process, where fi = (Ci , Di , Wi , Ai , qi ), for each i ∈ {0, . . . , n}. Let q be a transformation rule such that Kq ⊆ A0 and sig(q) = α. Let ψ = f0 , f1 , . . . , fn be an evolving interactive process such that fi = (Ci , Di , Wi , Ai , qi ), for each i ∈ {0, . . . , n}, where 1. f0 = (C0 , D0 , W0 , A0 , q), 2. con(φ) = con(ψ), 3. qi is trivial, for each i ∈ {1, . . . , n}, and 4. Wi is α-compatible, for each i ∈ {0, . . . , n}. Then ψ is a q-change of φ such that res(φ) = res(ψ) and st(φ) = st(ψ). ♦ Note that sre(φ) = A0 , A0 , . . . , A0 while sre(ψ) = A0 , A1 , A1 , . . . , A1 . Thus a possible change of the available set of reactions takes place only once in the transition from the first to the second instantaneous description. Therefore we refer to this lemma as the “one-change lemma”. Proof. [Lemma 6.4] Let, for each 0 ≤ j ≤ n, 22

fj = (Cj , Dj , Wj , Aj , qj ) and fj = (Cj , Dj , Wj , Aj , qj ). Since φ is stationary, each Aj equals A0 , and each qj is a trivial transformation rule. Also, for each j ≥ 1, each qj is trivial and so each Aj = A1 , while A0 = A0 and q0 = q. Since W0 is α-compatible, sig(q) = α implies that en q (W0 ). Since Kq ⊆ A0 and con(φ) = con(ψ), this implies that ψ is a q-change of φ. Therefore A1 = tr q,W0 (A0 ) = (A0 \ Kq ) ∪ out(q), and for each i ∈ {1, . . . , n}, Ai = A1 . We construct now a sequence φ(0) , φ(1) , . . . , φ(n) of evolving interactive processes in A as follows: (0)

(0)

(0)

(n)

(n)

(n)

φ(0) = f0 , f1 , . . . , fn (1) (1) (1) φ(1) = f0 , f1 , . . . , fn .. . φ(n) = f0 , f1 , . . . , fn where (a) φ(0) is defined as follows (see Figure 4): (0)

(0)

(0)

– (f0 , f1 , . . . , fn−1 ) = (f0 , f1 , . . . , fn−1 ) and (0)

– fn = (Cn , Dn , Wn , A0 , q), (b) φ(1) is defined as follows (see Figure 4): (1)

(1)

(1)

– (f0 , f1 , . . . , fn−2 ) = (f0 , f1 , . . . , fn−2 ) and (1)

(1)

– fn−1 = (Cn−1 , Dn−1 , Wn−1 , A0 , q), fn = (Cn , Dn , Wn , A1 , qn ), (c) for each 2 ≤ i ≤ n, φ(i) is defined as follows (see Figure 6 and Figure 7): (i)

(i)

(i)

– (f0 , f1 , . . . , fn−i−1 ) = (f0 , f1 , . . . , fn−i−1 ) and (i)

– fn−i = (Cn−i , Dn−i , Wn−i , A0 , q), (i)

 – fn−i+1 = (Cn−i+1 , Dn−i+1 , Wn−i+1 , A1 , qn−i+1 ), (i)

– fn−i+j = (Cn−i+j , res A1 (Wn−i+j−1 ), Cn−j+1 ∪ res A1 (Wn−i+j−1 ),  A1 , qn−i+j ), for each j ∈ {2, . . . , i}. 23

φ

C0

C1 . . . Cn−3

Cn−2

Cn−1

Cn

D0

D1 . . . Dn−3

Dn−2

Dn−1

Dn

W0

W1 . . . Wn−3

Wn−2

Wn−1

Wn

A0

A1 . . . An−3

An−2

An−1

An

q0

q1 . . . qn−3

qn−2

qn−1

qn

f0

f1 . . . fn−3

fn−2

fn−1

fn

all trivial

as in φ

φ(0)

(0)

C1 . . . Cn−3

(0)

D1 . . . Dn−3

C0

D0

(0)

W0

(0)

(0)

Cn−2

(0)

Cn−1 Cn =Cn

(0)

(0)

Dn−2

(0)

Dn−1 Dn =Dn

(0)

Wn−2

(0)

Wn−1 Wn =Wn

(0)

(0)

An−2

(0)

An−1 An =An

(0)

(0)

qn−2

(0)

qn−1

(0)

(0)

(0)

W1

. . . Wn−1

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

A1 . . . An−3

(0)

q1 . . . qn−3

f0

(0)

f1

(0)

. . . fn−3

fn−2

fn−1

(1) C0

(1) C1

(1) Cn−3

(1) Cn−2

(1) Cn−1

Cn

(1)

Dn−2

(1)

Dn−1

(1)

Dn

(1)

Wn−2

(1)

Wn−1

(1)

Wn

(1)

(1)

An−2 An−1 =A0

(1)

(1)

qn−2

(1)

fn−2

A0 q0

(0)

(0)

(0)

qn = q (0)

fn

as in φ

φ(1)

(1)

D0

(1)

W0

(1)

...

D1 . . . Dn−3 (1)

W1

. . . Wn−3

(1)

A1 . . . An−3

(1)

q1 . . . qn−3

(1)

f1

A0 q0 f0

(1)

(1)

(1)

(1)

. . . fn−3

(1)

q (1)

fn−1

(1) (1) (1)

A1 qn (1)

fn

as in φ

φ(2)

(2)

C1 . . . Cn−3

(2)

D1 . . . Dn−3

C0

D0

(2)

W0

(2)

A0

(2)

(2)

Cn−2

(2)

Cn−1

(2)

Cn

(2)

(2)

Dn−2

(2)

Dn−1

(2)

Dn

(2)

Wn−2

(2)

Wn−1

(2)

Wn

A1 . . . An−3 An−2 =A0

A1

A1

q

 qn−1

qn

(2)

fn−1

(2)

W1

. . . Wn−3

(2)

(2)

(2)

(2)

(2)

q1 . . . qn−3

(2)

f1

q0 f0

(2)

(2)

. . . fn−3

(2)

fn−2

(2)

(2) (2) (2)

(2)

fn

Figure 4: The initial stationary interactive process φ; a φ(0) change (the transformation (0) (0) rule q appears in column fn−0 = fn and A1 = tr q,Wn−1 (A0 ) = (A0 \ Kq ) ∪ out(q)); a (1) φ(1) change (the transformation rule q appears in column fn−1 and A1 = tr q,Wn−1 (A0 ) = (2) (A0 \ Kq ) ∪ out(q)); and a φ(2) change (the transformation rule q appears in column fn−2 ).

24

as in φ

φ(i−1)

(i−1)

(i−1)

(i−1)

Cn

(i−1)

(i−1)

(i−1)

Dn

(i−1)

(i−1)

(i−1)

Wn

C0 C1 . . . Cn−i−1 Cn−i Cn−i+1 Cn−i+2 . . . Cn−1

D0 D1 . . . Dn−i−1 Dn−i Dn−i+1 Dn−i+2 . . . Dn−1 W0 W1 . . . Wn−i−1 Wn−i Wn−i+1 Wn−i+2 . . . Wn−1 A0 A0 . . . q0

q1 . . .

A0

A0

A0

A1 . . .

qn−i−1

qn−i

q

qn−i+2 . . .

(i−1) (i−1) (i−1)

A1

(i−1)

A1

(i−1)

(i−1)

qn−1

qn

as in φ

φ

(i)

(i) Cn−i+1 (i) Dn−i+1 (i) Wn−i+1

C0 C1 . . . Cn−i−1 Cn−i D0 D1 . . . Dn−i−1 Dn−i W0 W1 . . . Wn−i−1 Wn−i A0 A0 . . . q0

q1 . . .

(i) Cn−i+2 (i) Dn−i+2 (i) Wn−i+2

...

(i) Cn−1 (i) Dn−1 (i) Wn−1

... ...

A0

A0

A1

A1 . . .

A1

qn−i−1

q

qn−i+1

qn−i+2 . . .

qn−1

(i)

(i)

(i)

Cn

(i)

Dn

(i)

Wn

A1

(i)

(i)

qn

Figure 5: Moving from φ(i−1) to φ(i) .

as in φ

φ

(n)

(n) C0 (n) D0 (n) W0 (n) A0

q (n)

f0

(n) C1 (n) D1 (n) W1

(n) C2 (n) D2 (n) W2

...

Cn

...

Dn

...

Wn

A1

A1

...

A1

q1

q2

...

qn

(n)

...

(n)

f1

f2

(n) (n) (n)

(n)

fn

(n)

(n)

Figure 6: A φ(n) change (the transformation rule q appears in column fn−n = f0 ).

25

as in φ

φ

(i)

(i) C0 (i) D0 (i) W0 (i) A0 (i) q0 (i)

f0

(i) C1 (i) D1 (i) W1 (i) A1 (i) q1

...

(i)

...

f1

(i)

... ... ... ...

fn−i+j

(i) Cn−i−1 (i) Dn−i−1 (i) Wn−i−1 (i) An−i−1 (i) qn−i−1

(i) Cn−i (i) Dn−i (i) Wn−i

(i) Cn−i+1 (i) Dn−i+1 (i) Wn−i+1

An−i =A0

A1

A1 . . .

A1

q

 qn−i+1

 qn−i+2 ...

qn

(i)

fn−i+1

(i)

(i)

fn−i−1 ⎡

(i)

fn−i

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ Cn−i+j ⎢ ⎢ ⎣

(i) Cn−i+2 (i) Dn−i+2 (i) Wn−i+2

(i)

. . . Cn

(i)

. . . Dn

(i)

fn−i+2 . . . ⎤ C Cn−i+j ⎥ n−i+j ⎥ res A1 (Wn−i+j−1 ) ⎥ Dn−i+j ⎥ ∪ res A1 (Wn−i+j−1 ) ⎥ ⎥ Wn−i+j ⎥ A1 ⎥ ⎦ An−i+j  qn−i+j

(i)

. . . Wn

(i)

fn

qn−i+j

Figure 7: The definition of φ(i) change, for 2 ≤ i ≤ n (the transformation rule q appears (i) in column fn−i ) and j ≥ 2.

As usual, for each i, j ∈ {0, . . . , n}, we use the notation (i)

(i)

(i)

(i)

(i)

(i)

fj = (Cj , Dj , Wj , Aj , qj ). Since Kq ⊆ A0 , it follows indeed from the definition of φ(0) , φ(1) , . . . , φ(n) that, for each i ∈ {1, . . . n}, φ(i) is an evolving interactive process. We also note (n) (n) (n) (see Figure 6) that f0 = f0 , f1 = f1 , . . . , fn = fn and so φ(n) = ψ. (i)

(i−1)

(i)

(i−1)

Claim 1. For all i ∈ {1, . . . , n}, Dn−i+2 = Dn−i+2 and Wn−i+2 = Wn−i+2 . Proof. [Claim 1] By the definition of the evolving interactive processes φ(i−1) and φ(i) , (i)

(i−1)

(i)

(i−1)

Dn−i+1 = Dn−i+1 and Wn−i+1 = Wn−i+1 . Thus 26

(i)

(i)

(i−1)

Dn−i+2 = res A1 (Wn−i+1 ) = res A1 (Wn−i+1 ). Since A1 = tr q,W i−1 (A0 ), by Corollary 4.7, n−i+1

(i−1)

(i−1)

(i−1)

res A1 (Wn−i+1 ) = res A0 (Wn−i+1 ) = Dn−i+2 . (i)

(i−1)

Consequently, Dn−i+2 = Dn−i+2 . Since the context sequence is the same for (i) (i−1) φ(i) and φ(i−1) , i.e., con(φ(i) ) = con(φ(i−1) ), we get also Wn−i+2 = Wn−i+2 . Hence the claim holds.

2

(Claim 1)

Claim 2. For each i ∈ {1, . . . , n}, res(φ(i) ) = res(φ(i−1) ) and st(φ(i) ) = st(φ(i−1) ). Proof. [Claim 2] Let i ∈ {1, . . . , n}. (1) By the definition of the evolving interactive processes φ(i) and φ(i−1) , (i) (i−1) (i) (i−1) Dn−i = Dn−i and Dn−i+1 = Dn−i+1 . Consequently, because con(φ(i) ) = (i) (i−1) (i) (i−1) con(φ(i−1) ), Wn−i = Wn−i and Wn−i+1 = Wn−i+1 . (2) Also, for each j ∈ {2, . . . , i}, (i)

(i−1)

sre(fn−i+j ) = sre(fn−i+j ) = A1 . Since by Claim 1, (i)

(i−1)

(i)

(i−1)

Dn−i+2 = Dn−i+2 and Wn−i+2 = Wn−i+2 , (i)

(i−1)

this implies that, for each j ∈ {0, . . . , i}, Dn−i+j = Dn−i+j and conse(i) (i−1) quently, because con(φ(i) ) = con(φ(i−1) ), Wn−i+j = Wn−i+j . It follows from (1) and (2), that, for each j ∈ {0, . . . , n}, res(φ(i) ) = res(φ(i−1) ) and st(φ(i) ) = st(φ(i−1) ). Hence the claim holds.

(Claim 2)

2

The lemma follows now from Claim 2 by the fact (mentioned already) that φ = ψ, where, as we proved already, ψ is a q-change of φ. (Lemma 6.4) 2 (n)

We need one more definition before we proceed to the proof of the invisibility theorem, Theorem 6.2. 27

μ0 μ1

: :

φ0 = f0,0 , f0,1 , f0,2 , . . . , f0,n φ1 = f1,1 , f1,2 , . . . , f1,n    = f0,1 , f0,2 , . . . , f0,n μ2 : φ 2 = f2,2 , . . . , f2,n   = f1,2 , . . . , f1,n .. .. . . μn−1 : φn−1 = fn−1,n−1 , fn−1,n   = fn−2,n−1 , fn−2,n

and and

    ψ0 = f0,0 , f0,1 , f0,2 , . . . , f0,n ,    ψ1 = f1,1 , f1,2 , . . . , f1,n ,

and

ψ2 =

  f2,2 , . . . , f2,n ,

.. . and ψn−1 =

  fn−1,n−1 , fn−1,n .

Figure 8: The construction of μ.

Definition 6.5. Let φ = f0 , f1 , . . . , fn be an evolving interactive process such that n ≥ 2. The left cut of φ, denoted by lcut(φ), is the evolving interactive process f1 , . . . , fn . ♦ Proof. [Theorem 6.2] Let then α, φ, and ψ be as in the statement of Theorem 6.2. The proof begins by constructing a sequence μ of pairs of evolving interactive processes, μ = μ0 , μ1 , . . . , μn−1 with μi = (φi , ψi ) for each i ∈ {0, . . . , n − 1}, where • φ0 = φ, • ψi is a q i -change of φi , for each i ∈ {0, . . . , n − 1}, and • φi = lcut(ψi−1 ), for each i ∈ {0, . . . , n − 1}. We will use the following notation. For each i ∈ {0, . . . , n − 1},    φi = fi,i , fi,i+1 , . . . , fi,n and ψi = fi,i , fi,i+1 , . . . , fi,n ,

where, for each k ∈ {i, i + 1, . . . , n},      fi,k = (Ci,k , Di,k , Wi,k , Ai,k , qi,k ) and fi,k = (Ci,k , Di,k , Wi,k , Ai,k , qi,k ).

The above construction is illustrated in Figure 8.

28

Consider now the sequence δ of pairs of result sequences corresponding to the sequence μ, i.e., δ = δ0 , δ1 , . . . , δn−1 with δi = (res(φi ), res(ψi )) for each i ∈ {0, . . . , n − 1}. Hence res(φ0 ) res(φ1 )

= D0,0 , D0,1 , . . . , D0,n = D1,1 , . . . , D1,n

res(φn−1 ) = Dn−1,n−1 , Dn−1,n

and and .. . and

res(ψ0 ) res(ψ1 )

   = D0,0 , D0,1 , . . . , D0,n ,   = D1,1 , . . . , D1,n ,

  res(ψn−1 ) = Dn−1,n−1 , Dn−1,n .

From the construction of μ it follows that: (1) By the one-change lemma (Lemma 6.4), for i ∈ {0, . . . , n − 1} ψi is a qi -change of φi and res(φi ) = res(ψi ), and (2) since, for i ∈ {0, . . . , n − 2}, φi = lcut(ψi ), we get    res(φi+1 ) = Di+1,i+1 , Di+1,i+2 , . . . , Di+1,n = Di,i+1 , Di,i+2 , . . . , Di,n .

(3) The evolving interactive processes φ and ψ from the statement of Theorem 6.2 are related to the sequence ψ0 , ψ1 , . . . , ψn−1 and to φ0 by the following equalities:  f 0 = f0,0 ,

 f 1 = f1,1 ,

  f 2 = f2,2 , . . . , f n−1 = fn−1,n−1 ,

 f n = fn,n ,

f0 = f0,0 ,

f1 = f0,1 ,

f2 = f0,2 , . . . , fn−1 = f0,n−1 ,

fn = f0,n ,

     f0,0 , f1,1 , f2,2 , . . . , fn−1,n−1 , fn,n ,

 fn,n

and ψ = where results from lcut(ψn−1 ) =    fn−1,n by replacing the rule component qn−1,n of fn−1,n by q n . This together with (1) and (2) implies that:  – D0 = D0,0 = D0,0 = D0 ,   – D1 = D1,1 = D1,1 = D0,1 = D0,1 = D1 ,    = D2,2 = D1,2 = D1,2 = D0,2 = D0,2 = D2 , – D2 = D2,2 .. – .    = Dn−1,n−1 = Dn−2,n−1 = . . . = D0,n−1 = – Dn−1 = Dn−1,n−1 D0,n−1 = Dn−1 , and    = Dn,n = Dn−1,n = Dn−1,n = . . . = D0,n = D0,n = Dn . – Dn = Dn,n

29

Therefore D0 = D0 , D1 = D1 , . . . , Dn = Dn . Consequently, res(φ) = res(ψ) and since con(φ) = con(ψ), this implies that also st(φ) = st(ψ). This reasoning is illustrated in Figure 9 where for each φi we show just the sequence of sets of reactions sre(φi ) and for each ψi we show sre(ψi ) and the rule sequence rul (ψi ). Hence the theorem holds.

↓

left cut of ψ0

A1 A1 . . .

A1

ψ1 :

A1 A2 . . . A2 A2 q 1 ←− − trivial −−→

↓

A1



φ1 :

A2

ψ2 :

A2 . . . A3 A3 q 2 ←− − trivial −−→

↓

A2



A2 . . .

...

μ1

left cut of ψ1

φ2 :

μ2

left cut of ψn−2

An−1 An−1

ψn−1 :

An−1 An q n−1

φn :

An

D0,n−1 D0,n

μn−1

D1,1

D2,2

Dn−1,n−1

=



φn−1 :

D0,0 D0,1 D0,2

=

ψ0 : A0 A1 A1 . . . A1 A1 q 0 ←− − trivial −−→

μ0

=

A0

=

A0



φ0 : A0 A0 A0 . . .

2

(Theorem 6.2)

Dn,n

Figure 9: An illustration of our reasoning that res(φ) = res(ψ). Here φn = lcut(ψn−1 ).

30

7. Example In this section we will consider an example which will facilitate an interpretation of the Invisibility Theorem related to evolution theory. Throughout this section we will use the following notation. Let l1 , l2 , l3 ≥ 1. (i)

(i)

• Z (i) = {z1 , . . . , zli }, for i ∈ {1, 2, 3}, are pairwise disjoint sets. • S = {x1 , x2 , x3 , y1 , y2 , y3 , z1 , z2 , z3 }∪Z (1) ∪Z (2) ∪Z (3) is a background set such that the set (Z (1) ∪Z (2) ∪Z (3) ) is disjoint with {x1 , x2 , x3 , y1 , y2 , y3 , z1 , z2 , z3 }. • A, B, H are sets of reactions over S such that A = B ∪ H, B

= { ({x1 , x2 , x3 } , {y1 , y2 , y3 } , {z1 , z2 , z3 }) }, and

H

= H (1) ∪ H (2) ∪ H (3) ,

where: (1)

H (1) = { ({x1 , z1 } , {x2 , x3 } , {zj }) : j ∈ {1, . . . , l1 } }, (2)

H (2) = { ({x2 , z2 } , {x1 , x3 } , {zj }) : j ∈ {1, . . . , l2 } }, and (3)

H (3) = { ({x3 , z3 } , {x1 , x2 } , {zj }) : j ∈ {1, . . . , l3 } }. • B  = { ({x1 } , {y1 } , {z1 }) , ({x2 } , {y2 } , {z2 }) , ({x3 } , {y3 } , {z3 }) }. We want to transform the set of reactions B into B  . We cannot do this in one step, by one transformation rule, as for such a hypothetical transformation rule q = (S, B, D, E) we would have D = B (because B contains only one reaction) and so it would not be true that Beeq(B \ D) (because B \ D = ∅) and consequently q could not be a transformation rule. The desired transformation can be accomplished by a sequence of two transformation rules: q 1 = (S, B, ∅, B  ) followed by q 2 = (S, B ∪ B  , B, ∅). Note that out(q 1 ) = B ∪ B  and out(q 2 ) = B  , so indeed the sequence q 1 , q 2 accomplishes a transformation of B into B  . Consider now a stationary interactive process ω: C0 C1 C2 . . . Cj−1 D0 D1 D2 . . . Dj−1 W0 W1 W2 . . . Wj−1 A0 A1 A2 . . . Aj−1 q0

q1

q2 . . . 31

qj−1

where j ≥ 2, • C0 = C1 = . . . = Cj−1 = {x1 , x2 , x3 }, and • A0 = A (and so A0 = A1 = . . . = Aj−1 = A). Consequently, • D1 = D2 = . . . = Dj−1 = {z1 , z2 , z3 }, • W0 = {x1 , x2 , x3 }, and W1 = W2 = . . . = Wj−1 = {x1 , x2 , x3 , z1 , z2 , z3 }. We extend now ω to an evolving interactive process π: C0 C1 . . . Cj−1 Cj Cj+1 Cj+2 . . . Cn D0 D1 . . . Dj−1 Dj Dj+1 Dj+2 . . . Dn W0 W1 . . . Wj−1 Wj Wj+1 Wj+2 . . . Wn A0 A1 . . . Aj−1 Aj Aj+1 Aj+2 . . . An q0

q1 . . .

qj−1

qj

qj+1

qj+2 . . .

qn

for some n ≥ j + 2, where • Cj = Cj+1 = Cj+2 = . . . = Cn = {x1 , x2 , x3 }, • qj = q 1 , qj+1 = q 2 , and qj+2 , . . . , qn are trivial transformation rules. Let A = Aj+1 and A = Aj+2 (thus A = B ∪ B  ∪ H and A = B  ∪ H). Hence • Aj+2 = Aj+3 = . . . = Aj+n = A . Since C0 = C1 = . . . = Cn and AeeqA and A eeqA , we note that • Dj−1 = Dj = . . . = Dn = {z1 , z2 , z3 } and • Wj−1 = Wj = . . . = Wn = {x1 , x2 , x3 , z1 , z2 , z3 } (as predicted by Theorem 6.2). Now we extend π to three different evolving interaction processes π1 , π2 , π3 as the context sequence con(π) will be extended in such a way that it will split into three different context sequences con(π1 ), con(π2 ), con(π3 ) because of three different continuations of con(π): 32

for some k ≥ 3, (1) (1) (1) Cn+1 = Cn+2 = . . . = Cn+k = {x1 }, (2) (2) (2) Cn+1 = Cn+2 = . . . = Cn+k = {x2 }, and (3) (3) (3) Cn+1 = Cn+2 = . . . = Cn+k = {x3 }. More specifically, for some k ≥ 3, (1) π1 is the evolving interactive process (1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

qn+k

C0 C1 . . . Cj Cj+1 Cj+2 . . . Cn Cn+1 Cn+2 . . . Cn+k D0 D1 . . . Dj Dj+1 Dj+2 . . . Dn Dn+1 Dn+2 . . . Dn+k W0 W1 . . . Wj Wj+1 Wj+2 . . . Wn Wn+1 Wn+2 . . . Wn+k A0 A1 . . . Aj Aj+1 Aj+2 . . . An An+1 An+2 . . . An+k q0

q1 . . .

qj

qj+1

qj+2 . . .

qn

(1)

qn+1

qn+2 . . .

(1)

where (1)

(1)

(1)

• Cn+1 = Cn+2 = . . . = Cn+k = {x1 }, (1)

(1)

(1)

• An = An+1 = An+2 = . . . = An+k = A , and (1)

(1)

(1)

• qn+1 , qn+2 , . . . , qn+k are trivial rules. Consequently, because x1 enables reaction ({x1 }, {y1 }, {z1 }) from B  and x1 inhibits the reactions from H (2) and H (3) , (1)

(1)

(1)

• Dn+2 = Dn+3 = . . . = Dn+k = {z1 } ∪ Z (1) , and (1)

(1)

(1)

(1)

• Wn+2 = Wn+3 = . . . = Wn+k = {x1 } ∪ Dn+k = {x1 , z1 } ∪ Z (1) . (2) π2 is the evolving interactive process (2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

qn+k

C0 C1 . . . Cj Cj+1 Cj+2 . . . Cn Cn+1 Cn+2 . . . Cn+k D0 D1 . . . Dj Dj+1 Dj+2 . . . Dn Dn+1 Dn+2 . . . Dn+k W0 W1 . . . Wj Wj+1 Wj+2 . . . Wn Wn+1 Wn+2 . . . Wn+k A0 A1 . . . Aj Aj+1 Aj+2 . . . An An+1 An+2 . . . An+k q0

q1 . . .

qj

qj+1

qj+2 . . .

where 33

qn

(2)

qn+1

qn+2 . . .

(2)

(2)

(2)

(2)

(2)

(2)

(2)

• Cn+1 = Cn+2 = . . . = Cn+k = {x2 }, • An+1 = An+2 = . . . = An+k = A , and (2)

(2)

(2)

• qn+1 , qn+2 , . . . , qn+k are trivial rules. Consequently, because x2 enables reaction ({x2 }, {y2 }, {z2 }) from B  and x2 inhibits the reactions from H (1) and H (3) , (2)

(2)

(2)

• Dn+2 = Dn+3 = . . . = Dn+k = {z2 } ∪ Z (2) , and (2)

(2)

(2)

(2)

• Wn+2 = Wn+3 = . . . = Wn+k = {x2 } ∪ Dn+k = {x2 , z2 } ∪ Z (2) . (3) π3 is the evolving interactive process (3)

(3)

(3)

(3)

(3)

(3)

(3)

(3)

(3)

(3)

(3)

(3)

(3)

qn+k

C0 C1 . . . Cj Cj+1 Cj+2 . . . Cn Cn+1 Cn+2 . . . Cn+k D0 D1 . . . Dj Dj+1 Dj+2 . . . Dn Dn+1 Dn+2 . . . Dn+k W0 W1 . . . Wj Wj+1 Wj+2 . . . Wn Wn+1 Wn+2 . . . Wn+k A0 A1 . . . Aj Aj+1 Aj+2 . . . An An+1 An+2 . . . An+k q0

q1 . . .

qj

qj+1

qj+2 . . .

qn

(3)

qn+1

qn+2 . . .

(3)

where (3)

(3)

(3)

(3)

(3)

(3)

• Cn+1 = Cn+2 = . . . = Cn+k = {x3 }, • An+1 = An+2 = . . . = An+k = A , and (3)

(3)

(3)

• qn+1 , qn+2 , . . . , qn+k are trivial rules. Consequently, because x3 enables reaction ({x3 }, {y3 }, {z3 }) from B  and x3 inhibits the reactions from H (1) and H (2) , (3)

(3)

(3)

• Dn+2 = Dn+3 = . . . = Dn+k = {z3 } ∪ Z (3) , and (3)

(3)

(3)

(3)

• Wn+2 = Wn+3 = . . . = Wn+k = {x3 } ∪ Dn+k = {x3 , z3 } ∪ Z (3) . We note that the three sets {x1 , z1 } ∪ Z (1) , {x2 , z2 } ∪ Z (2) , and {x3 , z3 } ∪ Z are pairwise disjoint, and so (3)

(1)

(1)

• the set of states {Wn+2 , . . . , Wn+k } (where each state equals {x1 , z1 } ∪ Z (1) ), 34

(2)

(2)

(3)

(3)

• the set of states {Wn+2 , . . . , Wn+k } (where each state equals {x2 , z2 } ∪ Z (2) ), and • the set of states {Wn+2 , . . . , Wn+k } (where each state equals {x3 , z3 } ∪ Z (3) ) are pairwise disjoint. Consider now stationary processes π1 , π2 , π3 which differ from interactive processes π1 , π2 , π3 by the fact that also qj and qj+1 are trivial transformation rules (while in π1 , π2 , π3 , we have qj = q 1 and qj+1 = q 2 ). This means that in all three interactive processes the set of reactions in each instantaneous description equals A = B ∪ H. We will have then: (1) in the interactive process π1 : (1 )

(1 )

(1 )

(1 )

• Cn+1 = Cn+2 = Cn+3 = . . . = Cn+k = {x1 }, (1 )

(1 )

(1 )

(1 )

• Dn+1 = {z1 , z2 , z3 }, Dn+2 = Z (1) , Dn+3 = . . . = Dn+k = ∅, (1 )

(1 )

(1 )

(1 )

(1 )

• Wn+1 = {x1 , z1 , z2 , z3 }, Wn+2 = {x1 } ∪ Dn+2 , Wn+3 = . . . = Wn+k = {x1 }. (2) in the interactive process π2 : (2 )

(2 )

(2 )

(2 )

• Cn+1 = Cn+2 = Cn+3 = . . . = Cn+k = {x2 }, (2 )

(2 )

(2 )

(2 )

• Dn+1 = {z1 , z2 , z3 }, Dn+2 = Z (2) , Dn+3 = . . . = Dn+k = ∅, (2 )

(2 )

(2 )

(2 )

(2 )

• Wn+1 = {x2 , z1 , z2 , z3 }, Wn+2 = {x2 } ∪ Dn+2 , Wn+3 = . . . = Wn+k = {x2 }. (3) in the interactive process π3 : (3 )

(3 )

(3 )

(3 )

• Cn+1 = Cn+2 = Cn+3 = . . . = Cn+k = {x3 }, (3 )

(3 )

(3 )

(3 )

• Dn+1 = {z1 , z2 , z3 }, Dn+2 = Z (3) , Dn+3 = . . . = Dn+k = ∅, (3 )

(3 )

(3 )

(3 )

(3 )

• Wn+1 = {x3 , z1 , z2 , z3 }, Wn+2 = {x3 } ∪ Dn+2 , Wn+3 = . . . = Wn+k = {x3 }.

35

There are various ways of interpreting organisms and species within the framework of evolving reaction systems. This topic is more suitable for a publication in a biology-related journal, but we will give now one such interpretation here. From a chemical point of view, a class of organisms F may be represented by a set of reactions G taking place within organisms in F. Given an evolving interactive process ρ with st(ρ) = W0 , . . . , Wn for some n ≥ 1, we say that G (and hence F) lives in ρ if G is enabled in each Wi , 0 ≤ i ≤ n. We note that in our example a sequential development pattern represented by π changes into a branching pattern of three processes π1 , π2 , π3 resulting from the branching of the environment/context from Cn into three (1) (2) (3) different contexts Cn+1 , Cn+1 , and Cn+1 . Then very soon (immediately after(1) (2) (3) wards, beginning with Wn+2 , Wn+2 , and Wn+2 , respectively) the three state sequences st(π1 ), st(π2 ), and st(π3 ) become disjoint, hence the groups of organisms living in π1 , π2 , π3 are disjoint. It is important to notice here that the three result sequences res(π1 ), res(π2 ), and res(π3 ) consist of nonempty sets only. Thus a speciation (a formation of new species) into three new species has happened. The fact that this has happened and then so quickly after the step n+1 is due to the fact that invisible changes (not observable in state sequences) were happening in the past in π (and these changes in general could have been happening over a long period of time). In our example these were changes from Aj to Aj+1 and from Aj+1 to Aj+2 . On the other hand in interactive stationary processes, π1 , π2 , π3 no reactions from the given constant set of reactions A0 = A are enabled from state n + 2 onwards, and so no organisms can live in π1 , π2 , and π3 from this state onwards. Hence we got here an extinction of species. The difference results from the fact that there was no silent/invisible evolution present in the past (i.e., in transitions from state j to state j + 1, and from state j + 1 to state j + 2). Presenting the theory of evolution, see e.g., [19], in a very simplified form, one can say that the Darwinian evolution is based on gradual changes: small changes of environment over time cause small changes in organisms. Therefore speciation happens gradually, very slowly. In the theory of punctuated evolution proposed by N. Eldredge and S.J. Gould, see e.g., [13, 19], evolution can happen rapidly when the environment branches into many possible environments (niches). This rapid evolution seems to contradict the principle 36

of (slow) gradualism. Within the framework of evolving reaction systems the principle of punctuated evolution may be reconciled with the principle of gradualism through the Invisibility Theorem. The rapid evolutionary changes following branchings of the environment could have been prepared for a very long time through “silent evolution” — changes which are not observable in phenotype (which is the set of observable characteristics). 8. Discussion In this paper we have introduced and investigated evolving interactive processes which generalize standard interactive processes of reaction systems by allowing the set of available reactions to evolve as a process progresses from state to state. The main technical focus of the paper is the Invisibility Theorem which allows for an evolution of a system which is not externally observable. We have also indicated a possible relationship between the Invisibility Theorem and the notion of punctuated evolution from evolution theory. Obviously, this is only a beginning of developing the framework of evolving reaction systems. A systematic investigation of this new framework could begin by investigating central research themes concerning reaction systems in the framework of evolving reaction systems. The topics/themes that come to mind include • properties of state sequences, see e.g., [5, 14, 16–18], • modularity, see e.g., [10], • duration, see e.g., [3], • minimizing resources, see e.g., [6, 18], • use of evolving reaction systems in investigating biological processes, see e.g., [1, 2, 8]. The notion of enabling equivalence introduced in this paper deserves a thorough systematic investigation. Results presented in Section 3 form a good starting point for such an investigation. An important topic, especially suited for evolving reaction systems, is concerned with control sequences: how properties of the sequence of available 37

sets of reactions (A0 , A1 , . . . , An ), such as e.g., periodicity, are reflected in the state sequence of evolving interactive processes. Acknowledgements The authors are indebted to the anonymous referees and to R.Brijder for valuable comments concerning this paper. References [1] S. Azimi, B. Iancu, I. Petre, Reaction system models for the heat shock response, Fundamenta Informaticae 131 (2014) 1–14. [2] S. Azimi, C. Panchal, E. Czeizler, I. Petre, Reaction systems models for the self-assembly of intermediate filaments, Annals of University of Bucharest LXII (2) (2015) 9–24. [3] R. Brijder, A. Ehrenfeucht, G. Rozenberg, Reaction systems with duration, in: J. Kelemen, A. Kelemenová (Eds.), Computation, Cooperation, and Life, vol. 6610 of Lecture Notes in Computer Science, Springer, 191– 202, 2011. [4] J. Dassow, G. Paun, Regulated Rewriting in Formal Language Theory, Springer, 1989. [5] A. Dennunzio, E. Formenti, L. Manzoni, A. E. Porreca, Ancestors, descendants, and gardens of Eden in reaction systems, Theoretical Computer Science 608 (2015) 16–26. [6] A. Ehrenfeucht, J. Kleijn, M. Koutny, G. Rozenberg, Minimal reaction systems, in: C. Priami, I. Petre, E. de Vink (Eds.), Transactions on Computational Systems Biology XIV SE - 5, vol. 7625 of Lecture Notes in Computer Science, Springer, 102–122, 2012. [7] A. Ehrenfeucht, J. Kleijn, M. Koutny, G. Rozenberg, Qualitative and quantitative aspects of a model for processes inspired by the functioning of the living cell, in: E. Katz (Ed.), Biomolecular Information Processing: From Logic Systems to Smart Sensors and Actuators, Wiley-VCH Verlag, 303–322, 2012.

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[8] A. Ehrenfeucht, J. Kleijn, M. Koutny, G. Rozenberg, Reaction systems: A natural computing approach to the functioning of living cells, in: Z. Hector (Ed.), A Computable Universe: Understanding and Exploring Nature as Computation, World Scientific, 189–208, 2012. [9] A. Ehrenfeucht, I. Petre, G. Rozenberg, Reaction systems: A model of computation inspired by the functioning of the living cell, in: S. Konstantinidis, N. Moreira, R. Reis, J. Shallit (Eds.), The Role of Theory in Computer Science, Essays dedicated to Janusz Brzozowski, World Scientific, 2017. [10] A. Ehrenfeucht, G. Rozenberg, Events and modules in reaction systems, Theoretical Computer Science 376 (1-2) (2007) 3–16. [11] A. Ehrenfeucht, G. Rozenberg, Reaction systems, Fundamenta Informaticae 75 (1) (2007) 263–280. [12] A. Ehrenfeucht, G. Rozenberg, Zoom structures and reaction systems yield exploration systems, International Journal of Foundations of Computer Science 25 (3) (2014) 275–306. [13] N. Eldredge, Time Frames: The evolution of punctuated equilibria, Princeton University Press, 1985. [14] E. Formenti, L. Manzoni, A. E. Porreca, Fixed points and attractors of reaction systems, in: A. Beckmann, E. Csuhaj-Varjú, K. Meer (Eds.), Language, Life, Limits, vol. 8493 of Lecture Notes in Computer Science, Springer, 194–203, 2014. [15] A. L. Lehninger, Bioenergetics: The Molecular Basis of Biological Energy Transformations, Addison-Wesley, 1971. [16] A. Salomaa, Functions and sequences generated by reaction systems, Theoretical Computer Science 466 (2012) 87–96. [17] A. Salomaa, On state sequences defined by reaction systems, in: R. L. Constable, A. Silva (Eds.), Logic and Program Semantics, vol. 7230 of Lecture Notes in Computer Science, Springer, 271–282, 2012. [18] A. Salomaa, Minimal and almost minimal reaction systems, Natural Computing 12 (3) (2013) 369–376. 39

[19] P. Whitfield, From So Simple a Beginning: The Book of Evolution, Macmillan, 1995.

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