Simulation Of Reaction Systems By The Strictly Minimal Ones
aa r X i v : . [ c s . L O ] F e b SIMULATION OF REACTION SYSTEMS BY THESTRICTLY MINIMAL ONES
WEN CHEAN TEH AND ADRIAN ATANASIU
Abstract.
Reaction systems, introduced by Ehrenfeucht and Rozenberg, areelementary computational models based on biochemical reactions transpiringwithin the living cells. Numerous studies focus on mathematical aspects ofminimal reaction systems due to their simplicity and rich generative power. In2014 Manzoni, Pocas, and Porreca showed that every reaction system can besimulated by some minimal reaction system over an extended background set.Motivated by their work, we introduce the concepts of strictly minimal andhybrid reaction systems. Using our new concepts, the result of Manzoni et al. isrevisited and strengthened. We also show that extension of the backgroundset by polynomially bounded many elements is not sufficient to guarantee theaforementioned simulation. Finally, an analogous result for strong simulation isobtained. Introduction
Reaction systems, introduced in 2007 by Ehrenfeucht and Rozenberg [5], areelementary computational models inspired by biochemical reactions taking placewithin the living cells. This study belongs to one of the diverse research linesinitiated in [3] that pertains to mathematical study of state transition functionsspecified by reaction systems, called rs functions. For a motivational survey onreaction systems, we refer the reader to Ehrenfeucht, Petre, and Rozenberg [4].Minimal reaction systems [2], where the number of resources in each reactionis minimal, have been relatively well-studied due to their simplicity. Salomaa [8]initiated the study on the generative power under composition of rs functionsspecified by minimal reaction systems and later we showed that not every rs function can be thus generated for the quarternary alphabet [13]. On the otherhand, Manzoni, Pocas, and Porreca [6] introduced the study of simulation byreaction systems and they showed that every rs function can be simulated bysome minimal reaction system over an extended background set. Other studies onmathematical properties of minimal reaction systems include [1, 9, 11, 15].This study refines and expands on the study of simulation by reaction systemsinitiated in [6]. We propose strictly minimal reaction systems as a new canonicalclass of reaction systems. We also introduce hybrid reaction systems, where thereactant set, inhibitor set, and product set of each reaction are allowed to containentities from different background sets. Then the result by Manzoni et al. is revis-ited and strengthened by showing that the extended background set can be fixed Key words and phrases.
Natural computing; simulation; reaction systems; universality ofminimal resources. ahead independent of the given rs function. Next, we show that the number ofextra resources needed in the fixed extended background set cannot be boundedpolynomially in terms of the size of the original background set. Finally, a strongerversion of simulation is studied and it will be shown that minimal reaction sys-tems are in fact rich enough to strongly simulate every rs function over a givenbackground set. 2. Preliminaries If S is any finite set, then the cardinality of S is denoted by ∣ S ∣ and the powerset of S is denoted by 2 S .From now onwards, unless stated otherwise, S is a fixed finite nonempty set. Definition 2.1. A reaction in S is a triple a = ( R a , I a , P a ) , where R a and I a are(possibly empty) disjoint subsets of S and P a is a nonempty subset of S . The sets R a , I a , and P a are the reactant set , inhibitor set , and product set respectively. Thepair ( R a , I a ) is the core of a . Definition 2.2. A reaction system over S is a pair A = (
S, A ) where S is calledthe background set and A is a (possibly empty) set of reactions in S . We saythat A is nondegenerate if R a and I a are both nonempty for every a ∈ A and A is maximally inhibited if I a = S / R a for every a ∈ A . Definition 2.3.
Suppose
A = (
S, A ) is a reaction system. The state transitionfunction res A ∶ S → S is defined byres A ( X ) = ⋃ { P a ∣ a ∈ A such that R a ⊆ X and I a ∩ X = ∅ } , for all X ⊆ S. If R a ⊆ X and I a ∩ X = ∅ , we say that the reaction a is enabled by X . Hence,res A ( X ) is the cumulative union of product sets of all reactions enabled by X . Forour purpose and without loss of generality, we may assume that distinct reactionsin A do not have the same core. Definition 2.4.
Every function f ∶ S → S is called an rs function over S . Wesay that f can be specified by a reaction system A over S if f = res A .Since every rs function over S can be canonically specified by a unique max-imally inhibited reaction system over S , it follows that the class of rs functionsover S is exactly the class of state transition functions over S . Definition 2.5. [2,15] Suppose
A = (
S, A ) is reaction system. Then A is minimal if ∣ R a ∣ ≤ ∣ I a ∣ ≤ a ∈ A .The elements in the reactant set or inhibitor set of a reaction a are called the resources of a . The classification of any reaction system according to the totalnumber of resources allowed in each of its reactions was initiated by Ehrenfeucht,Main, and Rozenberg [3]. From time to time, nondegeneracy has been a naturallyadopted convention. Hence, every minimal reaction system satisfies ∣ R a ∣ = ∣ I a ∣ = a ∈ A in the early studies. A characterization of rs functions that canbe specified by minimal reaction systems was obtained by Ehrenfeucht, Kleijn, IMULATION OF REACTION SYSTEMS BY THE STRICTLY MINIMAL ONES 3
Koutny, and Rozenberg [2]. Later the same characterization was extended in [15]to cover for degenerate reaction systems as well. We present this characterizationdue to its historical significance.
Theorem 2.6. [2, 15] Suppose f is an rs function over S . Then f = res A forsome (possibly degenerate) minimal reaction system A if and only if f satisfies thefollowing two properties: ● (Union-subadditivity) f ( X ∪ Y ) ⊆ f ( X ) ∪ f ( Y ) for all X, Y ⊆ S ; ● (Intersection-subadditivity) f ( X ∩ Y ) ⊆ f ( X ) ∪ f ( Y ) for all X, Y ⊆ S . The following definition of simulation was introduced by Manzoni et al. [6].
Definition 2.7.
Suppose f is an rs function over S and k is a positive integer.Suppose S ⊆ S ′ and A is a reaction system over S ′ . We say that f can be k -simulated by A if for every X ⊆ S , f n ( X ) = res kn A ( X ) ∩ S for all positive integers n. The following observation says that 1-simulation do not add to the expressivepower of reaction systems.
Proposition 2.8.
Suppose f is an rs function over S and S ⊆ S ′ . Suppose f can be -simulated by some reaction system A ′ = ( S ′ , A ′ ) over S ′ . Then f can bespecified by the reaction system A = (
S, A ) over S , where A = { ( R a , I a ∩ S, P a ∩ S ) ∣ a ∈ A ′ and R a ⊆ S } . Proof.
The definition of 1-simulation implies that f ( X ) = res A ′ ( X ) ∩ S for every X ⊆ S . Fix an arbitrary X ⊆ S . It suffices to show that res A ( X ) = res A ′ ( X ) ∩ S .Suppose x ∈ res A ( X ) . Then ( R a , I a ∩ S, P a ∩ S ) is enabled by X for some a ∈ A ′ with x ∈ P a ∩ S . It follows that a = ( R a , I a , P a ) is enabled by X because X ⊆ S and thus x ∈ res A ′ ( X ) ∩ S . Conversely, if x ∈ res A ′ ( X ) ∩ S . Then a = ( R a , I a , P a ) is enabled by X for some a ∈ A ′ with x ∈ P a . Hence, ( R a , I a ∩ S, P a ∩ S ) ∈ A isenabled by X . Since x ∈ P a ∩ S , it follows that x ∈ res A ( X ) . (cid:3) Manzoni et al. [6] showed that minimal reaction systems are rich enough for thepurpose of simulation. Their result serves as the main motivation for this study.We observe that the number of resources in each reaction of the reaction systemconstructed in their proof is actually one. Therefore, we introduce the followingdefinition before stating what they have actually shown.
Definition 2.9.
Suppose
A = (
S, A ) is reaction system. Then A is strictly minimal if ∣ R a ∪ I a ∣ ≤ a ∈ A . Theorem 2.10. [6] Suppose f is an rs function over S . Then there exists astrictly minimal reaction system B over some S ′ ⊇ S such that f can be -simulatedby B . IMULATION OF REACTION SYSTEMS BY THE STRICTLY MINIMAL ONES 4 Hybrid Reaction Systems
There are studies on mathematical properties of reaction systems, for example,the totalness of state transition functions and the functional completeness of thereaction systems as in Salomaa [7], where the properties do not depend on thenonempty product sets. More importantly, the reactant set, inhibitor set, andproduct set of each reaction in the reaction system constructed in the proof ofTheorem 2.10 appear to contain entities of different nature. These observationsmotivate our definition of hybrid reaction system, where the output elements areallowed to come from a different background set whenever a reaction is enabled.
Definition 3.1.
Suppose S and T are finite nonempty sets. An ( S, T ) -reaction isa triple of sets a = ( R a , I a , P a ) such that R a and I a are (possibly empty) disjointsubsets of S and P a is a nonempty subset of T . A hybrid reaction system over ( S, T ) is a triple A = ( S, T, A ) where S and T are the background sets and A is a(possibly empty) set of ( S, T ) -reactions.Obviously, a hybrid reaction system over ( S, T ) becomes a (normal) reactionsystem when S = T . Basic terminology of reaction systems carries over to hybridreaction systems analogously. Hence, the reader is assumed to know, for example,the definition of the state transition function res A and what it means for A to bemaximally inhibited when A is a hybrid reaction system. Furthermore, every rs function f ∶ S → T can be canonically specified by a unique maximally inhibitedhybrid reaction system over ( S, T ) .The following theorem says that every reaction system can be naturally de-composed into two strictly minimal hybrid reaction systems. This theorem isessentially extracted from the proof of Theorem 2.10. Theorem 3.2.
Suppose A = ( S, A ) is a reaction system. Let T = { ¯ a ∣ a ∈ A } where ¯ a is a distinguished symbol for each a ∈ A . Let C = { (∅ , { x } , { ¯ a }) ∣ a ∈ A and x ∈ R a } ∪ { ({ y } , ∅ , { ¯ a }) ∣ a ∈ A and y ∈ I a } and D = { (∅ , { ¯ a } , P a ) ∣ a ∈ A } . Then C = ( S, T, C ) and D = ( T, S, D ) are strictlyminimal hybrid reaction systems such that res A = res D ○ res C .Proof. Note that res D ( Y ) = ⋃ { P a ∣ ¯ a ∈ T / Y } for every Y ⊆ T . Therefore, itsuffices to show thatres C ( X ) = { ¯ a ∈ T ∣ a is not enabled by X } for all X ⊆ S because res A = res D ○ res C would then follow immediately.Suppose X ⊆ S and a ∈ A . By definition, a is not enabled by X if and onlyif R a ⊈ X or I a ∩ X ≠ ∅ . If x ∈ R a / X , then (∅ , { x } , { ¯ a }) ∈ C is enabled by X .Similarly, if y ∈ I a ∩ X , then ({ y } , ∅ , { ¯ a }) ∈ C is enabled by X . It follows that¯ a ∈ res C ( X ) whenever a is not enabled by X .Conversely, suppose ¯ a ∈ res C ( X ) . Then some c ∈ C such that P c = { ¯ a } is enabledby X . If c = (∅ , { x } , { ¯ a }) for some x ∈ R a , then x ∈ R a / X and so R a ⊈ X . Similarly,if c = ({ y } , ∅ , { ¯ a }) for some y ∈ I a , then y ∈ I a ∩ X and so I a ∩ X ≠ ∅ . It followsthat a is not enabled by X whenever ¯ a ∈ res C ( X ) . (cid:3) IMULATION OF REACTION SYSTEMS BY THE STRICTLY MINIMAL ONES 5
In view of the proof of Theorem 3.2, it is intriguing whether there are C and D such that res C ( X ) is the set of ¯ a such that a is enabled by X and res A = res D ○ res C .Trivially, we can take C = { ( R a , I a , { ¯ a }) ∣ a ∈ A } and D = { ({ ¯ a } , ∅ , P a ) ∣ a ∈ A } .However, it will only be interesting if such C exists where its complexity is lessthan A . By our next claim, this is not possible. Claim.
Suppose C = ( S, T, C ) is any hybrid reaction system such that res C ( X ) isthe set of ¯ a such that a is enabled by X for each X ⊆ S . Then for every reaction c ∈ C , if ¯ a ∈ P c , then R a ⊆ R c and I a ⊆ I c .Proof. Suppose c ∈ C and ¯ a ∈ P c . Clearly, c is enabled by R c and thus ¯ a ∈ res C ( R c ) .By the hypothesis, a is enabled by R c , implying that R a ⊆ R c . Similarly, c isenabled by S / I c and thus ¯ a ∈ res C ( S / I c ) . By the hypothesis again, a is enabled by S / I c , implying that ( S / I c ) ∩ I a = ∅ and thus I a ⊆ I c . (cid:3) Theorem 3.2 justifies the canonicalness of strictly minimal hybrid reaction sys-tems, as functions specified by them can generate every rs function f over S undercomposition. However, the hybrid reaction system C as in the theorem depends on A such that res A = f . Therefore, the next theorem is a variation of Theorem 3.2where the hybrid reaction system C is independent from f . This theorem is es-sentially implied by the proof of Theorem 4 in Salomaa [10], although over thereany reaction system is required to be nondegenerate. The main idea is to give aname to each subset of the background set. An alternative original proof of thisnext theorem can be found in [12]. Theorem 3.3.
Let T = { N X ∣ X ⊆ S } where N X is a distinguished symbol foreach X ⊆ S . Let C = { (∅ , { x } , { N X }) ∣ X ⊆ S and x ∈ X }∪{ ({ y } , ∅ , { N X }) ∣ X ⊆ S and y ∈ S / X } . Suppose f is an rs function over S . Let D = { (∅ , { N X } , f ( X )) ∣ X ⊆ S and f ( X ) ≠ ∅ } . Then C = ( S, T, C ) and D = ( T, S, D ) are strictly minimal hybrid reaction systemssuch that res D ○ res C = f .Proof. Let A = ( S, A ) be the canonical maximally inhibited reaction system suchthat f = res A , that is, where A = { ( X, S / X, f ( X )) ∣ X ⊆ S and f ( X ) ≠ ∅ } . Every X ⊆ S can be uniquely associated to the reaction a X = ( X, S / X, f ( X )) . Let N X be the distinguished symbol a X for each X ⊆ S . Then it can be verified that C = { (∅ , { x } , { ¯ a }) ∣ a ∈ A and x ∈ R a } ∪ { ({ y } , ∅ , { ¯ a }) ∣ a ∈ A and y ∈ I a } and D = { (∅ , { ¯ a } , P a ) ∣ a ∈ A } . Therefore, by Theorem 3.2, it follows thatres D ○ res C = res A = f . (cid:3) Using Theorem 3.3 and adapting the proof of Theorem 2.10, we now strengthenTheorem 2.10 by showing that the extended background set for the simulatingreaction system can be chosen ahead independent from the given rs function.Before that, we need a lemma. IMULATION OF REACTION SYSTEMS BY THE STRICTLY MINIMAL ONES 6
Lemma 3.4.
Suppose C = ( S, S ′ , C ) and D = ( T, T ′ , D ) are hybrid reaction sys-tems. Let A be the hybrid reaction system ( S ∪ T, S ′ ∪ T ′ , C ∪ D ) . Then res A ( X ) = res C ( X ∩ S ) ∪ res D ( X ∩ T ) , for all X ⊆ S ∪ T. Proof.
Suppose X ⊆ S ∪ T . Thenres A ( X ) = ⋃ c ∈ CR c ⊆ X,I c ∩ X =∅ P c ∪ ⋃ d ∈ DR d ⊆ X,I d ∩ X =∅ P d = ⋃ c ∈ CR c ⊆ X ∩ S,I c ∩( X ∩ S )=∅ P c ∪ ⋃ d ∈ DR d ⊆ X ∩ T,I d ∩( X ∩ T )=∅ P d = res C ( X ∩ S ) ∪ res D ( X ∩ T ) . (cid:3) Theorem 3.5.
There exists a fixed set S ′ ⊇ S such that every rs function over S can be -simulated by some strictly minimal reaction system over S ′ .Proof. Let T = { N X ∣ X ⊆ S } as in Theorem 3.3 and S ′ = S ∪ T . Then S ′ is a fixedbackground set extending S . Suppose f is an rs function over S . Let C and D beas in Theorem 3.3. Consider the reaction system A = ( S ′ , C ∪ D ) over S ′ . Clearly, A is strictly minimal. Claim.
For every integer n ≥ and every X ⊆ S ′ , res n A ( X ) ∩ S = ( res D ○ res C )( res n − A ( X ) ∩ S ) , where res A ( X ) = X .Proof of the claim. We argue by mathematical induction. For the base step, sup-pose X ⊆ S ′ . By Lemma 3.4, res A ( X ) = res C ( X ∩ S ) ∪ res D ( X ∩ T ) andres A ( X ) = res A ( res A ( X )) = res C ( res A ( X ) ∩ S ) ∪ res D ( res A ( X ) ∩ T ) . Since res C ∶ S → T , res D ∶ T → S , and S ∩ T = ∅ , it follows thatres A ( X ) ∩ S = res D ( res A ( X ) ∩ T ) = res D ( res C ( X ∩ S )) = ( res D ○ res C )( X ∩ S ) . Thus the base step is complete.For the induction step, suppose X ⊆ S ′ . Thenres n + A ( X ) ∩ S = res A ( res n − A ( X )) ∩ S = ( res D ○ res C )( res n − A ( X ) ∩ S ) . The last equality follows from the base step. The induction step is complete.The fact that f can be 2-simulated by A follows immediately from the nextclaim. Claim.
For every positive integer n and every X ⊆ S ′ , f n ( X ∩ S ) = res n A ( X ) ∩ S. IMULATION OF REACTION SYSTEMS BY THE STRICTLY MINIMAL ONES 7
Proof of the claim.
We argue by mathematical induction. By Theorem 3.3 andthe previous claim, res A ( X ) ∩ S = ( res D ○ res C )( X ∩ S ) = f ( X ∩ S ) for all X ⊆ S ′ ,thus the base step is done. For the induction step, suppose X ⊆ S ′ . By theprevious claim again, res ( n + )A ( X ) ∩ S = ( res D ○ res C )( res n A ( X ) ∩ S ) , which equals f ( f n ( X ∩ S )) = f n + ( X ∩ S ) by Theorem 3.3 and the induction hypothesis.Therefore, the proof is complete. (cid:3) Extension of Background Set is Necessary
In view of Theorem 3.5, the following theorem says that S ′ needs to properlyextend S if every rs function over S is to be k -simulated for some positive integer k by some strictly minimal reaction system over S ′ . Theorem 4.1. [6] Suppose ∣ S ∣ ≥ . There exists an rs function over S thatcannot be k -simulated for any positive integer k by any minimal reaction systemover S . The next lemma is a generalization of what was actually shown by the first halfof the proof of Theorem 4.1.
Lemma 4.2.
Suppose ∣ S ∣ = n ≥ and let X , X , . . . , X n be any enumeration ofall the subsets of S . Let f be the rs function over S defined by f ( X i ) = { X i + if ≤ i < n X n if i = n . Suppose S ′ is a finite background set extending S . Then f cannot be k -simulatedby any reaction system over S ′ whenever k > ∣ S ′∣ − ∣ S ∣ − .Proof. We argue by contradiction. Suppose A is an arbitrary reaction system over S ′ and k is an arbitrary integer such that k > ∣ S ′∣ − ∣ S ∣ − . Assume f can be k -simulatedby A . Consider the state sequence X , res A ( X ) , res A ( X ) , . . . generated by A with initial state X . Since k ⋅ ( n − ) + > ∣ S ′ ∣ , the following initial terms X , res A ( X ) , res A ( X ) , . . . , res k ( n − )+ A ( X ) cannot be all distinct subsets of S ′ . Hence, res k ( n − )A ( X ) is part of a cycle, saywith period p ≥
1. Thenres k ( n − + p )A ( X ) = res kp A ( res k ( n − )A ( X )) = res k ( n − )A ( X ) and thus res k ( n − + p )A ( X ) ∩ S = res k ( n − )A ( X ) ∩ S = f n − ( X ) = X n − . However, res k ( n − + p )A ( X ) ∩ S = f n − + p ( X ) = X n , which is a contradiction. (cid:3) The conclusion of Theorem 4.1 is true for ∣ S ∣ = S = { a, b } andconsider the rs function f defined by f ({ a }) = { b } , f ({ b }) = ∅ , f ( ∅ ) = S , and f ( S ) = S . By Lemma 4.2, f cannot be k -simulated by any reaction system over IMULATION OF REACTION SYSTEMS BY THE STRICTLY MINIMAL ONES 8 S whenever k >
1. Since f ( S ) / ⊆ f ({ a }) ∪ f ({ b }) , it follows that f is not union-subadditive and thus cannot be 1-simulated (equivalently, specified) by any mini-mal reaction system over S by Theorem 2.6.In view of Theorem 3.5, we now show that when the background set S isextended by a fixed finite number of elements, it is not generally sufficient to2-simulate every rs function over S by some strictly minimal reaction systemsover the extended background set. In fact, the following much stronger statementholds. Theorem 4.3.
No polynomial p has the property that for every set S , there existsa set S ′ ⊇ S with ∣ S ′ ∣ ≤ p (∣ S ∣) such that every rs function over S can be k -simulatedby some strictly minimal reaction system over S ′ for some positive integer k .Proof. First of all, notice that the rs function f as defined in Lemma 4.2 is uniquelydetermined by the corresponding sequence X , X , . . . , X ∣ S ∣ . Hence, by some sim-ple counting argument, there are 2 ∣ S ∣ ! distinct such rs functions over S . On theother hand, there are ( ∣ S ′ ∣ ) ∣ S ′ ∣+ distinct strictly minimal reaction systems over S ′ because there are 2 ∣ S ′ ∣ + ( ∅ , ∅ ) ) forreactions in such reaction systems.Fix a polynomial p . Suppose S ⊆ S ′ and ∣ S ′ ∣ ≤ p (∣ S ∣) . Since 2 p (∣ S ∣) ≥ ∣ S ′ ∣ > ∣ S ′∣ − ∣ S ∣ − (for ∣ S ∣ ≥ f cannot be k -simulated by any reactionsystem over S ′ whenever k > p (∣ S ∣) (in fact, whenever k > ∣ S ′∣ − ∣ S ∣ − ). Meanwhile, forevery positive integer k , by definition, there are at most ( p (∣ S ∣) ) p (∣ S ∣)+ rs functionsover S that can be k -simulated by some strictly minimal reaction system over S ′ .Therefore, at most 2 p (∣ S ∣) ⋅ ( p (∣ S ∣) ) p (∣ S ∣)+ rs functions over S can be k -simulatedby some strictly minimal reaction system over S ′ for some k ≤ p (∣ S ∣) . When ∣ S ∣ issufficiently large, 2 p (∣ S ∣) ⋅ ( p (∣ S ∣) ) p (∣ S ∣)+ = p (∣ S ∣)( p (∣ S ∣)+ ) < ∣ S ∣ ! , and thus it follows that not all of the 2 ∣ S ∣ ! rs functions over S as defined inLemma 4.2 can be k -simulated by some strictly minimal reaction system over S ′ for some positive integer k . (cid:3) Strong k -Simulation by Reaction Systems Now, we formally study a stronger version of k -simulation, which was first pro-posed in the conclusion section of [6]. Definition 5.1.
Suppose f is an rs function over S and k is a positive integer.Suppose S ⊆ S ′ and A is a reaction system over S ′ . We say that f can be strongly k -simulated by A iff f ( X ) = res k A ( X ) for all X ⊆ S .Some direct computation shows that the 2-simulating strictly minimal reac-tion system constructed in the proof of Theorem 3.5 generally does not strongly2-simulate the given rs function.The following is a reformulation of Theorem 3 in [10], which is a strong versionanalogue of Theorem 2.10. IMULATION OF REACTION SYSTEMS BY THE STRICTLY MINIMAL ONES 9
Theorem 5.2. [10] Suppose f is an rs function over S such that f ( ∅ ) = ∅ .Then there exists a minimal reaction system B over some S ′ ⊇ S such that f canbe strongly -simulated by B . Salomaa adopted the convention that every reaction system is nondegenerate.Relaxing this constraint, we strengthen Theorem 5.2, not only by having a fixedextended background set independent from the given rs function, but also bygeneralizing it to every rs function over S . First, we need a technical lemma,which is an adaptation of Theorem 3.3 to suit our purpose. Lemma 5.3.
Let T = { N X ∣ ∅ ≠ X ⊆ S } ∪ { ∗ , ◇ } , where N X is a distinguishedsymbol for each ∅ ≠ X ⊆ S . Let C = { ({ y } , ∅ , { N X }) ∣ ∅ ≠ X ⊆ S and y ∈ S / X } ∪ { ({ s } , ∅ , { ∗ }) ∣ s ∈ S } ∪ { ({ x } , { x ′ } , { N X }) ∣ X ⊆ S and x, x ′ ∈ X with x ≠ x ′ } ∪ {( ∅ , { ◇ } , { ◇ })} . Suppose f is an rs function over S . Let D = { ({ ∗ } , { N X } , f ( X )) ∣ ∅ ≠ X ⊆ S and f ( X ) ≠ ∅ } ∪ {({ ◇ } , { ∗ } , f ( ∅ ))} . Then C = ( S ∪ { ◇ } , T, C ) and D = ( T, S, D ) are hybrid minimal reaction systemssuch that ( res D ○ res C )( X ) = f ( X ) for all X ⊆ S .Proof. Note that res D ( res C ( ∅ )) = res D ({ ◇ }) = f ( ∅ ) . Suppose X is a nonemptysubset of S . It suffices to show that res C ( X ) = T /{ N X } because then only thereaction ({ ∗ } , { N X } , f ( X )) is enabled by res C ( X ) , provided f ( X ) ≠ ∅ , and thusres D ( res C ( X )) = f ( X ) .Suppose Y is a nonempty subset of S distinct from X . If X ⊈ Y , say x ∈ X / Y ,then the reaction ({ x } , ∅ , { N Y }) is enabled by X . Otherwise, if X ⊆ Y and so Y ⊈ X , then the reaction ({ y } , { y ′ } , { N Y }) is enabled by X for any y ∈ X (≠ ∅ ) and y ′ ∈ Y / X . In either case, N Y ∈ res C ( X ) .On the other hand, none of the reactions in C with product set being { N X } isenabled by X . Furthermore, { ∗ , ◇ } ⊆ res C ( X ) because X is a nonempty subset of S . Therefore, res C ( X ) = T /{ N X } as required. (cid:3) Theorem 5.4.
There exists a fixed set S ′ ⊇ S such that every rs function f over S can be strongly -simulated by some minimal reaction system over S ′ .Proof. Let T = { N X ∣ ∅ ≠ X ⊆ S } ∪ { ∗ , ◇ } as in Lemma 5.3 and S ′ = S ∪ T . Then S ′ is a fixed background set extending S . Suppose f is an rs function over S . Let C and D be as in Lemma 5.3. Consider the reaction system A = ( S ′ , C ∪ D ) over S ′ . Clearly, A is minimal.Suppose X ⊆ S . By Lemma 3.4,res A ( X ) = res C ( X ∩ ( S ∪ { ◇ })) ∪ res D ( X ∩ T ) = res C ( X ) ∪ res D ( ∅ ) = res C ( X ) . Hence, by Lemma 3.4 again,res A ( X ) = res A ( res C ( X )) = res C ( res C ( X ) ∩ ( S ∪ { ◇ })) ∪ res D ( res C ( X ) ∩ T ) . Note that res C ( X ) ∩ ( S ∪ { ◇ }) equals { ◇ } because res C ( X ) ⊆ T and ◇ ∈ res C ( X ) .It follows that res C ( res C ( X ) ∩ ( S ∪ { ◇ })) = res C ({ ◇ }) = ∅ . Therefore, res A ( X ) = res D ( res C ( X )) = f ( X ) by Lemma 5.3. (cid:3) IMULATION OF REACTION SYSTEMS BY THE STRICTLY MINIMAL ONES 10
Finally, we address a question related to Lemma 4.2. When ∣ S ′ ∣ = ∣ S ∣ + l , thelemma identifies certain rs functions that cannot be k -simulated by any reactionsystem over S ′ whenever k > ∣ S ∣+ l − ∣ S ∣ − > l . Does any of those rs functions behaveidentically for some k ≤ l ? The following theorem answers this negatively. Theorem 5.5.
Suppose ∣ S ′ ∣ = ∣ S ∣ + l for a nonnegative integer l . Then every rs function over S can be strongly k -simulated by some reaction system over S ′ whenever k ≤ l .Proof. Suppose f is an arbitrary rs function over S . Since f can be canonicallyspecified by a unique maximally inhibited reaction system over S , the case k = < k ≤ l . Let T = S ′ / S . Note that ∣ T ∣ = l ≥ k . Let ∅ = L , L , . . . , L k = T be any k distinct subsets of T . Consider the reactionsystem B = ( T, B ) , where B = { ( X ∪ T, S / X, f ( X )) ∣ X ⊆ S } ∪ { ( L i , T / L i , L i + ) ∣ ≤ i ≤ k − } ∪ ⋃ t ∈ T { ({ s } , { t } , { s }) ∣ s ∈ S } . We claim that f can be strongly k -simulated by B = ( S ′ , B ) .Suppose X is an arbitrary subset of S . It suffices to show that res i B ( X ) = X ∪ L i + for all 0 ≤ i ≤ k − k B ( X ) = res B ( res k − B ( X )) = res B ( X ∪ T ) = f ( X ) .We argue by induction. Trivially, res B ( X ) = X = X ∪ L . For the induction step,suppose 1 ≤ i ≤ k −
1. Then res i B ( X ) = res B ( res i − B ( X )) = res B ( X ∪ L i ) by theinduction hypothesis. Since L i ≠ T , it follows that res B ( X ∪ L i ) = X ∪ L i + . (cid:3) Conclusion and Open Problems
The reaction system rank of any rs function f over S is the smallest possiblesize of a set A of reactions such that f can be specified by the reaction system ( S, A ) . Through Theorem 3.2, it can be shown that the number of extra resourcesneeded to simulate a given rs function f by some strictly minimal reaction systemsis bounded by the reaction system rank of f . However, since the upper bound 2 ∣ S ∣ for reaction system rank is effectively attainable by rs functions over S (see [14]),in view of Theorem 3.5 and Theorem 4.3, it is intriguing but not surprising if nofixed S ′ ⊇ S exists with ∣ S ′ ∣ < ∣ S ∣ + ∣ S ∣ such that every rs function over S can be2-simulated by some strictly minimal reaction system over S ′ .In another direction, one can study the difference between k -simulation andstrong k -simulation. With respect to Theorem 5.4, one can ask whether the classof simulating reaction systems can be further restricted to the ones that are strictlyminimal. Additionally, would an extended background set S ′ of size ∣ S ∣ + ∣ S ∣ besufficient to strongly 2-simulate every rs function over S by some minimal reactionsystem over S ′ ? If either question has a negative answer, then this would mean thatstrong k -simulation is essentially weaker than k -simulation in terms of generativepower.As a conclusion, simulation of reaction systems and its strong version can befurther studied and compared from the following perspectives: IMULATION OF REACTION SYSTEMS BY THE STRICTLY MINIMAL ONES 11 (1) the complexity of the simulating reaction system;(2) the relative size of the extended background set;(3) the order of k -simulation, that is, the value of k .Finally, every hybrid reaction system over ( S, T ) can be viewed as a reactionsystem over S ∪ T . Hence, it is not our intention to generalize the study of re-action systems by introducing hybrid reaction systems. It is simply natural andconvenient to do so in this study. Acknowledgment
This work is an extension of that published in the proceedings paper [12].The first author acknowledges support of Fundamental Research Grant SchemeNo. 203.PMATHS.6711644 of Ministry of Education, Malaysia, and UniversitiSains Malaysia. Furthermore, this work is completed during his sabbatical leavefrom 15 Nov 2018 to 14 Aug 2019, supported by Universiti Sains Malaysia.
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School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM,Malaysia
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