Cause-efect (c-e) structures, a net-like algebraic formalism for describing and analysing systems, primarily parallel, may be adapted to work as Reaction Systems. This is acquired by a simple modification of the c-e structures' semantics Cause-efect structures (c-e) are objects of an algebraic calculus called a quasi-semiring1, for describing and analysing systems built up as nets. Among a number of similarities to Petri nets, the noticeable is graphic presentation of systems' dynamics. The complete presentation of c-e structures is in [Cza 2019].
(++)
+ = (+++) + = + (++++) + ( + ) = ( + ) + (∙ ) ) ∙ ( ∙ ) = ( ∙ ) ∙ (+∙ )
If ̸= ⇔ ̸= then ∙ ( + ) = ∙ + ∙
Algebraic system which obeys these axioms will be referred to as a quasi-semiring of formal polynomials. □ □
Definition 1.2 (cause-efect structure, carrier, set CE[X]) A cause-efect structure (c-e structure) over X is a pair = ⟨, ⟩ of total functions: : X → F [X] : X → F [X] (cause function; nodes occuring in () are causes of ) (efect function ; nodes occuring in () are efects of ) such that occurs in the formal polynomial () if occurs in (). Carrier of is the set ( ) = { ∈ X : () ̸= ∨ () ̸= }. is of finite carrier if |( )| < ∞ (|...| denotes cardinality). The set of all c-e structures over X is denoted by CE[X]. Since X is fixed, we write just CE. and are total, thus each c-e structure comprises all nodes from X, also the isolated ones - those from outside of its carrier. In presenting c-e structures graphically, only their carriers are drawn. A representation of a c-e structure = ⟨, ⟩ as a set of annotated nodes is {(()) : ∈ ( )}. is also presented as a directed graph with ( ) as set of vertices labelled with objects of the form (()) ( ∈ ( )); there is an edge from to if occurs in the polynomial (). Fig. 1 (a) and (b) depict two graphical presentations of the same c-e structure with carrier {, , , , , , , ℎ}; in (a) the encircled nodes comprise groups making products in formal polynomials in (b), where the sums of the products create families of the groups. This c-e structure is the set { subscript/superscript by node names is omitted.
, , ,
· +ℎ , +· +· , , , ℎ}. Sometimes the empty dant: used only for graphic presentation of c-e structures. {{}, {, }, {, }} and {{, }, {ℎ}}. (b) Notation by means of polynomials; the arrows are redun
Definition 1.3 (addition and multiplication, monomial c-e structure) ( ∙ )() = () ∙ () For c-e structures = ⟨ , ⟩, = ⟨ , ⟩ + = ⟨ + , + ⟩ = ⟨ + , + ⟩ ( + )() = () + () and similarly for ∙ = ( ∙ , ∙ ) = ( ∙ , ∙ ) and similarly for define: where where (The same symbols "+" and "∙ " are used for operations on c-e structures and formal polynomials). is a monomial c-e structure if each polynomial () and () is a monomial, i.e. does not comprise non-reducible (relative to equations in Definition 1.1) operation “+“. structures. Therefore the Proposition 1.1:
The set CE with addition, multiplication and a distinguished element denoted also by and understood as the empty c-e structure (, ), where is a constant function () = for all ∈ X, makes an algebraic system similar to that in Definition 1.1, the quasi-semiring of c-e CE[X], , ∈ X: Proposition 1.1 The system ⟨CE[X], +, ∙ , ⟩ obeys the following equations for all , , ∈ { {
, , ,
Apart from the representation of c-e structures as a set {(()) : ∈ ( )}, their linear notation is used as the so-called "arrow-expressions": , } is an arrow, denoted as → and, consequently, { , } ∙ { , }∙{
, }..... = denotes → → (equivalent to → → ), that is, the close cycle {, and arrows in particular, may be combined into “arrow expressions” representing some c-e }. Chains .........} is a chain, denoted as → → → .... . Bidirectional arrow ↔ ( → + → ) ∙ ( → ) ∙ ( → ). structures. For instance c-e structure {+,
∙ , ∙ , ∙ } may be written as (+) (++) (+∙ ) and post-set) + = + = + = (+++) + = + (++++) + ( + ) = ( + ) + □ □ ∙ ( + ) = ∙ + ∙ If () ̸= ⇔ () ̸= and () ̸= ⇔ () ̸= then ( ∙ ) (∙ ) ∙ = ∙ =
( → ) ∙ ( → ) = → ) ∙ = ∙ ) ∙ ( ∙ ) = ( ∙ ) ∙
Definition 1.4 (partial order ≤ ; substructure, set SUB[ ], firing component, set FC, pre-set For , ∈ CE let ≤ ⇔ = + ; ≤
is a partial order in CE. If ≤ then is a substructure of ; SUB[ ]= { : ≤ } is the set of all substructures of . For ⊆ CE: ∈ is minimal (wrt ≤ ) in if
∀ ∈ : ( ≤ ⇒ = ).
A minimal in CE∖{ } c-e structure = ⟨, ⟩ is a firing component if is a monomial c-e structure and () = ⇔ () ̸= for any ∈ (). The set of all firing components is denoted by FC, thus the set of all firing components of ∈CE is FC[ ] = SUB[ ] ∩ FC. Let for ∈ FC and ⊆ FC: ∙ = { ∈ () : () = } ∙ = { ∈ () : () = } ∙ ∙ = ∙ ∪ ∙ ∙ = ⋃︀ ∙
∈ ∙ = ⋃︀ ∙
∈ ∙ ∙ = ∙ ∪ ∙ (pre-set or causes of ) (post-set or efects of ) (pre-set or causes of ) (post-set or efects of ) □ Thus, the firing component is a connected graph, due to the required minimality. Elements of the pre-set are its causes and elements of the post-set are its efects .
Definition 1.5 (salvo - pairwise detached firing components; family FCS) Firing components and are detached if and only if ∙ ∙ ∩ ∙ ∙ = ∅. Any set ⊆ FC of pairwise detached firing components is called their salvo. The family of salvos is denoted by FCS. So, if ⊆ FC[ ] then FCS[ ] ⊆ 2FC[] for a c-e structure , denotes a collection of salvos in . The intention is that firing components in a salvo are capable of acting ("firing") simultaneously. This notion will be used in definition of parallel semantics of c-e structures. □ Definition 1.6 (state of elementary c-e structures)
A state is a subset of the set of nodes: ⊆ X. The set of all states: S = 2X. A node is active in the state if ∈ and passive otherwise. As in case of Petri nets, we say ” holds a token” when is active. Obviously, the state might be defined equivalently as a two-valued function : X →{0, 1}. □
Definition 1.7 (sequential and parallel semantics of elementary c-e structures) Sequential. For ∈FC[ ] , ∈ S, let [[]] ⊆ S × S be a binary relation defined as: (, ) ∈ [[]] if ∙ ⊆ ∧ ∙ ∩ = ∅ ∧ = (∖∙ ) ∪ ∙ ( transforms state into ). Semantics [[ ]] of ∈CE is: [[ ]] = ⋃︀ [[]]. [[ ]]* is its reflexive and transitive ∈FC [] closure, that is (, ) ∈ [[ ]]* if and only if = or there exists a sequence of states 0, 1, ..., with = 0, = and ( , +1) ∈ [[ ]] for = 0, 1, ..., − 1. State is reachable from in semantics [[ ]] and the sequence 0, 1, ..., is a computation performed by .
Parallel. For a salvo ∈FCS[ ], ̸= ∅, relations [[]] and [[ ]] are defined in the same way as [[]] and [[ ]] in the sequential case but with replaced with and FC[ ] replaced with FCS[ ]. Closure [[ ]]*, reachability and computation are defined as in the sequential case. □ Note that [[ ]] = ∅ if FC[ ] = ∅. Behaviour of elementary c-e structures may be thought of as a token game: if each node in a firing component’s pre-set holds a token and none in its post-set does, then remove tokens from the pre-set and put them in the post-set. This is illustrated in Fig. 2. Properties inferred from above definitions are proved in [Cza 2019].
2. Summary of extended cause-efect structures
The structure of extended c-e structures, their firing component in particular, is the same as in Definitions 1.1 - 1.4. The extensions consist in redefining the state, treating the pre and post sets of firing components as multisets, and redefining semantics. It is assumed that with a given c-e structure ∈CE[X] (i.e. already constructed by operations introduced in Definition 1.3) and the set of its firing components FC[ ] = SUB[ ]∩FC, some additional information is associated. The following extensions of elementary c-e structures with this information will be obtained: multi-valued nodes, capacity of nodes and coeficients of monomials in polynomials annotating nodes (counterparts of weight of arrows in Petri nets), in particular a coeficient which represents inhibiting. To this end, a notation for multisets is convenient: let N be the set of natural numbers including 0 and N = N ∪ {}, where the value means infinity, that is > for each ∈ N. A multiset over a base set is a (total) function : → N. If the set {: () ̸= 0} is finite then the linear-form notation is adopted for multisets , e.g.
( ) 2 0 3 1 is denoted by 2 ⊗ + 3 ⊗ + + ⊗ . A multiset is zero O, when O() = 0 for all . Addition, subtraction and multiplication on multisets is defined: ( +)() = ()+(), ( − )() = ()− () for () ≥ (), ( · )() = ()· (), comparison of multisets: ≤ if () ≤ () for all . Assume the customary arithmetic of : + = , − = , + = and additionally 0 − = 0.
Definition 2.1 (state of extended c-e structures) A state of extended c-e structure is a total function : ( ) → N, thus a multiset over ( ). The set of all states of is denoted by S. □ Definition 2.2. (weights of monomials and capacity of nodes) For a c-e structure = ⟨, ⟩ and its firing component ∈FC[ ], let with the pre-set ∙ and post-set ∙ of , some multisets ∙ : ∙ → N and ∙ : ∙ → N be given as additional information. The value ∙ () is a weight (or multiplicity) of monomial () and the value ∙ () - a weight (or multiplicity) of monomial (). For () = or () = assume respectively ∙ () = 0 or ∙ () = 0 and additionally let ∙ () = 0 for ∈/ ∙ and ∙ () = 0 for ∈/ ∙ . Let ( ) be a total function ( ) : ( ) → N∖{0}, assigning a capacity to nodes in the set ( ). A c-e structure endowed with such information is a c-e structure-with-weighted monomials and capacity of nodes. Note that this definition extends directly from the firing components onto their salvos. Indeed, For any non-empty salvo ∈FCS[ ] and any ∈ ∙ ∙ there exists exactly one firing component ∈ , with ∈ ∙ ∙ (because firing components in are pairwise detached). Thus one may define ∙ = ∙ and ∙ = ∙ . It should also be noticed that weights of cause or efect monomials in identical ifring components appearing in diferent c-e structures, may be diferent. □ An efect monomial () of a node ∈ ∙ , with weight ∙ (), is denoted by ∙ () ⊗ (). Similarly for a cause monomial () of a node ∈ ∙ with weight ∙ (): ∙ () ⊗ (). The coeficient representing weights will be abandoned if they are 1.
Definition 2.3 For a firing component ∈FC[ ], let ℎ[] = { ∈ ∙ : ∙ () = }, thus a set of nodes in the pre-set of , whose efect monomials () are of weight . The nodes in ℎ[] will play role of inhibiting nodes of firing component . In accordance with the note in Definition 1.5 (pairwise detached firing components in salvos), the concept of inhibiting nodes extends directly onto the salvos: ℎ[] = { ∈ ∙ : ∙ () = } where ∈FCS[ ]. The inhibiting nodes will be called inhibitors. □ Definition 2.4. (enabled firing components and enabled salvos) For a firing component ∈FC[ ] and state let us define the formula: []() if and only if: ∀ ∈ ℎ[] : () = 0∧ ∀ ∈ ∙ ∖ℎ[] : () > 0 ∧ ∙ () ≤ () ≤ ( )()∧ ∀ ∈ ∙ : ∙ () + () ≤ ( )() So, is enabled in the state if none of inhibiting nodes ∈ ∙ contains a token and each remaining node in ∙ contains, with no fewer tokens than is the weight of its efect monomial () and no more than capacity of each ∈ ∙ . Moreover, none of ∈ ∙ holds more tokens than their number when increased by the weight of the cause monomial (), exceeds capacity of . Evidently, for a salvo ∈FCS[ ], the formula []() is defined as above by replacing with and FC[ ] with FCS[ ]. □ Definition 2.5. (Sequential and parallel semantics of extended c-e structures) Sequential. For ∈FC[ ] , let [[]] ⊆ S × S be a binary relation defined as: (, ) ∈ [[]] if []() ∧ = ( − ∙ ) + ∙ ≤ ( ) ( transforms state into ). Semantics [[ ]] of ∈CE is [[ ]] = ⋃︀ [[]]. Closure [[ ]]* and reachability and ∈FC[] computation are defined as in Definition 1.7.
Parallel. For a salvo ∈FCS[ ], ̸= ∅, the relations [[]] and [[ ]] are defined in the same way as [[]] and [[ ]] in the sequential case, but with replaced with and FC[ ] replaced with FCS[ ]. Closure [[ ]]*, reachability and computation are defined as in the sequential case. □
An example of using inhibitor is in Fig. 3. The weight of one efect (i.e. subscript) monomial ⊗ of the node is , thus traversing arrow from to would require infinite number of tokens at , which is impossible. Thus, is the inhibiting node in firing component 0 and ordinary - in firing component 1. In accordance with aforesaid notation of multisets, the weighted monomials are denoted by ⊗ , where is the weight of - an unweighted monomial. The weight is skipped if = 1. The corresponding arrows are dashed in the figures if = . In all graphic presentations of c-e structures, nodes of unbounded capacity , for storing data (sets of tokens), will be drawn as bigger circles with double edge. Remaining nodes, the "control", are of capacity 1 and drawn as smaller circles.
3. Basic notions of reaction systems
This outline of reaction systems [Ehr 2007], [Ehr 2017] is limited to the primary definitions, not their extensions, generalizations and properties established in this theory. The outline serves only as a reference base for the next section concerned with a translation of reaction systems into a version of cause-efect structures. That is why the original (in the literature) denotation of certain constituents of reaction systems, has been somewhat changed, in order to avoid naming collision with the cause-efect structures. But the new presentation is completely equivalent to original and only tailored to the purpose of the next section. The reaction systems will be referred to as RE. The reaction systems as well as cause-efect structures, though devised to describe interactions or cooperation in a network of active objects, both are certain models of computing - in the broadest meaning of this word. Reaction systems are very inspiring new model of computing, if "computing" is to encompass interaction, not only "number crunching". That is why a certain attempt to translate the reaction systems into c-e structures is undertaken, though for a price of modifying semantics of the latter, but retaining their structure. There have been also other endeavours to relate reaction systems with some formal models, for instance [Kle 2011], [Dut 2018], [Bar 2018], [Gor 2018].
Definition 3.1 (reaction system) A reaction system is created of two sets: = (B, R) where B is a set called a background (intended to comprise the so-called entities), R is a set of reactions, each reaction ∈ R created of three sets = (, , ) where ⊆ B is a set of reactants, ⊆ B is a set of inhibitors with ∩ = ∅ and ⊆ B is a set of products. The initial state of the system is a set S0 ⊆ B. □ In the literature, all the sets in Definition 3.1 are required to be finite. Though the concept of "entity" has not been formally adopted in the definitions, for the purpose of this section it will be understood as an object associated in a given state with an element of the background, likewise a "token", an abstract object, of intuitive, informal status in the c-e structures or Petri nets. Hence, the set phrase "entity present in..." in writings on the reaction systems. Since the notion of "state" of reaction systems is analogous to that in elementary Petri nets (and elementary c-e structures), meaning of the word "entity" may be thought of as an element of state.
Definition 3.2 (state and enabled reactions) A state S of reaction system is a subset of its background: S ⊆ B. The intention is that S be the set of those background’s members, which comprise entities. A reaction = (, , ) is enabled in a state S if ⊆ S and ∩ S = ∅. □ Definition 3.3 (semantics of reaction systems - change of state) The result of a reaction in a state S is the set if is enabled at S and the empty set ∅ otherwise. This result is denoted by (S). The result of the reaction system in a state S is the union of all its reactions in this state: (S) = ⋃︀ (S) ∈R □ Note that on completion (if it exists) of reaction system work, the entities in the diference of sets S∖(S) disappear. Note also two features difering the reaction systems from cause-efect structures (or Petri nets): the lack of conflicts and possible absorption of reactants by products. These features will be taken into account in definition of semantics of reaction c-e structures in the next section.
Example 3.1 (test zero) The "test zero" task (Fig. 3) may be realized in the reaction system = (B, R) with B ={, , , }, R ={1, 2} where 1 = ({}, { }, {}), 2 = ({, }, ∅, {}). Result of this system’s work depends on the initial state S0 (i.e. what the environment supplies): if S0 = {, } then the result is {} ("not zero" - presence of entity at ), if S0 = {} then the result is {} ("zero" - absence of entity at ). Reactant initiates work of the system, while is tested for presence/absence of entity. Evolution of the system = ({, , , }, {1, 2}) with initial state {, } is the following: 1({, }) = ∅ (because {} ⊆ { , } and { } ∩ {, } ̸= ∅) 2({, }) = {} (because {, } ⊆ { , } and ∅ ∩ {, } = ∅) thus ({, }) = 1({, }) ∪ 2({, }) = {} ("not zero") Evolution of the system with initial state {} is the following: 1({}) = {} (because {} ⊆ { } and { } ∩ {} = ∅) 2({}) = ∅ (because {, } ⊈ {} and ∅ ∩ {} = ∅) ({}) = 1({}) ∪ 2({}) = {} ("zero") thus More complex example of evolution of a reaction system, that is its consecutive state changes, is in section 4.
4. Cause-efect structures working similarly to reaction systems
The objective is to build a system structurally identical with c-e structures but working like reaction systems. Let us call them "reaction c-e structures" and denote by RECE. The evident counterparts of some objects in reaction c-e structures and reaction systems are the following: nodes→← elements of background, ifring components →← reactions, causes in a firing component →← efects in a firing component →← inhibitors in a firing component →← tokens→← entities.
reactants in a reaction, products in a reaction,
inhibitors in a reaction, The state of reaction c-e structures is a restriction of the state introduced in Definition 2.1 with added "" to the range (codomain).
Definition 4.1 (state) A state of reaction c-e structure is a total function : ( ) → {0,1,}. The set of all states of is denoted by S. In the following, symbols 0 and 1 will be treated as logical values Definition 4.2 (weights of monomials and inconsistent firing components) pre-set ∙ and post-set ∙ of , some functions Given a c-e structure = ⟨, ⟩ and its firing component ∈
FC[ ], let along with the and ∙ () are diferent: (Definition 1.5) are not inconsistent. ∙ : ∙ → {0,1,} and ∙ : ∙
→ {0,1,} be given as additional information. The value ∙ () is called a weight of monomial () and the value ∙ () - a weight of monomial (). For () = or () = Additionally, ∙ () = 0 for ∈/ ∙ assume respectively ∙ () = 0 or ∙ () = 0.
and ∙ () = 0 for ∈/ ∙ . As formerly, is interpreted as a "disable signal" and used for defining inhibiting nodes. Firing components and are inconsistent if for a certain ∈ ∙ ∙ ∩ ∙ ∙ , weights ∙ () and ∙ () or ∙ () ∙ () ̸= ∙ () or ∙ () ̸= ∙ (). Note that detached and of false and true respectively and operations of propositional calculus on them will be applied. Moreover, operations ∨, ∧ on , are defined as:
0 ∨ = ∨ 0 = , 0 ∧ = ∧ 0 = 0, 1 ∨ = ∨ 1 = , 1 ∧ = ∧ 1 = 1, ∨ = ∧ = , ¬ = 0. As formerly, will be used for inhibiting actions. Interpretation of 0 and 1 as false and true is justified by absorption property of entities in reaction systems and will be made formal in Definition 4.6.
The lack of conflicts requires introducing for reaction c-e structures concept called here "volley", to difer it from "salvo" for c-e structures introduced in Definition 1.5. the parallel execution of the set of firing components Remark. Inconsistency of firing components must be avoided so that to obtain free of conflicts behaviour of the reaction c-e structures. The inconsistency is exemplified by the following c-e structure: = {0⊗ +, , } containing two firing components: = {0⊗ , }, = { }, thus ∙ () = 0, ∙ () = 1. In the initial state with () = 1, () = () = 0 = {, } yields undetermined state of node : it might be either 1 or 0. is a direct translation of the reaction system ({, , }, {, }, {}) with = ({}, ∅, {, }), = ({}, ∅, {}), which performs state transformation {} → {, , }. The same transformation of state is performed by the reaction c-e structure = {0⊗ +0⊗ , , }, leading to () = () = () = 1. Definition 4.3 (volley - simultaneous firing, family FCV, extension of weight functions) volleys is denoted by FCV. So, if ⊆ FC[ ] then FCV [ ] ⊆ Any set ⊆ FC without inconsistent firing components is called their volley. The family of 2FC[ ] for a c-e structure , denotes a collection of volleys in . The pre-set ∙ and post-set ∙ of a volley are defined as in Definition 1.4. Extension of the weight functions ∙ and ∙ onto the volley are defined else else ∙ ()
for arbitrary if it belongs to ∙ () for arbitrary if it belongs to □ □ This is a correct definition, since for any firing components and in : ∙ () = ∙ () if ∈ ∙ and ∙ () = ∙ () if ∈ ∙ . Functions ∙ and ∙ will be used in definition of reaction c-e structures semantics. □ Definition 4.4 As in Definition 2.3, for a firing component ∈FC[ ], the set ℎ[] = { ∈ ∙ : ∙ () = } comprises all nodes in the pre-set of , whose efect monomials () are of weight . This also extends onto the volleys: ℎ[] = { ∈ ∙ : ∙ () = } where ∈FCV [ ]. □ Definition 4.5 (enabled firing components and enabled volleys) For a firing component
∈ FC[ ] and state let the formula [](s) be defined as: ∀ ∈ ℎ[] : () = 0 ∧ ∀ ∈ ∙ ∖ℎ[] : () = 1 For a volley ∈FCV [ ], the formula []() is defined as above by replacing with and FC[ ] with FCV [ ].
In the "token game" metaphor, (or ) is enabled in the state if none of inhibiting nodes ∈ ∙ (or ∈ ∙ ) contains a token and each remaining node contains. As formerly, the inhibiting nodes will be called inhibitors. □ Definition 4.6 (semantics [[ ]] of reaction c-e structures) For a volley ∈FCV [ ], ̸= ∅ let [[]] ⊆ S × S be a binary relation defined as: (, ) ∈ [[]] if []() ∧ ∀ ∈ ( ) : () = (() ∧ ¬∙ ()) ∨ ∙ () (say: transforms state into ). Semantics [[ ]] of ∈RECE is ⋃︀ [[]], for any ∈FCV [ ] maximal volley , i.e. if ⊆ ′∈FCV [ ] and (, ) ∈ [[′]] then = ′. Closure [[ ]]* and reachability and computation are defined as in Definition 1.7. □ In Definition 4.6, if ∈ ∙ then ∙ () ̸= , because the volley is enabled in the state thus ∙ () is 0 or 1. If ∈/ ∙ ∙ then ∙ () = 0 and ∙ () = 0 hence ¬∙ () = 1, thus () = () (strictly, "=" means equivalence). Evidently, formula (() ∧ ¬∙ ()) ∨ ∙ () is equivalent to (() ⇒ ∙ ()) ⇒ ∙ (). Note also that description of semantics by means of this propositional formula is justified by the property of reaction systems: presence of token ("entity") at a node absorbs another token arriving in this node. Another property of reaction systems, the lack of conflicts between diferent reactions, takes place in reaction c-e structures due to the lack of inconsistent firing components in ∈FCV [ ].
Example 4.1 (reaction c-e structure assembling a chemical molecule) A description of creating some chemical molecules presents the reaction system
= ({, , , , , , , , }, {1, 2, 3, 4, 5, 6}), starting with initial state {, , } and with reactions defined as follows: 1 = ({, }, ∅, { }) 2 = ({, , }, ∅, { }) 3 = ({, }, ∅, { }) 4 = ({, }, ∅, {}) 5 = ({, , }, ∅, { }) 6 = ({, }, ∅, {}) contains molecule (methylidyne) contains molecule 22 (acetylene) contains molecule 23 (ethylenyl) contains molecule 24 (ethylene) contains molecule 25 (ethoxide) contains molecule 25 (ethanol) A diagram of the final result, that is the ethanol molecule, is in Fig. 4 and a certain translation of this reaction system into a reaction c-e structure is depicted in Fig. 5. Successive steps of the reaction system evolution are the following: 1({, , }) = { } (since {, } ⊆ { , , } and ∅ ∩ {, , } = ∅) 2({, , }) = 3({, , }) = 4({, , }) = 5({, , }) = 6({, , }) = ∅ thus ({, , }) = { } 1({, , , }) = { } (since {, } ⊆ { , , , } and ∅ ∩ {, , , } = ∅) 2({, , , }) = { } (since {, , } ⊆ { , , , } and ∅ ∩ {, , , } = ∅) 3({, , , }) = 4({, , }) = 5({, , , }) = 6({, , , }) = ∅ thus ({, , , }) = {, } 1({, , , , }) = { } (since {, } ⊆ { , , , , } and ∅ ∩ {, , , , } = ∅) 2({, , , , }) = { } (since {, , } ⊆ { , , , , } and ∅ ∩ {, , , , } = ∅) 3({, , , , }) = { } (since {, } ⊆ { , , , , } and ∅ ∩ {, , , , } = ∅) 4({, , , , }) = 5({, , , , }) = 6({, , , , }) = ∅ thus ({, , , , }) = {, , } 1({, , , , , }) = { } (since {, } ⊆ { , , , , , } and ∅∩{, , , , , } = ∅) 2({, , , , , }) = { } (since {, , } ⊆ { , , , , , } and ∅ ∩ {, , , , , } = ∅) 3({, , , , , }) = { } (since {, } ⊆ { , , , , , } and ∅∩{, , , , , } = ∅) 4({, , , , , }) = {} (since {, } ⊆ { , , , , , } and ∅∩{, , , , , } = ∅) 5({, , , , , }) = 6({, , , , , }) = ∅ thus ({, , , , , }) = {, , , } 1({, , , , , , }) = { } (since {, } ⊆ { , , , , , , } and ∅ ∩ {, , , , , , } = ∅) 2({, , , , , , }) = { } (since {, , } ⊆ { , , , , , , } and ∅ ∩ {, , , , , , } = ∅) 3({, , , , , , }) = { } (since {, } ⊆ { , , , , , , } and ∅ ∩ {, , , , , , } = ∅) 4({, , , , , , }) = {} (since {, } ⊆ { , , , , , , } and ∅ ∩ {, , , , , , } = ∅) 5({, , , , , , }) = { } (since {, , } ⊆ { , , , , , , } and ∅ ∩ {, , , , , , } = ∅) 6({, , , , , , }) = ∅) thus ({, , , , , , }) = {, , , , } 1({, , , , , , , }) = { } (since {, } ⊆ { , , , , , , , } and ∅ ∩ {, , , , , , , } = ∅) 2({, , , , , , , }) = { } (since {, , } ⊆ { , , , , , , , } and ∅ ∩ {, , , , , , , } = ∅) 3({, , , , , , , }) = { } (since {, } ⊆ { , , , , , , , } and ∅ ∩ {, , , , , , , } = ∅) 4({, , , , , , , }) = {} (since {, } ⊆ { , , , , , , , } and ∅ ∩ {, , , , , , , } = ∅) 5({, , , , , , , }) = { } (since {, , } ⊆ { , , , , , , , } and ∅ ∩ {, , , , , , , } = ∅) 6({, , , , , , , }) = {} (since {, } ⊆ { , , , , , , , } and ∅ ∩ {, , , , , , , } = ∅) thus ({, , , , , , , }) = {, , , , , } [Kle 2011] Kleijn J., Kountny M., Rozenberg G., Modelling Reaction Systems with Petri Nets, in: Proc. of the International Workshop on Biological Processes & Petri Nets (BioPPN 2011), pp. 36–52