# On Categories of Fuzzy Petri Nets

Authors: Arun K. Srivastava; S. P. Tiwari
Journal: Advances in Fuzzy Systems
 (2011)

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## Abstract

We introduce the concepts of fuzzy Petri nets and marked fuzzy Petri nets along with their appropriate morphisms, which leads to two categories of such Petri nets. Some aspects of the internal structures of these categories are then explored, for example, their reflectiveness/coreflectiveness and symmetrical monoidal closed structure.


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## Body

## 1. Introduction

Petri nets are a well-known model of concurrent systems [1]. A number of authors have been led to the study of the various categories of Petri nets and their appropriate morphisms [2–4], in the belief that the categorical study provides a tool to compare different models of Petri nets. At the same time, fuzzy Petri nets have also been studied by many authors in different ways [5, 6]. In this paper, taking a cue from [3], we introduce another concept of fuzzy Petri nets (although we model our definition on the lines of [2]). This, along with their appropriate morphisms, results in a category of fuzzy Petri nets (and also of marked fuzzy Petri nets). The structure of these categories is then studied on the lines similar to those in [2], showing that one of the categories is symmetric monoidal closed.


## 2. Category of Fuzzy Petri Nets

For categorical concepts used here, [7] may be referred. We begin by collecting some basic definitions.A Petri net is a bipartite graph, consisting of two kinds of nodes, namely, places and transitions, where arcs are either from a place to a transition or vice versa [3]. Graphically, the places are represented by circles, transitions by rectangles, and the arcs by arrows. We define a fuzzy Petri net as follows.Definition 1.
Afuzzy Petri net (in short, fPn) is 4-tuple N=(P,T,I-,I+), where P and T are sets, called the  set of places and  set of transitions, respectively, and I-,I+:P×T→[0,1], are functions, called the  incidence functions.We may interpretI- and I+ defined above as follows.For(p,t)∈P×T,I-(p,t) (resp., I+(p,t)) gives, the grade with which place p is related to transition t (resp., transition t is related to place p). Thus, I- and I+ describe some kind of fuzzy arcs between places and transitions.Definition 2.
An  fPn-morphism from an fPn (P1,T1,I1-,I1+) to fPn (P2,T2,I2-,I2+) is a pair (f,g) of functions, f:P2→P1 and g:T1→T2 such that the following two diagrams:
(1)
“hold”, by which it is meant that for all (p,t)∈P2×T1,I1-(f(p),t)≥I2-(p,g(t)) and I1+(f(p),t)≤I2+(p,g(t)).Remark 3.
Fuzzy Petri nets and fPn-morphisms form a category, denoted as FPN (the identity morphisms and the composition for this category are obvious to guess).Definition 4.
Theproduct of two fPn's N1=(P1,T1,I1-,I1+) and N2=(P2,T2,I2-,I2+) is the fPn N1×N2=(P̃1∪P̃2,T1×T2,I-,I+), where P̃1=P1×{0},P̃2=P2×{1} and I-,I+:(P̃1∪P̃2)×(T1×T2)→[0,1] are given by
(2)I-((p,n),(t1,t2))={I1-(p,t1)ifn=0,I2-(p,t2)ifn=1,I+((p,n),(t1,t2))={I1+(p,t1)ifn=0,I2+(p,t2)ifn=1,
for all (p,n)∈P̃1∪P̃2 and (t1,t2)∈T1×T2.Proposition 5.
Letπi:T1×T2→Ti,i=1,2, be the two projections and let ρi:Pi→P̃1∪P̃2,i=1,2, be the two injections. Then (ρi,πi):N1×N2→Ni,i=1,2, are fPn-morphisms.Proof.
To prove that(ρ1,π1):N1×N2→N1 is an fPn-morphism, we need to show that the following two diagrams hold.
(3)
The above diagrams hold because for all (p1,(t1,t2))∈P1×(T1×T2),I-(ρ1(p1),(t1,t2))=I-((p1,0),(t1,t2))=I1-(p1,t1)=I1-(p1,π1(t1,t2)) and I+(ρ1(p1),(t1,t2))=I+((p1,0),(t1,t2))=I1+(p1,t1)=I1+(p1,π1(t1,t2)). Similarly, one can prove that (ρ2,π2):N1×N2→N2 is also an fPn-morphism.Proposition 6.
The productN1×N2 of fPn's N1 and N2 is the categorical product of N1 and N2 in  FPN.Proof.
LetN′=(P′,T′,I′-,I′+) be an fPn together with fPn-morphisms (fi,gi):N′→Ni,i=1,2. We show that there exists a unique fPn-morphism (f,g):N′→N1×N2 such that (ρi,πi)∘(f,g)=(fi,gi),i=1,2, or equivalently that f∘ρi=fi and πi∘g=gi,i=1,2. For this purpose, we choose the following f and g. Let f:P̃1∪P̃2→P′ be the map given by
(4)f((p,n))={f1(p),ifn=0,f2(p),ifn=1,
and g=(g1,g2). We show that the diagrams
(5)
hold, that is, for all ((p,n),t)∈(P̃1∪P̃2)×T′, I′-(f((p,n)),t)≥I-(p,g(t)) and I′+(f((p,n)),t)≤I+(p,g(t)), that is, I1-(p,g1(t))≤I′-(f1(p),t), if n=0 and I2-(p,g2(t))≤I′-(f2(p),t), if n=1, for all t∈T′ as well as I1+(p,g1(t))≥I′+(f1(p),t), if n=0 and I2+(p,g2(t))≥I′+(f2(p),t), if n=1, for all t∈T′. But as (fi,gi):N′→Ni,i=1,2, are fPn-morphisms, the above inequalities hold, whereby the diagrams (5) hold. Thus, (f,g):N′→N1×N2 is an fPn-morphism. Also, the definitions of f and g are such that we obviously have f∘ρi=fi and πi∘g=gi,i=1,2.
To prove the uniqueness of(f,g), let there exist another fPn-morphism (f′,g′) such that (ρi,πi)∘(f′,g′)=(fi,gi),i=1,2, that is, f′∘ρi=fi and πi∘g′=gi,i=1,2. We then have f′∘ρ1=f∘ρ1,f′∘ρ2=f∘ρ2,π1∘g′=π1∘g, and π2∘g′=π2∘g, whereby f=f′ and g=g′. Thus, (f′,g′)=(f,g), proving the uniqueness of (f,g). Hence the product is a categorical product.Definition 7.
Thecoproduct of two fPn's N1=(P1,T1,I1-,I1+) and N2=(P2,T2,I2-,I2+) is the fPn N1⊕N2=(P1×P2,T̃1∪T̃2,I-,I+), where T̃1=T1×{0},T̃2=T2×{1}, and I-,I+:(P1×P2)×(T̃1∪T̃2)→[0,1] are given by
(6)I-((p1,p2),(t,n))={I1-(p1,t),ifn=0,I2-(p2,t),ifn=1,I+((p1,p2),(t,n))={I1+(p1,t),ifn=0,I2+(p2,t),ifn=1,
for all (p1,p2)∈P1×P2 and (t,n)∈T̃1∪T̃2.Similar to Propositions5 and 6, the following two propositions can also be proved.Proposition 8.
Letπi:P1×P2→Pi,i=1,2, be the two projections and ρi:Ti→T̃1∪T̃2,i=1,2, be the two injections. Then (πi,ρi):Ni→N1⊕N2,i=1,2, are fPn-morphisms.Proposition 9.
The coproductN1⊕N2 of fPn's N1 and N2 is the categorical coproduct of N1 and N2 in  FPN.Definition 10.
Given two fPn'sN1=(P1,T1,I1-,I1+) and N2=(P2,T2,I2-,I2+), we define two new fPn's N1⊗N2 and N1N2 as follows (for sets X and Y, YX shall denote the set of all functions from X to Y):  (1)
N1⊗N2=(P1T2×P2T1,T1×T2,I-,I+), where I-,I+:(P1T2×P2T1)×(T1×T2)→[0,1] are defined as(7)I-((α,β),(t1,t2))=∧{1,I1-(α(t2),t1)+I2-(β(t1),t2)}I+((α,β),(t1,t2))=∨{0,I2+(β(t1),t2)-I1+(α(t2),t1)},∀((α,β),(t1,t2))∈(P1T2×P2T1)×(T1×T2).(2)
N1N2=(T2×P1,T1T2×P2P1,I-,I+), where I-,I+:(T2×P1)×(T1T2×P2P1)→[0,1] are defined as(8)I-((t,p),(α,β))=∨{0,I1-(p,α(t))-I2-(β(p),t)}I+((t,p),(α,β))=∧{1,I1+(p,α(t))+I2+(β(p),t)}∀((t,p),(α,β))∈(T2×P1)×(T1T2×P2P1).Proposition 11.
The category FPN is a symmetric monoidal closed category (with the constructions in (1) and (2) above, respectively, giving the associated tensor product and hom-object).Proof.
For convenience, the notationNi is used to denote the fPn (Pi,Ti,Ii-,Ii+). We give a sketch of the proof of closedness of FPN. For this, the two functors -⊗N0,(-)N0:FPN→FPN are denoted, respectively, as F and G, which map any  FPN-morphism (f,g):N1→N2, respectively, to  FPN-morphisms, (f′,g′):F(N1)→F(N2), and (f′′g′′):G(N1)→G(N2), such that f′((λ,μ))=(f∘λ,μ∘g),g′((t2,t0))=(g(t2),t0),for all((λ,μ),(t2,t0))∈(P2T0×P0T2)×(T2×T0), and f′′((t0,p2))=(t0,f(p2)),g′′((ρ,ν))=(g∘ρ,ν∘f), for all ((t0,p2),(ρ,ν))∈(T0×P2)×(T1T0×P0P1).
It turns out thatG is a right adjoint to F; the associated unit of the adjunction, η:IFPN→GF, is given for each N1∈FPN, by ηN1=(FN1,GN1):N1→GF(N1), where FN1:T0×(P1T0×P0T1)→P1 and GN1:T1→(T1×T0)T0×P0(P1T0×P0T1), are such that FN1((t0,(λ,μ))=λ(t0), and GN1(t1)=(αt1,βt1), with αt1(t0)=(t1,t0) and βt1((λ,μ))=μ(t1), for all (t0,(λ,μ))∈T0×(P1T0×P0T1), and for all t1∈T1.
To establish the universality ofηN1, we need to produce, for any given N2∈FPN and FPN-morphism (f,g):N1→G(N2), a unique FPN-morphism (f*,g*):F(N1)→N2, such that the following diagram commutes.
We only describe(f*,g*), leaving out the verification of the commutativity of the above diagram and the uniqueness of (f*,g*). (f*,g*) is given by f*:P2→P1T0×P0T1 and g*:T1×T0→T2, such that f*(p2)=(f1p2,f2p2) and g*(t1,t0)=g1t1(t0), with f1p2(t0)=f(t0),f2p2(t1)=g2t1(p2), and g(t1)=(g1t1,g2t1).


## 3. Marked Fuzzy Petri Nets

A marked Petri net is a Petri net together with a function, calledmarking defined from the set of places to the set of natural numbers [2]. Marking at a particular place gives the number of tokens at that particular place. In this section, we introduce a concept of marked fuzzy Petri nets and thereby a category of marked fuzzy Petri nets.Definition 12.
An fPnN=(P,T,I-,I+), together with a function M:P→[0,1] (called a  fuzzy marking of N), is called a  marked fuzzy Petri net (in short, an mfPn) and is denoted as (N,M).Here, marking at a particular place may be interpreted as the degree of confidence to which a token can reside at that place.Definition 13.
Given an mfPn(N,M), a transition t∈T is said to  fire atM (or t is  enabled at M), if I-(p,t)≤M(p), for all p∈P.In an mfPn(P,T,I-,I+,M),I-:P×T→[0,1], for fixed t∈T induces a function It-:P→[0,1] such that for p∈P,It-(p), gives the degree of confidence to which a transition t∈T can fire at marking M. Thus, a transition t at a marking M of mfPn (N,M) can fire if the degree of confidence to which it fires does not exceed the degree of confidence to which a token can reside at places.Aftert firing at the fuzzy marking M, we get a new fuzzy marking Mt of N, given by Mt(p)=min{1,M(p)-I-(p,t)+I+(p,t)}, for all p∈P. We say that t∈T fires at M to yieldMt and denote this by M→tMt. Also, Mt is then said to be directly reachable from M through the transition t.Similar to [8], the marked fuzzy Petri net models of negation, disjunction, and conjunction of fuzzy proposition, can also be given. We illustrate these by following examples.Example 14.
Consider the following graphical representation of an mfPn, which gives the truth value of the negation of a fuzzy proposition. For this, take an mfPn withP={p1,p2} and T={t}. Given a fuzzy proposition, the initial marking M is so chosen that M(p1) is the truth value of the fuzzy proposition and M(p2)=0. Also, I-(p1,t) is so chosen that I-(p1,t)≤M(p1) (so that the transition t can fire) and we also take I+(p2,t) to be 1-M(p1). After the firing of the transition t, at marking M, the marking at p2 is given by Mt(p2)=min{1,M(p2)-I-(p2,t)+I+(p2,t)}=1-M(p1), the truth value of the negation of the fuzzy proposition.Example 15.
Similar to Example14, consider the following graphical representation of an mfPn, which gives the disjunction of truth values of two fuzzy propositions. For this, take an mfPn, with P={p1,p2,p3} and T={t}. Given two fuzzy propositions, the initial marking M is so chosen that M(p1) and M(p2) are the truth values of the given fuzzy propositions and M(p3)=0. Also, I-(p1,t) and I-(p2,t) are so chosen that I-(p1,t)≤M(p1) and I-(p2,t)≤M(p2) (so that the transition t can fire) and we also take I+(p3,t) to be M(p1)∨M(p2). After the firing of the transition t, at marking M, the marking at p3 is given by Mt(p3)=min{1,M(p3)-I-(p3,t)+I+(p3,t)}=M(p1)∨M(p2), the truth value of the disjunction of the fuzzy propositions.(Analogous to Example15, one can design mfPn, which gives the conjunction of the truth values of two fuzzy propositions.)


## 4. Category of Marked Fuzzy Petri Net

In this section, a category of marked fuzzy Petri net, inspired from [2], is introduced.Definition 16.
For mfPn's(P1,T1,I1-,I1+,M1) and (P2,T2,I2-,I2+,M2), a function f:P2→P1, (M1,M2) is said to be  f-ok if (M1∘f)(p)≤M2(p), for all p∈P2.Remark 17.
MFPN shall denote the category of all mfPn's, with mfPn-morphisms(f,g):(N1,M1)→(N2,M2) being the fPn-morphisms (f,g):N1→N2 such that (M1,M2) is f-ok.Proposition 18.
Let(N1,M1)=(P1,T1,I1-,I1+,M1) and (N2,M2)=(P2,T2,I2-,I2+,M2) be two mfPn's and let (f,g):(N1,M1)→(N2,M2) be an mfPn-morphism. Then for t1∈T1,g(t1) is enabled at M2, if t1 is enabled at M1.Proof.
As(f,g):N1→N2 is an mfPn-morphism, I1-(f(p2),t1)≥I2-(p2,g(t1)) and M2(p2)≥M1(f(p2)),  for all (p2,t1)∈P2×T1. Also, as t1 is enabled at M1, we have I1-(p1,t1)≤M1(p1), for all p1∈P1, whence I1-(f(p2),t1)≤M1(f(p2)),for allp2∈P2. But I2-(p2,g(t1))≤I1-(f(p2),t1)≤M1(f(p2))≤M2(p2), whereby, I2-(p2,g(t1))≤M2(p2), for allp2∈P2. Thus, g(t1) is enabled at M2.Proposition 19.
Let(N1,M1)=(P1,T1,I1-,I1+,M1) and (N2,M2)=(P2,T2,I2-,I2+,M2) be two mfPn's and (f,g):(N1,M1)→(N2,M2) be an mfPn-morphism. Then for t1∈T1,((M1)t1,(M2)g(t1)) is f-ok, if M1→t1(M1)t1.Proof.
From the above proposition, it is clear thatM2→g(t1)(M2)g(t1). Also, as (f,g):(N1,M1)→(N2,M2) is an mfPn-morphism, I1-(f(p2,t1))≥I2-(p2,g(t1)),I1+(f(p2,t1))≤I2+(p2,g(t1)), and M1(f(p2))≤M2(p2), for all p2∈P2. Consequently, for all (p2,t1)∈P2×T1,M1(f(p2))-I1-(f(p2),t1)+I1+(f(p2),t1)≤M2(p2)-I2-(p2,g(t1))+I2+(p2,g(t1)), whereby, min{1,M1(f(p2))-I1-(f(p2),t1)+I1+(f(p2),t1)}≤min{1,(M2(p2)-I2-(p2,g(t1))+I2+(p2,g(t1))}. Thus, (M1)t1(f(p2))≤(M2)g(t1)(p2)for all (p2,t1)∈P2×T1. Hence ((M1)t1,(M2)g(t1)) is f-ok, for t1∈T1.Definition 20.
Theproduct of two mfPn's (N1,M1) and (N2,M2) is the mfPn (N1×N2,M1⊕M2), where N1×N2 is the product of fPn's N1 and N2 and M1⊕M2:P̃1∪P̃2→[0,1] is given by
(9)(M1⊕M2)(p,n)={M1(p)ifn=0,M2(p)ifn=1,
for all (p,n)∈P̃1∪P̃2.Proposition 21.
The  FPN-morphisms(ρi,πi):N1×N2→Ni, given in Proposition 5 are  MFPN-morphisms from (N1×N2,M1⊕M2) to (Ni,Mi),i=1,2.Proof.
Since(M1⊕M2)(ρ1((p,n)))=(M1⊕M2)((p,0))=M1(p) and (M1⊕M2)(ρ2((p,n)))=(M1⊕M2)((p,1))=M2(p), (M1⊕M2,M1) and (M1⊕M2,M2) are ρ1-ok and ρ2-ok, respectively. Hence (ρi,πi),i=1,2, are MFPN-morphisms.Using Propositions5 and 21, the next proposition is evident.Proposition 22.
The product of mfPn's is the categorical product in MFPN.Definition 23.
Thecoproduct of two mfPn's (N1,M1) and (N2,M2) is the mfPn (N1⊕N2,M1∨M2), where N1⊕N2 is the coproduct of fPn's N1 and N2 and M1∨M2:P1×P2→[0,1] is given by (M1∨M2)(p1,p2)=M1(p1)∨M2(p2), for all (p1,p2)∈P1×P2.Similar to Propositions21 and 22, the following two propositions can also be proved.Proposition 24.
The FPN-morphisms(πi,ρi):Ni→N1⊕N2,i=1,2, given in Proposition 8 are  MFPN-morphisms from (Ni,Mi) to (N1⊕N2,M1∨M2).Proposition 25.
The coproduct of mfPn's is the categorical coproduct in MFPN.


## 5. Relationship between  FPN and  MFPN

There is an obvious functork1:MFPN→FPN, given by (N,M)↦N and (f,g)↦(f,g).We omit the easy verification of the following observations.Proposition 26.
There are full and faithful functorsk2,k3:FPN→MFPN, which, on objects, are respectively, given by N↦(N,0) and N↦(N,1), where  0 and  1, are respectively, the 0-valued and the 1-valued constant functions, and which leave the morphisms unchanged.It is easy to prove the following.Proposition 27.
The functork2 (resp., k3) is left adjoint (resp., right adjoint) to the functor k1.Thus, we have the following.Proposition 28.
The category FPN is isomorphic to a full reflective subcategory, and also to a full coreflective subcategory, of MFPN.


## 6. Conclusion

We note that nothing has been said about the symmetric monoidal closed structure of the category MFPN of marked fuzzy Petri nets. An obvious attempt to make  MFPN symmetric monoidal closed would appear to be as follows. Given mfPn's(N0,M0) and (N1,M1), the fPn's N1⊗N0 and N1N0 (cf. Definition 10) can be made mfPn's by taking their respective fuzzy markings to be M1⊗M0:P1T0×P0T1→[0,1] and M1M0:T0×P1→[0,1], defined as M1⊗M0((λ,μ))=∨{∨T0{M1(λ(t0))},∨T1{M0(μ(t1))}} and M1M0((t0,p1))=M1(p1), for all (λ,μ)∈P1T0×P0T1, for all(t0,p1)∈T0×P1. However, for each fixed mfPn (N0,M0), the resulting functors -⊗(N0,M0), (-)(N0,M0):MFPN→MFPN do not turn out to be adjoint. As an attempt to repair the above situation, M1⊗M0 may be redefined as M1⊗M0((λ,μ))=∨T0{M1(λ(t0))}, so that the functor (-)(N0,M0) does, now, turn out to be right adjoint to the (modified) functor -⊗(N0,M0). However, the symmetry of ⊗ in this modified setup is now lost (this situation is similar to the one noted in [2]). So there may be a different symmetric monoidal closed structure on  MFPN which we have not been able to find presently.



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