Open Petri Nets. John C. Baez

Open Petri Nets John C. Baez Department o Mathematics University o Caliornia Riverside CA, USA 9252 and Centre or Quantum Technologies National University o Singapore Singapore 7543 Jade Master Department o Mathematics University o Caliornia Riverside CA, USA 9252 email: baez@math.ucr.edu, jmast003@ucr.edu August 7, 208 Abstract The reachability semantics or Petri nets can be studied using open Petri nets. For us an open Petri net is one with certain places designated as inputs and outputs via a cospan o sets. We can compose open Petri nets by gluing the outputs o one to the inputs o another. Open Petri nets can be treated as morphisms o a category Open(Petri), which becomes symmetric monoidal under disjoint union. However, since the composite o open Petri nets is deined only up to isomorphism, it is better to treat them as morphisms o a symmetric monoidal double category Open(Petri). Various choices o semantics or open Petri nets can be described using symmetric monoidal double unctors out o Open(Petri). Here we describe the reachability semantics, which assigns to each open Petri net the relation saying which markings o the outputs can be obtained rom a given marking o the inputs via a sequence o transitions. We show this semantics gives a symmetric monoidal lax double unctor rom Open(Petri) to the double category o relations. A key step in the proo is to treat Petri nets as presentations o symmetric monoidal categories; or this we use the work o Meseguer, Montanari, Sassone and others. Introduction Petri nets are a simple and widely studied model o computation [5, 24], with generalizations applicable to many orms o modeling [8]. Recently more attention has been paid to a compositional treatment in which Petri nets can be assembled rom smaller open Petri nets [3, 4, 5, 7]. In particular, the reachability problem or Petri nets, which asks whether one marking o a Petri net

can be obtained rom another via a sequence o transitions, can be studied compositionally [25, 29, 3]. Here we seek to give this line o work a irmer ooting in category theory. Petri nets are closely tied to symmetric monoidal categories in two ways. First, a Petri net P can be seen as a presentation o a ree symmetric monoidal category F P, with the places and transitions o P serving to reely generate the objects and morphisms o F P. We show how to construct this in Section 2, ater reviewing a line o previous work going back to Meseguer and Montanari [23]. In these terms, the reachability problem asks whether there is a morphism rom one object o F P to another. Second, there is a symmetric monoidal category where the objects are sets and the morphisms are equivalence classes o open Petri nets. We construct this in Section 3, but the basic idea is very simple. Here is an open Petri net P rom a set X to a set Y : X 2 3 A B α C D Y 4 5 The yellow circles are places and the blue rectangle is a transition. The bold arrows rom places to transitions and rom transitions to places complete the structure o a Petri net. There are also arbitrary unctions rom X and Y into the set o places. These indicate points at which tokens could low in or out, making our Petri net open. We write this open Petri net as P : X Y or short. Given another open Petri net Q: Y Z: Y 4 5 E β γ F Z 6 the irst step in composing P and Q is to put the pictures together: X 2 3 A B α C D Y 4 5 E β γ F Z 6 At this point, i we ignore the sets X, Y, Z, we have a new Petri net whose set o places is the disjoint union o those or P and Q. The second step is to identiy 2

a place o P with a place o Q whenever both are images o the same point in Y. We can then stop drawing everything involving Y, and get an open Petri net QP : X Z: X 2 3 A B α C F β γ Z 6 Formalizing this simple construction leads us into a bit o higher category theory. The process o taking the disjoint union o two sets o places and then quotienting by an equivalence relation is a pushout. Pushouts are deined only up to canonical isomorphism: or example, the place labeled C in the last diagram above could equally well have been labeled D or E. This is why to get a category, with composition strictly associative, we need to use isomorphism classes o open Petri nets as morphisms. There are advantages to avoiding this and working with open Petri nets themselves. I we do this, we obtain not a category but a bicategory [29]. However, this bicategory is equipped with more structure. Besides composing open Petri nets, we can also tensor them via disjoint union: this describes Petri nets being run in parallel rather than in series. The result is a symmetric monoidal bicategory. Unortunately, the axioms or a symmetric monoidal bicategory are cumbersome to check directly. Double categories turn out to be more convenient. Double categories were introduced in the 960s by Ehresmann [3, 4]. More recently they have been used to study open dynamical systems [9], open electrical circuits and chemical reaction networks [0], open discrete-time Markov chains [9], and coarse-graining or open continuous-time Markov chains []. A 2-morphism in a double category can be drawn as a square: M X Y α g N X 2 Y 2. We call X, X 2, Y and Y 2 objects, and g vertical -morphisms, M and N horizontal -cells, and α a 2-morphism. We can compose vertical -morphisms to get new vertical -morphisms and compose horizontal -cells to get new horizontal -cells. We can compose the 2-morphisms in two ways: horizontally and vertically. This is just a quick sketch o the ideas; or ull deinitions see Appendix A. In Thm. 3 we construct a symmetric monoidal double category Open(Petri) with: 3

sets X, Y, Z,... as objects, unctions : X Y as vertical -morphisms, open Petri nets P : X Y as horizontal -cells, morphisms between open Petri nets as 2-morphisms. To get a eeling or morphisms between open Petri nets, a simple example may be helpul: there is a morphism rom this open Petri net: X A A α α B Y 2 to this one: X 2 Y 2 A α B 2 mapping both primed and unprimed symbols to unprimed ones. This describes a process o simpliying an open Petri net. There are also morphisms that include simple open Petri nets in more complicated ones, etc. The main goal o this paper is to describe the reachability semantics or open Petri nets as a map rom Open(Petri) to the double category o relations, Rel, which has: sets X, Y, Z,... as objects, unctions : X Y as vertical -morphisms, relations R X Y as horizontal -cells, squares R X Y X Y g S X 2 Y 2 X 2 Y 2 obeying ( g)r S as 2-morphisms. 4

In Petri net theory, a marking o a set X is a inite multisubset o X: we can think o this as a way o placing initely many tokens on the points o X. Let N[X] denote the set o markings o X. Given an open Petri net P : X Y, there is a reachability relation saying when a given marking o X can be carried by a sequence o transitions in P to a given marking o Y, leaving no tokens behind. We write the reachability relation o P as P N[X] N[Y ]. In Theorem 23 we show that the map sending P to P extends to a lax double unctor : Open(Petri) Rel. In Theorem 24 we go urther and show that this double unctor is symmetric monoidal. I the reader preers bicategories to double categories, they may be relieved to learn that any double category D gives rise to a bicategory H(D) whose 2-morphisms are those 2-morphisms o D o the orm X M Y α X Y N X Y. (The details are in De. 26.) In this manner, the reachability semantics gives rise to a map between symmetric monoidal bicategories. I the reader preers categories to bicategories, they may be urther relieved to learn that any bicategory can be reduced to a category by throwing out the 2-morphisms and working with isomorphism classes o morphisms. Using these ideas, the reachability semantics becomes a map between symmetric monoidal categories. However, only the double category ramework presents the reachability semantics in its ull glory. 2 From Petri Nets to Commutative Monoidal Categories In this section we treat Petri nets as presentations o symmetric monoidal categories. As we shall explain, this has already been done by various authors. Unortunately there are dierent notions o symmetric monoidal category, and also dierent notions o morphism between Petri nets, which combine to yield a conusing variety o possible approaches. Here we take the maximally strict approach, and work with commutative monoidal categories. These are just commutative monoid objects in Cat, so their associator: α a,b,c : (a b) c a (b c), 5

their let and right unitor: λ a : I a and even disturbingly their symmetry: a, ρ a : a I a, σ a,b : a b b a are all identity morphisms. The last would ordinarily be seen as going too ar, since while every symmetric monoidal category is equivalent to one with trivial associator and unitors, this ceases to be true i we also require the symmetry to be trivial. However, it seems that Petri nets most naturally serve to present symmetric monoidal categories o this very strict sort. Thus, we construct an adjunction between a category o Petri nets and a category o commutative monoidal categories, which we call CMC: Petri F U CMC. It seems Montanari and Meseguer were the irst to treat Petri nets as presentations o symmetric monoidal categories [23]. They constructed a closely related but dierent adjunction Petri F CatPetri. U Our category o Petri nets is a subcategory o theirs: our morphisms o Petri nets send places to places, while they allow more general maps that send a place to a ormal linear combination o places. On the other hand, their category CatPetri can be seen as the ull subcategory o CMC containing only commutative monoidal categories whose objects orm a ree commutative monoid. Further work by Degano, Meseguer, Montanari [2], Sassone [26, 27, 28], and Sassone and Sobociński [29] explores other variations on these themes. Resisting the temptation to dwell on the subtleties o this topic, we present our approach with no urther ado. Deinition. Let CommMon be the category o commutative monoids and monoid homomorphisms. Deinition 2. Let N: Set CommMon be the ree commutative monoid unctor, that is, the let adjoint o the unctor K : CommMon Set that sends commutative monoids to their underlying sets and monoid homomorphisms to their underlying unctions. We oten abuse language slightly and use N[X] to mean the underlying set o the ree commutative monoid on X. For any set X, N[X] is the set o ormal inite linear combinations o elements o X with natural number coeicients. The set X naturally includes in N[X], and or any unction : X Y, N[]: N[X] N[Y ] is the unique monoid homomorphism that extends. 6

Deinition 3. We deine a Petri net to be a pair o unctions o the ollowing orm: T s t N[S]. We call T the set o transitions, S the set o places, s the source unction and t the target unction. Deinition 4. A Petri net morphism rom the Petri net T s t N[S] to the Petri net T s N[S ] is a pair o unctions ( : T T, g : S S ) such t that the ollowing diagrams commute: T s N[S] T t N[S] N[g] T s N[S ] N[g] T t N[S ]. Deinition 5. Let Petri be the category o Petri nets and Petri net morphisms, with composition deined by (, g) (, g ) = (, g g ). Our deinition o Petri net morphism diers rom the earlier deinition used by Sassone [26, 27, 28] and Degano Meseguer Montanari [2]. The dierence is that our deinition requires that the homomorphism between ree commutative monoids come rom a unction between the sets o places. Deinition 6. A commutative monoidal category is a commutative monoid object internal to Cat. Explicitly, a commutative monoidal category is a strict monoidal category (C,, I) such that or all objects a and b and morphisms and g in C a b = b a and g = g. Note that a commutative monoidal category is the same as a strict symmetric monoidal category where the symmetry isomorphisms σ a,b : a b b a are all identity morphisms. Every strict monoidal unctor between commutative monoidal categories is automatically a strict symmetric monoidal unctor. This motivates the ollowing deinition: Deinition 7. Let CMC be the category whose objects are commutative monoidal categories and whose morphisms are strict monoidal unctors. Deinition 8. Let U : CMC Petri be the unctor that makes the ollowing 7

assignments on commutative monoidal categories and unctors: C Mor(C) C Mor(C ) s Ob(C) t s Ob(C ) t N[Ob(C)] N[] N[Ob(C )]. where s, t: Mor(C) Ob(C) map any morphism o C to its source and target, and similarly or s, t : Mor(C ) Ob(C ). Now we construct a let adjoint to U. For this we must construct the ree commutative monoidal category F P on a Petri net P = (s, t: T N[S]). We take the commutative monoid o objects Ob(F P ) to be N[S]. We construct the commutative monoid o morphisms Mor(F P ) as ollows. First we generate morphisms recursively: or every transition τ T we include a morphism τ : s(τ) t(τ); or any object a we include a morphism a : a a; or any morphisms : a b and g : a b we include a morphism denoted + g : a + a b + b to serve as their tensor product; or any morphisms : a b and g : b c we include a morphism g : a c to serve as their composite. Then we mod out by an equivalence relation on morphisms that imposes the laws o a commutative monoidal category, obtaining the commutative monoid Mor(F P ). Deinition 9. Let F : Petri CMC be the unctor that makes the ollowing assignments on Petri nets and morphisms: T s t s N[S] N[g] F P T N[S ] F P. t F (,g) Here F (, g): F P F P is deined on objects by N[g]. On morphisms, F (, g) is the unique map extending that preserves identities, composition, and the tensor product. Proposition 0. F is a let adjoint to U. Proo. We show that that or any Petri net P = (s, t: T N[S]) and commutative monoidal category C, there is a natural isomorphism hom CMC (F P, C) = hom Petri (P, UC). 8

Given a strict monoidal unctor we deine a morphism o Petri nets F : F P C T F T Mor(C) s t s t N[S] N[F] N[Ob(C)] using the act that T Mor(F P ). Conversely, given a Petri net morphism T Mor(C) s t s t N[S] N[g] N[Ob(C)] we deine G : F P C to be the unique strict monoidal unctor that equals g on objects in S N[S] = Ob(F P ) and equals on morphisms in T Mor(F P ). It is straightorward to veriy that these two operations are inverses. For naturality, it suices to show that or : P P and g : C C the ollowing square commutes hom(f P, C) hom(p, UC) hom(f,g) hom(f P, C ) hom(,ug) hom(p, UC ). This is also a straightorward calculation. 3 Open Petri Nets Our goal in this paper is to use the language o double categories to develop a theory o Petri nets with inputs and outputs that can be glued together. The irst step is to construct a double category Open(Petri) whose horizontal - morphisms are open Petri nets. For this we need a unctor L: Set Petri that maps any set S to a Petri net with S as its set o places, and we need L to be a let adjoint. 9

Deinition. Let L: Set Petri be the unctor deined on sets and unctions as ollows: X N[X] N[] Y N[Y ] where the unlabeled maps are the unique maps o that type. Lemma 2. The unctor L has a right adjoint R: Petri Set that acts as ollows on Petri nets and Petri net morphisms: T s t s N[S] N[g] S T N[S] S. t g Proo. For any set X and Petri net P = (s, t: T N[S]) we have natural isomorphisms hom Petri ( L(X), T s t N[S] ) ( = hompetri N[X], T = hom Set (X, S) = hom Set ( X, R( T t s N[S] ) ). s N[S] ) t We now introduce the main object o study: the double category Open(Petri). Since this is a symmetric monoidal double category, it involves quite a lot o structure. The deinition o symmetric monoidal double category can be ound in Appendix A. Theorem 3. There is a symmetric monoidal double category Open(Petri) or which: objects are sets vertical -morphisms are unctions horizontal -cells rom a set X to a set Y are open Petri nets P : X Y, that is, cospans in Petri o the orm i P o LX LY 0

2-morphisms α: P P are commutative diagrams LX i P o LY in Petri. L α o Lg LX i P LY. Composition o vertical -morphisms is the usual composition o unctions. Composition o horizontal -cells is composition o cospans via pushout: given two horizontal -cells P Q i o i 2 o 2 LX LY LY their composite is given by this cospan rom LX to LZ: LZ j P P + LY Q j Q P Q i o i 2 o 2 LX LY LZ where the diamond is a pushout square. The horizontal composite o 2-morphisms LX i P o LY LY i 2 Q o 2 LZ L α o LX i P LY Lg Lg β o 2 LY i 2 Q LZ Lh is given by LX j P i P + LY Q j Q o LZ L α+ Lg β j Q o 2 LX j P i P + LY Q LZ. Vertical composition o 2-morphisms is done using composition o unctions. The symmetric monoidal structure comes rom coproducts in Set and Petri. Proo. We construct this symmetric monoidal double category using the machinery o structured cospans [2]. The main tool is the ollowing lemma, which explains the symmetric monoidal structure in more detail: Lh

Lemma 4. Let A be a category with inite coproducts and X be a category with inite colimits. Given a let adjoint L: A X, there exists a unique symmetric monoidal double category L Csp(X), such that: objects are objects o A, vertical -morphisms are morphisms o A, a horizontal -cell rom a A to b A is a cospan in X o this orm: La x Lb a 2-morphism is a commutative diagram in X o this orm: La x Lb L h Lg Lc y Ld. Composition o vertical -morphisms is composition in A. Composition o horizontal -cells is composition o cospans in X via pushout: given horizontal -cells i x o i 2 y o 2 La Lb Lb their composite is this cospan rom La to Lc: j x x + Lb y j y Lc x y i o i 2 o 2 La Lb Lc where the diamond is a pushout square. The horizontal composite o 2-morphisms La i x o Lb Lb i 2 y o 2 Lc is given by L α o La i x Lb La j xi Lg x + Lb y Lg β o 2 Lb i 2 y Lc j yo 2 Lc Lh L α+ Lg β j y o 2 La j x i x + Lb y Lc. Lh 2

The vertical composite o 2-morphisms La i x o Lb L α o La i x Lb Lg La i x o Lb is given by L α Lg La i x o Lb La i x o Lb L( ) α α La i x o Lb. L(g g) The tensor product is deined using chosen coproducts in A and X. Thus, the tensor product o two objects a and a 2 is a + a 2, the tensor product o two vertical -morphisms a a 2 is b a + a 2 2 b 2 + 2 b + b 2, the tensor product o two horizontal -cells i o La i x 2 o Lb La 2 2 x 2 Lb 2 is L(a + a 2 ) i +i 2 o x + x 2 +o 2 L(b + b 2 ), and the tensor product o two 2-morphisms La i x o Lb La 2 i 2 x 2 o 2 Lb 2 L La α o i x Lb Lg L 2 La 2 α 2 o 2 i 2 x 2 Lb 2 Lg 2 3

is L(a + a 2 ) i +i 2 x + x 2 o +o 2 L(b + b 2 ) L( + 2) L(a + a 2) α +α 2 L(g +g 2) i +i 2 x + x o +o 2 2 L(b + b 2). The units or these tensor products are taken to be initial objects, and the symmetry is deined using the canonical isomorphisms a + b = b + a. Proo. This is [2, Cor. 3.0]. Note that we are abusing language slightly above. We must choose a speciic coproduct or each pair o objects in X and A to give LCsp(X) its tensor product. Given morphisms i : La x and i 2 : La 2 x 2, their coproduct is really a morphism i +i 2 : La +La 2 x +x 2 between these chosen coproducts. But since L preserves coproducts, we can compose this morphism with the canonical isomorphism L(a + a 2 ) = La + La 2 to obtain the morphism that we call i + i 2 : L(a + a 2 ) x + x 2 above. To apply this lemma to the situation at hand we need the ollowing result. Lemma 5. Petri has small colimits. Proo. A small diagram D : J Petri induces small diagrams D T : J Set on the transitions and D S : J Set on the places. We compute colimits in Petri pointwise, so colim D consists o the Petri net colim D T s t N[colim D S ] made into a cocone using these morphisms o Petri nets or each j J: D T (j) j colim D T s j t j σ τ N[D S (j)] N[g j] N[colim D S ]. Recall that we are abusing language slightly and using N[S] to mean the underlying set o the ree commutative monoid on S. All the above colimits are taken in Set, and σ and τ are the unique maps induced by the universal property o colim D T. Given a Petri net ( ) s P = T N[S] t 4

and a amily o Petri net morphisms {(h j, k j ): D(j) P }, we can prove there exists a unique morphism o Petri nets illing in the dashed arrows here: D T (j) s j t j N[D S (j)] j h j T s t N[S] N[k j] N[g j] colim D T σ τ N[colim D S ] The let dashed arrow arises rom the universal property o colim D T, while the right one is obtained by applying N to the arrow arising rom the universal property o colim D S. We now have all o the ingredients to apply Lemma 4 to the unctor L: Set Petri. Thm. 3 ollows rom realizing that Open(Petri) as described in the theorem is the symmetric monoidal double category L Csp(Petri). 4 Open Commutative Monoidal Categories In Section 2 we saw how a Petri net P gives a commutative monoidal category F P, and in Section 3 we constructed a double category Open(Petri) o open Petri nets. Now we construct a double category Open(CMC) o open commutative monoidal categories and a map Csp(F ): Open(Petri) Open(CMC). This can be seen as providing a unctorial semantics or open Petri nets in which any open Petri is mapped to the commutative monoidal category it presents. The reachability semantics or open Petri nets is based on this more undamental orm o semantics. The key is this commutative diagram o let adjoint unctors: Set L L Petri F CMC where L = F L sends any set to the ree commutative monoidal category on this set: L X has N[X] as its set o objects, and only identity morphisms. Using Lemma 4, we can produce two symmetric monoidal double categories rom this diagram. We have already seen one: Open(Petri) = L Csp(Petri). We now introduce the other: Open(CMC) = L Csp(CMC). 5

Theorem 6. There is a symmetric monoidal double category Open(CMC) or which: objects are sets vertical -morphisms are unctions horizontal -cells rom a set X to a set Y are open commutative monoidal categories C : X Y, that is, cospans in CMC o the orm i C o L X L Y where C is a commutative monoidal category and i, o are strict monoidal unctors, 2-morphisms α: C C are commutative diagrams in CMC o the orm L X i C o L Y L α o L g L X i C L Y. and the rest o the structure is given as in Lemma 4. Proo. To apply Lemma 4 to the unctor L : Set CMC we just need to check that CMC has inite colimits. For this one can construct the initial object, binary coproducts and coequalizers explicitly. Alternately, one can note that CMC is the Eilenberg Moore category o a monad on Cat and that Cat has small colimits, so CMC has small colimits too [6, Cor. 3.4.2]. The unctor F : Petri CMC induces a map sending open Petri nets to open commutative monoidal categories. This map is actually part o a symmetric monoidal double unctor, a concept deined in Appendix A. Lemma 7. There is a symmetric monoidal double unctor Csp(F ): Open(Petri) Open(CMC) that is the identity on objects and vertical -morphisms, and makes the ollowing assignments on horizontal -cells and 2-morphisms: LX i P o LY L X F i F P F o L Y L α Lg L LX i P LY L X F i F P L Y. o F α F o L g 6

Proo. This ollows rom the theory o structured cospans. More generally, suppose A is a category with inite coproducts and X, X are categories with inite colimits. Suppose there is a commuting triangle o let adjoints A L X L Then Lemma 4 gives us symmetric monoidal double categories L Csp(X) and L Csp(X ), and Thm. 4.2 o [2] gives a symmetric monoidal double unctor F X. Csp(F ): L Csp(X) L Csp(X ) that is the identity on objects and vertical morphisms, and acts as ollows on horizontal -cells and 2-morphisms: La i x o Lb L a F i F x F o L b L α Lg L F α La i x Lb L a F i F x L b. o F o L g 5 The Double Category o Relations Using the language o unctorial semantics, Open(Petri) can be thought o as a syntax, and reachability as a choice o semantics. To implement this, we show that the reachability relation o a Petri net can be deined or open Petri nets in a way that gives a lax double unctor rom Open(Petri) to the double category o relations constructed by Grandis and Paré [6, Sec. 3.4]. Here we recall this double category and give it a symmetric monoidal structure. This double category, which we call Rel, has: sets as objects, unctions : X Y as vertical -morphisms rom X to Y, relations R X Y as horizontal -cells rom X to Y, squares R X Y X Y g S X 2 Y 2 X 2 Y 2 obeying ( g)r S as 2-morphisms. 7

The last item deserves some explanation. A preorder is a category such that or any pair o objects a, b there exists at most one morphism α: x y. When such a morphism exists we usually write x y. Similarly there is a kind o double category or which given any rame M X Y g there exists at most one 2-morphism N X 2 Y 2 M X Y α g N X 2 Y 2 illing this rame. Following [] we call this a degenerate double category. Our deinition o the 2-morphism in Rel will imply that this double category is degenerate. Composition o vertical -morphisms in Rel is the usual composition o unctions, while composition o horizontal -cells is the usual composition o relations. Since composition o relations obeys the associative and unit laws strictly, Rel will be a strict double category. Since Rel is degenerate, there is at most one way to deine the vertical composite o 2-morphisms R X Y X Y α g R X Y X Y S X 2 Y 2 X 2 Y 2 = βα g g β g T X 3 Y 3 X 3 Y 3 T X 3 Y 3 X 3 Y 3 so we need merely check that a 2-morphism βα illing the rame at right exists. This amounts to noting that ( g)r S, ( g )S T = ( g )( g)r T. 8

Similarly, there is at most one way to deine the horizontal composite o 2- morphisms R X Y X Y R Y Z Z R R X Z X Z α g α h = α α h S X 2 Y 2 X 2 Y 2 S Y 2 Z 2 Z 2 S S X 2 Z 2 X 2 Z 2 so we need merely check that a iller α α exists, which amounts to noting that ( g)r S, (g h)r S = ( h)(r R) S S. Theorem 8. There exists a strict double category Rel with the above properties. Proo. We use the deinition o double category in Appendix A (De. 25), which introduces two concepts not mentioned so ar: the category o objects and the category o arrows. We deine the category o objects Rel 0 to have sets as objects and unctions as morphisms. We deine the category o arrows Rel to have relations as objects and squares R X X 2 X X 2 g S Y Y 2 Y Y 2 with ( g)r S as morphisms. The source and target unctors S, T : Rel Rel 0 are clear. The identity-assigning unctor u: Rel 0 Rel sends a set X to the identity unction X and a unction : X Y to the unique 2-morphism X X X Y Y Y The composition unctor : Rel Rel0 Rel Rel acts on objects by the usual composition o relations, and it acts on 2-morphisms by horizontal composition as described above. These unctors can be shown to obey all the axioms o a double category. In particular, because Rel is degenerate, all the required equations between 2-morphisms, such as the interchange law, hold automatically. 9

Next we make Rel into a symmetric monoidal double category. To do this, we irst give Rel 0 = Set the symmetric monoidal structure induced by the cartesian product. Then we give Rel a symmetric monoidal structure as ollows. Given linear relations R X Y and R 2 X 2 Y 2, we deine R R 2 = {(x, x 2, y, y 2 ) : (x, y ) R, (x 2, y 2 ) R 2 } X X 2 Y Y 2. Given two 2-morphisms in Rel : R X Y X Y R X Y X Y α g α g S X 2 Y 2 X 2 Y 2 S X 2 Y 2 X 2 Y 2 there is at most one way to deine their product R R (X X ) (Y Y ) X X Y Y α α g g S S (X 2 X 2 ) (Y2 Y 2 ) X 2 X 2 Y 2 Y because Rel is degenerate. To show that α α exists, we need merely note that ( g)r S, ( g )R S = ( g g )(R R ) S S. Theorem 9. The double category Rel can be given the structure o a symmetric monoidal double category with the above properties. Proo. We have described Rel 0 and Rel as symmetric monoidal categories. The source and target unctors S, T : Rel Rel 0 are strict symmetric monoidal unctors. We must also equip Rel with two other pieces o structure. One, called χ, says how the composition o horizontal -cells interacts with the tensor product in the category o arrows. The other, called µ, says how the identityassigning unctor u relates the tensor product in the category o objects to the tensor product in the category o arrows. These are deined as ollows. Given our horizontal -cells R X Y, R 2 Y Z, S X 2 Y 2, S 2 Y 2 Z 2, 2 20

the globular 2-isomorphism χ: (R 2 S 2 )(R S ) (R 2 R ) (S 2 S ) is the identity 2-morphism (R 2 S 2)(R S ) X X 2 Z Z 2 (R 2R ) (S 2S ) X X 2 Z Z 2 The globular 2-isomorphism µ: u(x Y ) u(x) u(y ) is the identity 2- morphism X Y X Y X Y X Y X Y X Y All the commutative diagrams in the deinition o symmetric monoidal double category (Des. 29 and 30) can be checked straightorwardly. In particular, all diagrams o 2-morphisms commute automatically because Rel is degenerate. 6 The Reachability Semantics Now we explain how Open(Petri) provides a compositional approach to the reachability problem. In particular, we prove that the reachability semantics deines a lax double unctor which is symmetric monoidal. : Open(Petri) Rel Deinition 20. Let P be a Petri net (s, t: T N[S]). A marking o P is an element m N[S]. Given a transition τ T, a iring o τ is a tuple (τ, m, n) such that n = m s(τ) + t(τ). We say that a marking n is reachable rom a marking m i or some k there is a sequence o markings m = m,..., m k = n and irings {(τ i, m i, m i+ )} k i=. In particular, taking k =, any marking is reachable rom itsel with no irings. Given two markings o a Petri net, the problem o deciding whether one is reachable rom the other is called the reachability problem. In 984 Mayr showed that the reachability problem is decidable [22]. However, it is a very hard problem: in 976 Lipton had showed that it requires at least exponential 2

space, and in act any EXPSPACE algorithm can be reduced in polynomial time to a Petri net reachability problem [20]. There is a close connection between reachability and the ree commutative monoidal category on a Petri net constructed in Thm. 9. Proposition 2. I m and n are markings o a Petri net P, then n is reachable rom m i and only i there is a morphism : m n F P. Proo. I n is reachable rom m, there is a sequence o markings m = m,..., m k = n and irings {(τ i, m i, m i+ )} k i=. For each iring (τ i, m i, m i+ ) there is a morphism in F P given by τ i + mi s(τ i) : m i m i+. Taking the composite o these morphisms gives a morphism : m n in F P. Conversely, i : m n is a morphism in F P, it can be obtained by composition and addition (that is, the tensor product) rom morphisms arising rom the basic transitions and symmetry morphisms. Because + is a unctor, we have the interchange law ( g ) + ( 2 g 2 ) = ( + 2 ) (g + g 2 ) whenever, g and 2, g 2 are pairs o composable morphisms in F P. We can use this inductively to simpliy into a composite o sums. I : a b and 2 : a 2 b 2 are morphisms in F P, the interchange law also tells us that + 2 = ( a ) + ( b2 2 ) = ( + b2 ) ( a + 2 ). This act allows us to inductively simpliy to a composite o sums each containing one transition. The actors in this composite correspond to irings that make n reachable rom m. (Here we allow the possibility o an empty composite, which corresponds to an identity morphism.) Deinition 22. We deine the reachability relation o an open Petri net to be the relation LX i P o LY P = {(x, y) N[X] N[Y ] o(y) is reachable rom i(x)} N[X] N[Y ]. Note that P depends on the whole open Petri net P : X Y, not just its underlying Petri net P. By Prop. 2, P = {(x, y) N[X] N[Y ] h: F (i)(x) F (o)(y)}. Here F (i)(x) and F (o)(y) are objects o the category F P, and the reachability relation holds i there is a morphism in F P rom the irst o these to the second. 22

Theorem 23. There is a lax double unctor : Open(Petri) Rel, called the reachability semantics, that sends any object X to the underlying set o the ree commutative monoid N[X], which we denote simply as N[X], any vertical -morphism : X Y to the underlying unction o N[], any horizontal -cell, that is, any open Petri net to the reachability relation P. LX i P o LY, any 2-morphism α: P P, that is any commuting diagram LX i P o LY to the square L α Lg LX i P o LY, P X Y N[X] N[Y ] N[] P X Y N[g] N[X ] N[Y ]. Proo. We construct as the composite G Csp(F ) where Csp(F ): Open(Petri) Open(CMC) is the double unctor constructed in Prop. 7 and G: Open(CMC) Rel is deined as ollows. Recall that we have categories o objects Open(CMC) 0 = Rel 0 = Set. We deine G 0 : Open(CMC) 0 Rel 0 to be the composite K N where N: Set CommMon is the ree commutative monoid unctor and K : CommMon Set is its right adjoint, sending any commutative monoid to its underlying set. We deine G : Open(CMC) Rel as ollows: L X i C o L Y G C N[X] N[Y ] N[X] N[Y ] L α L g N[] N[g] i o L X C L Y G C N[X ] N[Y ] N[X ] N[Y ]. 23

where or brevity we write N[X] to mean the set K(N[X]). We deine G C to be the relation {(x, y) L X L Y h: i(x) o(y)} N[X] N[Y ] and G α to be the inclusion (N[] N[g])G C G C. To see that this inclusion is well-deined, suppose (x, y) G C. Then there exists a morphism h: i(x) o(y) in C. We thus have a morphism α(h): α(i(x)) α(o(y)) in C. However, on objects we have α i = i L = i N[] and similarly α o = o N[g], so α(h): i (N[](x)) o (N[g](y)). It ollows that (N[] N[g])(x, y) G C. Next we prove that G is a lax double unctor. First note that by construction we have the ollowing equalities: S G = G 0 S, T G = G 0 T. Next we need the composition comparison required by De. 27. compose C : X Y and D : Y Z in Open(CMC): Suppose we j C C + L Y D j D C D i o i 2 o 2 L X We need to prove that We have L Y G (D) G (C) G (D C). L Z. G (D C) = {(x, z) L X L Z h: j C i (x) j D o 2 (z)}. On the other hand, G C = {(x, y) L X L Y m: i (x) o (y)} and G D = {(y, z) L Y L Z n: i 2 (y) o 2 (z)} which compose to give the relation G D G C = {(x, z) L X L Z y (x, y) G C and (y, z) G D}. 24

Suppose (x, z) G D G C. Then there exist morphisms m: i (x) o (y) in C and n: i 2 (y) o 2 (z) in D. By commutativity o the pushout square, j C o = j D i 2. Thereore, the codomain o j C (m) is j C o (y) = j D i 2 (y), which is also the domain o j D (n). This allows us to orm the composite j D (n) j C (m): j C i (x) j D o 2 (z). Thus (x, z) G (D C) as desired. We also need the identity comparison required by De. 27. Thus, we need U G0(X) G (U X ) or any set X. By deinition, U X Open(CMC) is the cospan L X L X L X. Because L X has no non-identity morphisms, G maps this to the identity relation on the set N[X]. On the other hand, G 0 (X) = N[X] and U G0(X) is the identity relation on this set. So, the desired inclusion is actually an equality. Finally, because Rel is a degenerate double category, the composition and identity comparisons or G are trivially natural transormations. For the same reason, the diagrams in De. 27 expressing compatibility with the associator, let unitor, and right unitor also commute trivially. It ollows that G is a lax double unctor. To complete the proo, one simply computes the composite = G Csp(F ) and checks that it matches the description in the theorem statement. The reachability semantics is only lax: given two open Petri nets P : X Y and Q: Y Z, the composite o Q and P is in general a proper subset o (QP ). To see this, take P to be this open Petri net: X A α B Y 2 β C D 3 4 and take Q to be this: Y 2 3 B C γ Z 4 D δ E 5 25

Then their composite, QP : X Z, looks like this: X A α β B C D γ δ E Z 5 We have P = {(n, n, 0, 0) n N} N N 3 since tokens starting at A can only move to B, and similarly Q = {(0, 0, n, n) n N} N 3 N. It ollows that On the other hand Q P = {(0, 0)} N N. (QP ) = {(n, n) n N} N N since in the composite open Petri net QP tokens can move rom A to E. The point is that tokens can only accomplish this by leaving the open Petri net P, going to Q, then returning to P, then going to Q. The composite relation Q P only keeps track o processes where tokens leave P, move to Q, and never reenter P. On the other hand, the reachability semantics is maximally compatible with running Petri nets in parallel: Theorem 24. The reachability semantics : Open(Petri) Rel is symmetric monoidal. Proo. Because Csp(F ) is symmetric monoidal it suices to show that G: Open(CMC) Rel is symmetric monoidal. This is simpliied by that act that Rel is a degenerate double category. Following De. 3, it suices to show that G 0 : (Set, +) (Set, ) is symmetric monoidal, G : Open(CMC) Rel is symmetric monoidal, we have equations o monoidal unctors S G = G 0 S, T G = G 0 T, 26

the composition and unit comparisons are monoidal natural transormations. First, recall that G 0 = K N. Since N: Set CommMon is a let adjoint it preserves inite coproducts. Since K : CommMon Set is a right adjoint is preserves inite products. However, inite products in CommMon are also inite coproducts. Thus, G 0 maps inite coproducts to inite products, and is thus a symmetric monoidal unctor rom (Set, +) to (Set, ). Next, suppose we are given two open commutative monoidal categories L X i C o L Y, L X i C L Y. Their tensor product is L (X + X ) i+i C + C o+o L (Y + Y ). The set o objects o L (X + X ) is naturally isomorphic to N[X] N[X ], and similarly or L (Y + Y ), so we have natural isomorphisms G (C + C ) = {((x, x, y, y ) N[X] N[X ] N[Y ] N[Y ] h: i(x) o(y) and h : i (x ) o (y )} = G (C) G (C ). Using this act one can check that G is symmetric monoidal. One can check that the equations S G = G 0 S and T G = G 0 T are equations o monoidal unctors, and the composition and unit comparisons o G are trivially monoidal natural transormations because Rel is degenerate. 7 Conclusions The ideas presented here can be adapted to handle timed Petri nets, colored Petri nets with guards, and other kinds o Petri nets. One can also develop a reachability semantics or open Petri nets that are glued together along transitions as well as places. We hope to treat some o these generalizations in uture work. It would be valuable to have (QP ) = Q P, since then the reachability relation or an open Petri net could be computed compositionally, not merely approximated rom below using Q P (QP ). We conjecture that (QP ) = Q P i P and Q are one-way open Petri nets. Here an open Petri net LX i P o LY is one-way i no place in the image o i appears in the target t(τ) o any transition τ o P, and no place in the image o o appears in the source s(τ) o any transition τ o P. One-way open Petri nets should be the horizontal o 27

-cells in a ull sub-double category OneWay(Petri) o Open(Petri), and we conjecture that the reachability semantics restricts to an actual (not merely lax) double unctor : OneWay(Petri) Rel. Acknowledgements We would like to thank Kenny Courser or help with double categories and or a careul reading o this paper. We also thank Daniel Cicala, Joe Moeller, Christina Vasilakopoulou and Christian Williams or many insightul conversations. A Double Categories What ollows is a brie introduction to double categories. A more detailed exposition can be ound in the work o Grandis and Paré [6, 7], and or monoidal double categories the work o Shulman [30]. We use double category to mean what earlier authors called a pseudo double category. Deinition 25. A double category is a category weakly internal to Cat. More explicitly, a double category D consists o: a category o objects D 0 and a category o arrows D, source and target unctors an identity-assigning unctor and a composition unctor S, T : D D 0, U : D 0 D, : D D0 D D where the pullback is taken over D T D0 S D, such that S(U A ) = A = T (U A ), S(M N) = SN, T (M N) = T M, natural isomorphisms called the associator α N,N,N : (N N ) N N (N N ), the let unitor and the right unitor λ N : U T (N) N N, ρ N : N U S(N) N 28

such that S(α), S(λ), S(ρ), T (α), T (λ) and T (ρ) are all identities and such that the standard coherence axioms hold: the pentagon identity or the associator and the triangle identity or the let and right unitor [2, Sec. VII.]. I α, λ and ρ are identities, we call D a strict double category. Objects o D 0 are called objects and morphisms in D 0 are called vertical - morphisms. Objects o D are called horizontal -cells o D and morphisms in D are called 2-morphisms. A morphism α: M N in D can be drawn as a square: A M B α g N C D. where = Sα and g = T α. I and g are identities we call α a globular 2-morphism. These give rise to a bicategory: Deinition 26. Let D be a double category. Then the horizontal bicategory o D, denoted H(D), is the bicategory consisting o objects, horizontal -cells and globular 2-morphisms o D. We have maps between double categories, and also transormations between maps: Deinition 27. Let A and B be double categories. A double unctor F : A B consists o: unctors F 0 : A 0 B 0 and F : A B obeying the ollowing equations: S F = F 0 S, T F = F 0 T, natural isomorphisms called the composition comparison: φ(n): F (N) F (N ) and the identity comparison: φ A : U F0(A) F (N N ) F (U A ) whose components are globular 2-morphisms, such that the ollowing diagram commmute: 29

a diagram expressing compatibility with the associator: (F (N) F (N )) F (N ) φ(n,n ) F (N N ) F (N ) α F (N) (F (N ) F (N )) φ(n,n ) F (N) F (N N ) φ(n N,N ) F ((N N ) N ) F (α) φ(n,n N ) F (N (N N )) two diagrams expressing compatibility with the let and right unitors: F (N) U F0(A) ρ F (N) F (N) φ A F (ρ N ) F (N) F (U A ) φ(n, U A ) F (N U A ) U F0(B) F (N) φ B F (U B ) F (N) λ F (N) φ(u B, N) F (N) F (λ N ) F (U B N). I the 2-morphisms φ(n, N ) and φ A are identities or all N, N A and A A 0, we say F : A B is a strict double unctor. I on the other we drop the requirement that these 2-morphisms be invertible, we call F a lax double unctor. Deinition 28. Let F : A B and G: A B be lax double unctors. A transormation β : F G consists o natural transormations β 0 : F 0 G 0 and β : F G (both usually written as β) such that S(β M ) = β SM and T (β M ) = β T M or any object M A, β commutes with the composition comparison, and β commutes with the identity comparison. Shulman deines a 2-category Dbl o double categories, double unctors, and transormations [30]. This has inite products. In any 2-category with inite products we can deine a pseudomonoid [], which is a categoriication o the concept o monoid object. For example, a pseudomonoid in Cat is a monoidal category. 30

Deinition 29. A monoidal double category is a pseudomonoid in Dbl. Explicitly, a monoidal double category is a double category equipped with double unctors : D D D and I : D where is the terminal double category, along with invertible transormations called the associator: let unitor: and right unitor: A: ( D ) ( D ), L: ( D I) D, R: (I D ) D satisying the pentagon axiom and triangle axioms. This deinition neatly packages a large quantity o inormation. Namely: D 0 and D are both monoidal categories. I I is the monoidal unit o D 0, then U I is the monoidal unit o D. The unctors S and T are strict monoidal. is equipped with composition and identity comparisons χ: (M N ) (M 2 N 2 ) (M M 2 ) (N N 2 ) µ: U A B (UA U B ) making three diagrams commute as in De. 27. The associativity isomorphism or is a transormation between double unctors. The unit isomorphisms are transormations between double unctors. Deinition 30. A braided monoidal double category is a monoidal double category equipped with an invertible transormation β : τ called the braiding, where τ : D D D D is the twist double unctor sending pairs in the object and arrow categories to the same pairs in the opposite order. The braiding is required to satisy the usual two hexagon identities [2, Sec. XI.]. I the braiding is sel-inverse we say that D is a symmetric monoidal double category. In other words: D 0 and D are braided (resp. symmetric) monoidal categories, the unctors S and T are strict braided monoidal unctors, and 3

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