An axiomatization of the Token Game based onPetri Algebras

Eric Badouel1, Jules Chenou2, Goulven Guillou3

1: INRIA Rennes Bretagne Atlantique,Campus Universitaire de Beaulieu, F35042Rennes Cedex, France

2: Faculte des Sciences, Universite de Douala, B.P. 24157 Douala, Cameroon3: LISYC/EA 3883, UFR des Sciences et Techniques, Universite de Bretagne

Occidentale. 6, avenue Le Gorgeu, B.P. 817. F29285 Brest Cedex, Franceeric.badouel@inria.fr, chenouj@yahoo.fr, goulven.guillou@univ-brest.fr

Abstract. The firing rule of Petri nets relies on a residuation opera-tion for the commutative monoid of non negative integers. We identify aclass of residuated commutative monoids, called Petri algebras, for whichone can mimic the token game of Petri nets to define the behaviour ofgeneralized Petri nets whose flow relations and place contents are val-ued in such algebraic structures. The sum and its associated residuationcapture respectively how resources within places are produced and con-sumed through the firing of a transition. We show that Petri algebrascoincide with the positive cones of lattice-ordered commutative groupsand constitute the subvariety of the (duals of) residuated lattices gener-ated by the commutative monoid of non negative integers. We howeverexhibit a Petri algebra whose corresponding class of nets is strictly moreexpressive than the class of Petri nets. More precisely, we introduce aclass of nets, termed lexicographic Petri nets, that are associated withthe positive cones of the lexicographic powers of the additive group ofreal numbers. This class of nets is universal in the sense that any netassociated with some Petri algebra can be simulated by a lexicographicPetri net. All the classical decidable properties of Petri nets however (ter-mination, covering, boundedness, structural boundedness, accessibility,deadlock, liveness ...) are undecidable on the class of lexicographic Petrinets. Finally we turn our attention to bounded nets associated with Petrialgebras and show that their dynamics can be reformulated in term ofMV-algebras.

1 Introduction

The Petri net model is a graphical and mathematical modeling tool that, sinceits introduction in the early sixties, have come to play a pre-eminent role in theformal study of concurrent discrete-event dynamic systems. Petri nets are naturalextensions of automata in which states are distributed (they can be representedas vectors of ”local states”) and for which the occurrence of an event relieson local conditions. More precisely a Petri net (P, T, Pre, Post) consists of afinite set P of places, a finite set T of transitions (disjoint from P ), and flow

relations Pre, Post : P × T → N. Places can contain some tokens representingthe resources available in this place for the current configuration. Such a (global)configuration of a Petri net is given as a vector M : P → N, called marking,indicating the number of tokens available in each place. Tokens are consumedand produced by the firing of transitions according to the so-called token game:

M [t〉M ′ ⇔ ∀p ∈ P M(p) w Pre(p, t)and M ′(p) = (M(p)− Pre(p, t)) + Post(p, t)

The token game of Petri nets says that in order for a transition t to fire inmarking M it should be the case that each place contains enough resources asit is expressed by the condition M(p) w Pre(p, t). Then the firing of transition tproceeds in two stages: a consumption of resources (Pre(p, t) tokens are removedfrom place p) followed by a production of resources (Post(p, t) tokens are addedto place p). The notation M [t〉M ′ expresses the fact that transition t is allowedto fire in marking M and that firing t in marking M produces the new markingM ′. Numerous techniques, supported and automated by software tools, can beused to verify that some required properties are met for systems specified usingPetri nets. For instance the following problems have an effective solution on theclass of Petri nets:

Reachability Is a given marking reachable (from a fixed initial marking) througha sequence of firable transitions?

Coverability Is a given marking M covered by some reachable marking M ′?(i.e. ∀p ∈ P M(p) ≤M ′(p))

Place-boundedness Does there exists an upper bound on the number of tokensthat a given place can hold in reachable markings?

Boundedness Are all places bounded on the set of reachable markings? (i.e. isthe Petri net equivalent to a finite state machine?)

Deadlock Does there exist a reachable marking in which no transition can fire?Liveness A transition t is live if for every reachable marking M there exists at

least one marking, reachable from M , in which t can fire.

By combining these basic properties we can express complex static and dynamicrequirements for the system. The ability, through the token game rule, to graph-ically visualise the behaviour of a Petri net helps the practitioner to uncovershortcomings in the specification or errors in the model. The Petri net techniquethus promotes an iterative modelling process in which the model can incremen-tally be constructed by alternatively modify and re-analyse it.

Numerous extensions of this basic model of Petri nets have been introducedover the years. Some of them are high level nets that allows for more compactrepresentations but does not increase the expressive power of Petri nets: thesehigh level nets can be unfolded into equivalent, even though in general muchlarger, Petri nets. Some extensions however change more dramatically the se-mantics of the original model. For instance timing constraints may be added,as in timed Petri nets or stochastic Petri nets for the purpose of enabling per-formance analysis. With continuous Petri nets the discrete state transition rule

is replaced by a notion of trajectory using a continuum of intermediate states.In Fuzzy Petri nets one has a possibilistic measure of the firing of a transitionin the given marking thus enabling to deal with incertainty. Our purpose inthis paper is to put forward an axiomatization of the token game of Petri nets.More precisely we identify a class of commutative residuated monoids, calledPetri algebras, for which one can mimic the token game of Petri nets to definethe behaviour of generalized Petri nets whose flow relations and place contentsare valued in such algebraic structures. The sum and its associated residuationcapture respectively how resources within places are produced and consumedthrough the firing of a transition. The class of usual Petri nets is associated withthe commutative monoid of non negative integers. We show that Petri alge-bras coincide with the positive cones of lattice-ordered commutative groups andconstitute the subvariety of the (duals of) residuated lattices generated by thecommutative monoid of non negative integers. The basic Petri net model is thusassociated with the generator of the variety of Petri algebras which shows thatthese extended nets share all algebraic properties of Petri nets, in particular theyhave the same equational and inequational theory. We however exhibit a Petri al-gebra whose corresponding class of nets is strictly more expressive than the classof Petri nets, i.e. their class of marking graphs is strictly larger. More precisely,we introduce a class of nets, termed lexicographic Petri nets, that are associatedwith the positive cones of the lexicographic powers of the additive group of realnumbers. This class of nets is proved to be universal in the sense that any netassociated with some Petri algebra can be simulated by a lexicographic Petrinet. All the classical decidable properties of Petri nets however (termination,covering, boundedness, structural boundedness, accessibility, deadlock, liveness...) are proved to be undecidable on the class of lexicographic Petri nets. Thisnegative result put emphasize on the fact that algebraic considerations alone arenot sufficient to capture the intrinsic nature of Petri nets. This might give usan understanding on the fact that all past attempts to find a strict extensionof Petri nets with the same decidable properties did fail. Finally we turn ourattention to bounded nets associated with Petri algebras and show that theirdynamics can be reformulated in term of MV-algebras.

2 An axiomatization of the token game

2.1 Playing the token game

In order to obtain an axiomatization of the token game of Petri nets we representthe marking of a net as a map M : P →

⊔p∈P Ap that associates with each

place p ∈ P the local value of the current configuration M(p) ∈ Ap in this place.Content of places are resources that are consumed and produced according tothe token game. Thus we assume that each place p ∈ P is associated with acommutative divisibility monoid Ap = (Ap,⊕, 0), i.e. a monoid such that

the relation a w b ⇔ ∃c · a = b⊕ c is an order relation (1)

The constant 0 represents the absence of resource and the binary operator ⊕the accumulation of resources in places. Immediate consequences of condition(1) are the following:

a⊕ b w a, b0 v aa⊕ b = 0 ⇒ a = b = 0

Moreover we need to have a residuation operation such that a b representsthe residual resource obtained by substracting b to a when b v a. Thus thefollowing should hold true:

b v a ⇒ a = (a b)⊕ b (2)

Usual Petri nets corresponds to the situation where, for every place p, Ap =(N,+, 0) is the commutative monoid of non negative integers with the truncateddifference nm = max (0;n−m) as residuation. This operation is characterizedby the universal property that for every non negative integer n, m and p

n+m w p ⇔ n w pm

Up to the reversal of the order relation, it is a commutative residuated monoidi.e. a commutative monoid (A,⊕, 0) with an order relation ≤ and a residuationoperation which is a right adjoint to the addition, in the sense that

a⊕ b ≤ c ⇔ a ≤ c b (3)

It follows immediately from this definition that a commutative monoid is resid-uated if and only if its addition is order preserving in each argument and theinequation a⊕ b ≤ c has a largest solution for a (namely c b). In particular theresidual is uniquely determined by the addition and the order relation. When themonoid is a divisibility monoid the order relation itself is defined in terms of theaddition and thus the whole structure is characterized by its monoid reduct. Thefollowing result is known in the context of residuated lattices (i.e. if we furtherassume that each pair of elements admits a least upper bound and a greatestlower bound) but the proof given below, shows that it is also valid even if themonoid is not assumed to be lattice-ordered.

Proposition 1. If (A,⊕, 0,v) is a co-residuated commutative monoid, i.e. amonoid equipped with a residuation operation such that

a⊕ b w c ⇔ a w c b (4)

where 0 is the least element, then the following three conditions are equivalent

(i) It is a divisibility monoid: a w b ⇔ ∃c · a = b⊕ c(ii) It is an upper semi-lattice with: a t b = (a b)⊕ b(iii) b v a ⇒ a = (a b)⊕ b

Lemma 1. Let (A,⊕, 0,v) be a commutative ordered monoid where the neutralelement is also the least element for the order relation, and let be some binaryoperation on A; then the universal property

a⊕ b w c ⇔ a w c b

is equivalent to the conjunction of the following properties:(i) (a⊕ b) b v a(ii) (a b)⊕ b w a, b(iii) a (a b) v a, b(iv) monotonies: a v a′ ∧ b v b′ ⇒ a⊕ b v a′ ⊕ b′ ∧ a b′ v a′ b

Proof. Let us assume that is left adjoint to the addition, i.e. the universalproperty (4) holds. Then from a⊕ b w a⊕ b it follows that (a⊕ b) b v a andfrom a b w a b if follows that (a b)⊕ b w a. Combining the commutativityof addition with the adjunction if follows that a w c b ⇔ b w c a and thenin particular b w a (a b). Since the neutral element is also the least elementfor the order relation we deduce a a = 0 (because a ⊕ 0 w a ⇒ 0 w a a)and then a⊕ b w b follows from a w b b = 0 and a⊕ b w a by commutativity.From a b w 0 = b b it follows that (a b) ⊕ b w b by adjunction. Similarlya (a b) v a follows from a b w 0 = a a by the equivalence a w c b ⇔b w c a. Thus conditions (i) up to (iii) are established. Suppose a v a′ thenfrom (a⊕ b) b v a v a′ it follows by adjunction that a ⊕ b v a′ ⊕ b andthen b v b′ ⇒ a ⊕ b v a ⊕ b′ by commutativity of the sum. By monotonyof the sum one can now deduce a v a′ v (a′ b) ⊕ b v (a′ b) ⊕ b′ and thena b′ v a′ b when a v a′ ∧ b v b. For the converse direction, let us nowassume that the conditions (i) up to (iv) are satisfied. Then a ⊕ b w c entailsthat a w (a⊕ b) b w c b and in the converse direction a w c b entailsa⊕ b w (c b)⊕ b w c.

We see that under the assumptions of this lemma, each pair a, b of elements hasan upper bound (a b)⊕ b and a lower bound a (a b). Proposition (1) statesthat the divisibility of the monoid is equivalent of the fact that the former is theleast upper bound at b . We will see below that the cancellability of the monoidis equivalent to the fact that the inequality in equation (i) in the above lemmais an equality and it entails that a (a b) is the greatest lower bound a u b.

Proof (of Proposition 1). The implications (ii) ⇒ (iii) ⇒ (i) are immediate.It remains to prove that (i) ⇒ (ii). Let us therefore assume that (A,⊕, 0) is acommutative co-residuated divisibility monoid. The assumption of Lemma (1)are met and we can therefore deduce that (a b) ⊕ b w a, b . Suppose thatc w a, b is some upper bound of a and b. By divisiblity there exist some x andy such that c = a⊕ x = b⊕ y and by adjunction it follows that y w (a⊕ x) band thus c = b ⊕ y w b ⊕ ((a⊕ x) b) w b ⊕ (a b) (by monotony of ⊕ and of w.r.t. its first argument and using the identity a⊕ x w a ).

In view of the above discussion we are let to set the following

Definition 1. A Petri pre-structure is a commutative co-residuated divisibil-ity monoid i.e. a commutative monoid equipped with a residuation operation(M,⊕, 0,) satisfying the conditions (1) and (4).

The firing of a transition proceeds in two stages: a consumption of resourcesin the input places followed by a production of resources in the output places.More precisely, the transition relation M [t〉M ′ stating that transition t can firein marking M and leads, when it is fired, to the new marking M ′ is given by:

M [t〉M ′ ⇔ ∀p ∈ P M(p) w Pre(p, t) ∧M ′(p) = (M(p)Pre(p, t))⊕Post(p, t)

A net is called homogeneous if all the algebras Ap are identical. We willstick to homogeneous nets until Section 3 where it will be noticed that the”multi-sorted” case add in fact no extra generality. By the way we also restrictour attention in this paper to commutative algebras. With non commutativemonoids it would be possible [3] for example to take fifo nets [27] into account.

2.2 Associativity of the firing rule

For any non empty sequence of transitions u = a0 . . . an−1 ∈ T+we let M [u〉M ′state the existence of markingsM = M0,M1, . . .,Mn = M ′ such thatMi [ai〉Mi+1

for every 0 ≤ i< n. Moreover we set M [ε〉M where ε ∈ T ∗ is the empty se-quence and M an arbitrary marking. We use M [u〉 (respectively [u〉M ′) as ashorthand for ∃M ′ M [u〉M ′ (resp. ∃M M [u〉M ′). If a, b ∈ T are transitionsin a (n usual) Petri net we have the following equivalences

M [ab〉 ⇔ M w Pre(a) and (M − Pre(a)) + Post(a) w Pre(b)⇔ M w max (Pre(a);Pre(a) + (Pre(b)− Post(a)))⇔ M w Pre(a) + max (0;Pre(b)− Post(a))⇔ M w Pre(a) + (Pre(b) Post(a))

This suggest to let for any sequences u, v ∈ T ∗

Pre(uv) = Pre(u)⊕ (Pre(v) Post(u))

and symmetrically

Post(uv) = (Post(u) Pre(v))⊕ Post(v)

For these definitions to make sense however, it remains to show that they donot depend upon the specific chosen decomposition w = uv; otherwise stated,the product defined on A×A by

(x, y)⊗ (x′, y′) = (x⊕ (x′ y), (y x′)⊕ y′)

should be associative.

Theorem 1. For any Petri pre-structure, the following conditions are equiva-lent:

(i) Operation ⊗ is associative,(ii) the identity (b⊕ c) a = (b a)⊕ (c (a b)) holds,(iii) the monoid is cancellable: a⊕ b = a⊕ c ⇒ b = c, and(iv) the identity (a⊕ b) b = a holds.

The remaining part of this section is devoted to the proof of this result. Asalready noticed the divisibility of the monoid (i.e. a v b ⇔ ∃c · b = a ⊕ c)entails the following identities:

(div1) : a⊕ b w a(div2) : 0 v a(div3) : a⊕ b = 0 ⇒ a = b = 0

the second of which states that the neutral element for ⊕ is the least element.This property in conjunction with condition (4) has the following interestingconsequences as it can be readily verified:

(int1) : a v b ⇔ a b = 0(int2) : a 0 = a(int3) : 0 a = 0(int4) : a a = 0(int5) : a b v a

The equivalence between the cancellability of the monoid and the identity

(a⊕ b) b = a (5)

is well-known for residuated lattices and it holds also trivially in our case: on oneside ((a⊕ b) b)⊕ b = (a⊕ b)t b = a⊕ b shows that the identity (a⊕ b) b = afollows from cancellability, conversely if this identity holds then a⊕ b = a⊕ c ⇒b = (a⊕ b) a = (a⊕ c) a = c. The following proposition establishes theequivalence between the first two statements in Theorem (1).

Proposition 2. In a Petri pre-structure associativity of ⊗ is equivalent to theidentity

(b⊕ c) a = (b a)⊕ (c (a b)) (8)

Proof. For the one hand we compute (x, y)⊗ [(x′, y′)⊗ (x”, y”)] =

= (x, y)⊗ [x′ ⊕ (x” y′), (y′ x”)⊕ y”] == (x⊕ [(x′ ⊕ (x” y′)) y], [y (x′ ⊕ (x” y′))]⊕ (y′ x”)⊕ y”)

and, on the other hand [(x, y)⊗ (x′, y′)]⊗ (x”, y”) =

= [x⊕ (x′ y), (y x′)⊕ y′]⊗ (x”, y”) == (x⊕ (x′ y)⊕ [x” ((y x′)⊕ y′)], [((y x′)⊕ y′) x”]⊕ y”)

Associativity of ⊗ is equivalent to the conjunction of both following identities

x⊕ [(x′ ⊕ (x” y′)) y] = x⊕ (x′ y)⊕ [x” ((y x′)⊕ y′)][((y x′)⊕ y′) x”]⊕ y” = [y (x′ ⊕ (x” y′))]⊕ (y′ x”)⊕ y”

which are in fact equivalent to each other by commutativity of ⊕, and they canbe both rewriten as:

x⊕ [(y ⊕ (z t)) u] = x⊕ (y u)⊕ [z ((u y)⊕ t)] (6)

By letting x = t = 0, u = a, y = b and z = c in the preceding identity it comesthat

(b⊕ c) a = (b a)⊕ (c (a b)) (8)

Conversely, the latter suffices to establish identity (6) since:

x⊕ [(y ⊕ (z t)) u] = x⊕ (y u)⊕ [(z t) (u y)] By (8)= x⊕ (y u)⊕ [z (t⊕ (u y)]

The last equality comes from the identity (a b)c = a(b⊕ c) which followsfrom the following equivalences

(a b) c v d ⇔ a b v c⊕ d ⇔ a v b⊕ c⊕ d ⇔ a (b⊕ c) v d

Proposition 3. A Petri pre-structure satisfying the identity (b⊕ c) a = (ba) ⊕ (c (a b)) has a cancellable monoid reduct (a ⊕ b = a ⊕ c ⇒ b = c) andit satisfies the following equivalence

a w b⊕ c ⇔ a w b and a b w c (7)

Proof. By (int1) a w b⊕ c is equivalent to (b⊕ c) a = 0, that is to say, by (8),to (b a)⊕ (c (a b)) = 0. By (div3) this latter condition is equivalent to theconjunction of ba = 0 with c (ab) = 0, and by (int1), to the conjunction ofa w b with a b w c. Now the conjunction of (7) with (4) allows us to concludethat

a w b ⇒ [a = b⊕ c ⇔ c = a b]which establish the cancellability of the monoid: Let us assume a ⊕ b = a ⊕ c,let d be this value, then d w a and b = d a = c.

Cancellability of the monoid is then a necessary condition if we want to ensureassociativity of the firing rule. Thus we let

Definition 2. A Petri algebra is a Petri pre-structure with a cancellable monoidreduct; i.e. it is a co-residuated cancellable commutative monoid.

Proposition 4. A Petri algebra is a lower semi-lattice with au b = a (a b)

Proof. By Lemma (1) we have that a (a b) v a, b . Let c v a, b be somelower bound of a and b. By monotony of residuation one has a b v a c fromwhich it follows that (a b) ⊕ c v (a c) ⊕ c = a t c = a (since c v a). Thenby cancellability c = (c⊕ (a b)) (a b) v a (a b) .

We recall that a commutative residuated lattice (see [5, 24]) consists of a set Aequipped with a structure of commutative monoid (A,⊕, 0), of a lattice structure(A,∨,∧) and of a residuation operation adjoint to the addition in the sensethat

a⊕ b ≤ c ⇔ a ≤ c b

where ≤ is the lattice order (i.e. a ≤ b ⇔ b = a ∨ b). By Propositions (1)and (4) the dual of a Petri algebra is a commutative residuated lattice. Thusthe following identities, which are obtained by dualizing1 known identities ofcommutative residuated lattices, hold (and moreover can be straightforwardlychecked):

(RL1) a⊕ (b u c) = (a⊕ b) u (a⊕ c)(RL2) (a t b) c = (a c) t (b c)(RL3) a (b u c) = (a b) t (a c)(RL4) (a b)⊕ b w a(RL5) a (a b) v b(RL6) a⊕ (b c) w (a⊕ b) c(RL7) (c b)⊕ (b a) w c a(RL8) c b w (c a) (b a)(RL9) b a w (c a) (c b)(RL10) c b w (c⊕ a) (b⊕ a)(RL11) (c a) b = c (a⊕ b) = (c b) a

as well as the monotony property asserting that x ⊕ y is increasing w.r.t. botharguments and x y is increasing w.r.t. x and decreasing w.r.t. y .

The last stage toward the proof of Theorem (1) is given by the followingproposition

Proposition 5. Petri algebras satisfy the following identity

(b⊕ c) a = (b a)⊕ (c (a b)) (8)

the proof of which follows from a sequence of intermediate results starting fromthe following property:

Lemma 2. In any Petri algebra1. If a w b then (i) a = b⊕ c⇔ a b = c and (ii) a w b⊕ c⇔ a b w c2. If a w c then (i) ab = c⇔ ac = aub and (ii) ab w c⇔ ac w aub3. If a w b and a w c then (i) a = b ⊕ c ⇔ a b = c ⇔ a c = b and (ii)

a w b⊕ c⇔ a b w c⇔ a c w b4. a b = (a t b) b and b = (a t b) (a b)5. b = (a u b)⊕ (b a) and b a = b (a u b)

Proof. 1.(ii): If a w b⊕c then ab w (c⊕b)b = c by cancellability. Conversely,let us assume that ab w c, from a w b it follows from Prop.1 that a = (ab)⊕b1 We inverse the order relation, which means that we replace ≤,∨, ∧ respectively byw, u, t in every identity valid for the class of commutative residuated lattices.

and thus a w c ⊕ b. 1.(i) follows from 1.(ii) together with the adjunction :a v b⊕ c⇔ a b v c

2.(i): If ab = c then ac = a(ab) = aub. Conversely, let us assume thatac = aub, from a w c it follows that c = auc = a(ac) = a(aub) = ab.Actually a (a u b) = [by RL3] (a a) t (a b) = [by int4] 0 t (a b) =[by div2] a b

2.(ii): (Similar to 2.(i)) If ab w c then ac w a(ab) = aub. Conversely,let us assume that ac w aub, from a w c it follows that c = auc = a(ac) va (a u b) = a b.

3: Follows from 1.4: Follows from 3.(i). since at b = (a b)⊕ b, at b w a w a b and at b w b.5: Follows from 3.(i). since a u b = b (b a), b w a u b and b w b a.

Lemma 3. The following conditions are equivalent in any Petri algebra

1. (b⊕ c) a = (b a)⊕ (c (a b))2. c (a b) = ((b⊕ c) a) (b a)3. [(b⊕ c) a] [c (a b)] = b a

Proof. Let α = (b ⊕ c) a, β = b a and γ = c (a b). One has α w β bymonotony and (div1), and α w γ follows from (RL8) and identity (5):

(b⊕ c) a w ((b⊕ c) b) (a b) = c (a b)

Thus by Lemma 2.(3.i) if follows that α = β ⊕ γ ⇔ α β = γ ⇔ α γ = β.

Lemma 4. In any Petri algebra

1. b w c ⇒ a⊕ (b c) = (a⊕ b) c2. c w b ⇒ (a⊕ b) c = a (c b)

Proof. If b w c then (b c)⊕ c = b by Prop.1 and thus a⊕ (b c) = [By 5] (a⊕(b c) ⊕ c) c = (a ⊕ b) c. If c w b then c (c b) = c u b = b and thusa⊕b = a⊕(c(cb)) = (a⊕c)(cb) using the preceding point with c w cb.It follows (a⊕b)c = ((a⊕c) (cb))c = [By RL11] ((a⊕c)c) (cb) =[By 5] a (c b).

Lemma 5. The following identity holds in any Petri algebra

c (a b) = ((b⊕ c) a) (b a)

Proof. ((b⊕ c) a) (b a) =

= [(b⊕ c) ((a u b)⊕ (a b))] (b a) Lemma 2.(5)= [((b⊕ c) (a u b)) (a b)] (b a) RL11= [(c⊕ (b (a u b)) (a b)] (b a) Lemma 4.(1) since b w a u b= ((c⊕ (b a)) (a b)) (b a) Lemma 2.(5)= ((c⊕ (b a)) (b a)) (a b) RL11= c (a b) (5)

The proof of Proposition (5) and therefrom of Theorem (1) follows from lemma(3) and lemma (5).

Corollary 1. Petri algebras satisfy the following equivalence

a w b⊕ c ⇔ a w b and a b w c (9)

Proof. Follows from Proposition (3) and Theorem (1)

Identity (8) is an internalization of (9) using the axiomatization of the orderrelation: a v b⇔ a b = 0.

2.3 Additional properties of the firing rule

Let us consider a net over a Petri algebra A, then we can inductively define theapplications Pre, Post : P ×T ∗ → A by letting ϕ(p, u) = (Pre(p, u), Post(p, u))where ϕ(p,−) : T ∗ → A × A is the unique monoid morphism such that theimages ϕ(p, t) = (Pre(p, t), Post(p, t)) of the generators t ∈ T be given by theflow relations of the net. Then

Pre(p, ε) = Post(p, ε) = 0Pre(p, uv) = Pre(p, u)⊕ (Pre(p, v) Post(p, u))Post(p, uv) = (Post(p, u) Pre(p, v))⊕ Post(p, v)

Theorem 2. The generalized transition relation M [u〉M ′ stating the existenceof a sequence u of transitions leading from M to M ′ is given by any of the threefollowing equivalent conditions

1. ∀p ∈ P M(p) w Pre(p, u) and M ′(p) = (M(p) Pre(p, u))⊕ Post(p, u)2. ∀p ∈ P M ′(p) w Post(p, u) and M(p) = (M ′(p) Post(p, u))⊕ Pre(p, u)3. ∀p ∈ P M(p) w Pre(p, u) ; M ′(p) w Post(p, u) and M(p) Pre(p, u) =

M ′(p) Post(p, u)

Proof. On the one hand we have to show that the three above conditions areequivalent, and on the other hand that they do correspond to the generalizedtransition relation M [u〉M ′. To simplify the presentation of the proof, we adoptvectorial notations using the product algebra AP which satisfies the same prop-erties that A for the corresponding pointwise defined operations. It amounts toconsider a net with only one place whose algebra is the product of the algebrasof the original net.

Concerning the former point we thus have to establish the equivalence be-tween the following assertions.

1. M w Pre(u) and M ′ = (M Pre(u))⊕ Post(u)2. M ′ w Post(u) and M = (M ′ Post(u))⊕ Pre(u)3. M w Pre(u) ; M ′ w Post(u) and M Pre(u) = M ′ Post(u)For that purpose we recall the equivalence (9)

a w b⊕ c ⇔ a w b and a b w c

which allows us to conclude that

[M ′ w (MPre(u))⊕Post(u)] ⇔ [M ′ w Post(u) and M ′Post(u) wMPre(u)]

which, in turn, in conjunction with

[(M Pre(u))⊕ Post(u) wM ′] ⇔ [M ′ Post(u) vM Pre(u)]

that itself is an instance of condition (4) proves the equivalence between as-sertions (1) and (3) above. The equivalence between assertions (2) and (3) areproven similarly.

As far as the second point is concerned let us temporarily denote M [u]M ′

the relation given by any of the three assertions that we have proved to beequivalent. We recall the identities

Pre(ε) = Post(ε) = 0Pre(uv) = Pre(u)⊕ (Pre(v) Post(u))Post(uv) = (Post(u) Pre(v))⊕ Post(v)

We deduce that M [ε]M ′ ⇔ M = M ′ ⇔ M [ε〉M ′ where ε ∈ T ∗ is the emptyword, and by definition M [t]M ′ ⇔ M [t〉M ′ for t ∈ T any transition. It remainsto prove that M [uv]M ′ ⇔ ∃M” · M [u]M” and M”[v]M ′ in order to be ableto deduce that {M [u]M ′ | u ∈ T ∗} is the transitive closure of the elementarytransition relation {M [t〉M ′ | t ∈ T} and that it consequently coincides with{M [u〉M ′ | u ∈ T ∗}. For that purpose, let us observe that

(0,M)⊗ (Pre(u), Post(u)) = (Pre(u)M, (M Pre(u))⊕ Post(u))

and thus

M [u]M ′ ⇔ (0,M ′) = (0,M)⊗ (Pre(u), Post(u))

(we recall that a w b ⇔ b a = 0). By definition of the maps Pre and Postone has

(Pre(uv), Post(uv)) = (Pre(u), Post(u))⊗ (Pre(v), Post(v))

which allows us to conclude, on the basis of the associativity of ⊗, that (∃M” ·M [u]M” and M”[v]M ′)⇒M [uv]M ′. In order to establish the converse directionit remains to verify that M [uv] ⇔ (M [u] and M ∗ u[v]) where M ∗ u stands for(M Pre(u)) ⊕ Post(u) and where M [u] stands for ∃M ′ · M [u]M ′ that isto say M w Pre(u). By definition of ⊗ one has M [uv] ⇔ M w Pre(uv) =Pre(u) ⊕ (Pre(v) Post(u)), which by (7) is equivalent to [M w Pre(u) andM Pre(u) w Pre(v) Post(u)] which, in turn by adjunction (equation 4) isequivalent to [M w Pre(u) and (MPre(u))⊕Post(u)) w Pre(v)], i.e. M [u]M ′

and M ∗ u[v].

2.4 The variety of Petri algebras

We have so far identified the set of conditions that should be fulfilled by Petrialgebras so that we can play the token game and the resulting firing rule isassociative. To sum up, such a structure is a dual of a commutative residuatedlattice whose joins and meets are given by the formulas atb = a (ab) (Prop.1) and a u b = b ⊕ (a b) (Prop. 4). Moreover since the a Petri algebra is adivisibility monoid, this lattice is integral in the sense that the neutral elementfor the sum is also the least element of the lattice. Finally the underlying monoidis cancellable and this condition is equivalent to the identity (5).

Proposition 6. A Petri algebra consists of a set A equipped with a structure ofa commutative monoid (A,⊕, 0) and a structure of a lattice (A,t,u) such that:

1. the neutral element is the least element of the lattice: 0 v a .2. Sum admits an adjoint characterized by the universal property: a⊕ b w

c ⇔ a w c b3. (a⊕ b) b = a.4. a u b = a (a b).5. a t b = b⊕ (a b).

Proof. It remains to prove that a structure satisfying the conditions stated inthe proposition is a Petri algebra. It is a co-residuated commutative monoid(Condition 2). Integrality (Condition 1) together with Condition (5) implies byProp. (1) that it is a divisibility monoid which is a cancellable monoid due toCondition (3).

Recall that a commutative GMV-algebra ([4, 21]) is a commutative residuatedlattice such that a∨ b = a ((a b)∧ 0); it also satisfies a∧ b = ((a b)∧ 0)⊕ b.A GMV-algebra is said to be integral when the neutral element is the greatestelement of the lattice (hence the least element for the inverse order relationv). Thus we conclude that Petri algebras coincide with the (duals of) integral,cancellative and commutative GMV-algebras. These algebras form a subvariety ofthe variety of residuated lattices and the following result is a direct consequenceof [24, Theorem 5.6 and corollaries].

Theorem 3. Petri algebras coincide with the positive cones of lattice-orderedabelian groups. Moreover lattice-ordered abelian groups constitute the subvarietyof lattice-ordered groups generated by the group Z of integers, and their positivecones (i.e. Petri algebras) is the subvariety of residuated lattices generated by N.

To give a better understanding of this result we recall that if (G,+, 0) is anabelian group equipped with an order relation compatible with its addition (i.e.addition is increasing w.r.t. both of its arguments) then the positive cone P ={x ∈ G | x ≥ 0} is a submonoid of G which is zerosumfree (x + y = 0 ⇒ x =y = 0). Conversely any zerosumfree submonoid P of G induces a compatibleorder relation on G whose corresponding positive cone is P : x v y ⇔ ∃z ∈P · y = x + z . Of course a submonoid of a group is necessarily cancellable. Ifmoreover G is a lattice-ordered group, meaning that each pair {a; b} admits a

join at b and a meet au b for the corresponding order relation, then the positivecone is residuated with a b = (a− b) t 0 (where a − b = a + (−b)) and isalso lattice-ordered. For instance the monoid (N,+, 0) is the positive cone of thegroup (Z,+, 0) for the usual order relation on integers; this is a total order andthus in particular a lattice and therefore the preceeding considerations apply. Weknow that through linear algebra techniques this group of integers plays a centralrole in the study of the structural properties of Petri nets ([34]: the fundamentalequation, linear invariants, siphons, traps ...) and as well in the algorithmicimplementation of Petri nets synthesis based on the theory of regions [2].

3 Lexicographic Petri nets

3.1 Subdirectly irreducible Petri algebras

We define a (generalized) Petri net as a structure N = (P, T, Pre, Post, M0)where P is a finite set of places with a Petri algebra Ap associated with each placep ∈ P , T is a finite set of transitions disjoint from P and Pre, Post : P × T →⊔p∈P Ap, the flow relations, are such that ∀p ∈ P ∀t ∈ T Pre(p, t), Post(p, t) ∈

Ap. A marking is a map M : P →⊔p∈P Ap that associates with each place p ∈ P

the local value of the current configuration M(p) ∈ Ap in this place. M0 is somefixed marking, called the initial marking. The transition relation M [t〉M ′ statingthat transition t can fire in marking M and leads, when it is fired, to the newmarking M ′ is given by:

M [t〉M ′ ⇔ ∀p ∈ P M(p) w Pre(p, t) ∧M ′(p) = (M(p)Pre(p, t))⊕Post(p, t)

This relation can be extended inductively to sequences u ∈ T ∗ of transitions byletting M [ε〉M for every marking M and M [t · u〉M ′ if and only if there existssome markingM ′′ such thatM [t〉M ′′ andM ′′ [u〉M ′ for every t ∈ T and u ∈ T ∗.We letReach(N ) = {M | ∃u ∈ T ∗ M0 [u〉M} stand for the set of reachable mark-ings, and the marking graph of a generalized net N = (P, T, Pre, Post, M0) isthen the labelled graph ΓN = (V,Λ, v0) whose set of vertices is given by theset V = Reach(N ) of reachable markings with v0 = M0 and whose set of arcsΛ ⊆ V ×T ×V is the restriction of the transition relation to the set of reachablemarkings: Λ = {(M, t,M ′) |M, M ′ ∈ V ∧ M [t〉M ′}.

Definition 3. Two generalized Petri nets are termed equivalent when they haveisomorphic marking graphs.

We immediately see that a place p whose type Ap is a subalgebra of a productof Petri algebras (Ap ⊆ A1 × · · · × An) can be replaced by n places p1, . . . , pnwith respective types A1, . . . , An without changing the marking graph (at leastup to isomorphism). A classical result of universal algebra due to Birkhoff saysthat any algebra of a variety is a subdirect product of subdirectly irreduciblealgebras 2. Thus we can assume without loss of generality that all algebras

2 An algebra A is a subdirect product of a family of similar algebras {Ai | i ∈ I} if Ais a subalgebra of the product algebra

∏i∈I Ai such that for every index i ∈ I the

Ap are subdirectly irreducible algebras in the variety of Petri algebras. Nowany M(p) belongs to the subalgebra of Ap generated by the set {M0(p)} ∪⋃t∈T {Pre(p, t), Post(p, t)}. Thus:

Theorem 4. Every generalized Petri net is equivalent to a generalized Petrinet all of whose types are finitely generated subalgebras of subdirectly irreduciblePetri algebras.

Let Irr (V ) denote the set of subdirectly irreducible algebras of a variety V , thenif V is a subvariety of W one has Irr(W )∩ V = Irr(V ); using the fact that thesubdirectly irreducible commutative GMV-algebras are chains (simply orderedsets) we deduce that

Proposition 7. subdirectly irreducible Petri algebras are chains.

An algebra is subdirectly irreducible if and only if it admits a least non trivialcongruence [5]. Now we know [6, 7] that the congruences of Petri algebras arein bijective correspondance with their convex submonoids. On the one directionwe can associate each congruence θ of a Petri algebra A with the class of theneutral element which is a convex submonoid Mθ = [0]θ of A. Conversely weassociate each such monoid M to the congruence θM = {(a, b) ∈ A2 | b a, a b ∈ M}. The correspondances θ 7→ Mθ et M 7→ θM are inverses to eachother and they establish an isomorphism between the lattice of congruences ofA and the lattice of the convex submonoids of A. Moreover for every a ∈ A, theprincipal congruence generated by the equation a = 0 corresponds to the convexsubmonoid generated by a. A Petri algebra is then subdirectly irreducible if andonly if it admits a least non trivial convex submonoid. Let us assume that A isa totally ordered Petri algebra. Let

M(x) = {y ∈ A | ∃k ∈ N · y v k · x = x⊕ · · · ⊕ x︸ ︷︷ ︸k times

}

denote the principal convex submonoid generated by x ∈ A. M(x) is non-trivialif and only if x 6= 0. Now if x is some element of a convex submonoid M of A,one necessarily have M(x) ⊆M ; thus a minimal convex submonoid is principaland is generated by any of its non null elements. Since A is totally orderedand x ≤ y ⇒ M(x) ⊆ M(y) we deduce that A admits at most one minimalnon-trivial submonoid. M(x) is minimal if and only if y � x ⇒ y = 0 whererelation � is given by

y � x ⇔ ∀k ∈ N · k · y @ x

morphism pi : A→ Ai, obtained by composition of the embedding of A into∏

i∈I Ai

with the canonical projection∏

i∈I Ai → Ai, is surjective. The set {(Ai, pi) | i ∈ I}is a representation of A as subdirect product of the algebras Ai. Symetrically such arepresentation can be given as a family of surjective morphisms pi : A→ Ai jointlyinjective, that is to say such that the product 〈pi〉 : A →

∏i∈I Ai is injective. An

algebra A is say to be subdirectly irreducible if for every representation {(Ai, pi) | i ∈I} of A as a subdirect product of algebras Ai there exists at least one index i forwhich the corresponding morphism pi is an isomorphism.

Otherwise stated y � x if and only if y @ x and M(y) is strictly includedin M(x). Therefore A has no non trivial minimal submonoid if and only if forevery x ∈ A \ {0} one can find some y ∈ A \ {0} such that y � x . Underthat condition one can form an infinite strictly decreasing chain thus provingthat the order relation � is not well-founded3. Conversely if this order is well-founded then any non-empty subset of A, and thus in particular A \ {0} if A isnot trivial, admits a least element for this order which shows the existence of aminimal non-trivial submonoid. We thus have established the following:

Theorem 5. A Petri algebra is subdirectly irreducible if and only if it is a chainand the order relation y � x ⇔ ∀k ∈ N · k · y @ x is well-founded.

3.2 Lexicographic Petri nets

The lexicographic product G ◦H of two ordered groups G and H is the productgroup G×H equipped with the lexicographic order relation:

(x, y) ≤G◦H (x′, y′) ⇔ x <G x′ or (x = x′ and y ≤H y

′)

If G and H are totally ordered abelian groups then the same holds for theirlexicographic product. This product is associative and we can define inductivelyLn (G) = (Gn)+ for every simply ordered abelian group G and integer n ∈ Nby letting G0 = {0} be the trivial group and Gn+1 = Gn ◦ G and where G+

denote the positive cone of group G. The group Gn naturally embedds into Gmwhen n ≤ m; the projective limit of this sequence of embeddings is the group Gωwhose elements are the infinite sequences of elements in G, with componentwisecomposition and the lexicographic order relation defined as follows: u ≤lex v ⇔u <lex v or u = v where

u <lex v ⇔ ∃n ∈ N ∀m ≤ n um = vm and un <G vn

The inductive limit, or ”union”⋃n<ω Gn, is the subgroup of Gω consisting of

the sequences u of finite support (supp(u) = sup {k ∈ N | uk 6= 0} < ω) with Gnidentified with the subgroup of u ∈ Gω such that supp(u) ≤ n.

Definition 4. The set Lex (G) of lexicographic Petri nets based on a totallyordered abelian group G is the set of (homogeneous) generalized Petri nets of type(Gω)

+. Lex (G, n) ⊆ Lex (G) is the set of n-dimensional lexicographic Petri nets

with type Ln (G) = (Gn)+ ⊆ (Gω)+

, i.e. all flow arc inscriptions and initial placecontents, and hence all place contents in every accessible marking, are elementin (Gn)

+.

3 We recall that an order relation is well-founded if it satisfies any of the followingequivalent conditions:

1. every decreasing infinite sequence is stationary,2. every stricly decreasing sequence is finite,3. every non-empty subset admits a least element.

An ordinal, or well-ordered set, is a set with a total and well-founded order.

If K and L are subclasses of generalized Petri nets we let K . L when every net inK is equivalent to some net in L. This is a pre-order relation, we let ≈ denote itsassociated equivalence relation and � the corresponding strict relation: K � Lwhen every net in K is equivalent to some net in L but there exists some netin L not equivalent to any net in K. Notice that Lex(G, n) . Lex(H,m) whenG ⊆ H and n ≤ m; and that Lex(Z, 1) is the class of Petri nets.

Lemma 6. Any finitely generated subalgebra of a subdirectly irreducible Petrialgebra A is isomorphic to a subalgebra of the positive cone of some finite powerof the additive group of real numbers: A ⊆ (Rn)+.

Proof. A is a chain and we can partition its set of generators into G = G1∪· · ·∪Gn such that (i) ∀i < j ≤ n ai ∈ Gi ∧ aj ∈ Gj ⇒ai � aj and (ii) all elementin Gi are incomparable with respect to �: ai, a

′i ∈ Gi ⇒∃k ∈ N ai v k · a′i. Let

Ai be the subalgebra of A generated by Gi. Ai is a simply ordered archimedean(a� b⇒ a = 0) divisibility monoid (Archimedean systems of Magnitudes) andthus is isomorphic to an ordered submonoid of the additive monoid of positivereal numbers ([5, 19]), hence Ai ⊆ R+. Now we check that A is a subalgebra ofthe lexicographic product of the Ai’s: A ⊆ An ◦ · · · ◦A1. For each decompositiona =

∑g∈G λg · g of an element a ∈ A in terms of the generators (λg ∈ N)

we obtain a related decomposition a = an ⊕ · · · ⊕ a1 with ai ∈ Ai by lettingai =

∑g∈Gi

λg · g. Let us verify that such a decomposition is unique. We shallassume n = 2, the proof extends by induction for larger n. Thus we assumea1 ⊕ a2 = a′1 ⊕ a′2 with ai, a

′i ∈ Ai . Since the order relation is total and the

addition is monotonic we may assume that a1 v a′1 and a′2 v a2. Then

a1 ⊕ a2 = a′1 ⊕ a′2 ⇔ a′1 = (a1 ⊕ a2) a′2 By Lemma 2⇔ a′1 = a1 ⊕ (a2 a′2) By Lemma 4⇔ a′1 a1 = a2 a′2 By Lemma 2

Now a′1 a1 ∈ A1 and a2 a′2 ∈ A2 and the only way that these elements canbe comparable is when they are null, i.e. a1 = a′1 and a2 = a′2 as required. Welet ϕi : A → Ai be the map such that ϕi(a) = ai where a = an ⊕ · · · ⊕ a1 withai ∈ Ai . These maps are monoid morphisms, we let ϕ :A→ An ◦ · · · ◦A1 denotethe monoid morphism obtained by collecting them: ϕ(a) = (ϕn(a), . . . , ϕ1(a)).By construction ϕ is an injection with left-inverse ψ : An ◦ · · · ◦ A1 → A givenby ψ (xn, . . . , x1) = xn ⊕ · · · ⊕ x1. Thus ϕ induces an isomorphism between themonoid reduct of A and a submonoid of the monoid reduct of An ◦ · · · ◦A1. Nowby definition of the lexicographic order we have that this injection ϕ preservesand reflects the respective order relations. Since residuation in a Petri algebrais characterized by the monoid structure and the order relation, we deduce thatthe image of A by ϕ is a subalgebra of An ◦ · · · ◦A1.

By Theorem 4 we deduce the following result.

Theorem 6. Every generalized Petri net is equivalent to some lexicographicPetri net, more precisely:

GenPetri ≈ Lex (R)

We have in fact proved a more precise result, namely that every generalizedPetri net is equivalent to a finite dimensional lexicographic Petri net based onthe additive group of real numbers, i.e. GenPetri ≈

⋃n<ω Lex (R, n).

Proposition 8. Petri = Lex(Z, 1) � GenPetri

Proof. We provide an example showing that Lex(Z, 1) � Lex(Z, 2), i.e. thatlexicographic Petri nets based on the group of integers of dimension 2 are alreadystrictly more expressive than the class of Petri nets. Let us consider the net of

Fig. 1. A two dimensional lexicographic Petri net

type

L2 = (Z ◦ Z)+ = {(n,m) | (n = 0 and m ≥ 0) or (n > 0 and m ∈ Z)}

depicted in Fig. 1. This net has a unique place P = {p} and two transitionsT = {a; b} with the following flow relations Pre(p, a) = (1, 0), Pre(p, b) = (0, 1)and Post(p, a) = Post(p, b) = (0, 0). From the initial marking (1, 0) transition acan fire once (1, 0) [a〉 (0, 0) and transition b can fire an infinite number of timeleading to the infinite firing sequence

(1, 0) [b〉 (1,−1) [b〉 (1,−2) . . . [b〉 (1,−n) . . .

and there are no other transitions in the marking graph of the net. Supposethere exists some Petri net with an isomorphic marking graph. Since transitionb can fire an infinite number of time, and

M0 [bn〉Mn ⇒ ∀p ∈ P Mn(p) = M0(p)− n× (Pre(p, b)− Post(p, b))

we deduce that for every place p it is the case that Post(p, b) w Pre(p, b) andthus Mn(p) w M0(p) . By monotony of the firing rule, any transition that canfire in the initial marking M0 can also fire in any of the markings Mn obtainedby firing b. Transition a is in contradiction with this property.

By generalizing on the above we could establish that

n < m < ω ⇒ Lex(Z, n) � Lex(Z,m) � Lex(Z)

it can also be shown that

Lex(Z, n) ≈ Lex(Q, n) � Lex(R, n)

3.3 Lexicographic Petri nets and Petri nets with inhibitor arcs

It appears to be difficult to obtain strict extensions of the class of Petri nets thatpreserve all of its decidable properties. Many of these extensions, like the classof Petri nets with inhibitor arcs, are indeed Turing-powerful.

Definition 5. A Petri net with inhibitor arcs is a structure N = (P, T, Pre, Post, M0)where P , T , Post, and M0 (initial marking) are defined as for the usual Petrinets. Pre is a map from P × T to N ∪ {ς} where ζ is a special symbol. Wesay that we have an inbitor arc from place p to transition t when Pre(p, t) = ς.Transition t is firable in marking M ∈ NP when

∀p ∈ P [Pre(p, t) ∈ N ∧ M(p) w Pre(p, t)] ∨ [Pre(p, t) = ς ∧ M(p) = 0]

i.e. they are enough resources in the input places •t = {p ∈ P | Pre(p, t) > 0}of transition t and all of its inhibiting places ◦t = {p ∈ P | Pre(p, t) = ς} areempty. Then the firing of transition t leads to a new marking M ′ defined asfollows:

∀p ∈ P M ′(p) =

{M(p)− Pre(p, t) + Post(p, t) if Pre(p, t) ∈ NM(p) + Post(p, t) if Pre(p, t) = ς

We let M [t〉M ′ express the fact that transition t can fire in marking M andleads to marking M ′ when it fires. This relation can be extended inductively tosequences u ∈ T ∗ of transitions by letting M [ε〉M for every marking M andM [t · u〉M ′ if and only if there exists some marking M ′′ such that M [t〉M ′′and M ′′ [u〉M ′ for every t ∈ T and u ∈ T ∗. The set of reachable markingsis Reach(N ) = {M | ∃u ∈ T ∗ M0 [u〉M}, and the marking graph of a Petrinet with inhibitor arcs N = (P, T, Pre, Post, M0) is the labelled graph ΓN =(V,Λ, v0) whose set of vertices is given by the set V = Reach(N ) of reachablemarkings with v0 = M0 and whose set of arcs Λ ⊆ V × T × V is the restrictionof the transition relation to the set Λ = {(M, t,M ′) |M, M ′ ∈ V ∧ M [t〉M ′}of reachable markings.

Fig. 2. A Petri net with inhibitor arcs and its marking graph

Figure (2) displays a Petri net with inhibitor arcs and its marking graph.Inhibitor arcs restrict the firing of transitions but have no effect on the content

of places when transitions are allowed to occur. This means that the markinggraph of a Petri net with inhibitor arcs N is a subgraph of the marking graphof the induced Petri net N ◦ obtained by removing the inhibitor arcs.

Fig. 3. A translation from Petri nets with inhibitor arcs into lexicographic Petri nets

Theorem 7. Lexicographic Petri nets are a strict extension of the class of Petrinets with inhibitor arcs.

Proof. The translation of a Petri net with inhibitor arcs N into an equivalentlexicographic Petri net N , illustrated in Fig. (3), consists in splitting every placep with initial marking m ∈ N of the original net into two places denoted p andp′ with initial markings (0,m) ∈ (Z ◦ Z)

+and (1,−m) ∈ (Z ◦ Z)

+respectively.

The restriction of this lexicographic Petri net to set of places p of the first familyis isomorphic to N ◦, i.e. the original net in which the inhibitor arcs have beenremoved. We thus have a graph morphism ΓN → ΓN◦ sending a marking of thelexicographic Petri net to its restriction to the set of places of the original netN . Now the set of reachable markings of N satisfies the following property:

K(M) ≡ ∀p ∈ P ∃mp ∈ N M(p) = (0, mp) ∧ M(p′) = (1,−mp)

This property holds for the initial marking by definition and is preserved by thefiring of a transition due to the fact that

x− y = max (0, x− y)−max (0, y − x)

Therefore this graph morphism is injective and one has a pair of graphs monomor-phisms:

ΓNι↪→ ΓN◦

ι′←↩ ΓN

The inhibitor arcs in N and the places of the second family in N (i.e. places ofthe form p′ for p ∈ P ) serve the same purpose: cutting off a subset of transitionsin ΓN◦ . It just remains to show that they cut off exactly the same transitions.Place p′ inhibits the firing of transition t in some marking M satisfying propertyK if and only if (1,−mp) @ Pre(p′, t); this is possible only if Pre(p′, t) = (1, 0)in N i.e. Pre(p, t) = ς in N (there is an inhibitor arc from p to t) and mp 6= 0and these conditions are also clearly sufficient conditions:

(1,−mp) v Pre(p′, t) ⇔ Pre(p, t) = ς ∧ mp 6= 0

which means there exists for the associated marking in N an inhibitor arc froma non-empty place to that transition. Figure (4) shows the lexicographic Petrinet associated with the Petri net with inhibitor arcs of Fig. (2).

Fig. 4. A lexicographic Petri net equivalent to the Petri net with inhibitor arcs of Fig.2

In order to show that lexicographic Petri nets are strictly more expressivethan Petri nets with inhibitor arcs, it suffices to observe that there exists noPetri net with inhibitor arcs whose language is the language of the lexicographicPetri net of Fig. (5) i.e. the prefix closure of b∗ + ab. Suppose on the contrary

Fig. 5. A lexicographic Petri net with no equivalent Petri net with inhibitor arcs

that such a Petri net with inhibitor arcs exists. Since transition b can fire anarbitrary number of time in the initial marking it should be the case that forevery place p either there exists an inbitor arc from p to b and no output arc

from b to p or Post(p, b) w Pre(p, b). Thus if follows that b can fire an arbitrarynumber of times in every marking at which it can fire once; which contradict thefact that after the firing of transition a transition b can fire only once.

Corollary 2. Reachability, Coverability, Place-boundedness, Boundedness, Dead-lock and Liveness are undecidable for the class of lexicographic Petri nets.

4 Bounded nets

4.1 Nets associated with MV-algebras

A net is bounded if we can find an upper bound on the possible values of placesin any accessible marking. Let us start our study on the algebraization of thedynamics of bounded nets by the following observation.

Proposition 9. Any non trivial Petri algebra is an unbounded lattice.

Proof. Let us consider a bounded and non trivial Petri algebra. Let∞ 6= 0 standfor its greatest element. One has

1. ∞∞ = 0 because more generally from x x v 0 ⇔ x v 0⊕ x = x itfollows x x = 0.

2. ∞ 0 =∞ because ∞ 0 v x ⇔ ∞ v 0⊕ x = x.3. ∞⊕ x =∞ because ∞⊕ x w y ⇔ ∞ w y x.4. x∞ = 0 because x∞ v y ⇔ x v y ⊕∞ =∞.Then the identity (a ⊕ b) b = a is not satisfied for a = b = ∞ since the

left-hand side evaluates to 0.

However we can enforce boundedness by modifying the rule of the token game.Let us consider first the case of the usual Petri nets: assume that each placep ∈ P is associated with a capacity kp ∈ N and that we want to ensure that thevalue of a place p of a Petri net is bounded from above by its capacity kp. Forthat purpose we modify the firing rule as follows (where all computations areperformed in Z)

M [t〉M ′ ⇔ ∀p ∈ P{M(p) w Pre(p, t) ∧ (M(p)− Pre(p, t)) + Post(p, t) v kpM ′(p) = (M(p)− Pre(p, t)) + Post(p, t)

this rule can be reformulated as:

M [t〉M ′ ⇔ ∀p ∈ P{Pre(p, t) vM(p) v (kp + Pre(p, t))− Post(p, t)M ′(p) = (M(p)− Pre(p, t)) + Post(p, t)

Petri algebras are the positive cones G+ of lattice-ordered abelian groups G =(G,+, 0,t,u). It is an algebra with the following operations:

– the sum: restriction of the group operation x⊕ y = x+ y– the truncated difference: x y = (x− y) t 0

Let k w 0 be some element of this positive cone; we suppose that it is a strongunit in the sense that ∀g ∈ G ∃n ∈ N · n · k w g. We can, by modifying thefiring rule, ensure that the values of places stay within the interval I = [0, k] ={g ∈ G | 0 v g v k}. This interval with induced order is a bounded lattice thatcan be equipped with the following operations:

– the truncated sum: x� y = (x+ y) u k– the truncated difference: x y = (x− y) t 0– the product: x • y = ((x+ y)− k) t 0– the implication: x→ y = ((y + k)− x) u k– the negation: ¬x = x→ 0 = k x

The following properties are met:

1. It is a bounded lattice: 0 v x v k2. (I,�, 0) is an abelian monoid whose unit is an absorbing element for the

product: 0 • x = 0.3. (I, •, k) is an abelian monoid whose unit k is an absorbing element for the

sum: k � x = k.4. Negation is an involution (¬¬x = x) such that k = ¬0 (hence 0 = ¬k) andx • y = ¬(¬x� ¬y) (hence x� y = ¬(¬x • ¬y))

5. x v y ⇔ (x→ y) = k ⇔ (x y) = 06. • is a residuated monoid: x • y v z ⇔ x v y → z7. � is a co-residuated monoid: x� y w z ⇔ x w z y8. (y → z) = z � ¬y and z y = z • ¬y9. x t y = (y → x)→ x = (x→ y)→ y = x� (y x) = y � (x y)

10. x u y = x • (x→ y) = y • (y → x) = x (x y) = y (y x)

Such a structure is called an MV-algebra. MV-algebras are generalizations ofboolean algebras used in the algebraic analysis of Lukasiewicz infinite-valuedpropositional logic and this class of algebras admits several equivalent definitions[14, 31]. For instance an MV-algebra is a bounded commutative residuated lattice(x • y v z ⇔ x v y → z) such that x t y = (y → x) → x. An MV-algebrais also an abelian monoid (A, •, 1) together with an involution ¬ : A → A suchthat if we let x� y = ¬(¬x • ¬y), 0 = ¬1 and (x→ y) = ¬x� y then x • 0 = 0and (y → x)→ x = (x→ y)→ y.

Definition 6. An MV-Petri net is a structure N = (P, T, Pre, Post, M0)where P is a finite set of places with an MV-algebra Ap associated with eachplace p ∈ P (the unit kp of Ap is called the capacity of place p), T is a finiteset of transitions disjoint from P , and Pre, Post : P × T →

⊔p∈P Ap, the flow

relations, are such that ∀p ∈ P ∀t ∈ T Pre(p, t), Post(p, t) ∈ Ap. A markingis a map M : P →

⊔p∈P Ap that associates with each place p ∈ P the local value

of the current configuration M(p) ∈ Ap in this place. M0 is some fixed marking,called the initial marking. The transition relation M [t〉M ′ stating that transi-tion t can fire in marking M and leads, when it is fired, to the new marking M ′

is given by:

M [t〉M ′ ⇔ ∀p ∈ P{Pre(p, t) vM(p) v Post(p, t)→ Pre(p, t)M ′(p) = (M(p) Pre(p, t))� Post(p, t) (10)

This relation can be extended inductively to sequences u ∈ T ∗ of transitions byletting M [ε〉M for every marking M and M [t · u〉M ′ if and only if there existssome marking M ′′ such that M [t〉M ′′ and M ′′ [u〉M ′ for every t ∈ T and u ∈T ∗. We let Reach(N ) = {M | ∃u ∈ T ∗ M0 [u〉M} stand for the set of reachablemarkings, and the marking graph of an MV-Petri net N = (P, T, Pre, Post, M0)is then the labelled graph ΓN = (V,Λ, v0) whose set of vertices is given by theset V = Reach(N ) of reachable markings with v0 = M0 and whose set of arcsΛ ⊆ V ×T ×V is the restriction of the transition relation to the set of reachablemarkings: Λ = {(M, t,M ′) |M, M ′ ∈ V ∧ M [t〉M ′}.

Bounded Petri nets A bounded Petri net N where place p ∈ P has capacitykp is an MV-Petri net where the MV-algebra associated with place p is the[0, kp] interval of the group Z of integers. Indeed under the assumptions thatPre(p, t), Post(p, t) and M(p) are elements of the interval [0, kp] one has M(p) v(kp + Pre(p, t)) − Post(p, t) if and only if M(p) v Post(p, t) → Pre(p, t) =[(kp + Pre(p, t))− Post(p, t)] u kp and one has M ′(p) = (M(p) Pre(p, t)) �Post(p, t) = [[(M(p)− Pre(p, t)) t 0] + Post(p, t)] u kp = (M(p) − Pre(p, t)) +Post(p, t) when both Pre(p, t) vM(p) and (M(p)−Pre(p, t)) +Post(p, t) v kpare satisfied (i.e. Pre(p, t) vM(p) v (kp + Pre(p, t))− Post(p, t)).

One-safe nets The boolean algebra 2 = {0, 1} is an MV-algebra where x�y =x t y and x • y = x u y. Let P be the set of places of a net and B = 2P = ℘(P )the corresponding product structure, the firing rule of MV-Petri nets can bereformulated as:

M [a〉M ′ ⇔ M ⊇ Pre(a) ∧M∩Post(a) ⊆ Pre(a) ∧M ′ = (M\Pre(a))∪Post(a)

which is the usual firing rule of one-safe nets.

One-safe Fuzzy nets If we replace the boolean algebra 2 = {0, 1} by theinterval [0, 1] of the additive group of real numbers, i.e. with x�y = min(1, x+y)and x • y = max(0, x + y − 1) then we obtain the firing rule of some kind ofone-safe ”fuzzy” nets:

M [a〉M ′ ⇔ ∀p ∈ P{Pre(a, p) vM(p) v Pre(a, p) + 1− Post(a, p)M ′(p) = min(1,M(p) + Post(a, p)− Pre(a, p))

4.2 Complementary places

Mundici [31] proved that MV-algebras coincide with [0, k] intervals of abelianlattice-ordered groups where k is a strong unit. More precisely if G = (G,+, 0) isan abelian lattice-ordered group with strong unit k then Γ (G, k) = (A,�, 0, •, k,¬)where G = [0, k] = {x ∈ G | 0 v x v k}, x�y = (x+y)uk, x•y = ((x+y)−k)t0,and ¬x = k − x is an MV-algebra such that the restriction of the order relationof the group on the unit interval [0, k] coincides with the order relation of the

MV-algebra: x v y ⇔ xy = 0 (where xy = x•¬y = (x− y)t0). Moreover Γextends into an equivalence between the respective categories of abelian lattice-ordered groups with strong unit and MV-algebras, i.e. Γ induces a bijectivecorrespondence between isomorphism classes of abelian lattice-ordered groupswith strong unit and isomorphism classes of MV-algebras. Now we have seenthat Petri algebras corresponds bijectively, up to isomorphism, to the positivecones of abelian lattice-ordered groups, we thus have a Petri algebra canonicallyassociated with each MV-algebra. The following result shows that, by using com-plementary places, we can simulate an MV-Petri net by a generalized Petri netdefined on the associated Petri algebra.

Theorem 8. Any MV-Petri net can be simulated by a generalized Petri net.

Lemma 7. For every elements a, b, and c in an MV-algebra one has

b v ¬a v b� c ⇒ (a� b) • c = (a • (c� b))� (b • c) (11)

Proof. Since every MV-algebra is a subdirect product of totally ordered MV-algebras we can assume without loss of generality that the given MV-algebra istotally ordered. Condition b v ¬a is equivalent to a • b = 0 i.e. a + b v k inthe underlying group. Condition ¬a v b � c is equivalent to a � b � c = k, i.e.a+ b+ c w k in the underlying group. On the one hand

(a� b) • c = ((a+ b) u k) • c = (a+ b) • c = (a+ b+ c− k) t 0 = a+ b+ c− k

On the other hand

a • (c� b) = a • [(c+ b) u k]= (a+ [(c+ b) u k]− k) t 0

=

{a if c+ b w ka+ c+ b− k if c+ b v k

and

b • c = (b+ c− k) t 0 =

{b+ c− k if b+ c w k0 if b+ c v k

and thus (a� b) • c = (a • (c� b))� (b • c) as required.

The above property may be rephrased as:

b v ¬a v b� ¬c (= c→ b) ⇒ (a� b) c = (a (c b))� (b c) (12)

and thus appears as a weakening of equation (8) which ensures the associativityof the firing rule of generalized Petri nets.

Proof (of Theorem 8). The translation of a bounded net N into an equivalentgeneralized Petri net N , illustrated in Fig. (6), consists in splitting every placep with initial marking m ∈ Ap of the bounded net into two places denoted p

and p′ with initial markings m ∈ (Gp)+ and kp −m ∈ (Gp)+respectively where

Fig. 6. Translation from bounded nets to generalized Petri nets

(Gp, kp) is the lattice-ordered abelian group with strong unit associated with theMV-algebra Ap ' Γ (Gp, kp). I.e. one has

Pre(p, t) = Pre(p, t)Post(p, t) = Post(p, t)Pre(p′, t) = Post(p, t) Pre(p, t)Post(p′, t) = Pre(p, t) Post(p, t)

First one has x v m v y → x = ¬y � x if and only if m w x and ¬m wy • ¬x = y x; hence transition t can fire in marking M in N if and only if thehomonymous transition t can fire inN in the markingM such thatM(p) = M(p)and M(p′) = ¬M(p) = kp−M(p). Finally M [t〉M ′ in N if and only if M [t〉M ′in N due to the fact that

x v m v y → x ⇒ ¬ [(m x)� y] = [¬m (y x)]� (x y)

which follows from (12) since ¬ [(m x)� y] = (¬m� x) •¬y = (¬m� x) y .

5 Conclusion

5.1 Contribution of the paper

In this paper we have put forward an axiomatization of the token game of Petrinets by identifying a class of commutative residuated monoids, called Petri alge-bras, for which one can generalize the rule of token game of Petri nets to definethe behaviour of generalized Petri net whose flow relation and place contents arevalued in such algebras. This rule is given by

M [t〉M ′ ⇔M w Pre(t) ∧ M ′ = (M Pre(t))⊕ Post(t)

where the sum and its associated residuation capture respectively how resourceswithin places are produced and consumed through the firing of a transition. Theabove rule extends to sequences of transitions by letting

Pre(uv) = Pre(u)⊕ (Pre(v) Post(u))Post(uv) = (Post(u) Pre(v))⊕ Post(v)

subject to the condition that the associated binary operation

(x, y)⊗ (x′, y′) = (x⊕ (x′ y), (y x′)⊕ y′)

be associative. We prove this latter condition to be equivalent to the cancella-bility of the sum which is also characterized by the identity:

(b⊕ c) a = (b (a c))⊕ (c a)

The firing relation can then be expressed as

M [u〉M ′ ⇔ (0,M ′) = (0,M)⊗ (Pre(u), Post(u))

from which the reversibility of the firing rule follows, namely the three followingstatements are proved equivalent:

1. M w Pre(u) and M ′ = (M Pre(u))⊕ Post(u)2. M ′ w Post(u) and M = (M ′ Post(u))⊕ Pre(u)3. M w Pre(u) ; M ′ w Post(u) and M Pre(u) = M ′ Post(u)We have proved that Petri algebras coincide with the positive cones of lattice-

ordered abelian groups. We know that lattice-ordered abelian groups constitutethe subvariety of lattice-ordered groups generated by the group Z of integers, andthat their positive cones (i.e. Petri algebras) form the subvariety of residuatedlattices generated by N. We have introduced a class of nets, termed lexicographicPetri nets, that are associated with the positive cones of the lexicographic powersof the additive group of real numbers. This class of nets is proved to be universalin the sense that any generalized Petri net can be simulated by a lexicographicPetri net. All the classical decidable properties of Petri nets however (termi-nation, covering, boundedness, structural boundedness, accessibility, deadlock,liveness ...) are proved to be undecidable on the class of lexicographic Petri nets,even if restricted to integer coefficients, by showing that this class is a strictextension of the class of Petri nets with inhibitor arcs.

Finally we have turned our attention to bounded nets associated with Petrialgebras and have shown that their dynamics can be reformulated in term of MV-algebras and that any such MV-Petri net can be simulated by some generalizedPetri net.

5.2 Comparaison with related works

This study is not the first attempt to generalize Petri nets using cones of groups[8, 25, 26], however the originality of the present work is to derive such structuresfrom an axiomatization of the token game of Petri nets. Various axiomatizationsof Petri nets token game have also been introduced using variants of linearlogic. Linear logic, introduced by J.-Y. Girard [23], is generally described asa resource concious logic because resources are consumed or produced underrules of inference like tokens in Petri nets. In [9, 22, 29] the authors deal with amultiplicative fragment of linear logic. In [22, 29] each transition is consideredas one axiom added to linear logic, when [9] translates a Petri net into a unique

linear logic formula. These works provide new tools for analysing Petri nets orworking with them and concern accessibility as well as composition mechanismsand concurrency. In [28, 29] the correspondance between Petri nets and linearlogic is given by an isomorphism between two categories in which places taketheir value in a free commutative monoid. In [30] one finds generalisations ofthe algebra of commutative monoids, they suggest also abelian groups but theyconsider this choice too restrictive. G. Winskel and U. Engberg [17, 18] use asemantic approach and consider Petri nets as models of intuitionistic linear logic.Their work is based on the fact that a Petri net induces a preorder on its setof markings. This preorder is compatible with the binary operation of the freemonoid over the set of places of the net. Thus, this monoid is preordered andone can construct a quantale from any Petri net as the set of the downwardclosed subsets of this preorder. Quantales [33] are distributive lattices used inparticular in the phase semantics [23] of linear logic. This allows G. Winskel andU. Engberg to give an interpretation of many connectives of linear intuitionisticlogic. Another approach [32] using PCLL, the partially commutative linear logic,an intuistionistic logic which contains both commutative and non commutativeconnectives, permits to represent step transitions of Petri nets where a stepis a serie-parallel pomset of events. The greatest part of these works considera marking as a multiset of places or as an element of the free commutativemonoid generated by the set of places, i.e. a map from P , the set of places, to N.Henceforth they don’t propose an algebraic generalisation of Petri nets in oursense, and the token game is exactly that of classical Petri nets.

5.3 Directions for further research

We have put the basis for a uniform presentation of various families of Petri netsby recasting them as particular instances of a generic class of Petri nets para-metric in algebraic structures representing some concrete notion of resources.However the present approach, centered on the notion of resources, is probablytoo concrete to be of practical interest in many situations. For instance eventhough one can describe continuous Petri nets in this framework the obtainedsemantics is too much extensional. We see two directions that can be used inorder to derive more abstract representations for the behaviour of these genericnets. First, one can abstract of the flow arc inscriptions. Such an inscription takesits value in some algebra of abstract properties. A precondition then appears asa guard stating that some property has to be satisfied by the resources con-tained in the corresponding place and a postcondition is interpreted as addingresources to enforce some property. Unfortunately, it seems that the inductivedefinitions of the flow relations Pre and Post strangely forces the algebra of flowarc inscriptions to be a Petri algebra. If one want to continue in this direction,one has to leave them and consequently lose the three characterizations of thegeneralized transition relation given in Theorem 2. Second, one can abstract onthe firing relation itself by giving the measure in some adequate semiring of the”firability” of a transition in some marking.

A second line of research is to recast the theory of regions [1, 2] in the contextof nets over Petri algebras. For that purpose the marking graphs of these gen-eralized Petri nets should also be presented as transition relations enriched oversuch structures. It will probably be easier to solve that problem for MV-Petrinets; indeed an MV-algebra gives rise to two De Morgan dual semi-rings, oneof which is used for the definition of the firing relation of the nets while theirmarking graphs can be expressed as transition systems with coefficients in theother semiring.

Finally, we have hinted at the fact that MV-Petri nets might be used forthe description of uncertainty in dynamic systems. Several definitions of fuzzyPetri nets and of possibilistic Petri nets have been introduced already in thelitterature [12, 13, 11]. These definitions were generally put forward in an ad hoc,and sometimes informal, manner to meet with very specific applications andwithout paying the attention to the underlying algebraic structures. As a resultit seems that even now there is no general agreement on which definition shouldbe favored.

Acknowledgements

The first author thanks Nikolaos Galatos for fruitful discussions on totally or-dered commutative GMV algebras. We also thank Marek Bednarczyk for a care-ful reading of a draft of that paper and for pointing us an inaccurate statement.The second author was supported by a grant from the University of Douala,Cameroon.

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