On the Decidability of Behavioral Equivalences for (P,P)-PRS
aa r X i v : . [ c s . L O ] J a n On the decidability of behavioral equivalences for(P,P)-PRS
Irina A. Lomazova
HSE UniversityMoscow, [email protected]
Vladimir A. Bashkin
Yaroslavl State UniversityYaroslavl, Russiav [email protected]
Abstract —We study resource similarity and resource bisimi-larity – congruent restrictions of the bisimulation equivalencefor the (P,P)-class of Process Rewrite Systems (PRS). Both theseequivalences coincide with the bisimulation equivalence for (1,P)-subclass of (P,P)-PRS, which is known to be decidable. Whileit has been shown in the literature that resource similarity isundecidable for (P,P)-PRS, decidability of resource bisimilarityfor (P,P)-PRS remained an open question.In this paper, we present an algorithm for checking resourcebisimilarity for (P,P)-PRS. We show that although both resourcesimilarity and resource bisimilarity are congruences and have afinite semi-linear basis, only the latter is decidable.
Index Terms —process rewrite systems, bisimulation equiva-lence, congruence, decidability, Petri nets, resource bisimilarity
I. I
NTRODUCTION
General background.
The notion of process equivalencecan be formalized in many different ways [1]. One of themost important is bisimulation equivalence [2], [3], capturingthe main features of an observable behaviour of a system. As arule, bisimulation equivalence is a relation on sets of states ofa labelled transition systems (LTS). Two states are bisimilar,if they are undistinguishable modulo systems behavior (theinterleaving semantics are assumed).In [4], an elegant formalism of process rewrite systems (PRS) is defined, allowing to represent a (possibly infinite)LTS by a finite set of rewrite rules using algebraic operationsof sequential and parallel composition. Many natural and well-studied classes of infinite-state systems are contained in thePRS hierarchy (Fig. 1): basic process algebra (BPA) are ( , S )-PRS, basic parallel processes (BPP) are ( , P )-PRS, pushdownautomata (PDA) are ( S , S )-PRS, Petri nets are ( P , P )-PRS andprocess algebra (PA) are ( , G )-PRS.The PRS hierarchy is strict and represents a number ofinteresting decidability borders. One of them is the decidabilityof bisimilarity. It is known that bisimilarity is decidable for thelower left (“sequential”) part of the PRS hierarchy diagram: • ( S , S ): decidable [5] and Ackerman-hard [6]; • ( , S ): decidable [7], PSPACE-hard [8] and EXPTIME-hard [9]; • ( , P ): decidable [10] and PSPACE-complete [11].On the other hand, bisimilarity is undecidable for the upperright (“parallel”) part of the diagram. Janˇcar [12] proved thisfor Petri nets (( P , P )-PRS) and hence for both classes above them — ( P , G ) and ( G , G ). The decidability of bisimilarity forthe “central” fragment of PRS hierarchy (( , G ) and ( S , G ))is still open. It is known that bisimulation is decidable fornormed ( , G )-PRS [13], but there are some negative resultson the weaker notions of bisimulation for generic ( , G )-PRS[14] and even simpler normed models [15].The class of ( P , P )-PRS is of particular interest in thecontext of bisimulation checking, because it corresponds to thestandard and well-studied notion of labelled place/transition(P/T) nets as originally proposed by Petri [16]. Other importantformalism that is equivalent to ( P , P )-PRS are Vector AdditionSystems with States (VASS). Unfortunately, the fundamentalundecidability result from [12] shows that bisimilarity is tooweak to be efficiently checked on the ( P , P )-PRS. We needeither to simplify the model (for example, using only ( , P )-PRS or one-counter Petri nets) or to strengthen the relationitself. In this paper the latter way is studied. Related work.
In [17], an equivalence on places of a Petrinet is defined which when lifted to the reachable markingspreserves their token game and distribution over the places.This structural equivalence is called “strong bisimulation”in [17] because it allows to generate a non-trivial subset ofmarking (state) bisimilarity.The name place bisimulation comes from [18] where theOlderog’s notion is improved and used for nets reduction.Place bisimulation relates “equivalent” places in a net, whereplaces are equivalent if they give rise to bisimilar markings(states). More precisely, place bisimulation is defined by a so-called weak transfer property : places p and q are bisimilarif any transition using p can be successfully imitated bysome other transition using q , and vice versa (w.r.t. arccardinality). In [19], a polynomial algorithm for the largestplace bisimulation is given. The equivalences on the set ofplaces are further explored in [20]–[22]. In particular, it hasbeen proven (using Janˇcar’s technique from [23] that a weakerrelation of largest correct place fusion is undecidable. Places p and q can be correctly fused if for any marking M the twomarkings M + p and M + q are bisimilar. Not related to the more common use of terms “strong bisimulation” and“weak bisimulation” for fundamental state bisimulation and state bisimulationin LTS with invisible (silent) transitions. eparately, in [24], a similar equivalence relation, called structural bisimilarity , is defined for labeled Place-TransitionNets. This relation is defined on the set of all Petri net vertices(both places and transitions) and also uses a kind of weaktransfer property.In [25], [26] a notion of team bisimulation is definedfor ( , P )-PRS (BPP nets). Two distributed systems, eachcomposed of a team of sequential, non-cooperating agents(the tokens in the BPP net), are equivalent if it is possibleto match each sequential component of the first system withteam-bisimilar sequential component of the other system (asin sports, where competing teams have the same number ofequivalent player positions). On ( P , P )-PRS, team bisimulationcoincides with place bisimulation from [18].In [27], a structural equivalence is defined on Petri netresources. A resource in a Petri net is a part of its marking.In PRS notation resources are subterms of process terms. Tworesources are similar if replacing one of them by another in anyprocess term does not change the observable process behavior(modulo bisimulation).The correct place fusion equivalence from [22] turns outto be a special case of the resource similarity, namely theplace fusion is the resource similarity for one-token resourcesin ( P , P )-PRS. Hence, the resource similarity is undecidablein ( P , P )-PRS. On the other hand, the resource similarity isa congruence and can be generated by a finite semilinearbasis. A special type of minimal basis, called ground basis , ispresented in [27]. This ground basis is explicitly defined andcan be effectively computed for any finite relation.Also, in [27], a stronger equivalence on the set of a Petrinet resources, called resource bisimulation , is defined. Anequivalence of resources is called a resource bisimulation ifits additive and transitive closure is a bisimulation.In [29], [30] properties of these resource equivalences arestudied. In particular it is shown that resource bisimulationis defined by a weak transfer property, similar to the weaktransfer property of place bisimulation. It is proved also thatthere exists the largest resource bisimulation, called resourcebisimilarity . The questions of whether the resource bisimilarityis decidable and whether it is strictly weaker than the resourcesimilarity remained open [29]–[31]. Our contribution.
In this paper we give answers to theseopen questions. Based on the well-known “tableau method”(see [32], [33]), we construct an algorithm for checkingthe resource bisimilarity for a ( P , P )-PRS. Thus it is provedthat resource bisimularity is decidable for ( P , P )-PRS. Thisimplies that the resource bisimilarity is strictly weaker thanthe resource similarity. We give also an example of a ( P , P )-PRS, where two similar resources are not resource bisimilar.Hence another decidability border for the bisimulation-induced structural equivalences on ( P , P )-PRS is obtained:between resource bisimulation and resource similarity. Both ofthem have a compact semilinear representation (ground basis)but only resource bisimulation is decidable. Not related to the notion of “resource bisimulation” from [28].
II. P
RELIMINARIES By N we denote the set of natural numbers. A multiset m over a set S is a mapping m : S → N , i.e. a multiset maycontain several copies of the same element.For two multisets m, m ′ over S the inclusion relation andthe sum of two multisets are defined as usual: m ⊆ m ′ iff ∀ s ∈ S : m ( s ) ≤ m ′ ( s ) , and ∀ s ∈ S : m + m ′ ( s ) = m ( s ) + m ′ ( s ) .A multiset m is finite iff the set supp ( m ) = { s ∈ S | m ( s ) > } , called a support of m , is finite. By M ( S ) wedenote the set of all finite multisets over S . For a finitemultiset m by | m | we denote its cardinality, i. e., the numberof elements taking multiple copies into account.When a set S is finite, multisets over S can be encodedas Parikh vectors over N n . Let all elements of S be arrangedin some order. A Parikh vector for m ∈ M ( S ) is a vector φ ( m ) = ( k , k , . . . , k n ) of length n = | m | , where k i is thenumber of occurrences in m of its i -th element. Thus for m , m ∈ M ( S ) the Parikh vector of their sum m + m is computed as a pointwise sum of φ ( m ) and φ ( m ) . A. Labeled Transition Systems
As a rule, semantics of processes is described using labeledtransition system . Definition 1:
Let
Act = { a, b, . . . } be a finite set of actions . A (labeled) transition system (LTS) is a tuple LTS =( S , Act , → ) , where S = { p, r, s, . . . } is a set of states , →⊆ S × Act × S is a transition relation .A state in a labeled transition system represents the systembehavior starting from it, and is also called a process . Inwhat follows we will not distinguish states and processes.A transition relation defines the system dynamics. We write s a → s ′ instead of ( s, a, s ′ ) ∈→ , and s a → s ′ represents a stepchanging state s to state s ′ by firing action a .A labeled transition system can be represented as a finiteor infinite oriented graph, where nodes are labeled with statesand arcs are labeled with actions, and an arc a goes from anode s to a node s ′ iff s a → s ′ . B. Bisimulation Equivalence
A binary relation R ⊆ S × S over some set S is an equiva-lence iff it is reflexive, symmetric, and transitive. Equivalencesover algebraic expressions can also satisfy the congruenceproperty.Let L be a set of expressions. For e, e , e ∈ L , let e [ e /e ] denote a substitution of e for a subexpression e in e . An equivalence R over L is called a congruence ifffor all e, e , e ∈ L , such that ( e , e ) ∈ R , we have ( e [ e /e ] , e ) ∈ R , i. e., replacing a subexpression with anequivalent one in any expression e produces an expressionequivalent to e .Now we present the basic definitions and concepts relatedto bisimulation equivalence. Definition 2:
Let
LTS = ( S , Act , → ) be a labeled transitionsystems, and R ⊆ S × S be a relation over the set of itsstates. We say that R conforms the transfer property iff forach ( s, p ) ∈ R and for every step s a → s ′ , there exists animitating step p a → p ′ such that ( s ′ , p ′ ) ∈ R .The transfer property can be illustrated by the followingdiagram: s R p ↓ a ↓ as ′ R p ′ Definition 3:
A relation R is called a process bisimulation iff both R and R − conform the transfer property.Processes s and p are called bisimilar (bisimulation equiv-alent), written s ∼ p , iff there exists bisimulation relation R such that ( s, p ) ∈ R . The relation ∼ is called bisimilarity .Thus, given a LTS, the bisimilarity is the union of all itsbisimulations.It is easy to prove that each bisimulation is an equivalence.Bisimilarity is also an equivalence, and is itself a bisimulation.In other words, bisimilarity is the largest process bisimulation.The stratified bisimulation relations [34] ∼ k ⊆ S × S for k ∈ N are defined as follows: • s ∼ p for all s, p ∈ S ; • s ∼ k +1 p iff for each a ∈ Act if s a → s ′ then there is p ′ ∈ S s.t. p a → p ′ and s ′ ∼ k p ′ ; and if p a → p ′ then thereis s ′ ∈ S s.t. s a → s ′ and s ′ ∼ k p ′ . It is known [34] that for all image-finite labelled transitionsystems s ∼ p iff s ∼ k p for all k ∈ N . Another way to define bisimilarity is to use its gamecharacterization [12], [35], [36]. A bisimulation game is agame between two players, Attacker and Defender. Given twoprocesses s and p , Attacker tries to prove s p , and Defender— to refute him, and thus to prove s ∼ p .The game goes in rounds. In each round the players changethe current pair of states (processes) ( s, p ) . Attacker chooseseither s or p, an action a and performs a move s a → s ′ or p a → p ′ , depending on whether he chose s or p. Defenderresponds by choosing the other process (either p or s ) andperforms a simulating move p a → p ′ or s a → s ′ under the sameaction a. The pair ( s ′ , p ′ ) becomes the new current pair ofstates.A play is finite if one of the players gets stuck (cannot makea move); the player who gets stuck loses the play and the otherplayer is the winner. If the play is infinite then Defender isthe winner.It holds that s ∼ p if Defender has a winning strategy inthe bisimulation game starting with the pair ( s, p ) , and s p if Attacker has a winning strategy in the corresponding game. C. Process Rewrite Systems
Let
Act = { a, b, . . . } be an infinite set of action names and Const = { A, B, . . . } be an infinite set of process constants . Process expressions are built from constants using binaryoperations of parallel composition (indicated by a vertical bar Finite-state machines ( , ) ( , P ) BPP ( P , P ) Petri netsBPA ( , S ) PA ( , G ) ( P , G ) PANPDA ( S , S ) PAD ( S , G ) PRS ( G , G ) ❥ ❥ ❥✙ ✙✙ ✙✙ ✙❥ ❥ ❥ Fig. 1. Hierarchy of PRS-classes k ) and sequential composition (indicated by a dot). Formally,process expressions are defined by the following grammar. E ::= A | ε | E.E | E k E, where A ranges over Const , and ’ ε ’ is the empty process.By E we denote the set of all process terms.The parallel composition is supposed to be associative andcommutative, sequential composition — associative, and ’ ε ’— a unit for both operations, i. e., E k ε = E and ε.E = E.ε = E . We will not distinguish between expressions whichmatch up to the above properties. Definition 4: [4] A process rewrite system (PRS) ∆ is afinite set of basic rewrite rules of the form E a −→ E , where E , E are process expressions, and E = ε .For a given PRS ∆ , by Act (∆) and
Const (∆) we denotefinite sets of actions and, accordingly, constants occurring inthe rules from ∆ .The semantics of a PRS-system ∆ is defined as a labeledtransition system LTS (∆) = ( E , Act , → ) , where states areprocess expressions from E , and the transition relation → isdefined by the following rules of inference (SOS rules): R : ( E a −→ E ) ∈ ∆( E a −→ E ) R : E a −→ E ′ E k E a −→ E ′ k E R : E a −→ E ′ E k E a −→ E k E ′ R : E a −→ E ′ E .E a −→ E ′ .E [6] distinguishes the following classes of PRS-terms: — atomic terms (e.g., A ). S — atoms and their sequential compositions (e.g., A.B.A ). P — atoms and their parallel compositions (e.g., A k B k B ). G — all terms.PRS systems are classified according to the type of rules used: ( α, β ) -PRS is defined as a class of systems with the rulesof the form α a −→ β ∈ ∆ , where α = ε , β may be ε , and α, β ∈ { , S , P , G } .It is proven in [4] that the class ( , )-PRS coincides withthe class of finite-state machines, ( , P ) corresponds to processalgebra BPP, ( , S ) — process algebra BPA, ( P , P ) — Petrinets, ( S , S ) — push-down automata.R. Mayr has proven that reachability is decidable for allPRS-systems. Y Z ♠ ♠♠ b ac ✻✻✻ ✲ ❄❄❄✛ ✲ X a → ZY b → X k Z k XZ k Z c → Y Fig. 2. Petri net and the equivalent ( P , P )-PRS III. R
ESOURCE S IMILARITY AND R ESOURCE B ISIMULATION FOR ( P , P )-S YSTEMS
We study bisimilation equivalences for ( P , P )-systems. Pro-cess expressions in ( P , P )-systems are built with the onlyoperation of parallel composition. Since parallel compositionis commutative and associative, expressions in ( P , P )-systemsare invariant under grouping and permutation and can beconsidered as multisets of constants.( P , P )-systems are equivalent to classical Petri nets (calledalso P/T-nets) [16]. For a ( P , P )-PRS ∆ , each constant from Const (∆) corresponds to a Petri net place named with thisconstant, and a basic rewriting rule δ ∈ ∆ is represented bya transition labeled by Act ( δ ) and a flow relation – multiplearcs connecting places and transitions. Thus for each basicrewriting rule δ = E a → E there is a transition t , labeledwith a , and for each constant A ∈ Const ( E ) there is an arcgoing from the place (named with) A to t with the multiplicityequal to the number of occurrences of A in E . Symmetrically,for each constant A ∈ Const ( E ) there is a similar arc goingfrom t to the place (named with) A .Process expressions in ( P , P )-systems are multisets of con-stants/places and correspond to Petri net markings (states),where for each constant the number of its occurrences equalsto the number of tokens residing in the place named with thisconstant. Then firing a transition in a Petri net is equivalent tothe application of the corresponding basic rule in the ( P , P )-system.We extend the standard Petri net notation to ( P , P )-systems,and for a basic rewriting rule δ = E a → E by • δ we denotethe multiset of constants occurring in E . Symmetrically, by δ • we denote the multiset of constants in E .In a graphical representation of a Petri net places arerepresented by circles named with constants, transitions areboxes labeled with actions, a current marking (a multiset ofconstants) m is designated by putting m ( A ) tokens into eachplace (named with) A ∈ Const .Fig. 2 illustrates the described correspondence between( P , P )-PRSs and Petri nets. Further we will use the samenotations for the corresponding elements of PRSs and Petrinets, i. e., we will not distinguish Petri net places and PRSconstants, Petri net markings and PRS process expressions etc.It has been proven by P. Jan ˇ c ar [12] that marking(state/process expression) bisimularity is undecidable for Petri 10 cent ShopBought 5 cent ♠ ♠♠ ♠ b b b ❄❄❄ ❄❄ ❄ t t t ❄❄✴ ✇❘ ✠✠❘ ✠ Fig. 3. Buying goods for 20 cents nets. However, bisimilarity is decidable for ( , P )-systems,which correspond to conflict-free Petri nets, i.e. Petri netswhere each transition has not more than one input place. An-other interpretation of ( , P )-systems is BPP process algebras[37].Note, that for ( , P )-systems bisimilarity equivalence is acongruence, while this is not true for a more general classof ( P , P )-systems. The algorithm for checking bisimilarity oftwo ( , P )-terms [10] is essentially based on its congruenceproperty. Hence one can suppose that congruence ensuresdecidability of the bisimilation relations. We will explore thisissue.Since states in ( P , P )-systems are multisets of constants,the congruence property in this case can be reformulated asfollows: Definition 5:
For a ( P , P )-system ∆ , an equivalence relation R ⊆ M ( Const (∆)) ×M ( Const (∆)) is a congruence iff for all r, s, w ∈ M ( Const (∆)) , ( r, s ) ∈ R implies ( r + w, s + w ) ∈ R . A. Resource Similarity
The relation of resource similarity , introduced in [27] forPetri nets, is a congruent strengthening of the bisimulationequivalence. Intuitively, a resource in a Petri net is a part ofits marking. Since a marking is a multiset of places/constants,formally, a resource is a submarking, and the definition ofa resource coincides with the definition of a marking, sincesubmarking is a marking itself. Markings and resources aredifferentiated because of their different substantial interpreta-tion. Resources are parts of markings which may or may notprovide this or that kind of net behavior, e. g., in the Petrinet example on Fig. 3 from [31], two ten-cent coins form aresource — enough to buy an item of goods.Two resources are similar for a given Petri net if replacingone of them by another in any marking does not change theobservable net behavior. In other words, resource similarity isa congruent relation, which preserves visible behavior up tobisimilarity.Now we define the resource similarity formally. Recall that’ ∼ ’ denotes the state bisimilarity relation. Definition 6:
Let ∆ be a ( P , P )-system. A resource in ∆ is a multiset of constants from Const (∆) . Two resources r nd s in ∆ are called similar (denoted r ≈ s ) iff for all w ∈ M ( Const (∆)) we have ( r + w ) ∼ ( s + w ) .Below we will give some properties of the similarity relationfrom [27].First of all, it is easy to check that the resource similarity isan equivalence, i. e., it is reflexive, symmetric, and transitive.Also, the relation has the following important properties. Proposition 1:
Let ∆ be a ( P , P )-system, r, s, r ′ , s ′ — itsresources. Then1) r ≈ s ⇒ s ∼ r ;2) r ≈ s & r ′ ≈ s ′ ⇒ ( r + r ′ ) ≈ ( s + s ′ ) .Thus, resource similarity implies process bisimulation, inother words it is a strengthening of process bisimilarity: ( ≈ ) ⊆ ( ∼ ) . The second statement says that resource similarity is closedunder addition of resources. It is also closed under transitivityas an equivalence.Let R AT denote the additive-transitive closure (AT-closure)of the relation R (the minimal congruence, containing R ). Theresource similarity ≈ coincides with its AT-closure ( ≈ ) AT .Let B be a binary relations. A relation B ′ is called an AT-basis of B iff ( B ′ ) AT = B AT . An AT-basis B ′ is called minimal iff there is no B ′′ ⊂ B ′ such that ( B ′′ ) AT = B AT .When a relation is closed under addition and transitivity, itis completely determined by its AT-basis. So, when a relationhas a finite AT-basis, it has a convenient finite representation.It was proved [27] that the resource similarity has a finite AT-basis. For constructing this basis a special well-quasi orderingon resources is used.Recall that a well-quasi-ordering (a wqo) over a set S is a quasi-ordering ≤ such that, for any infinite sequence x , x , x , . . . , in S , there exist indices i < j with x i ≤ x j .The classical property of wqo’s states that the set of minimalw.r.t. a wqo elements is finite.Now we define a special wqo on pairs of Petri net resources,and minimal elements in this wqo will form a minimal AT-basis for the resource similarity relation.Note that for a given Petri net its markings (and resources)are multisets over a finite set of places and can be encodedas Parikh vectors – integer vectors of a fixed length. So,by Higman’s lemma [38] sequence (coordinate-wise) partialordering ≤ on the set of Petri net markings is a wqo.This ordering is very helpful for analysis of Petri nets aswell-structured transition systems [39] and, in particular, fortechniques based on constructing a coverability tree/graph.However, for constructing AT-basis for resource similarityrelation we need another ordering. Definition 7:
Let S be a finite set, B ⊆ M ( S ) × M ( S )) bea binary relation on the set of multisets over S . We define apartial order ⊑ on the set B of pairs of multisets as follows:1) For identity pairs let ( r , r ) ⊑ ( r , r ) def ⇔ r ⊆ r ;
2) For two non-identity pairs, the maximal identity partsand the addend pairs of disjoint multisets are comparedseparately: ( r + o , r + o ′ ) ⊑ ( r + o , r + o ′ ) def ⇔ def ⇔ o ∩ o ′ = ∅ & o ∩ o ′ = ∅ & r ⊆ r & o ⊆ o & o ′ ⊆ o ′ .
3) An identity pair and a non-identity pair are alwaysincomparable.It is easy to prove that the partial ordering ⊑ is a wqo, sincefor each pair of multisets r , r ∈ M ( S ) we have r ⊑ r implies r ≤ r , and ≤ is a wqo.Let then B s denote the set of all elements of B AT , whichare minimal with respect to ⊑ . Theorem 1: [27] Let S be a finite set, B ⊆ M ( S ) × M ( S )) be a symmetric and reflexive binary relation on the set ofmultisets over S . Then B s is an AT-basis of B AT and B s isfinite.We call B s the ground basis of B .Now consider ( P , P )-system ∆ . Resources of ∆ are multisetsover the finite set Const (∆) . The resource similarity for ∆ is a symmetric and reflexive binary relation on the set ofits resources, closed under addition and transitivity. Hence,by Theorem 1 the resource similarity relation ∼ has a finiteground basis ∼ S , such that ( ∼ S ) AT = ( ∼ ) , and ∼ S consistsof all minimal pairs of ∼ w.r.t. the ordering ⊑ .Thus, the resource similarity is finitely-based. However, theplace fusion (studied in [21], [22]) is a special case of resourcesimilarity for resources of capacity one. Place fusion wasproved to be undecidable, and this implies undecidability ofresource similarity.Note, that a resource similarity may be not a bisimulation,since it may not satisfy the transfer property. B. Resource Bisimilarity
The resource similarity relation is a strengthening of themarking bisimilarity, which is closed under addition of re-sources and transitivity, i.e. if a relation B is a resourcesimilarity, then B = B AT . We know also that, being finitelybased, resource similarity is undecidable, and therefore itsfinite basis cannot be computed effectively. Then looking fora computable approximation of the marking bisimilarity onemore relation on Petri net markings (resources) was definedin [27]. Definition 8:
Let ∆ be a ( P , P )-system. An equivalencerelation B ⊆ M ( Const (∆)) × M ( Const (∆)) on the set ofresources of ∆ is called a resource bisimulation iff B AT is aprocess bisimulation.First of all we recall several important properties of resourcebisimulations [27].Let ∆ be a ( P , P )-system. Then) if B , B are resource bisimulations for ∆ then B ∪ B is a resource bisimulation for ∆ ;2) there exists the largest resource bisimulation, denoted by ≃ , such that for every resource bisimulation B we have B ⊆ ( ≃ ) .3) for a given ( P , P )-system the resource bisimilarity is astrengthening of the resource similarity relation, i.e. foreach two resources r and s : r ≃ s ⇒ r ≈ s. So, we have that the relation ≃ is a symmetric and reflexiverelation, closed w.r.t. transitivity and addition of resources.Then by Theorem 1 above the maximal resource bisimulationhas a finite ground AT-basis, consisting of minimal w.r.t. ⊑ ordering. The main consequence of this for our research isthat for each ( P , P )-system, its largest resource bisimulationcan be represented by a finite number of pairs.The AT-closure of a resource bisimulation is a markingbisimulation, and marking bisimulations are characterized bythe transfer property. It was proved in [27] that resource bisim-ulations can be characterized by a weak variant of the transferproperty, when only ’relevant’ markings are considered for atransition t . The weak variant of the transfer property for placebisimulation was defined in [19]. The following definitiongeneralizes it for arbitrary resources. Definition 9:
Let ∆ be a ( P , P )-system. We say that arelation B ⊆ M ( Const (∆)) × M ( Const (∆)) satisfies the weak transfer property iff for all ( r, s ) ∈ B , for all δ ∈ ∆ ,s.t. • δ ∩ r = ∅ , there exists an imitating step γ ∈ ∆ , s.t. Act ( δ ) = Act ( γ ) and, writing m for • δ ∪ r and m for • δ − r + s , we have m δ → m ′ and m γ → m ′ with ( m ′ , m ′ ) ∈ B AT .The weak transfer property can be represented by thefollowing diagram: r ≈ B s • δ ∪ r • δ − r + s ↓ δ ↓ ( ∃ ) γ : Act ( δ ) = Act ( γ ) m ′ ∼ B AT m ′ The following theorem gives an important characterizationof resource bisimulation relations.
Theorem 2: [27] Let ∆ be a ( P , P )-system. A relation B ⊆M ( Const (∆)) × M ( Const (∆)) is a resource bisimulation iff B is an equivalence and it satisfies the weak transfer property.Summarizing the above results, we note that given a ( P , P )-system ∆ , • if a relation R is a resource similarity, then R is acongruence and R strengthens the process bisimilarity; • R is a resource bisimilation iff R is a bisimulation and R is a congruence; ♠ ♠ a ab ✛ ✛✲✲ X Y
Fig. 4. The resource similarity does not coincide with the state bisimularity: X ∼ Y , but X Y • the resource bisimilarity is the largest resource bisimula-tion.We now look at how the three relations, i. e., processbisimilarity, resource similarity, and resource bisimilarity arerelated to each other. Theorem 3:
1) For each ( , P )-system, process bisimilarity, resourcesimilarity and resource bisimilarity coincide: ( ∼ ) = ( ≈ ) = ( ≃ ) .2) There exists a ( P , P )-system for which ( ∼ ) = ( ≈ ) .3) There exists a ( P , P )-system for which ( ≈ ) = ( ≃ ) . Proof:
1. For ( , P ) systems process bisimilarity is acongruence, hence it coincides with the resource similarity andis closed under addition and transitivity. Then ( ∼ ) AT = ( ∼ ) ,and ∼ is a resource bisimilarity.2. Resource similarity is stronger than the state bisimulation: m ≈ m ′ ⇒ m ∼ m ′ . The converse implication is not true.Fig. 4 gives an example of the Petri net, where states X and Y are bisimular – for both of them the only possible step is a . But X and Y are not similar resources, since states X + X and Y + X are not bisimular – the former can fire b , whilethe latter cannot. In other words ( ≈ ) ⊂ ( ∼ ) . Note, that eachtransition in the Petri net on Fig. 4 has not more than twoincoming arcs, i. e., this net belongs to ( P , P )-PRS.3. Fig. 5 gives an example of a Petri net, where for tworesources X and Y we have X ≈ Y , but X Y .Indeed, for each resource u there is a winning Defenderstrategy in the Bisimulation Game for the pair of states ( X + u, Y + u ) . When e. g., Attacker fires a by moving a tokenfrom X to X and Z ∩ u = ∅ , Defender answers by firingthe transition without output arcs, and removes a token from Y . If Z ∩ u = ∅ , Defender answers by moving a token from Y to Y . Note that the Defender’s answer depends on the addedresource u . It is not difficult to check that a ’clever’ Defender,who counts tokens in Z , always wins in this game.However, there is no possibility to ensure the weak transferproperty for any relation which includes ( X , Y ) . The stepfrom X to X cannot be imitated by the step from Y to Y ,since when Z is empty the step b from Y cannot be imitatedon the other side. Also, the step from X to X cannot beimitated by the step from Y to Y or by firing t , since afterthese steps the ’Y side’ cannot imitate steps b and c , while the’X side’ can fire b and then c if Z has at least one token.Therefor, similar resources can be not resource bisimulardue to the locality of the weak transfer property (characterizingresource bisimilarity), while for resource similarity, bisimula-tion can be provided in different ways for different ‘contexts’. ♠♠ ♠♠ ♠♠ ♠ ♠ c cb bb ba aa aa a ❄ ❄❄ ❄❄ ❄❄ ❄❄ ❄❄ ❄❄ ❄ X Y X Y X Y X Z Y ✙ ❥s ✰✙ ◆ Fig. 5. The resource similarity does not coincide with the resource bisimi-larity: X ≈ Y , but X Y Remark 1:
Note that in the proof we used only very simple ( P , P ) systems with at most two atoms on each side of PRStransition rule.Theorem 3 implies that for ( P , P )-systems we have strictinclusions for three bisimulation equality relations: ( ∼ ) ⊂ ( ≈ ) ⊂ ( ≃ ) . The state bisimilarity ( ∼ ) and the resource similarity ( ≈ ) are undecidable. Further we prove that the resource bisimilar-ity ( ≃ ) is decidable. C. Resource Bisimilation Game and Stratified ResourceBisimilarity
Here we define
Resource bisimilarity game based on theweak transfer property. Similar to the classical Bisimulationgame, there are two players: Attacker and Defender.Let ∆ be a ( P , P )-system, and r, s ∈ M ( Const (∆)) betwo resources in ∆ . Attacker wants to prove that r s byfinding an example of a violation of the weak transfer property.Defender tries not to let him do this.The game starts with a given pair of resources ( r, s ) . Theplay goes in rounds, and one round goes as follows. Attackerchooses a resource (either r , or s ) and a transition t in N .Let the resource r be selected, and let the selected transition t be labeled with the action a . Attacker fires t in the marking ( • t ∪ r ) . Let ( • t ∪ r ) t → r ′ . The Defender should answer withfiring some transition t ′ , such that l ( t ) = l ( t ′ ) , in the marking ( • t − r + s ) . If Defender cannot answer, Attacker wins. IfDefender answers with ( • t − r + s ) t ′ → s ′ , the play continuesfor a new round with the pair ( r ′ , s ′ ) .It is easy to note, that Attacker can always do a step, since heis allowed to add resources for firing any transitions. Attackerwins the play, if at some round Defender cannot answer.Defender wins, if the play goes infinitely long. Proposition 2:
In the Resource Bisimilarity Game for a pair ( r, s ) Defender has a universal winning strategy iff r ≃ s , andAttacker has a universal winning strategy iff r s . Proof:
Let Defender has a winning strategy. This strategycan be represented as an infinite tree, where nodes are pairs of resources, ( r, s ) is the root node, and a node ( r ′′ , s ′′ ) is achild of a node ( r ′ , s ′ ) iff a round starting with ( r ′ , s ′ ) can endwith ( r ′′ , s ′′ ) according to the winning strategy of Defender.The strategy is universal, i. e., it contains a Defender’s answerto any step of Attacker.Consider the set R of all nodes of this winning strategytree. R is a binary relation. It is symmetric, since Attacker canchoose any resource from the input pair. It satisfies the weaktransfer property by construction of the tree. Then extendedwith the identity relation I = { ( r, r ) | r ∈ M ( Const (∆)) } ,by Theorem 1 R ∪ I is a resource bisimulation. Thus r ≃ s .Let now r ≃ s . A universal winning strategy tree forDefender is constructed as follows. We start with the root ( r, s ) . Then, due to the weak transfer property, Defender cananswer to any step of Attacker so that the game comes to anew pair of bisimilar resources. This pair is added as a childof ( r, s ) . Since the number of rules in ( P , P )-system is finite,the node ( r, s ) gets a finite number of children. Then the sameprocedure is repeated for new children going breadth-first.Now, an Attacker’s universal winning strategy can be rep-resented as a finite Attacker’s strategy tree, where for eachleaf Attacker can fire a transition to which Defender cannotanswer. When Defender has a universal winning strategy, auniversal winning strategy for Defender cannot exist, and viceversa.Now we define stratified resource bisimilarity, similar to theclassical stratified (state) bisimilarity. Definition 10:
Let ∆ be a ( P , P )-system, and r, s ∈M ( Const (∆)) be two resources in ∆ . Let k ∈ N . We saythat r and s are resource bisimular up to k rounds , denoted r ≃ k s , iff in the resource bisimilarity game for ( r, s ) Defendercan hold out k rounds.Stratified resource bisimilarity has some nice properties.First of all, it is easy to note that for each k , r ≃ k s is an equivalence, ( r ≃ k s ) ⊆ ( r ≃ k +1 s ) ,and by the definition of resource bisimilarity we have ( ≃ ) = ∩ k ∈ N ( r ≃ k s ) .The next proposition states that stratified resource bisimi-larity is also a congruence. Proposition 3:
Let ∆ be a ( P , P )-system. For all r, s, w ∈M ( Const (∆)) , if r ≃ k s then ( r + w ) ≃ k ( s + w ) . Proof:
Let r ≃ k s . Then for any Attacker’s firingtransition t , such that ( • t ∪ r ) t → u , Defender can answerwith t ′ , such that l ( t ) = l ( t ′ ) , and for some process term u ′ , ( • t − r + s ) t ′ → u ′ , and u ≃ k − u ′ , i.e. Defender can win around and he is to hold out for more k − rounds. (Similarly,if Defender chose s instead of r .)Now consider Attacker’s firing transition t for a processterm ( r + w ) . Let (( • t \ r ) ∩ w ) = w ′ , and w = w ′ + w ′′ ,then ( • t ∪ ( r + w )) = ( • t ∪ r ) + w ′′ . Hence, the Defender cananswer with (( • t − r + s ) + w ′′ ) t ′ → ( u ′ + w ′′ ) , and u ≃ k − u ′ .Therefore, one can start another round, and this procedure canbe continued for k rounds. ♠ ♠ a b a b ❄ ❄ X Y Z ✢ ❫
Fig. 6. Petri net example
Corollary 1:
Let r ≃ k s and r ′ ≃ k s ′ . Then ( r + r ′ ) ≃ k ( s + s ′ ) , i.e. stratified resource bisimilarity is closed underaddition. Proof:
By Proposition 3, r ≃ k s implies ( r + r ′ ) ≃ k ( s + r ′ ) , and r ′ ≃ k s ′ implies ( s + r ′ ) ≃ k ( s + s ′ ) .Then by transitivity of the resource bisimilarity, we have ( r + r ′ ) ≃ k ( s + s ′ ) . Corollary 2:
For each k , stratified resource bisimilarity ≃ k has a finite AT-basis. Proof:
For every k , stratified resource bisimilarity ≃ k isclosed under addition and transitivity, hence, ( ≃ k ) AT = ( ≃ k ) .The relation ≃ k is also an equivalence. Then by Theorem 1, ≃ k has a finite AT-basis.Stratified resource bisimilarity can be considered as anapproximation of the resource bisimilarity: ( ≃ ) ⊇ ( ≃ ) ⊇ . . . ( ≃ k ) ⊇ · · · ⊇ ( ≃ ) . For each k , the resource bisimilarity up to k rounds canbe represented by its finite AT-basis. An AT-basis for ( ≃ k ) can be effectively computed, since the number of rounds inthe Resource bisimilarity game is limited. However, the nextexample shows that computing AT-bases for ( ≃ k ) does nothelp in computing an AT-basis for the resource bisimilarity. Example 1:
As an example consider the Petri net in Fig. 6.This Petri net is conflict free, each transition has exactlyone input place. So, here resource bisimilarity coinsides withclassical (state) bisimilarity.For this Petri net the AT-basis for ≃ consists of X + Y ≃ Z , X ≃ X , Y ≃ Y , and Z ≃ Z , since adding moretokens does not effect the execution of the first round.The AT-basis for ≃ consists of X + 2 Y ≃ Z , and X ≃ X , Y ≃ Y , Z ≃ Z .Similarly, AT-basis for ≃ k consists of kX + kY ≃ k kZ ,and kX ≃ k ( k + 1) X , kY ≃ k ( k + 1) Y , kZ ≃ k ( k + 1) Z .This example shows that a sequence ( ≃ ) ⊇ ( ≃ ) ⊇ . . . ( ≃ k ) ⊇ . . . may not stabilize, and its limit ≃ may benot reached at any round.IV. A LGORITHM FOR C HECKING R ESOURCE B ISIMILARITY FOR ( P , P )-PRS The problem:
Given a ( P , P )-system ∆ and two resources r , s ∈ M ( Const (∆)) we need to check whether they areresorce bisimilar, i.e. whether r ≃ s .The algorithm we present here to solve this problem followsthe well-known tableau technique (see e. g., [33]) adaptedto the resource bisimilarity relation and its weak transferproperty. We start with some definitions that we will need later. Definition 11:
Let S be a finite set, B ⊆ M ( S ) × M ( S )) be a binary relation on the set of multisets over S .1) For a pair ( r, s ) ∈ B , define its difference rate dr asfollows: • for an identity pairs dr ( r, r ) = 0 ; • for a non-identity pair ( r + o, r + o ′ ) , where o ∩ o ′ = ∅ , dr measures difference of two multisets: dr ( r + o, r + o ′ ) = | o + o ′ | .2) For two pairs ( r, s ) , ( r ′ , s ′ ) ∈ B , define ( r, s ) ⊳ ( r ′ , s ′ ) iff dr ( r, s ) ≤ dr ( r ′ , s ′ ) Obviously, ⊳ is a partial order on B , moreover, it iswell-founded, i. e., there cannot be infinite strictly decreasingchains, since natural numbers are well-founded.The other ordering, which will be used in the algorithm forchecking resource bisimilarity, is the wqo ⊑ (see Definition 7).Let ∆ be a ( P , P )-system. Recall that ∆ is a set of basicrules. We define a relation ⇒ ⊆ ( B × ∆ × B ) , whichrepresents one round of the Resource bisimilarity game. Wewrite ( r, s ) δ ⇒ ( r ′ , s ′ ) for (( r, s ) , δ, ( r ′ , s ′ )) ∈ ⇒ , where δ ∈ ∆ , and ( r, s ) , ( r ′ , s ′ ) ∈ B . Definition 12:
Let ∆ be a ( P , P )-system. For δ ∈ ∆ , and r, s, r ′ , s ′ ∈ M ( Const (∆)) , the pair ( r ′ , s ′ ) is a δ -child of ( r, s ) , denoted ( r, s ) δ ⇒ ( r ′ , s ′ ) , iff there exists γ ∈ ∆ , suchthat Act ( δ ) = Act ( γ ) , • δ ∪ r δ → r ′ , and ( • δ − r + s ) γ → s ′ .Note, that a pair ( r, s ) can have many children. By next δ ( r, s ) we denote the set of all δ -children of ( r, s ) . Thisset is always finite, since ∆ contains a finite number of rules.Now we come to describing the algorithm. The algorithmis based on the nondeterministic procedure of constructing afinite proof tree , called also a tableau . Nodes of this tree arelabeled by pairs of resources.We start at a root labeled with the given pair ( r , s ) and build the proof tree depth-first, gradually adding childrento each node. Identity nodes of the form ( r, r ) are called successful terminal nodes (leaves).There are two types of steps, generating new children:EXPAND and REDUCE.To apply an EXPAND step to a node ( r, s ) , you need foreach δ ∈ ∆ , nondeterministically choose exactly one pair ofresources from next δ ( r, s ) and exactly one pair of resourcesfrom next δ ( s, r ) . All these pairs are then added to the tree aschildren of the node ( r, s ) . Thus, in the regular case, if ∆ consists of k basic rules, the node ( r, s ) gets k children.If for some δ either next δ ( r, s ) , or next δ ( s, r ) is empty, thenthe node ( r, s ) is marked as a fail terminal leaf, and the wholetree is marked as FAIL.To apply a REDUCE step to a node ( r, s ) , you need tocheck if there is a node ( r ′ , s ′ ) on the path from the root tothe node ( r, s ) , such that ( r ′ , s ′ ) ⊑ ( r, s ) . If there is no suchnode, the step is not applicable. Let ( r ′ , s ′ ) be such a node,i. e., for some r , r , s , r ′ , r ′ , s ′ we have r = r + r , s = r + s , r ∩ s = ∅ , similarly, r ′ = r ′ + r ′ , s ′ = r ′ + s ′ , r ′ ∩ s ′ = ∅ , and ′ ⊆ r , r ′ ⊆ r , s ′ ⊆ s .Let also | r ′ | ≤ | s ′ | . Then the REDUCE step adds to thenode ( r, s ) exactly one child ( r, r + ( s − s ′ ) + r ′ ) , i. e., wesubstitute r ′ for s ′ .Note that after such a step the difference rate of the childis strictly less than the difference rate of the parent. Indeed, dr ( r, s ) = dr ( r + r , r + s ) = | r | + | s | , and dr ( r, r +( s − s ′ ) + r ′ ) = dr ( r + r , r + ( s − s ′ ) + r ′ ) = | ( r − r ′ ) | + | ( s − s ′ ) | < | r | + | s | .EXPAND and REDUCE steps are performed in the fol-lowing order. Initially, there is the only root node, and anEXPAND step is applied to it. If there is no FAIL, to eachnew child a REDUCE step is applied iteratively until either itis not applicable, or a successful terminal node is reached.When a REDUCE step is not applicable and the node is nota successful terminal node, a EXPAND step is applied, afterwhich REDUCE steps are iteratively applied to each new node.The procedure terminates, when either all current leaves inthe tree are successful terminal leaves, or at least one of thecurrent leaves is a fail terminal leaf.If there exists a successful proof tree (a tree with all leavesbeing successful terminal nodes), we enclose r ≃ s , otherwise r s .The following theorem asserts the correctness of the algo-rithm. Theorem 4:
Let ∆ be a ( P , P )-system and r , s ∈M ( Const (∆)) be its two resources. Let T ( r , s ) be theset of all proof trees constructed by applying the abovenondetermenistic algorithm to the pair of resources ( r , s ) in ∆ . Then1) the set T ( r , s ) is finite, and each proof tree T ∈ T ( r , s ) is finite;2) r ≃ s iff there exists a successful proof tree T ∈ T ( r , s ) . Proof:
1) First of all, any proof tree is finitely branching,since the number of basic rules in ∆ is finite. Then anEXPAND step cannot be applied infinitely many times withoutreductions, since the ordering ⊑ is a wqo. In turn, a REDUCEstep also cannot be applied in a row an infinite number of timeswithout expanding, since each application of a REDUCE stepdecreases the difference rate of the resource pair, and the order ⊳ is well-founded.Finally, we show that EXPAND and REDUCE steps cannotalternate infinitely many times. Suppose, a proof tree has aninfinite branch. Consider a sequence ( r , s ) , ( r , s ) , . . . ofresource pairs along this branch, for which an EXPAND stepis applied immediately after a REDUCE step. Since for each ( r i , s i ) in this sequence a REDUCE step cannot be applied,we have ( r i , s i ) ( r j , s j ) for all i ≤ j in this sequence – acontradiction with the fact that the ordering ⊑ is a wqo.This proves that any branch in a proof tree is finite. Hence,by K¨onig’s lemma, each proof tree is finite.Since the set next δ ( r, s ) is finite for any ( r, s ) , and the set ofbasic rules in ∆ is also finite, there are only a limited numberof nondeterministic choices when an EXPAND step is applied. Hence, the number of proof trees constructed by the algorithmis finite.2) Let r ≃ s O . We will prove that then there exists a prooftree T ∈ T ( r , s ) with the root ( r , s ) such that for eachnode ( r, s ) in T we have r ≃ s . For this it is sufficient to provethat for each current node ( r, s ) in a proof tree, if r ≃ s , andall its ancestors are also pairs of bisimular resources in ∆ ,then it is possible to make a step from ( r, s ) such that foreach its new child ( r ′ , s ′ ) we have r ′ ≃ s ′ .Let ( r, s ) be a current nonterminal node in a proof tree, r ≃ s , and the REDUCE rule is not applicable to it. Then byTheorem 2 the weak transfer property is valid for ( r, s ) , i. e.,for each δ there exists a pair ( r ′ , s ′ ) , such that ( r, s ) δ ⇒ ( r ′ , s ′ ) and r ′ ≃ s ′ . Since the relation ( ≃ ) is symmetric, we have alsothat there exists a pair ( s ′′ , r ′′ ) , such that ( s, r ) δ ⇒ ( s ′′ , r ′′ ) and r ′′ ≃ s ′′ . Thus, the EXPAND step can be applied to ( r, s ) bychoosing the above variants.Consider now the case when the REDUCE rule is applicableto ( r, s ) . Then there exists its ancestor ( r ′ , s ′ ) such that ( r ′ , s ′ ) ⊑ ( r, s ) . This implies r ′ ⊆ r and s ′ ⊆ s , and theresult of applying the REDUCE step is ( r − r ′ + s ′ , s ) , orsymmetrically ( r, s − s ′ + r ′ . Since the relation ≃ is closedunder addition and transitivity, from r ≃ s and r ′ ≃ s ′ wesuccessively have: ( r ′ , s ′ ) , (( r − r ′ ) + s ′ , ( r − r ′ ) + r ′ ) , (( r − r ′ ) + s ′ , r ) , ( r, s ) , ( r − r ′ + s ′ , s ) .The symmetrical case is proved in a similar way.Let us now prove the opposite. Let T ∈ T ( r , s ) be asuccessful proof tree. To prove by contradiction assume that r s . Since ( ≃ ) = ∩ k ∈ N ( r ≃ k s ) , there exists a minimal k such that r k s .Consider a node ( r, s ) in T for which k is the minimalnumber such that r k s . If the EXPAND step was applied tothis node in T , then it has at least one child ( r ′ , s ′ ) such that r ′ k − s ′ .If the REDUCE step with an ancestor node ( r ′′ , s ′′ ) wasapplied to the node ( r, s ) , then due to the minimality of k ,we have r ′′ ≃ k s ′′ , since the node ( r ′′ , s ′′ ) is at least oneapplication of the EXPAND step ’closer’ to the root ( r , s ) than the node ( r, s ) .The child of ( r, s ) via the REDUCE step is the node ( r − r ′′ + s ′′ , s ) , or symmetrically ( r, s − s ′′ + r ′′ ) . Consider the firstcase. By Proposition 3, r ′′ ≃ k s ′′ implies that r = (( r − r ′′ ) + r ′′ ) ≃ k (( r − r ′′ ) + s ′′ ) . It cannot be that ( r − r ′′ + s ′′ ) ≃ k s ,since otherwise by the transitivity of the relation ≃ we wouldget r ≃ k s , which is not true. Hence, the child of ( r, s ) viathe REDUCE step also does not belong to the relation r ≃ k s .Thus we can construct a path from the root to a leaf so thatfor each node ( r, s ) on this path we have r s . This is acontradiction with the fact that each branch in T ends with asuccessful (identity) leaf.. C ONCLUSIONS
In this paper we investigate decidability issues of twocongruent strengthenings of the bisimulation equivalence forthe ( P , P )-subclass of Process Rewrite Systems (PRS), whichis equivalent to classical Petri nets (P/T-nets), as well as VectorAddition Systems with States (VASS). These equivalencesare resource similarity and resource bisimularity defined in[27], both of which are congruences, and replacing a resource(subterm/submarking) in a ( P , P )-PRS term, respectively, in aPetri net marking, by a similar resource does not change theobservable system behavior.Resource similarity and resource bisimilarity are subrela-tions of the state bisimilarity for ( P , P )-PRS. An importantproperty of both of these equivalences is that they, in contrastto state bisimularity, are finitely-based. So, these equivalencescan be used for deducing bisimular states, and the latter can beused for increasing the efficiency of verification by reducingthe state space of the system. Another application of resourceequivalences is a Petri net reduction [40].However, the resource similarity is undecidable, and de-cidability of the resource bisimilarity, as well as whether theresource bisimilarity is a proper subrelation of the resourcesimilarity, remained an open question [27], [31].We have shown that the resource bisimilarity is decidablefor (P,P)-PRS. This implies that the resource bisimilarity is aproper subrelation of the resource similarity. We also give anexample of a ( P , P )-PRS, where two similar resources are notresource bisimilar.An interesting question for further research: is it possible toeffectively compute a finite AT-basis of the research bisimilar-ity equivalence for a given ( P , P )-PRS. Note that this will alsoanswer the question of the computability of a finite AT-basisof the state bisimilarity for ( , P )-PRS, since for ( , P )-PRS,the resource and state bisimilarity coincide.The answer to this question for the resource similarityequivalence in ( P , P )-PRS is obviously ’no’, since the resourcesimilarity is undecidable for ( P , P )-PRS.A CKNOWLEDGMENT
This work has been supported by the Basic Re-search Program at the National Research University HigherSchool of Economics and by Research Project AAAA-A16-116070610022-6 at Yaroslavl State University.R
EFERENCES[1] R. van Glabbeek, “The linear time - branching time spectrum,” in
Hand-book of Process Algebra
Theoret-ical Computer Science , P. Deussen, Ed. Berlin, Heidelberg: SpringerBerlin Heidelberg, 1981, pp. 167–183.[3] R. Milner,
Communication and Concurrency . USA: Prentice-Hall, Inc.,1989.[4] R. Mayr, “Process Rewrite Systems,”
Information and Computa-tion
SIAM Journal on Computing ,vol. 34, no. 5, pp. 1025–1106, 2005. [Online]. Available:https://doi.org/10.1137/S0097539700377256[6] P. Janˇcar, “Equivalences of pushdown systems are hard,” in
Foundationsof Software Science and Computation Structures , A. Muscholl, Ed.Berlin, Heidelberg: Springer Berlin Heidelberg, 2014, pp. 1–28.[7] O. Burkart, D. Caucal, and B. Steffen, “An elementary bisimulationdecision procedure for arbitrary context-free processes,” in
MathematicalFoundations of Computer Science 1995 , J. Wiedermann and P. H´ajek,Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995, pp. 423–433.[8] J. Srba, “Strong bisimilarity and regularity of Basic Process Alge-bra is PSPACE-hard,” in
Automata, Languages and Programming ,P. Widmayer, S. Eidenbenz, F. Triguero, R. Morales, R. Conejo, andM. Hennessy, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg,2002, pp. 716–727.[9] S. Kiefer, “BPA bisimilarity is EXPTIME-hard,”
Information ProcessingLetters
CONCUR’93 , E. Best, Ed.Berlin, Heidelberg: Springer Berlin Heidelberg, 1993, pp. 143–157.[11] P. Janˇcar, “Strong bisimilarity on basic parallel processes in PSPACE-complete,” in , 2003, pp. 218–227.[12] ——,
Decidability questions for bisimilarity of Petri netsand some related problems . Berlin, Heidelberg: SpringerBerlin Heidelberg, 1994, pp. 581–592. [Online]. Available:http://dx.doi.org/10.1007/3-540-57785-8 173[13] Y. Hirshfeld and M. Jerrum, “Bisimulation equivalence is decidable fornormed Process Algebra (extended abstract),” in
Automata, Languagesand Programming , J. Wiedermann, P. van Emde Boas, and M. Nielsen,Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 1999, pp. 412–421.[14] J. Srba, “Undecidability of weak bisimilarity for PA-processes,” in
Developments in Language Theory , M. Ito and M. Toyama, Eds. Berlin,Heidelberg: Springer Berlin Heidelberg, 2003, pp. 197–209.[15] Q. Yin, Y. Fu, C. He, M. Huang, and X. Tao, “Branching bisimilaritychecking for PRS,” in
Automata, Languages, and Programming , J. Es-parza, P. Fraigniaud, T. Husfeldt, and E. Koutsoupias, Eds. Berlin,Heidelberg: Springer Berlin Heidelberg, 2014, pp. 363–374.[16] C. A. Petri, “Kommunikation mit automaten,” PhD thesis, Bonn: Insti-tute f¨ur Instrumentelle Mathematik, 1962.[17] E.-R. Olderog, “Strong bisimilarity on nets: A new concept for compar-ing net semantics,” in
Linear Time, Branching Time and Partial Order inLogics and Models for Concurrency , J. W. de Bakker, W. P. de Roever,and G. Rozenberg, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg,1989, pp. 549–573.[18] C. Autant, Z. Belmesk, and P. Schnoebelen, “Strong bisimilarity on netsrevisited,” in
Parle ’91 Parallel Architectures and Languages Europe ,E. H. L. Aarts, J. van Leeuwen, and M. Rem, Eds. Berlin, Heidelberg:Springer Berlin Heidelberg, 1991, pp. 717–734.[19] C. Autant and P. Schnoebelen,
Place bisimulations in Petri nets . Berlin,Heidelberg: Springer Berlin Heidelberg, 1992, pp. 45–61. [Online].Available: http://dx.doi.org/10.1007/3-540-55676-1 3[20] C. Autant, W. Pfister, and P. Schnoebelen, “Place bisimulations for thereduction of labeled Petri nets with silent moves,” in
Proc. 6th Int. Conf.on Computing and Information, Peterborough, Canada , 1994.[21] W. Quivrin-Pfister, “Des bisimulations de places pour la r´eduction desr´esaux de Petri,” PhD thesis, I.N.P. de Grenoble, France, 1995.[22] P. Schnoebelen and N. Sidorova,
Bisimulation and the Reduction of PetriNets . Berlin, Heidelberg: Springer Berlin Heidelberg, 2000, pp. 409–423. [Online]. Available: http://dx.doi.org/10.1007/3-540-44988-4 23[23] P. Janˇcar, “Undecidability of bisimilarity for Petri nets and some relatedproblems,”
Theor. Comput. Sci. , vol. 148, no. 2, pp. 281–301, Sep. 1995.[Online]. Available: https://doi.org/10.1016/0304-3975(95)00037-W[24] M. Voorhoeve, “Structural Petri net equivalence,” Technische Univer-siteit Eindhoven, Tech. Rep. 9607, 1996.[25] R. Gorrieri, “Team bisimilarity, and its associated modal logic, for BPPnets,”
Acta Informatica , pp. 1–41, 2020.[26] ——, “A study on team bisimulations for BPP nets,” in
Application andTheory of Petri Nets and Concurrency , R. Janicki, N. Sidorova, and. Chatain, Eds. Cham: Springer International Publishing, 2020, pp.153–175.[27] V. A. Bashkin and I. A. Lomazova, “Petri nets and resource bisimula-tion,”
Fundamenta Informaticae , vol. 55, no. 2, pp. 101–114, 2003.[28] F. Corradini, R. De Nicola, and A. Labella, “Models of nondeter-ministic regular expressions,”
Journal of Computer and System Sci-ences
Resource Similarities inPetri Net Models of Distributed Systems . Berlin, Heidelberg:Springer Berlin Heidelberg, 2003, pp. 35–48. [Online]. Available:http://dx.doi.org/10.1007/978-3-540-45145-7 4[30] ——,
Similarity of Generalized Resources in Petri Nets . Berlin,Heidelberg: Springer Berlin Heidelberg, 2005, pp. 27–41. [Online].Available: http://dx.doi.org/10.1007/11535294 3[31] I. A. Lomazova, “Resource equivalences in Petri nets,” in
Applicationand Theory of Petri Nets and Concurrency , W. van der Aalst and E. Best,Eds. Cham: Springer International Publishing, 2017, pp. 19–34.[32] P. Janˇcar, “Selected ideas used for decidability and undecidability ofbisimilarity,” in
Lecture Notes in Computer Science (including subseriesLecture Notes in Artificial Intelligence and Lecture Notes in Bioinfor-matics) , 2008.[33] L. Aceto, A. Ingolfsdottir, and J. Srba, “The algorithmics of bisimilarity,”in
Advanced Topics in Bisimulation and Coinduction , 2011.[34] M. Hennessy and R. Milner, “Algebraic laws for nondeterminism andconcurrency,”
J. ACM , vol. 32, no. 1, pp. 137–161, Jan. 1985. [Online].Available: https://doi.org/10.1145/2455.2460[35] W. Thomas, “On the Ehrenfeucht-Fra¨ıss´e game in theoretical computerscience,” in
TAPSOFT’93: Theory and Practice of Software Develop-ment , M. C. Gaudel and J. P. Jouannaud, Eds. Berlin, Heidelberg:Springer Berlin Heidelberg, 1993, pp. 559–568.[36] C. Stirling, “Local model checking games (extended abstract),” in
CONCUR ’95: Concurrency Theory , I. Lee and S. A. Smolka, Eds.Berlin, Heidelberg: Springer Berlin Heidelberg, 1995, pp. 1–11.[37] S. Christensen, “Decidability and decomposition in process algebras,”Ph.D. dissertation, University of Edinburgh, UK, 1993. [Online]. Avail-able: http://hdl.handle.net/1842/410[38] G. Higman, “Ordering by divisibility in abstract algebras,”
Proceedingsof The London Mathematical Society , pp. 326–336, 1952.[39] A. Finkel and P. Schnoebelen, “Well-structured transitionsystems everywhere!”
Theoretical Computer Science