Decidability of Branching Bisimulation on Normed Commutative ContextFree Processes
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Abstract
We investigate normed commutative contextfree processes (Basic Parallel Processes). We show that branching bisimilarity admits the bounded response property: in the Bisimulation Game, Duplicator always has a response leading to a process of size linearly bounded with respect to the Spoiler’s process. The linear bound is effective, which leads to decidability of branching bisimilarity. For weak bisimilarity, we are able merely to show existence of some linear bound, which is not sufficient for decidability. We conjecture however that the same effective bound holds for weak bisimilarity as well. We suppose that further elaboration of novel techniques developed in this paper may be sufficient to demonstrate decidability.
Keywords
Branching bisimulation equivalence Commutative contextfree graphs Equivalence checking Basic parallel processes1 Introduction
Bisimulation equivalence (bisimilarity) is a fundamental notion of equivalence of processes, with many natural connections to logic, games and verification [12, 15]. This paper is a continuation of the active line of research focusing on decidability and complexity of decision problems for bisimulation equivalence on various classes of infinite systems [14].
Over this class of graphs, we focus on bisimulation equivalence as the primary type of semantic equality of processes. It is known that strong bisimulation equivalence is decidable [2] and PSPACEcomplete [10, 13]; and is polynomial for normed processes [7]. Dramatically less is known about weak bisimulation equivalence, that abstracts from the silent εtransitions: we only know that it is semidecidable [4] and that it is decidable in polynomial space over a very restricted class of totally normed processes [5]. The same applies to branching bisimulation equivalence, a variant of weak bisimulation that respects faithfully branching of equivalent processes. The only nontrivial decidability result known by now for weak bisimulation equivalence is proved in [16], it applies however to a very restricted subclass.^{1}
During last two decades decidability of weak bisimulation over contextfree processes became an established longstanding open problem. This paper is a significant step towards solving this problem in affirmative.
It is well known that bisimulation equivalences have an alternative formulation, in terms of Bisimulation Game played between Spoiler (aiming at showing nonequivalence) and Duplicator (aiming at showing equivalence) [15]. One of the main obstacles in proving decidability of weak (or branching) bisimulation equivalence is that Duplicator may do arbitrarily many silent transitions in a single move, and thus the size of the resulting process is hard to bound.
In this paper we investigate branching bisimilarity over normed commutative contextfree processes. Our main technical result is the proof of the following bounded response property, formulated as Theorem 6 in Sect. 3: if Duplicator has a response, then he also has a response that leads to a process of size linearly bounded with respect to the other (Spoiler’s) process. Importantly, we obtain an effective bound on the linear coefficient, which enables us to prove (Theorem 7) decidability of branching bisimulation equivalence. The proof of Theorem 6 is quite complex and involves a lot of subtle investigations of combinatorics of BPP transitions, the main purpose being elimination of unnecessary silent transitions.
A major part of the proof works for weak bisimulation equally well (and, as we believe, also for any reasonable equivalence that lies between the two equivalences). However, for weak bisimulation we can merely show existence of the linear coefficient witnessing the bounded response property, while we are not able to obtain any effective bound. Nevertheless we strongly believe (and conjecture) that a further elaboration of our approach will enable proving decidability of weak bisimulation equivalence. In particular, we actually reprove decidability of weak bisimilarity in the subclass investigated in [16].
This paper is the full and improved version of the extended abstract [3].
2 Preliminaries
The commutative contextfree processes (known also as Basic Parallel Processes [1]) are determined by the following ingredients (called a process definition): a finite set V of variables, a finite set A of letters, and a finite set of transition rules, each of the form \(X\stackrel {\zeta }{\longrightarrow } \alpha\) where X is a variable, ζ∈A∪{ε} and α is a finite multiset of variables.
A process, is any finite multiset of variables, thus a mapping that assigns a finite nonnegative multiplicity to each variable, and may be understood as the parallel composition of a given number of copies of respective variables. In particular the empty process, denoted ε, is the empty multiset. For any W⊆V we denote by W ^{⊗} the set of all processes where only variables from W occur, that is, W ^{⊗} is the set of all finite multisets over W.
The transitions labeled by letters from A are observable transitions, while the transition relation \(\stackrel {\varepsilon }{\longrightarrow }\) models silent steps. This distinction will play a role when defining bisimulation equivalence below. The transition relation \(\stackrel {\varepsilon }{\longrightarrow }\) will be written \(\stackrel {}{\longrightarrow }\), and the reflexive and transitive closure of \(\stackrel {\varepsilon }{\longrightarrow }\) will be denoted \(\stackrel {}{\Longrightarrow }\). Thus \(\alpha \stackrel {}{\Longrightarrow }\beta\) if a process β can be reached from α by a sequence of \(\stackrel {\varepsilon }{\longrightarrow }\) transitions.
Example 1
Remark 2
Commutative contextfree processes are precisely labeled communication free Petri nets, where the places are variables and transitions \(X\stackrel {\zeta }{\longrightarrow }\alpha\) are firing rules. A process represents a marking, i.e., a finite multiset of places.
The standard references for introduction of branching and weak bisimilarity are [19] and [12], respectively. For an exhaustive taxonomy of various notion of bisimulation respecting silent moves, the classical reference is [18].
To simplify definitions, we conveniently postulate from now on that every process α, including the empty process ε, has the silent transition \(\alpha \stackrel {}{\longrightarrow }\alpha\).
Definition 3
(Branching Bisimilarity)

if \(\alpha \stackrel {\zeta }{\longrightarrow }\alpha'\) then \(\beta \stackrel {}{\Longrightarrow }\beta''\stackrel {\zeta }{\longrightarrow }\beta'\) such that α B β″ and α′ B β′.
In the proofs we will use the characterization of bisimilarity in terms of Bisimulation Game [12, 15]. The game is played by two players, Spoiler and Duplicator, over an arena consisting of all pairs of processes, and proceeds in rounds. Each round starts with a Spoiler’s move followed by a Duplicator’s response. In position (α,β), Spoiler chooses one of processes, say α, and one transition \(\alpha \stackrel {\zeta }{\longrightarrow } \alpha'\). As a response, Duplicator has to do a sequence of transitions of the form \(\beta \stackrel {}{\Longrightarrow }\beta''\stackrel {\zeta }{\longrightarrow }\beta'\), and then Spoiler chooses whether the next starts from (α,β″) or (α′,β′).
If one of players gets stuck, the other wins. Otherwise the play is infinite and in this case it is Duplicator who wins. A wellknown fact is that two processes are branching bisimilar iff Duplicator has a winning strategy in the game that starts in these two processes.
According to the winning condition of Bisimulation Game, if the two processes are branching bisimilar in some position of the game then, whatever move is played by Spoiler, there is always a Duplicator’s response so that the two resulting pairs of processes are branching bisimilar again. Any such Duplicator’s response move will be called matching in the sequel. We will also say that the Spoiler’s move is matched by some Duplicator’s response.
Weak Bisimilarity

if \(\alpha \stackrel {\zeta }{\longrightarrow }\alpha'\) then \(\beta \stackrel {}{\Longrightarrow }\stackrel {\zeta }{\longrightarrow }\stackrel {}{\Longrightarrow }\beta'\) such that α′ B β′.
Example 4
3 Decidability via Bounded Response Property
Definition 5
(Bounded Response Bisimulations)
Then we define in a standard way cbranching bisimilarity (cweak bisimilarity), denoted ≃_{ c } (≈_{ c }, respectively), as the greatest cbranching bisimulation (cweak bisimulation, respectively).
Let the size of a process definition be the sum of sizes of all production rules. Our main technical result is an efficient estimation of the constant c in Definition 5, with respect to the size of a process definition:
Theorem 6
(Bounded Response Property of ≃)
Given a normed process definition, one can compute \(c \in\mathbb{N}\) such that branching bisimilarity ≃, over the transition graph induced by the process definition, is a cbranching bisimulation.
The proof of Theorem 6 is deferred to Sects. 4–6.
In consequence of the theorem, a Spoiler’s winning strategy, seen as a tree, becomes finitely branching. This observation leads directly to decidability:
Theorem 7
Branching bisimilarity ≃ is decidable over normed commutative contextfree processes.
Proof
The decision procedure starts with computing \(c \in\mathbb{N}\), according to Theorem 6, such that branching bisimilarity coincides with cbranching bisimilarity. Then we run two semidecision procedures (along the lines of [11]): the positive one for branching bisimilarity and the negative one for cbranching bisimilarity.
For the positive side we use a standard semilinear representation of bisimulation, knowing that each congruence, including ≃, is semilinear [6, 9]. The algorithm guesses a baseperiod representation of a semilinear set and then checks validity of a Presburger formula that says that this set is a branching bisimulation containing the input pair of processes.
For the negative side, we observe that due to Theorem 6 Duplicator has only finitely many possible answers to each Spoiler’s move. Thus, if Spoiler wins then its winning strategy may be represented by a finitelybranching tree. Furthermore, by König Lemma this tree is finite. The algorithm thus simply guesses a finite Spoiler’s strategy. This can be done effectively: for given β,β′,β″ and ζ it is decidable if \(\beta \stackrel {}{\Longrightarrow }\beta'' \stackrel {\zeta }{\longrightarrow } \beta'\), as the \(\stackrel {}{\Longrightarrow }\) relation is effectively semilinear [4]. □
For weak bisimilarity we obtain a result weaker than Theorem 6, as we are not able to prove that the coefficient c is computable:
Theorem 8
(Bounded Response Property of ≈)
For every normed process definition, there is \(c \in\mathbb{N}\) such that weak bisimilarity ≈, over the transition graph induced by the process definition, is a cweak bisimulation.
Theorem 8 follows, similarly as Theorem 6, from our results in Sects. 4–6. We note that Theorem 8 does not imply decidability of weak bisimilarity. In Sect. 3.1 we investigate in detail the possible impact on decidability.
3.1 Approximations
In order to discuss in detail the actual impact of Theorems 6 and 8 on decidability, we now define standard approximating hierarchies for branching and weak bisimilarity. For convenience we use a new symbol ≡ to stand for any of the two equivalences, ≃ or ≈. Thus, whenever we state a property of ≡, it should be understood as the property of both ≃ and ≈.
One may define a sequence of approximating relations ≡^{ m } as follows. Every pair of processes belongs to the first approximant ≡^{0}. Every consecutive approximant ≡^{ m+1} contains those pairs of processes that satisfy branching/weak bisimulation expansion wrt. ≡^{ m }. Thus the relation ≡^{ m } relates two processes iff Duplicator has a strategy to survive at least m rounds of the Bisimulation Game.
Let \(c \in\mathbb{N}\) be an arbitrary natural number. In an entirely analogous way one defines approximants \(\equiv _{c}^{m}\), describing m steps in the variant of Bisimulation Game where Duplicator is obliged to play under the size restrictions (1) or (2), respectively. We have the following ωstabilization result applying both to branching and weak bisimilarity:
Proposition 9
(ωApproximation)
For every \(c \in\mathbb{N}\), \(\equiv _{c}\, = \cap_{m < \omega} \equiv _{c}^{m}\).
Proof
As usual, one proves that \(\cap_{m < \omega} \equiv _{c}^{m}\) is a cbranching/weak bisimulation, similarly as for strong bisimilarity for imagefinite systems. □
This may be surprising at first sight, in view of Theorems 6 and 8. Theorems 6 and 8 say jointly that ≡ coincides with ≡_{ c }, for some c:
Corollary 10
For every normed process definition there is \(c \in\mathbb{N}\) with ≡ = ≡_{ c }.
Thus for every process definition one obtains the following seeming contradiction. On one hand, ≡ is not equal to the limit ∩_{ m<ω }≡^{ m }. On the other hand, By Proposition 9 and Corollary 10, ≡ is equal to \(\cap_{m < \omega} \equiv _{c}^{m}\), for some \(c \in \mathbb{N}\). The following example serves as an illustration:
Example 11

they have the same number of occurrences of P;

Q occurs in both, or in none of them; and

in the latter case, the number of occurrences of A is the same.
In consequence \(P \not \approx _{1} PQ\). In agreement with Proposition 9 it is not true that \(P \approx _{1}^{n} PQ\) for all n, for instance \(P \not \approx _{1}^{3} PQ\).
3.2 Decidability of Weak Bisimilarity?
Observe that ≡_{ c } need not be an equivalence relation in general. However, due to Theorems 6 and 8 we know that for every process definition, for sufficiently large c the relation ≡_{ c } is an equivalence indeed. We claim the following:
Theorem 12
For every normed process definition, if ≡_{ c } is an equivalence then it is decidable.
The proof is entirely analogous to the proof of Theorem 7. The equivalence assumption is necessary for the correctness of the positive semiprocedure: the assumption guarantees existence of a semilinear bisimulation. Furthermore, there is an algorithm whose input is a process definition, a constant c and a pair of processes, and it answers whether the pair is related by ≡_{ c }.
3.3 Proof Strategy
The computable estimation of the coefficient c is derived in Sect. 5, in the case of branching bisimilarity. Finally, in Sect. 6 we show how the bounds (3) are used to prove Theorem 6. Section 6 contains also the proof of Theorem 8.
As observed e.g. in [16], a crucial obstacle in proving decidability is so called generating transitions of the form \(X \stackrel {}{\longrightarrow }XY\), as they may be used by Duplicator to reach silently XY ^{ m } for arbitrarily large m. A great part of our proofs is an analysis of combinatorial complexity of generating transitions and, roughly speaking, elimination of ‘unnecessary’ generations.
Weak Bisimilarity
Branching bisimilarity is more discriminating than weak bisimilarity. The whole development of Sect. 4 is still valid if weak bisimilarity is considered in place of branching bisimilarity. Furthermore, except one single case, the entire proof of estimation of the coefficient in Sect. 5 remains valid too. Interestingly, this single case is obvious under assumptions of [16], thus our proof remains valid for weak bisimilarity over the subclass studied there. We conjecture that the single missing case is provable for weak bisimilarity and thus Theorem 6 holds for weak bisimilarity just as well. This would imply decidability.
4 Normal Form by Squeezing
The results of this section are quite general and apply equally well to branching bisimilarity ≃ and to weak bisimularity ≈. (This will not be however the case in later sections.) We will thus continue using the symbol ≡ to stand for either ≃ or ≈ in this section. Actually the only place where we need to distinguish between weak and branching bisimilarity is Lemma 31 that speaks of matching Duplicator’s responses. Furthermore, we claim that all the results of Sect. 4 apply equally well to intermediate notions of bisimulation, lying between branching and weak bisimilarity, as introduced in [19].
4.1 Normal Forms
In this section we develop a framework useful for the proofs of Theorems 6 and Theorem 8, to be given in the following sections.
By the bisimulation class of a process α we mean the set of all processes β with β≡α. Note that this definition applies both bisimilarities, as ≡ instantiates either with ≃ or ≈. An important role in our development will be played by normal forms of processes that identify the bisimulation classes uniquely. The normal forms are defined using the linear wellfounded order ⪯ on processes, as defined in Definition 22 in Sect. 4.3 below. We prefer to postpone the definition of ⪯, in order to avoid inessential technical details at this early stage.
Definition 13
(Normal Form)
For any process α let nf(α) denote the smallest process with respect to ⪯, bisimilar to α.
Clearly α≡nf(α) and thus we conclude that bisimulation equivalence is characterized by syntactic equality of normal forms:
Lemma 14
α≡β if and only if nf(α)=nf(β).
4.2 Decreasing Transitions
We will often use the following easy observation, that actually holds for unnormed processes as well.
Lemma 15
If \(\alpha \stackrel {}{\Longrightarrow }\beta \stackrel {}{\Longrightarrow }\alpha'\) and α≡α′ then β≡α.
Proof
Immediate using Definition 3. If Spoiler plays from α, Duplicator uses its response from α′, precomposed with \(\beta \stackrel {}{\Longrightarrow }\alpha'\). On the other hand, if Spoiler plays from β, Duplicator moves \(\alpha \stackrel {}{\Longrightarrow }\beta\) and then copies the Spoiler’s transition. □
A transition \(\alpha \stackrel {\zeta }{\longrightarrow }\beta\) is norm preserving if α=β and norm reducing if α=β+1. In the sequel we will pay special attention to norm preserving εtransitions. Therefore we write \(\alpha \stackrel {}{\longrightarrow }_{0} \beta\), respectively \(\alpha \stackrel {}{\Longrightarrow }_{0}\beta\), to emphasize that the transitions are norm preserving. Note that in the definition of branching bisimulation (Definition 3) one could equivalently use \(\stackrel {}{\Longrightarrow }_{0}\) instead of \(\stackrel {}{\Longrightarrow }\), as all the transitions performed in a Duplicator’s response, except possibly the last one, are necessarily normpreserving. Similarly, the \(\stackrel {}{\Longrightarrow }\) transitions in Lemma 15 are actually \(\stackrel {}{\Longrightarrow }_{0}\) transitions, i.e. whenever \(\alpha \stackrel {}{\Longrightarrow }\alpha'\) and α≡α′ then the transitions from α to α′ are necessarily normpreserving.
Definition 16
We call the transition \(\alpha \stackrel {\zeta }{\longrightarrow } \beta\) decreasing if either ζ∈A and the transition is normreducing; or ζ=ε and the transition is norm preserving.
Note that every variable has a sequence of decreasing transitions leading to the empty process ε.
Lemma 17
(Decreasing Response)
Whenever α≡β and \(\alpha \stackrel {\zeta }{\longrightarrow } \alpha'\) is decreasing then any Duplicator’s matching sequence of transitions from β contains exclusively decreasing transitions.
Proof
Follows from the following simple observations: ≡ is norm preserving; for a≠ε, the transition relation \(\stackrel {a}{\longrightarrow }\) may decrease the norm by at most one; the transition relation \(\stackrel {\varepsilon }{\longrightarrow }\) never decreases the norm. □
Due to Lemma 15, instantiated to single variables, we may assume wlog. that there are no two distinct variables X,Y with \(X\stackrel {}{\Longrightarrow }_{0} Y\stackrel {}{\Longrightarrow }_{0} X\). Indeed, since reachability via the \(\stackrel {}{\Longrightarrow }_{0}\) transitions is decidable [4], in a preprocessing one may eliminate such pairs X,Y. Relying on this assumption, we may define a partial order induced by decreasing transitions.
Definition 18
For variables X,Y, let X>_{decr} Y if there is a sequence of decreasing transitions leading from X to Y. Let > denote an arbitrary total order extending >_{decr}.
Note that we do not assume that X>Y⇒norm(X)≥norm(Y). Indeed, the order > may be chosen in arbitrary way. In Sect. 4 it is only relevant to have some fixed linear order on variables. In the next sections we will alternate between different orders, but only those extending >_{decr}.
Directly from the definition of > we deduce:
Lemma 19
(Decreasing Transition)
If a decreasing transition \(X_{1}^{a_{1}}\cdot \ldots \cdot X_{n}^{a_{n}} \stackrel {\zeta }{\longrightarrow } X_{1}^{b_{1}}\cdot \ldots \cdot X_{n}^{b_{n}}\) is performed by X _{ k }, say, then b _{1}=a _{1},…,b _{ k−1}=a _{ k−1}.
Consider a norm preserving silent transition rule \(X \stackrel {}{\longrightarrow }_{0} \delta\). If X appears in δ, i.e. \(\delta= X \bar{\delta}\), we call the transition rule generating. We use the name generating also for a transition \(\alpha \stackrel {}{\longrightarrow }_{0} \beta\) induced by a generating transition rule. Note that generating transitions are decreasing.
Lemma 20
(Decreasing Transition cont.)
If a decreasing transition, as in Lemma 19, is not generating then b _{ k }=a _{ k }−1.
Following [16], we say that X generates Y if \(X \stackrel {}{\Longrightarrow }_{0} XY\). Thus if \(X \stackrel {}{\Longrightarrow }_{0} X\bar{\delta}\) then X generates every variable that appears in \(\bar{\delta}\). In particular, X may generate itself. Note that each generated variable is of norm 0. More generally, we say that α generates β if \(\alpha \stackrel {}{\Longrightarrow }_{0} \alpha \beta \). This is the case precisely iff every variable occurring in β is generated by some variable occurring in α.
We write α⊑β if there is some γ such that αγ=β (⊑ is thus the multiset inclusion of processes). As an easy implication of Lemma 15 we obtain:
Lemma 21
If α generates β then \(\alpha \equiv \alpha \bar{\beta}\) for any \(\bar{\beta} \sqsubseteq \beta\).
Lemma 21 will be useful in the sequel, as a tool for eliminating unnecessary transitions and thus decreasing the size of a resulting process.
4.3 Unambiguous Processes
Once we have a fixed ordering on variables, a process \(X_{1}^{a_{1}}\cdot \dots \cdot X_{n}^{a_{n}}\) may be equivalently presented as a sequence of exponents \((a_{1}, \dots, a_{n}) \in\mathbb{N}^{n}\). In this perspective, ⊑ is the pointwise order. The sequence presentations induce additionally the lexicographic order on processes, denoted ⪯.
Definition 22
The same may be written briefly using concatenation: α≺β if \(\alpha= \gamma \cdot X^{a}_{k}\cdot \alpha'\), \(\beta= \gamma \cdot X^{b}_{k}\cdot \beta'\), and a<b.
For instance, the decreasing nongenerating transitions \(\alpha \stackrel {\zeta }{\longrightarrow } \beta\) always go strictly down the lexicographical order, i.e. α≻β.
We will exploit the fact that the order ⪯ is total, and thus each bisimulation class exhibits the least element. The least process in the bisimulation class of α will serve as the normal form of α, denoted nf(α) (cf. Definition 13 in Sect. 4.1).
The sequence presentation allows us to speak naturally of prefixes of a process: the kprefix of \(X_{1}^{a_{1}}\cdot \dots \cdot X_{n}^{a_{n}}\) is the process \(X_{1}^{a_{1}}\cdot \dots \cdot X_{k}^{a_{k}}\), for k=0…n.
We now introduce one of the core notions used in the proof: unambiguous processes and their greatest extensions.
Definition 23
(Unambiguous Processes)
Note that being kunambiguous is a property of the kprefix: a process is kunambiguous iff its kprefix is so.
Observe that an unambiguous process is necessarily the least one wrt. ⪯ in its bisimulation class, as the definition disallows the equivalence (4) to hold for a _{ i }=b>c. On the other hand, it is not immediately clear whether the opposite implication holds, i.e. whether every bisimulation class contains some unambiguous process. In the sequel we will show that this is actually the case.
Example 24
Note that a prefix of a kunambiguous process is kunambiguous as well. Moreover, kunambiguous processes are downward closed wrt. ⊑: whenever α⊑β and β is kunambiguous, then α is kunambiguous as well.
Directly by Definition 23, if \(\gamma= X_{1}^{a_{1}}\cdot \ldots \cdot X_{k1}^{a_{k1}}\) is (k−1)unambiguous then it is automatically kunambiguous (in fact junambiguous for any j≥k). This corresponds to a _{ k }=0; we will be especially interested in the greatest value of a _{ k } possible, as formalized in the definition below.
Definition 25
(The Greatest Extension)
The greatest kextension of a (k−1)unambiguous process γ∈{X _{1}…X _{ k−1}}^{⊗} is that process among kunambiguous processes \(\gamma \cdot X_{k}^{a}\) that maximizes a.
Clearly the greatest extension does not need exist in general, as illustrated below.
Example 26
Consider the processes from Example 24. The process X _{1} is the greatest 1extension of the empty process as \(X_{1}^{2}\) is not 1unambiguous. X _{1} is also its own greatest 2extension. Furthermore, X _{1} does not have the greatest 3extension. Indeed, \(X_{1} X_{3}^{a}\) is not bisimilar to \(X_{1} X_{3}^{b}\), for a≠b, therefore \(X_{1} X_{3}^{a}\) is 3unambiguous for any a.
Definition 27
(Unambiguous Prefix)
By an unambiguous prefix of a process \(X_{1}^{a_{1}}\cdot \dots \cdot X_{n}^{a_{n}}\) we mean any kprefix \(X_{1}^{a_{1}}\cdot \dots \cdot X_{k}^{a_{k}}\) that is kunambiguous, for k=0…n. The maximal unambiguous prefix is the one that maximizes k.
Example 28
For the process definition from Example 24, the maximal unambiguous prefix of \(X_{1}\cdot X_{2}^{2}\) is X _{1}, and the maximal unambiguous prefix of \(X_{1}^{2}\cdot X_{2}\) is the empty process.
4.4 Squeezes
The following lemma is fundamental for our subsequent development. The rough idea is as follows. For an unambiguous α consider a sequence of decreasing transitions from α⋅β, for an arbitrary β. The resulting process is necessarily of the form α′β′, where α′ is obtained from α by a subsequence of transitions, and β′ is obtained from β by the remaining subsequence of transitions. The lemma says that if α′β′ is still bisimilar to αγ for some γ then up to bisimilarity, the same process is reached by the latter subsequence of transitions.
Lemma 29
Proof

α and β′ are separated (in other words, β′ contains only variables which are smaller than all variables from α with respect to >), and

α′≡α.
Let’s prove the first item first. Observe that each variable X occurring in the process β′ is an effect of a sequence of decreasing transitions originating from some variable Y occurring in the process β, hence Y≥X. Thus α and β′ are separated since α and β are.
The following simple example illustrates the reasoning in the proof above.
Example 30
Then Lemma 29 says that P⋅A ^{8}≡P. Indeed, this must be true as P generates A. This simple example has an advantage of being general enough: in Lemma 29, all variable occurrences in α′ that do not belong to α are actually generated by α.
Lemma 31
Proof
Lemma 32
Proof
The lemma to follow applies Lemma 32 to a special kind of unambiguous processes, namely to the greatest kextensions \(\gamma \cdot X_{k}^{a}\) of unambiguous processes γ.
Lemma 33
(Squeezing Out)
Proof
By δ,δ′, etc. we denote below processes from {X _{ k+1}…X _{ n }}^{⊗}.
Definition 34
If a (k−1)unambiguous process γ∈{X _{1}…X _{ k−1}}^{⊗} has the greatest kextension, say \(\gamma \cdot X_{k}^{a}\), then any δ∈{X _{ k+1}…X _{ n }}^{⊗} satisfying (12) is called a γsqueeze of X _{ k }.
By the very definition, X _{ k } has a γsqueeze only if γ has the greatest kextension. Lemma 33 shows the opposite: if γ has the greatest kextension then X _{ k } has a γsqueeze, that may depend in general on γ and k. The squeeze is however not uniquely determined and in fact X _{ k } may admit many different γsqueezes. In the sequel assume that for each (k−1)unambiguous γ∈{X _{1}…X _{ k−1}}^{⊗} and X _{ k }, some γsqueeze of X _{ k } is chosen; this squeeze will be denoted by δ _{ k,γ }.
Example 35
Consider again the process definition from Example 1, with the order P > Q > A. Observe that Q ^{2}≡Q which means that Q ^{2} is not an unambiguous process. If we fix γ=P ^{3}, say, then ne possible γsqueeze of Q is the empty one: P ^{3}⋅Q ^{2}≡P ^{3}⋅Q⋅ε, but there are many others, for instance P ^{3}⋅Q ^{2}≡P ^{3}⋅Q⋅A ^{ n }, for any n≥0. The same squeezes are fine for any other γ∈{P}^{⊗}.
Definition 36
(Squeezing Step)
By Lemma 33 and by substitutivity of ≡ we conclude that α≡squeeze(α) and if α is not unambiguous then squeeze(α)≺α.
4.5 Bounds on Normal Forms
For an arbitrary partial order \(\unlhd \) on processes, a process α is called \(\unlhd \)minimal if it is minimal with respect to \(\unlhd \) in its bisimulation class (in other words, there is no β◁α with β≡α). In the sequel in this section we will often refer to ⊑minimal processes, and to ⪯minimal ones. Clearly in every bisimulation class there is exactly one ⪯minimal process, the normal form of processes from that class. In the following sections we will use the notion of \(\unlhd \)minimality also for other orders than ⪯ and ⊑.
For a process α, by a \(\unlhd \)minimization of α we mean any \(\unlhd \)minimal process β with β≡α and \(\beta \unlhd \alpha\). In particular, if α is \(\unlhd \)minimal then it is its own minimization, in fact the unique one. Clearly, if the order is wellfounded then every process has some \(\unlhd \)minimization. All orders considered in this paper are refinements of the lexicographical order ⪯ and are thus wellfounded.
Due to Lemma 33 we learn that every bisimulation class contains an unambiguous processes:
Lemma 37
A process α is unambiguous if and only if it is ⪯minimal.
Proof
One implication does not refer to squeezes. Suppose α is not the least process in its bisimulation class. That is, for some i≤n we have \(\alpha = \gamma \cdot X_{i}^{a}\cdot \bar{\alpha}\) and there is some \(\beta = \gamma \cdot X_{i}^{b}\cdot \bar{\beta} \equiv \alpha\) with b<a. Thus, according to the definition, α is not unambiguous.
The other implication is easily provable building on the development of this section. If α is not unambiguous then it is not the least one in its bisimulation class wrt. ⪯ as α≡squeeze(α) and squeeze(α)≺α. □
It follows that the squeezing step, applied in a systematic manner sufficiently many times on a process α, leads to the normal form process nf(α).
Lemma 38
(Normal Form via Squeezing)
Let α be an arbitrary process. Then consecutive applications of the squeezing step eventually stabilize at nf(α), i.e. for some m≥0, squeeze^{ m }(α)=nf(α).
Finally we formulate lower and upper bounds on the size of nf(α), with respect to the size of α, that will be crucial for the proof of Theorem 6. The first one, stated in Lemma 40, applies uniquely to ⊑minimal processes. The following lemma is the technical preparation:
Lemma 39
If α is ⊑minimal then \(\text {size}(\alpha) \leq \text {size}(\bar{\alpha})\), for any ⊑minimization \(\bar{\alpha}\) of squeeze(α).
Proof
Lemma 40
(Lower Bound)
If α is ⊑minimal then size(nf(α))≥size(α).
Proof

the minimizationsqueezing step: replace α by any ⊑minimization of squeeze(α).
Contrarily to Lemma 40, the upper bound holds for all processes.
Lemma 41
(Upper Bound)
There is a constant c, depending only on the process definition, such that size(nf(α))≤c⋅size(α) for any process α.
Proof
Concerning the upper bound, in the following section we demonstrate a sharper result, with the constant c estimated effectively. However, the estimation will be only shown for branching bisimilarity.
5 Effective Bound on Normal Form
In this section we only consider branching bisimilarity ≃. In particular, the notion of normal form is understood with respect to ≃. Fix an arbitrary process definition and denote its size by d.
Contrarily to the previous section, where the linear order > on variables was fixed, in this section we consider all linear orders on variables that extend >_{decr} (cf. Definition 18); such orders we call briefly admissible. Note however that the whole development of Sect. 4 strongly depends on the choice of >. In particular, the normal form of a process may change if one changes the order. Thus in this section we will have to be careful enough to explicitly specify the order we use, whenever we apply any notation or result of Sect. 4.
The following lemma is the main result of this section. The lemma will be used in Sect. 6 for the proof of Theorem 6.
Lemma 42
(Upper Bound)
For every admissible order >, and for every process α, size(nf(α))≤d ^{ n−1}⋅size(α).
Lemma 43
For every k∈{1…n}, for every admissible order > and (k−1)unambiguous γ∈{X _{1}…X _{ k−1}}^{⊗} that has the greatest kextension, the variable X _{ k } has a γsqueeze δ with dsize(δ)≤dsize(X _{ k }).
All the rest of this section is devoted to the proof of Lemma 43.
5.1 Proof of Lemma 43
The proof is by induction on n−k. The induction basis is for k=n. Whatever an admissible order is chosen, if k=n then it trivially holds that dsize(δ)≤dsize(X _{ k }), as the only possible γsqueeze δ of X _{ n } is the empty process, whose weighted size is 0.
For the induction step, fix some k and an admissible order >, assuming that the lemma holds for all greater values of k, for all admissible orders.

a>0,

a=0 and X _{ k } has a γsqueeze δ such that \(X_{k} \stackrel {}{\Longrightarrow }_{0} \delta\),

a=0 and X _{ k } has no γsqueeze δ such that \(X_{k} \stackrel {}{\Longrightarrow }_{0} \delta\).
Case 1: a>0
In this case we will not refer to the induction assumption at all.
The idea behind that proof is based on the fact that a>0, and thus, roughly speaking, X _{ k } does not vanish during squeezing. From this we deduce that variables generated by X _{ k } do not appear in some squeeze δ of X _{ k }. Bounding the number of occurrences of other variables in δ is an easy conclusion from Claim 44, formulated below.
Claim 44
The variable X _{ k } has a γsqueeze η such that \(\gamma \cdot X_{k}^{a+1} \stackrel {}{\Longrightarrow }_{0} \gamma \cdot X_{k}^{a}\cdot \eta\).
Proof
Remark 45
Actually it follows easily that \(X_{k} \stackrel {}{\Longrightarrow }_{0} \eta\). We will however not need this property in the remaining part of the proof.
Case 2.1: a=0 and X _{ k } Has a γSqueeze δ such that \(X_{k} \stackrel {}{\Longrightarrow }_{0} \delta\)
This is the only case that we are not able to adapt to weak bisimilarity.
 1.
>′ is kconsistent with >, and
 2.
all variables generated by X _{ k } are smaller with respect to >′ than all variables not generated by X _{ k }.
To make the notation more readable, we will constantly use symbols α,α′, etc. for processes containing exclusively variables smaller than X _{ k } that are not generated by X _{ k }, and symbols β,β′, etc. for those containing exclusively variables generated by X _{ k }.
Lemma 46
\(\text {nf}(\gamma \cdot \omega)=\gamma \cdot \alpha \cdot \bar{\beta}\) for some process \(\bar{\beta}\).
Proof
The process ω, being the righthand side of a transition rule, is of size smaller than d. Hence ωY is a γsqueeze of size at most d. This completes the proof of Case 2.1.
Case 2.2: a=0 and X _{ k } Has no γSqueeze δ such that \(X_{k} \stackrel {}{\Longrightarrow }_{0} \delta\)
We start with a lemma that only holds under assumptions of Case 2.2:
Lemma 47
No ⊑minimal γsqueeze of X _{ k } contains a variable generated by X _{ k }.
Proof
 1.
>′ is kconsistent with >,
 2.
all variables generated by X _{ k } are smaller with respect to >′ than all variables not generated by X _{ k }, and
 3.
the orders > and >′ coincide on variables not generated by X _{ k }.
From now on we work with the order >′ instead of >, and thus indexing of variables, squeezes, normal forms, etc. are implicitly understood to be defined with respect to that order.
Let nf(γ⋅X _{ k })=γ⋅α. By Lemma 47 we know that no variable appearing in α is generated by X _{ k }. We are aiming at showing that the dsize(α)≤dsize(X _{ k }).
Case 2.2 is the only one which requires referring to the induction assumption. We will invoke the induction assumption for variables smaller than X _{ k }, and the same admissible order >′ on variables. To this aim, we will start by considering Bisimulation Game starting with a decreasing nongenerating Spoiler’s move, as outlined below.
Remark 48
Lemma 43 is formulated for ≃ but the major part of the proof either works for weak bisimilarity directly, or may be adapted. The only case that we can not adapt to weak bisimilarity is Case 2.1. Importantly, under the restriction of [16] the proof of this subcase is straightforward.
The restriction on a process definition assumed in [16] is the following: whenever a variable X generates (some variable) then every silent transition rule of X is generating, i.e. of the form \(X \stackrel {}{\longrightarrow }X\cdot \omega\). In short words, generators can not vanish silently.
We claim that our proof, after slight adaptations in Cases 1 and 2.2, shows decidability of weak bisimilarity in the subclass of [16].
6 Proof of the Bounded Response Property
This section contains finally the proofs of two main results announced in Sect. 3, namely Theorem 6 and Theorem 8. The two theorems state the bounded response property for branching and weak bisimilarity, respectively; moreover the former one claims a response of an effectively bounded size.
6.1 Proof of Theorem 8
6.2 Proof of Theorem 6
From now on we focus on branching bisimilarity only. Compared to weak bisimilarity, the case of branching bisimilarity is slightly more subtle. As previously, consider a fixed process definition and a fixed admissible order on variables.
Lemma 49
If α is \(\unlhd \)minimal and \(\alpha \stackrel {}{\Longrightarrow }_{0} \beta \simeq \alpha\) then α⊑β.
Proof
From now on, the remaining part of Sect. 6 is devoted to proving Theorem 6, using Lemmas 40 and 42 together with Lemma 49.
We can not simply extend this response analogously as for weak bisimilarity, and we have to estimate the size of the process β _{2} resulting from the last transition. The basic idea of the proof is essentially to eliminate some unnecessary generation done by the transitions \(\beta \stackrel {}{\Longrightarrow }_{0} \beta_{1}\), without affecting executability of the transition \(\beta_{1} \stackrel {\zeta }{\longrightarrow } \beta_{2}\).
Thus from now on we consider a pair \(\alpha \simeq \bar{\beta}\), with \(\bar{\beta}\) a \(\unlhd \)minimal process, instead of arbitrary α≃β, together with a matching Duplicator’s response (39). Note that by Lemma 49 we know that \(\bar{\beta} \sqsubseteq \beta_{1}\).
As the third and the last step of the proof, we claim that β _{1} and β _{2} may be replaced by processes of size bounded, roughly, by the sum of sizes of \(\bar{\beta}\) and \(\bar{\beta}_{2}\).
Claim 50
We will use now an intuitive coloring argument. Let us color uniquely every variable occurrence in β _{1} and let every transition preserve the color of the lefthand side variable of a transition rule that is used. Obviously at most \(\text {size}(\bar{\beta}_{2})\) of these colors will still be present in \(\bar{\beta}_{2}\), name them surviving colors. Suppose the \(\beta_{1} \stackrel {\zeta }{\longrightarrow } \beta_{2}\) transition be induced by a transition rule \(X \stackrel {\zeta }{\longrightarrow } \delta\) and color the occurrence of X in β _{1} involved in this transition, say, brown.
By Lemma 15 we have \(\beta'_{1} \simeq \beta_{1}\) and \(\beta'_{2} \simeq \beta_{2}\). Clearly there is a sequence \(\beta_{1} \stackrel {}{\Longrightarrow }_{0} \beta'_{1}\), that simply cancels superfluous variable occurrences, hence the condition (42) is fulfilled.
Finally we obtain the size estimation \(\text {size}(\beta'_{1}) \leq \text {size}(\bar{\beta}) + \text {size}(\bar{\beta}_{2}) + 1\) as in \(\beta'_{1}\) there can be at most \(\text {size}(\bar{\beta}_{2}) + 1\) surviving and brown colored variables occurrences, except for those that come from \(\bar{\beta}\). This easily implies the required size estimation for \(\text {size}(\beta'_{2})\). Thus the required condition (43) holds.
Footnotes
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