Branching Bisimilarity Research Articles

Dormancy-aware timed branching bisimilarity with an application to communication protocol analysis

A variant of the standard notion of branching bisimilarity for processes with discrete relative timing is proposed which is coarser than the standard notion. Using a version of ACP (Algebra of Communicating Processes) with abstraction for processes with discrete relative timing, it is shown that the proposed variant allows of both the functional correctness and the performance properties of the PAR (Positive Acknowledgement with Retransmission) protocol to be analyzed. In the version of ACP concerned, the difference between the standard notion of branching bisimilarity and its proposed variant is characterized by a single axiom schema.

Branching bisimulation semantics for quantum processes

The qCCS model proposed by Feng et al. provides a powerful framework to describe and reason about quantum communication systems that could be entangled with the environment. However, they only studied weak bisimulation semantics. In this paper we propose a new branching bisimilarity for qCCS and show that it is a congruence. The new bisimilarity is based on the concept of ϵ-tree and preserves the branching structure of concurrent processes where both quantum and classical components are allowed. Furthermore, we present a polynomial time equivalence checking algorithm for the ground version of our branching bisimilarity.

Rooted Divergence-Preserving Branching Bisimilarity is a Congruence for Guarded CCS

Branching bisimilarity is a well-known equivalence relation for labelled transition systems. Based on this equivalence relation, with an additional simple rootedness condition, a congruence relation for calculus of communication system (CCS) processes can be obtained. However, neither branching bisimilarity nor the corresponding congruence relation preserves divergence, and it is still a question whether, based on a divergence-preserving variant of branching bisimilarity, a divergence-preserving congruence relation for CCS processes can be obtained by introducing the same simple rootedness condition. In this article, we present a partial solution by showing that rooted divergence-preserving branching bisimilarity is preserved under the usual CCS operators, including prefixing, summation, parallel composition, relabelling, restriction, and (weakly) guarded recursion.

Probabilistic weak bisimulation and axiomatization for probabilistic models

Recently a branching bisimilarity congruence has been established for randomized process models in a model independent approach. In this paper, we propose a probabilistic weak bisimilarity for randomized CCS based on the notion of weak ϵ-tree and show that it is a congruence. We also provide a complete axiomatization for the finite fragment of the language. Both the definition of weak bisimilarity and the proof of congruence respect model independency and thus can be easily generalized to a randomized extension of any interactive model.

Linear parallel algorithms to compute strong and branching bisimilarity

We present the first parallel algorithms that decide strong and branching bisimilarity in linear time. More precisely, if a transition system has n states, m transitions and vert Act vert action labels, we introduce an algorithm that decides strong bisimilarity in mathcal {O}(n+vert Act vert ) time on max (n,m) processors and an algorithm that decides branching bisimilarity in mathcal {O}(n+vert Act vert ) time using up to max (n^2,m,vert Act vert n) processors.

Divergence-Preserving Branching Bisimilarity

This note considers the notion of divergence-preserving branching bisimilarity. It briefly surveys results pertaining to the notion that have been obtained in the past one-and-a-half decade, discusses its role in the study of expressiveness of process calculi, and concludes with some suggestions for future work.

Team equivalences for finite-state machines with silent moves

Finite-state machines, a simple class of finite Petri nets, were equipped in [16] with an efficiently decidable, truly-concurrent, bisimulation-based, behavioral equivalence, called team equivalence, which conservatively extends classic bisimulation equivalence on labeled transition systems and which is checked in a distributed manner. This paper addresses the problem of defining variants of this equivalence which are insensitive to silent moves. We define (rooted) weak team equivalence and (rooted) branching team equivalence as natural transposition to finite-state machines of Milner's weak bisimilarity [25] and van Glabbeek and Weijland's branching bisimilarity [12] on labeled transition systems. The process algebra CFM [15] is expressive enough to represent all and only the finite-state machines, up to net isomorphism. Here we first prove that the rooted versions of these equivalences are congruences for the operators of CFM, then we show some algebraic properties, and, finally, we provide finite, sound and complete, axiomatizations for them.

Compositional model checking with divergence preserving branching bisimilarity is lively

• Divergence-preserving branching Bisim, is a congruence for LTS network composition. • We present a method to safely decompose an LTS network in components. • We discuss parallel composition with alphabet synchronisation. • We prove that alphabet synchronisation is not required for DPBB to be a congruence. • We demonstrate the effectiveness of compositionally constructing state spaces. Compositional model checking approaches attempt to limit state space explosion by iteratively combining the behaviour of the components in a concurrent system and reducing the result modulo an appropriate equivalence relation. In this article, we consider Labelled Transition Systems (LTSs), in which transitions are labelled by actions, to describe component behaviour, and LTS networks to combine the behaviour of all components in a system. For an equivalence relation to be useful for the compositional model checking of LTS networks, it should be a congruence for the parallel composition operator that is used to combine component behaviour. Such an operator may define synchronisations between the actions of component transitions. An equivalence relation preserving both safety and liveness properties is divergence-preserving branching bisimilarity (DPBB). It has long been generally assumed that DPBB is a congruence for parallel composition. Fokkink, Van Glabbeek and Luttik recently proposed a congruence format that implies that this is the case. In parallel, we were the first to prove, by means of the Coq proof assistant, that DPBB is a congruence for the parallel composition of two LTS networks with synchronisation on transition labels. In the current article, we also consider an instance of our parallel composition operator that is both associative and commutative, which are two essential properties for the compositional construction of state spaces. Furthermore, we show that DPBB is a congruence for LTS networks in which many LTSs are composed in parallel at once with support for multi-party synchronisation. Additionally, we discuss how to safely decompose an existing LTS network into components such that their recomposition is equivalent to the original LTS network. Finally, to demonstrate the effectiveness of compositional model checking with intermediate DPBB reductions, we discuss the results we obtained after having conducted a number of experiments.

A complete axiomatization of weighted branching bisimulation

We propose an axiomatization for weighted branching bisimulation over a weighted process algebra with positive rational weights including zero and show that this axiomatization is both sound and complete. Our proof of soundness and completeness are inspired by similar results by Milner for strong and weak bisimulation and by van Glabbeek for branching bisimulation. We also show that the claim that weighted branching bisimilarity is an equivalence relation indeed holds true. As auxiliary results, we give two alternative characterizations of weighted branching bisimulation, one in terms of weighted stuttering transitions and another in terms of a relative branching base which can be seen as a linear basis from which we can construct all weighted stuttering transitions.

Raiders of the lost equivalence: Probabilistic branching bisimilarity

Probabilistic branching bisimilarity allows us to compare process behaviour with respect to their branching structure and probabilistic features, while abstracting away from those computation steps that can be marked as irrelevant to the analysis at hand. To the best of our knowledge, in the context of nondeterministic probabilistic processes, no proof of probabilistic branching bisimilarity being an equivalence has been provided so far. Since, as happened in the fully nondeterministic case, some researchers are using such a result by taking it for granted, we decided to dedicate this note to a formal proof of it. More precisely, we extend and adapt the proof strategy used by Basten in the fully nondeterministic case. Thus, we introduce the probabilistic analogue to the notion of semi-branching bisimilarity and to van Glabbeek and Weijland's Stuttering Lemma to prove that probabilistic branching bisimilarity is indeed an equivalence relation.

Probabilistic divide & congruence: Branching bisimilarity

Since the seminal paper by Bloom, Fokkink and van Glabbeek, the Divide and Congruence technique allows for the derivation of compositional properties of nondeterministic processes from the SOS-based decomposition of their modal properties. In an earlier paper, we extended their technique to deal also with quantitative aspects of process behavior: we proved the (pre)congruence property for strong (bi)simulations on processes with nondeterminism and probability. In this paper we further extend our decomposition method to favor compositional reasoning with respect to probabilistic weak semantics. In detail, we consider probabilistic branching and rooted probabilistic branching bisimilarity, and we propose logical characterizations for them. These are strongly based on the modal operator 〈ε〉 which combines quantitative information and weak semantics by introducing a sort of probabilistic lookahead on process behavior. Our enhanced method will exploit distribution specifications, an SOS-like framework defining the probabilistic behavior of processes, to decompose this particular form of lookahead. We will show how we can apply the proposed decomposition method to derive congruence formats for the considered equivalences from their logical characterizations.

Distribution of Behaviour into Parallel Communicating Subsystems

The process of decomposing a complex system into simpler subsystems has been of interest to computer scientists over many decades, for instance, for the field of distributed computing. In this paper, motivated by the desire to distribute the process of active automata learning onto multiple subsystems, we study the equivalence between a system and the total behaviour of its decomposition which comprises subsystems with communication between them. We show synchronously- and asynchronously-communicating decompositions that maintain branching bisimilarity, and we prove that there is no decomposition operator that maintains divergence-preserving branching bisimilarity over all LTSs.

State-Based Encoding of Large Asynchronous Controllers

State encoding is one of the fundamental problems in the synthesis of asynchronous controllers. The requirement for a correct hazard-free implementation imposes severe constraints on the way encoding signals can be inserted in the specification of a controller. Even though some specification formalisms, such as burst-mode machines or signal transition graphs, enable the specification of behaviors at the event level, the state encoding methods that provide the best good-quality solutions work at the state level. This imposes a severe limitation on the size of the controllers that can be handled by these methods. This paper proposes a method to solve the encoding problem for large asynchronous controllers using state-based methods. It is based on an iterative process of projection and re-composition that reduces the size specification by hiding signals, partially solves the encoding problem at the state level, and re-composes the original specification using a synchronous product. The process iterates until all encoding conflicts have been solved. The method is proved to preserve the behavior of the specification (branching bisimilarity) and shown to be capable of providing good-quality solutions for controllers of more than 100 signals and 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sup> states.

Fair testing and stubborn sets

Partial order methods alleviate state explosion by considering only a subset of actions in each constructed state. The choice of the subset depends on the properties that the method promises to preserve. Many methods have been developed ranging from deadlock-preserving to CTL $$^*$$ -preserving and divergence-sensitive branching bisimilarity preserving. The less the method preserves, the smaller state spaces it constructs. Fair testing equivalence unifies deadlocks with livelocks that cannot be exited and ignores the other livelocks. It is the weakest congruence that preserves whether or not the system may enter a livelock that it cannot leave. We prove that a method that was designed for trace equivalence also preserves fair testing equivalence. We demonstrate its effectiveness on a protocol with a connection and data transfer phase. This is the first practical partial order method that deals with a practical fairness assumption.

A generic framework for checking semantic equivalences between pushdown automata and finite-state automata

For a given process equivalence, we say that a process g is fully equivalent to a process f of a transition system T if g is equivalent to f and every reachable state of g is equivalent to some state of T. We propose a generic method for deciding full equivalence between pushdown processes and finite-state processes applicable to every process equivalence satisfying certain abstract conditions. Then, we show that these conditions are satisfied by bisimulation-like equivalences (including weak and branching bisimilarity), weak simulation equivalence, and weak trace equivalence, which are the main conceptual representatives of the linear/branching time spectrum. The list of particular results obtained by applying our method includes items which are first of their kind, and the associated upper complexity bounds are essentially optimal.

Multi-core symbolic bisimulation minimisation

We introduce parallel symbolic algorithms for bisimulation minimisation, to combat the combinatorial state space explosion along three different paths. Bisimulation minimisation reduces a transition system to the smallest system with equivalent behaviour. We consider strong and branching bisimilarity for interactive Markov chains, which combine labelled transition systems and continuous-time Markov chains. Large state spaces can be represented concisely by symbolic techniques, based on binary decision diagrams. We present specialised BDD operations to compute the maximal bisimulation using signature-based partition refinement. We also study the symbolic representation of the quotient system and suggest an encoding based on representative states, rather than block numbers. Our implementation extends the parallel, shared memory, BDD library Sylvan, to obtain a significant speedup on multi-core machines. We propose the usage of partial signatures and of disjunctively partitioned transition relations, to increase the parallelisation opportunities. Also our new parallel data structure for block assignments increases scalability. We provide SigrefMC, a versatile tool that can be customised for bisimulation minimisation in various contexts. In particular, it supports models generated by the high-performance model checker LTSmin, providing access to specifications in multiple formalisms, including process algebra. The extensive experimental evaluation is based on various benchmarks from the literature. We demonstrate a speedup up to 95times for computing the maximal bisimulation on one processor. In addition, we find parallel speedups on a 48-core machine of another 17times for partition refinement and 24times for quotient computation. Our new encoding of the reduced state space leads to smaller BDD representations, with up to a 5162-fold reduction.

Executable Behaviour and the π-Calculus (extended abstract)

Reactive Turing machines extend classical Turing machines with a facility to model observable interactive behaviour. We call a behaviour executable if, and only if, it is behaviourally equivalent to the behaviour of a reactive Turing machine. In this paper, we study the relationship between executable behaviour and behaviour that can be specified in the pi-calculus. We establish that all executable behaviour can be specified in the pi-calculus up to divergence-preserving branching bisimilarity. The converse, however, is not true due to (intended) limitations of the model of reactive Turing machines. That is, the pi-calculus allows the specification of behaviour that is not executable up to divergence-preserving branching bisimilarity. Motivated by an intuitive understanding of executability, we then consider a restriction on the operational semantics of the pi-calculus that does associate with every pi-term executable behaviour, at least up to the version of branching bisimilarity that does not require the preservation of divergence.

A polynomial time algorithm for checking regularity of totally normed process algebra

A polynomial algorithm for the regularity problem of weak and branching bisimilarity on totally normed process algebra (PA) processes is given. Its time complexity is \(\mathcal{O}(n^3 + mn)\), where n is the number of transition rules and m is the maximal length of the rules. The algorithm works for totally normed basic process algebra (BPA) as well as basic parallel process (BPP).

Property-dependent reductions adequate with divergence-sensitive branching bisimilarity

When analyzing the behavior of finite-state concurrent systems by model checking, one way of fighting state space explosion is to reduce the model as much as possible whilst preserving the properties under verification. We consider the framework of action-based systems, whose behaviors can be represented by labeled transition systems (Ltss), and whose temporal properties of interest can be formulated in modal μ-calculus (Lμ). First, we determine, for any Lμ formula, the maximal set of actions that can be hidden in the Lts without changing the interpretation of the formula. Then, we define Lμdsbr, a fragment of Lμ which is adequate w.r.t. divergence-sensitive branching bisimilarity. This enables us to apply the maximal hiding and to reduce the Lts on-the-fly using divergence-sensitive τ-confluence during the verification of any Lμdsbr formula. The experiments that we performed on various examples of communication protocols and distributed systems show that this reduction approach can significantly improve the performance of on-the-fly verification.

Reactive Turing machines

We propose reactive Turing machines (RTMs), extending classical Turing machines with a process-theoretical notion of interaction, and use it to define a notion of executable transition system. We show that every computable transition system with a bounded branching degree is simulated modulo divergence-preserving branching bisimilarity by an RTM, and that every effective transition system is simulated modulo the variant of branching bisimilarity that does not require divergence preservation. We conclude from these results that the parallel composition of (communicating) RTMs can be simulated by a single RTM. We prove that there exist universal RTMs modulo branching bisimilarity, but these essentially employ divergence to be able to simulate an RTM of arbitrary branching degree. We also prove that modulo divergence-preserving branching bisimilarity there are RTMs that are universal up to their own branching degree. We establish a correspondence between executability and finite definability in a simple process calculus. Finally, we establish that RTMs are at least as expressive as persistent Turing machines.