Weak Bisimulation Equivalence Research Articles

Compositional equivalences based on open pNets

Establishing equivalences between programs is crucial both for verifying correctness of programs and for justifying optimisations and program transformations. There exist several equivalence relations for programs, and bisimulations are among the most versatile of these equivalences. Among bisimulations one distinguishes strong bisimulation that requires that each action of a program is simulated by a single action of the equivalent program, and weak bisimulation that allows some of the actions to be invisible, and thus not simulated.pNet is a generalisation of automata that model open systems. They feature variables and hierarchical composition. Open pNets are pNets with holes, i.e. placeholders that can be filled later by sub-systems. However, there is no standard tool for defining the semantics of an open system in this context. This article first defines open automata that are labelled transition systems with parameters and holes. Relying on open automata, it then defines bisimilarity relations for the comparison of systems specified as pNets. We first present a strong bisimilarity for open pNets called FH-bisimilarity. Next we offer an equivalence relation similar to the classical weak bisimulation equivalence, and study its properties. Among these properties we are interested in compositionality: if two systems are proven equivalent they will be indistinguishable by their context, and they will also be indistinguishable when their holes are filled with equivalent systems. We identify sufficient conditions to ensure compositionality of strong and weak bisimulation. The contributions of this article are illustrated using a transport protocol as running example.

Analyzing divergence in bisimulation semantics

Some bisimulation based abstract equivalence relations may equate divergent systems with non-divergent ones, examples including weak bisimulation equivalence and branching bisimulation equivalence. Thus extra efforts are needed to analyze divergence for the compared systems. In this paper we propose a new method for analyzing divergence in bisimulation semantics, which relies only on simple observations of individual transitions. We show that this method can verify several typical divergence preserving bisimulation equivalences including two well-known ones. As an application case study, we use the proposed method to verify the HSY collision stack to draw the conclusion that the stack implementation is correct in terms of linearizability with lock-free progress condition.

Branching-Time Model Checking Gap-Order Constraint Systems

We consider the model checking problem for Gap-order Constraint Systems (GCS) w.r.t. the branching-time temporal logic CTL, and in particular its fragments EG and EF. GCS are nondeterministic infinitely branching processes described by evolutions of integer-valued variables, subject to Presburger c onstraints of the form x − y ≥ k, where x and y are variables or constants and k ∈ ℕ is a non-negative constant. We show that EG model checking is undecidable for GCS, while EF is decidable. In particular, this implies the decidability of strong and weak bisimulation equivalence between GCS and finite-state systems.

On the tradeoff between compositionality and exactness in weak bisimilarity for integrated-time Markovian process calculi

Integrated-time Markovian process calculi rely on actions whose durations are quantified by exponentially distributed random variables. The Markovian bisimulation equivalences defined so far for these calculi treat exponentially timed internal actions like all the other actions, because each such action has a nonzero duration and hence can be observed if it is executed between a pair of exponentially timed noninternal actions. However, no difference may be noted, at stationary state, between a sequence of exponentially timed internal actions and a single exponentially timed internal action, if their expected durations and execution probabilities coincide, a fact exploited in Hillston's weak isomorphism. We show that Milner's approach can be adapted on the basis of this fact, so to derive a weak bisimulation equivalence for integrated-time Markovian process calculi, up to a tradeoff between compositionality and exactness inherent to the Markovian setting. The resulting weak Markovian bisimulation equivalence induces a pseudo-aggregation that is exact at stationary state for all the considered processes, but turns out to be a congruence only over sequential processes. To achieve compositionality over concurrent processes, we need to enhance the abstraction capability of the equivalence in the presence of interleaved computations. However, the corresponding pseudo-aggregation turns out to be exact at stationary state only for a subset of concurrent processes. In addition to this tradeoff, we present, for the first equivalence, a sound and complete axiomatization over sequential processes, which is instrumental to characterize pseudo-aggregations, and a polynomial-time equivalence-checking algorithm, which can be exploited for the compositional minimization of concurrent processes.

Computing strong and weak bisimulations for psi-calculi

We present a symbolic transition system and strong and weak bisimulation equivalences for psi-calculi, and show that they are fully abstract with respect to bisimulation congruences in the non-symbolic semantics. A procedure which computes the most general constraint under which two agents are bisimilar is developed and proved correct.A psi-calculus is an extension of the pi-calculus with nominal data types for data structures and for logical assertions representing facts about data. These can be transmitted between processes and their names can be statically scoped using the standard pi-calculus mechanism to allow for scope migrations. Psi-calculi can be more general than other proposed extensions of the pi-calculus such as the applied pi-calculus, the spi-calculus, the fusion calculus, or the concurrent constraint pi-calculus.Symbolic semantics are necessary for an efficient implementation of the calculus in automated tools exploring state spaces, and the full abstraction property means the symbolic semantics makes exactly the same distinctions as the original.

On cool congruence formats for weak bisimulations

In TCS 146, Bard Bloom presented rule formats for four main notions of bisimulation with silent moves. He proved that weak bisimulation equivalence is a congruence for any process algebra defined by WB cool rules, and established similar results for rooted weak bisimulation (Milner’s “observational congruence”), branching bisimulation and rooted branching bisimulation. This study reformulates Bloom’s results in a more accessible form and contributes analogues for (rooted) η -bisimulation and (rooted) delay bisimulation. Moreover, finite equational axiomatisations of rooted weak bisimulation equivalence are provided that are sound and complete for finite processes in any RWB cool process algebra. These require the introduction of auxiliary operators with lookahead, and an extension of Bloom’s formats for this type of operator with lookahead. Finally, a challenge is presented for which Bloom’s formats fall short and further improvement is called for.

On the complexity of checking semantic equivalences between pushdown processes and finite-state processes

Simulation preorder/equivalence and bisimulation equivalence are the most commonly used equivalences in concurrency theory. Their standard definitions are often called strong simulation/bisimulation, while weak simulation/bisimulation abstracts from internal τ -actions. We study the computational complexity of checking these strong and weak semantic preorders/equivalences between pushdown processes and finite-state processes. We present a complete picture of the computational complexity of these problems and also study fixed-parameter tractability in two important input parameters: x, the size of the finite control of the pushdown process, and y, the size of the finite-state process. All simulation problems are generally EXPTIME-complete and only become polynomial if both parameters x and y are fixed. Weak bisimulation equivalence is PSPACE-complete, but becomes polynomial if and only if parameter x is fixed. Strong bisimulation equivalence is PSPACE-complete, but becomes polynomial if either parameter x or y is fixed.

Behavioural equivalence for a graph rewriting model of concurrent programs

This paper presents a formal model of concurrent systems based on graph rewriting to represent scopes of communication channel names precisely. A bipartite directed acyclic graph represents a concurrent system consists of a number of processes and messages. Each process or message corresponds to a source node of the graph. Names of communication channel in the system are sink nodes. The edges of the graph represent the scopes of the names in the system. The operational semantics of the system is given as a labelled transition system. The model presented here makes it possible to represent local names that their scope is not nested. The author defines the weak bisimulation equivalence relation that two systems are equivalent in their behaviours. The author shows that the equivalence relation is a congruence relation wrt prefix, new-name, replication and composition.

Complete Axiomatization for Divergent-Sensitive Bisimulations in Basic Process Algebra with Prefix Iteration

We study the divergent-sensitive spectrum of weak bisimulation equivalences in the setting of process algebra. To represent the infinite behavior, we consider the prefix iteration extension of a fragment of Milner's CCS. The prefix iteration operator is a variant on the binary version of the Kleene star operator obtained by restricting the first argument to be an atomic action and allows us to capture the notion of recursion in a pure algebraic way. We investigate four typical divergent-sensitive weak bisimulation equivalences, namely divergent, stable, completed and divergent stable weak bisimulation equivalences from an axiomatic perspective. A lattice of distinguishing axioms is developed and thus pure equational axiomatizations for these congruences are obtained. A large part of the current paper is devoted to a considerable complicated proof for completeness. This work, to some extent, sheds light on distinct semantics of divergence.

Refining the Undecidability Border of Weak Bisimilarity

Weak bisimilarity is one of the most studied behavioural equivalences. This equivalence is undecidable for pushdown processes (PDA), process algebras (PA), and multiset automata (MSA, also known as parallel pushdown processes, PPDA). Its decidability is an open question for basic process algebras (BPA) and basic parallel processes (BPP). We move the undecidability border towards these classes by showing that the equivalence remains undecidable for weakly extended versions of BPA and BPP. In fact, we show that the weak bisimulation equivalence problem is undecidable even for normed subclasses of BPA and BPP extended with a finite constraint system.

Axiomatising divergence

When a process is capable of executing an unbounded number of non-observable actions it is said to be divergent . Different capabilities of an observer to identify this phenomena along the execution leads to different divergent sensitive semantics. This paper develops sound and complete axiomatisations for the divergence sensitive spectrum of weak bisimulation equivalence. The axiomatisations separates the axioms concerning recursion and those that capture the essence of diverging behaviour.

Comparative branching-time semantics for Markov chains

This paper presents various semantics in the branching-time spectrum of discrete-time and continuous-time Markov chains (DTMCs and CTMCs). Strong and weak bisimulation equivalence and simulation pre-orders are covered and are logically characterized in terms of the temporal logics Probabilistic Computation Tree Logic (PCTL) and Continuous Stochastic Logic (CSL). Apart from presenting various existing branching-time relations in a uniform manner, this paper presents the following new results: (i) strong simulation for CTMCs, (ii) weak simulation for CTMCs and DTMCs, (iii) logical characterizations thereof (including weak bisimulation for DTMCs), (iv) a relation between weak bisimulation and weak simulation equivalence, and (v) various connections between equivalences and pre-orders in the continuous- and discrete-time setting. The results are summarized in a branching-time spectrum for DTMCs and CTMCs elucidating their semantics as well as their relationship.

Stepwise Development of Process-Algebraic Specifications in Decorated Trace Semantics

Process algebras are convenient formalisms to develop specifications stepwise. This can be done with the help of partially defined states in a specification. When refining the specification, new transitions are added to partially defined states. At every step, it is verified with the help of special preorders, refinement relations, that the step leads towards a desired goal. This approach has already been introduced in the case, where the verification is based on weak bisimulation equivalence. We show in this article that refinement relations can also be developed in decorated trace semantics. Moreover, the intuitive picture seems to be simpler in trace-based than in bisimulation-based semantics. The algorithms to compute the new refinement relations are exponential in the worst case, but behave quite well in practical cases.

Weak bisimilarity and regularity of context-free processes is EXPTIME-hard

We show that checking weak bisimulation equivalence of two context-free processes (also called BPA-processes) is EXPTIME-hard, even under the condition that the processes are normed. Furthermore, checking weak regularity (finiteness up to weak bisimilarity) for context-free processes is EXPTIME-hard as well. Adding a finite control of the minimal non-trivial size of 2 to the BPA process already makes weak bisimilarity undecidable.

Weak Bisimilarity and Regularity of Context-Free Processes is EXPTIME-hard

We show that checking weak bisimulation equivalence of two context-free processes (also called BPA-processes) is EXPTIME-hard, even under the condition that the processes are normed. Fur- thermore, checking weak regularity (finiteness up to weak bisimilarity) for context-free processes is EXPTIME-hard as well. Adding a finite control of the minimal non-trivial size of 2 to the BPA process already makes weak bisimilarity undecidable.

Complete inference systems for weak bisimulation equivalences in the π-calculus

Proof systems for weak bisimulation equivalences in the π -calculus are presented, and their soundness and completeness are shown. Two versions of π -calculus are investigated, one without and the other with the mismatch operator. For each version of the calculus proof systems for both late and early weak bisimulation equivalences are studied. Thus there are four proof systems in all. These inference systems are related in a natural way: the inference system for early equivalence is obtained from the one for late equivalence by replacing the inference rule for input prefix, while the inference system for the version of π -calculus with mismatch is obtained by adding a single inference rule for mismatch to the one for the version without it. The proofs of the completeness results rely on the notion of symbolic bisimulation .

Proof Techniques for Cryptographic Processes

Contextual equivalences for cryptographic process calculi, like the spi-calculus, can be used to reason about correctness of protocols, but their definition suffers from quantification over all possible contexts. Here, we focus on two such equivalences, namely may-testing and barbed equivalence, and investigate tractable proof methods for them. To this aim, we design an enriched labelled transition system, where transitions are constrained by the knowledge the environment has of names and keys. The new transition system is then used to define a trace equivalence and a weak bisimulation equivalence that avoid quantification over contexts. Our main results are soundness and completeness of trace and weak bisimulation equivalence with respect to may-testing and barbed equivalence, respectively. They lead to more direct proof methods for equivalence checking. The use of these methods is illustrated with a few examples concerning implementation of secure channels and verification of protocol correctness.

On the formalization of architectural types with process algebras

Architectural styles play an important role in software engineering as they convey codified principles and experience which help the construction of software systems with high levels of efficiency and confidence. We address the problem of formalizing and analyzing architectural styles in an operational setting by introducing the intermediate abstraction of architectural type. We develop the concept of architectural type in a process algebraic framework because of its modeling adequacy and the availability of means, such as Milner's weak bisimulation equivalence, which allow us to reason compositionally and efficiently about the well formedness of architectural types.

Petri Nets and Regular Processes

We consider the following problems: (a) Given a labelled Petri net and a finite automaton, are they equivalent?; (b) Given a labelled Petri net, is it equivalent to some (unspecified) finite automaton? These questions are studied within the framework of trace and bisimulation equivalences, in both their strong and weak versions. (In the weak version a special τ action—likened to an ε-move in automata theory—is considered to be nonobservable.) We demonstrate that (a) is decidable for strong and weak trace equivalence and for strong bisimulation equivalence, but undecidable for weak bisimulation equivalence. On the other hand, we show that (b) is decidable for strong bisimulation equivalence, and undecidable for strong and weak trace equivalence, as well as for weak bisimulation equivalence.

A theory of weak bisimulation for Core CML

Concurrent ML (CML) is an extension of Standard ML of New Jersey with concurrent features similar to those of process algebra. In this paper, we build upon John Reppy's reduction semantics for CML by constructing a compositional operational semantics for a fragment of CML, based on higher-order process algebra. Using the operational semantics we generalise the notion of weak bisimulation equivalence to build a semantic theory of CML. We give some small examples of proofs about CML expressions, and show that our semantics corresponds to Reppy's up to weak first-order bisimulation.