Parallel Composition Operator Research Articles

Distribution-based limited fuzzy bisimulations for nondeterministic fuzzy transition systems

Bisimulation techniques have been adopted and developed in fuzzy transition systems (FTSs) to compare their behavior. Recently, distribution-based bisimulation has been introduced to nondeterministic fuzzy transition systems (NFTSs) to measure the similarity between two distributions which can be regarded as generalized states. However, the new bisimulation does not consider the similarity of states. In terms of limited approximate bisimilarity, in this article, a notion of distribution-based k-limited bisimulation is proposed for NFTSs, which can measure the similarity between two distributions in the neighboring subgraphs (subsystems) and take the similarity of states into account, where k is a natural number. This article discusses some properties of limited bisimulation and the nonexpansiveness of typical parallel composition operators with respect to the limited bisimulation which makes compositional verification possible. Then a fixed point characterization of the limited bisimulation is investigated and an algorithm is given to compute the degree of k-limited similarity between two distributions. Moreover, a fuzzy modal logic interpreted on distributions is adopted to characterize the limited bisimulation.

Approximate bisimulations for fuzzy-transition systems

Bisimulation techniques have been recently introduced to fuzzy transition systems to compare their behavior. When studying bisimulation, one of the most popular methods is relational lifting. In this paper, we first realize the relational lifting through a function S˜. The new relational lifting method relaxes the condition of lifting relation from using weight functions to function S˜ defined by residua (implications) in complete residuated lattices. We first discuss some properties of the lifting operation, and then define an α-bisimulation under S˜, where α∈[0,1], for nondeterministic fuzzy transition systems (NFTSs). The new bisimulation can handle the case that two states are equivalent intuitively but are simply treated as distinguishable by some behavioral measures, and so it is much more natural and robust. Then we provide some properties of α-bisimulation under S˜. The parallel composition operator of NFTSs is addressed and we show that α-bisimilarity under S˜ is nonexpansive with respect to the operator, which makes compositional verification possible. In addition, we give a fixed-point characterization of α-bisimulation under S˜ and compute α-bisimilarity under S˜. Finally, we introduce a real-valued logic to characterize α-bisimilarity under S˜.

On the Axiomatisability of Parallel Composition

This paper studies the existence of finite equational axiomatisations of the interleaving parallel composition operator modulo the behavioural equivalences in van Glabbeek's linear time-branching time spectrum. In the setting of the process algebra BCCSP over a finite set of actions, we provide finite, ground-complete axiomatisations for various simulation and (decorated) trace semantics. We also show that no congruence over BCCSP that includes bisimilarity and is included in possible futures equivalence has a finite, ground-complete axiomatisation; this negative result applies to all the nested trace and nested simulation semantics.

Coordination and Control of Hierarchically Organized Interacting Agents

The coordination and control of hierarchically organized interacting agents is an important issue in many applications, e.g., harbor or warehouse automation. A formalism of agents as hierarchical input/output automata is proposed. A system of interacting agents is modeled as the parallel composition of their automata. We extend the usual parallel composition operation of I/O automata with a hierarchical composition operation for refining abstract tasks into lower-level subtasks. We provide an algorithm to synthesize hierarchically organized controllers to coordinate the agents' interactions in order to drive the system toward desired states. Our main contribution regards the formal definition, the representation, the theorems about its properties (i.e., the parallel and hierarchical composition are distributive operations), and the synthesis algorithm, proved to be complete and correct.

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.

Keep it fair: Equivalence and composition

Fairness assumptions are commonly used to filter system behaviors, thereby distinguishing between realistic and unrealistic executions. This allows for key arguments in correctness proofs of distributed systems, which would not be possible otherwise. Our first contribution is an equivalence spectrum in which fairness assumptions are preserved. Although the identified equivalences allow for reasoning about correctness incorporating fairness assumptions, this does not necessarily allow for the lifting of arguments from sequential processes to parallel compositions employing arbitrary synchronization mechanisms. Our second contribution is, therefore, an analysis of parallel composition operators and their synchronization mechanisms in this respect.

Toward an algebraic theory of systems

We propose the concept of a system algebra with a parallel composition operation and an interface connection operation, and formalize composition-order invariance, which postulates that the order of composing and connecting systems is irrelevant, a generalized form of associativity. Composition-order invariance explicitly captures a common property that is implicit in any context where one can draw a figure (hiding the drawing order) of several connected systems, which appears in many scientific contexts. This abstract algebra captures settings where one is interested in the behavior of a composed system in an environment and wants to abstract away anything internal not relevant for the behavior. This may include physical systems, electronic circuits, or interacting distributed systems.One specific such setting, of special interest in computer science, are functional system algebras, which capture, in the most general sense, any type of system that takes inputs and produces outputs depending on the inputs, and where the output of a system can be the input to another system. The behavior of such a system is uniquely determined by the function mapping inputs to outputs. We consider several instantiations of this very general concept. In particular, we show that Kahn networks form a functional system algebra and prove their composition-order invariance.Moreover, we define a functional system algebra of causal systems, characterized by the property that inputs can only influence future outputs, where an abstract partial order relation captures the notion of “later”. This system algebra is also shown to be composition-order invariant and appropriate instantiations thereof allow to model and analyze systems that depend on time.

Distribution-Based Behavioral Distance for Nondeterministic Fuzzy Transition Systems

Modal logics and behavioral equivalences play an important role in the specification and verification of concurrent systems. In this paper, we first present a new notion of bisimulation for nondeterministic fuzzy transition systems, which is distribution based and coarser than state-based bisimulation appeared in the literature. Then, we define a distribution-based bisimilarity metric as the least fixed point of a suitable monotonic function on a complete lattice, which is a behavioral distance and is a more robust way of formalizing behavioral similarity between states than bisimulations. We also propose an on-the-fly algorithm for computing the bisimilarity metric. Moreover, we present a fuzzy modal logic and provide a sound and complete characterization of the bisimilarity metric. Interestingly, this characterization holds for a class of fuzzy modal logics. In addition, we show the nonexpansiveness of a typical parallel composition operator with respect to the bisimilarity metric, which makes compositional verification possible.

Bisimilar and logically equivalent programs in PDL with parallel operator

In standard Propositional Dynamic Logic (PDL) literature, the semantics is given by Labeled Transition Systems, where for each program π we associate a binary relation Rπ. Process Algebras also give semantics to process (terms) by means of Labeled Transition Systems. In both formalisms, PDL and Process Algebra, the key notion to compare processes is bisimulation. In PDL, we also have the notion of logic equivalence, that can be used to prove that two programs π1 and π2 are logically equivalent ⊢〈π1〉φ↔〈π2〉φ. Unfortunately, logic equivalence and bisimulation do not match in PDL. Bisimilar programs are logic equivalent but the converse does not hold.This paper proposes a semantics and an axiomatization for PDL that makes logically equivalent programs also bisimilar. This allows for developing Dynamic Logics to reasoning about CCS specification. As in CCS the bisimulation is the main tool to establish equivalence of programs, it is very important that these two relations coincide. We propose a new Propositional Dynamic Logic with a new non-deterministic choice operator, PDL+. We prove its soundness, completeness, finite model property and EXPTIME-completeness for the satisfiability problem.We also add to PDL+ the parallel composition operator (PPDL+) and prove its soundness and completeness. We establish that the satisfiability problem for PPDL+ is in 2-EXPTIME. Finally, we define some fragments of PPDL+ and prove its EXPTIME-completeness.

Taming Multirelations

Binary multirelations generalise binary relations by associating elements of a set to its subsets. We study the structure and algebra of multirelations under the operations of union, intersection, sequential, and parallel composition, as well as finite and infinite iteration. Starting from a set-theoretic investigation, we propose axiom systems for multirelations in contexts ranging from bi-monoids to bi-quantales.

An algebraic theory of interface automata

We formulate a compositional specification theory for interface automata, where a component model specifies the allowed sequences of input and output interactions with the environment. A trace-based linear-time refinement is provided, which is the weakest preorder preserving substitutivity of components, and is weaker than the classical alternating simulation defined on interface automata. Since our refinement allows a component to be refined by refusing to produce any output, we also define a refinement relation that guarantees safety and progress. The theory includes the operations of parallel composition to support the structural composition of components, logical conjunction and disjunction for independent development, hiding to support abstraction of interfaces, and quotient for incremental synthesis of components. Our component formulation highlights the algebraic properties of the specification theory for both refinement preorders, and is shown to be fully abstract with respect to observation of communication mismatches. Examples of independent and incremental component development are provided.

Partial Model Checking using Networks of Labelled Transition Systems and Boole an Equation Systems

Partial model checking was proposed by Andersen in 1995 to verify a temporal logic formula compositionally on a composition of processes. It consists in incrementally incorporating into the formula the behavioural information taken from one process - an operation called quotienting - to obtain a new formula that can be verified on a smaller composition from which the incorporated process has been removed. Simplifications of the formula must be applied at each step, so as to maintain the formula at a tractable size. In this paper, we revisit partial model checking. First, we extend quotienting to the network of labelled transition systems model, which subsumes most parallel composition operators, including m-among-n synchronisation and parallel composition using synchronisation interfaces, available in the ELOTOS standard. Second, we reformulate quotienting in terms of a simple synchronous product between a graph representation of the formula (called formula graph) and a process, thus enabling quotienting to be implemented efficiently and easily, by reusing existing tools dedicated to graph compositions. Third, we propose simplifications of the formula as a combination of bisimulations and reductions using Boolean equation systems applied directly to the formula graph, thus enabling formula simplifications also to be implemented efficiently. Finally, we describe an implementation in the CADP (Construction and Analysis of Distributed Processes) toolbox and present some experimental results in which partial model checking uses hundreds of times less memory than on-the-fly model checking.

On projecting processes into session types

We define session types as projections of the behaviour of processes with respect to the operations processes perform on channels. This calls for a parallel composition operator over session types denoting the simultaneous access to a channel by two or more processes. The proposed approach allows us to define a semantically grounded theory of session types that does not require the linear usage of channels. However, type preservation and progress can only be guaranteed for processes that never receive channels they already own. A number of examples show that the resulting framework validates existing session-type theories and unifies them to some extent.

An automata-based approach to synthesize untimed operating procedures in batch chemical processes

Systematic synthesis of untimed operating procedures has always been considered as an important design issue for batch chemical processes. An automaton-based method is developed in the present study to perform this task automatically. On the basis of the proposed methodical model-building principles, two distinct types of automata can be constructed to characterize the plant behaviors and control specifications, respectively. An admissible supervisor can be produced by applying the parallel composition operation with these models. For the purpose of identifying the most efficient operation procedures, the supervisor can then be integrated with a set of auxiliary automata to set the operation target(s) and, also, to impose upper limits on the total numbers of actuator actions and operation steps. Three examples are presented to demonstrate the feasibility and correctness of the proposed approach.

An automata based method for online synthesis of emergency response procedures in batch processes

Rapid response to remove (or reduce) the detrimental effects of accidents has always been an important safety issue for the chemical industries. A systematic strategy is presented in this paper to synthesize emergency response procedures in any given batch system. Specifically, two distinct sets of automata are first constructed offline to model the plant behaviors and the control specifications, respectively. On the basis of these automata, an admissible supervisor can be synthesized online for a diagnosed failure-induced system state by applying the parallel composition operation. For the purpose of identifying an efficient operating procedure to steer the system away from hazardous conditions while still maintaining an acceptable production rate, an additional set of auxiliary automata can be augmented with this supervisor to set the operation targets and to limit the total number of actuator actions. Two examples are presented in this paper to demonstrate the feasibility of the proposed approach.

Propositional Dynamic Logic with Storing, Recovering and Parallel Composition

This work extends Propositional Dynamic Logic (PDL) with parallel composition operator and four atomic programs which formalize the storing and recovering of elements in data structures. A generalization of Kripke semantics is proposed that instead of using set of possible states it uses structured sets of possible states. This new semantics allows for representing data structures and using the five new operator one is capable of reasoning about the manipulation of these data structures. The use of the new language (PRSPDL) is illustrated with some examples. We present sound and complete set of axiom schemata and inference rules to prove all the valid formulas for a restricted fragment called RSPDLo.

Incremental verification of component-based timed systems

We are interested in the incremental development, by integration of components, of component-based timed systems, and in particular, in the preservation of their properties during such a development process. We model timed components with timed automata. Their composition is achieved with the classic parallel composition operator for timed automata. The specifications of these timed systems are expressed with the timed linear logic Metric Interval Temporal Logic (MITL). To guarantee the preservation of properties during an incremental development process, we propose to use τ-simulation relations, adapted for timed systems. First, we extend the classic notion of τ-simulation with timed aspects. As in the untimed case, this relation, called timed τ-simulation, preserves safety properties. To preserve more properties, in particular liveness ones, we present another relation, called divergence-sensitive and stability-respecting (DS) timed τ-simulation. This last relation preserves all MITL properties (and thus liveness ones), but also strong non-zenoness and deadlock-freedom. Moreover, as we put ourselves in a component-based framework, we study if the relations are appropriate to the use of the composition operator that we consider. For this purpose, we study if the relations are compatible with this operator, and if composability and compositionality hold. These three properties are a way to reduce the cost of the verification of the preservation, or even to get it for free. It results that the timed τ-simulation is appropriate with the classic operator since the properties hold without any assumption. However, this is not the case for the DS timed τ-simulation. We implemented the algorithmic verification of the simulations in a tool called Verification of Simulation for Timed Automata (VeSTA). The structure of the tool was inspired from the one of the OPEN-KRONOS tool. This allows, as additional feature, to connect the models considered in VeSTA to the modules of the verification platform OPEN-CAESAR. We show the interest of our method by applying it on a case study, concerning a production cell example.

Causal semantics for the algebra of connectors

The Algebra of Connectors $\mathcal{AC}(P)$ is used to model structured interactions in the BIP component framework. Its terms are connectors, relations describing synchronization constraints between the ports of component-based systems. Connectors are structured combinations of two basic synchronization protocols between ports: rendezvous and broadcast. In a previous paper, we have studied interaction semantics for $\mathcal{AC}(P)$ which defines the meaning of connectors as sets of interactions. This semantics reduces broadcasts into the set of their possible interactions and thus blurs the distinction between rendezvous and broadcast. It leads to exponentially complex models that cannot be a basis for efficient implementation. Furthermore, the induced semantic equivalence is not a congruence. For a subset of $\mathcal{AC}(P)$ , we propose a new causal semantics that does not reduce broadcast into a set of rendezvous and explicitly models the causal dependency relation between ports. The Algebra of Causal Interaction Trees $\mathcal{T}(P)$ formalizes this subset. It is the set of the terms generated from interactions on the set of ports P, by using two operators: a causality operator and a parallel composition operator. Terms are sets of trees where the successor relation represents causal dependency between interactions: an interaction can participate in a global interaction only if its father participates too. We show that causal semantics is consistent with interaction semantics; the semantic equivalence on $\mathcal{T}(P)$ is a congruence. Furthermore, it defines an isomorphism between $\mathcal{T}(P)$ and a subset of $\mathcal{AC}(P)$ . Finally, we define for causal interaction trees a boolean representation in terms of causal rules. This representation is used for their manipulation and simplification as well as for synthesizing connectors.

Component simulation-based substitutivity managing QoS and composition issues

Several scientific bottlenecks have been identified in existing component-based approaches. Among them, we focus on the identification of a relevant abstraction for the component expression and verification of properties like substitutivity: when is it possible to formally accept or reject the substitution of a component in a composition? This paper suggests integer weighted automata to tackle this problem when considering a new factor — Quality of Service (QoS). Four notions of simulation-based substitutivity managing QoS aspects are proposed, and related complexity issues on integer weighted automata are investigated. Furthermore, the paper defines composition operators: sequential, strict-sequential and parallel compositions, bringing path costs into the analysis. New results on the compatibility of proposed substitutivity notions w.r.t. sequential and parallel composition operators are established.

A non-interleaving process calculus for multi-party synchronisation

We introduce the wire calculus. Its dynamic features are inspired by Milner's CCS: a unary prefix operation, binary choice and a standard recursion construct. Instead of an interleaving parallel composition operator there are operators for synchronisation along a common boundary and non-communicating parallel composition. The (operational) semantics is a labelled transition system obtained with SOS rules. Bisimilarity is a congruence with respect to the operators of the language. Quotienting terms by bisimilarity results in a compact closed category.