Process Algebra Operators Research Articles

The quest for minimal quotients for probabilistic and Markov automata

One of the prevailing ideas in applied concurrency theory and verification is the concept of automata minimization with respect to strong or weak bisimilarity. The minimal automata can be seen as canonical representations of the behaviour modulo the bisimilarity considered. Together with congruence results wrt. process algebraic operators, this can be exploited to alleviate the notorious state space explosion problem. In this paper, we aim at identifying minimal automata and canonical representations for concurrent probabilistic models. We present minimality and canonicity results for probabilistic and Markov automata modulo strong and weak probabilistic bisimilarity, together with the corresponding minimization algorithms. We also consider weak distribution bisimilarity, originally proposed for Markov automata. For this relation, the quest for minimality does not have a unique answer, since fanout minimality clashes with state and transition minimality. We present an SMT approach to enumerate fanout-minimal models.

ULTraS at work: Compositionality metaresults for bisimulation and trace semantics

The ULTraS metamodel can be instantiated to a large number of well established models, including labeled transition systems together with their probabilistic, deterministic timed, and stochastic timed extensions. Several metaequivalences have been defined on the ULTraS metamodel, which can be instantiated to well known behavioral equivalences for those specific models. However, new equivalences pop up, instead of the widely accepted ones, in the case of processes featuring probabilities and internal nondeterminism. Most importantly, the properties of the metaequivalences have not been investigated yet. Focusing on bisimulation and trace semantics, we first show that, by simply introducing the notion of resolution in the ULTraS theory, and exchanging the order of certain universal quantifiers in the definition of the metaequivalences, it is possible to retrieve the behavioral equivalences not captured before, as well as to keep the new ones. We then study the compositionality, with respect to typical process algebraic operators, of the two bisimulation metaequivalences and of the two trace metaequivalences respectively arising from the two different quantification orders. The congruence metaresults for parallel composition confirm the existence of a foundational difference in the compositionality of bisimulation and trace semantics when internal nondeterminism is present, which had recently emerged in the specific setting of probabilistic and nondeterministic processes.

SOS specifications for uniformly continuous operators

Behavioral metric semantics provide formal notions to compare probabilistic systems, giving a notion of behavioral distance characterizing how far the behavior of two systems is apart. Compositional reasoning over probabilistic systems with respect to behavioral metric semantics requires the language operators to be uniformly continuous, which ensures that a limited change in the behavior of a subsystem implies a smooth and limited change in the behavior of the whole system. We consider a hierarchy of uniform continuity properties for process algebra operators and, for each of these properties, we propose a Structural Operational Semantics specification format (namely a set of syntactical constraints on the form of the SOS rules) ensuring that the property is satisfied by construction by all operators captured by the format.

Compositional bisimulation metric reasoning with Probabilistic Process Calculi

We study which standard operators of probabilistic process calculi allow for compositional reasoning with respect to bisimulation metric semantics. We argue that uniform continuity (generalizing the earlier proposed property of non-expansiveness) captures the essential nature of compositional reasoning and allows now also to reason compositionally about recursive processes. We characterize the distance between probabilistic processes composed by standard process algebra operators. Combining these results, we demonstrate how compositional reasoning about systems specified by continuous process algebra operators allows for metric assume-guarantee like performance validation.

Formal refinement of extended state machines

In a traditional formal development process, e.g. using the B method, the informal user requirements are (manually) translated into a global abstract formal specification. This translation is especially difficult to achieve. The Event-B method was developed to incrementally and formally construct such a specification using stepwise refinement. Each increment takes into account new properties and system aspects. In this paper, we propose to couple a graphical notation called Algebraic State-Transition Diagrams (ASTD) with an Event-B specification in order to provide a better understanding of the software behaviour. The dynamic behaviour is captured by the ASTD, which is based on automata and process algebra operators, while the data model is described by means of an Event-B specification. We propose a methodology to incrementally refine such specification couplings, taking into account new refinement relations and consistency conditions between the control specification and the data specification. We compare the specifications obtained using each approach for readability and proof complexity. The advantages and drawbacks of the traditional approach and of our methodology are discussed. The whole process is illustrated by a railway CBTC-like case study. Our approach is supported by tools for translating ASTD's into B and Event-B into B.

On recursive operations over logic LTS

Recently, in order to mix algebraic and logic styles of specification in a uniform framework, the notion of a logic labelled transition system (Logic LTS or LLTS for short) has been introduced and explored. A variety of constructors over LLTS, including usual process-algebraic operators, logic connectives (conjunction and disjunction) and standard temporal modalities (always and unless), have been given. However, no attempt has been made so far to develop the general theory concerning (nested) recursive operations over LLTS and a few fundamental problems are still open. This paper intends to study this issue in a pure process-algebraic style. A few fundamental properties, including precongruence and the uniqueness of consistent solutions of equations, will be established.

Syntax and semantics of the compositional interchange format for hybrid systems

Different modeling formalisms for timed and hybrid systems exist, each of which addresses a specific set of problems, and has its own set of features. These formalisms and tools can be used in each stage of the embedded systems development, to verify and validate various requirements.The Compositional Interchange Format (CIF), is a formalism based on hybrid automata, which are composed using process algebraic operators. CIF aims to establish interoperability among a wide range of formalisms and tools by means of model transformations and co-simulation, which avoids the need for implementing many bilateral translators.This work presents the syntax and formal semantics of CIF. The semantics is shown to be compositional, and proven to preserve certain algebraic properties, which express our intuition about the behavior of the language operators. In addition we show how CIF operators can be combined to implement widely used constructs present in other timed and hybrid formalisms, and we illustrate the applicability of the formalism by developing several examples.Based on the formal specification of CIF, an Eclipse based simulation environment has been developed. We expect this work to serve as the basis for the formal definition of semantic preserving transformations between various languages for the specification of timed and hybrid systems.

Computing Distances between Probabilistic Automata

We present relaxed notions of simulation and bisimulation on Probabilistic Automata (PA), that allow some error epsilon. When epsilon is zero we retrieve the usual notions of bisimulation and simulation on PAs. We give logical characterisations of these notions by choosing suitable logics which differ from the elementary ones, L with negation and L without negation, by the modal operator. Using flow networks, we show how to compute the relations in PTIME. This allows the definition of an efficiently computable non-discounted distance between the states of a PA. A natural modification of this distance is introduced, to obtain a discounted distance, which weakens the influence of long term transitions. We compare our notions of distance to others previously defined and illustrate our approach on various examples. We also show that our distance is not expansive with respect to process algebra operators. Although L without negation is a suitable logic to characterise epsilon-(bi)simulation on deterministic PAs, it is not for general PAs; interestingly, we prove that it does characterise weaker notions, called a priori epsilon-(bi)simulation, which we prove to be NP-difficult to decide.

On the expressive power of process interruption and compensation

The investigation into the foundational aspects of linguistic mechanisms for programming long-running transactions (such as thescopeoperator of WS-BPEL) has recently renewed the interest in process algebraic operators that, due to the occurrence of a failure,interruptthe execution of one process, replacing it with another one called thefailure handler. We investigate the decidability of termination problems for two simple fragments of CCS (one with recursion and one with replication) extended with one of two such operators, theinterruptoperator of CSP and thetry-catchoperator for exception handling. More precisely, we consider the existential termination problem (existence of one terminated computation) and the universal termination problem (all computations terminate). We prove that, as far as the decidability of the considered problems is concerned, under replication there is no difference between interrupt and try-catch (universal termination is decidable while existential termination is not), while under recursion this is not the case (existential termination is undecidable while universal termination is decidable only for interrupt). As a consequence of our undecidability results, we show the existence of an expressiveness gap between a fragment of CCS and its extension with either the interrupt or the try-catch operator.

Extending statecharts with process algebra operators

This paper describes an adaptation of statecharts to take advantage of process algebra operators like those found in CSP and EB3. The resulting notation is called algebraic state transition diagrams (ASTDs). The process algebra operators considered include sequence, iteration, parallel composition, and quantified synchronization. Quantification is one of the salient features of ASTDs, because it provides a powerful mechanism to precisely and explicitly define cardinalities in a dynamic model. The formal semantics of ASTDs is expressed using the operational style typically used in process algebras. The target application domain is the specification and implementation of information systems.

Through Modeling to Synthesis of Security Automata

We define a set of process algebra operators, that we call controller operators, able to mimic the behavior of security automata introduced by Schneider in [Schneider, F. B., Enforceable security policies, ACM Transactions on Information and System Security 3 (2000), pp. 30–50] and by Ligatti and al. in [Bauer, L., J. Ligatti and D. Walker, More enforceable security policies, in: I. Cervesato, editor, Foundations of Computer Security: proceedings of the FLoC'02 workshop on Foundations of Computer Security (2002), pp. 95–104]. Security automata are mechanisms for enforcing security policies that specify acceptable executions of programs.Here we give the semantics of four controllers that act by monitoring possible un-trusted component of a system in order to enforce certain security policies. Moreover, exploiting satisfiability results for temporal logic, we show how to automatically build these controllers for a given security policy.

Behavioral complexity indicators for process algebra: The NKS approach

Several techniques for the experimental investigation of the computing power of various formal models, from cellular automata to Turing machines, have been proposed by S. Wolfram with his NKS (New Kind of Science). Visual complexity indicators reveal the ‘internal shapes’ of computations, and may expose constant, periodic, nested/fractal, pseudo-random, and even more sophisticated dynamics. In this paper we investigate visual complexity indicators for process algebra. With its emphasis on reactive, continuously observable behavior, as opposed to input/output behavior, process algebra might appear as an ideal candidate for NKS-style investigations; however, this formal model is in some sense more elaborate than the simple formalisms addressed in NKS, and poses specific problems, such as the presence of both events and states, and the explosive nature of nondeterminism. We consider a set of process algebraic operators and prove its Turing universality by showing that they can emulate any elementary cellular automaton, including n. 110, which is itself universal. The correctness of the emulation is supported by an original visual indicator, which is then used for exploring various subclasses of algebraic expressions and their emergent features. Based on this indicator, and even by restricting to deterministic computations, we have detected and measured, by a data compression technique, the emergence of randomness in a subclass of expressions which is provably not universal. Besides providing a suggestive visualization of the relative strengths of operator subsets, we believe that results of this type, both of theoretical and of experimental nature, are desirable in light of one of the key NKS conjectures, according to which random-like behavior would be a witness of computational universality.

Non-bisimulation-based Markovian behavioral equivalences

The behavioral equivalence that is typically used to relate Markovian process terms and to reduce their underlying state spaces is Markovian bisimilarity. One of the reasons is that Markovian bisimilarity is consistent with ordinary lumping. The latter is an aggregation for Markov chains that is exact, hence it guarantees the preservation of the performance characteristics across Markovian bisimilar process terms. In this paper we show that two non-bisimulation-based Markovian behavioral equivalences – Markovian testing equivalence and Markovian trace equivalence – induce at the Markov chain level an aggregation strictly coarser than ordinary lumping that is still exact. We then show that only Markovian testing equivalence may constitute a useful alternative to Markovian bisimilarity, as it turns out to be a congruence with respect to the typical process algebraic operators, while Markovian trace equivalence is not a congruence with respect to parallel composition.

Service Oriented Computing: A New Challenge for Process Algebras

Service Oriented Computing is emerging as a reference model for a new class of distributed computing technologies such as Web Services and the Grid. We discuss three main aspects of Service Oriented Computing (loose coupling, communication latency, and open endedness), and we relate them with traditional process algebra operators. We also indicate some new issues, raising from the combination of these three aspects, that require the investigation of suitable new process algebra operators.

Service oriented computing from a process algebraic perspective

Service oriented computing is emerging as a reference model for a new class of distributed computing technologies such as Web Services and the Grid. We discuss three main aspects of service oriented computing (loose coupling, communication latency, and open endedness), and we relate them with traditional process algebra operators. We also indicate some new issues, raising from the combination of these three aspects, that require the investigation of suitable new process algebra operators.

Well-behaved flow event structures for parallel composition and action refinement

Flow event structures were introduced as a model for giving semantics to process algebras. However, it turned out that certain restrictions have to be made to make them suitable for this purpose. In this paper, we investigate subclasses of flow event structures which are both suited for the process algebraic composition operators, and for action refinement as a means of regarding processes on different levels of abstraction. First, suitable subclasses are characterised. Then two specific subclasses are proposed. The larger class generalises the one from Castellani and Zhang (Theoret. Comput. Sci. 179 (1997) 203–215), which is not suitable for action refinement. The smaller one is still sufficiently expressive for dealing with all standard process algebras and action refinement.

Process algebra with action dependencies

In this paper, we present a process algebra with a minimal form of semantics for actions given by dependencies . Action dependencies are interpreted in the Mazurkiewicz sense: independent actions should be able to commute, or (from a different perspective) should be unordered, whereas dependent actions are always ordered. In this approach, the process algebra operators are used to describe the conceptual behavioural structure of the system, and the action dependencies determine the minimal necessary orderings and thereby the additionally possible parallelism within this structure. In previous work on the semantics of specifications using Mazurkiewicz dependencies, the main interest has been on linear time. We present in this paper a branching time semantics, both operationally and denotationally. For this purpose, we introduce a process algebra that incorporates, besides some standard operators, also an operator for action refinement. For interpreting the operators in the presence of action dependencies, a new concept of partial termination has to be developed. We show consistency of the operational and denotational semantics; furthermore, we give a axiomatisation of bisimilarity, which is complete for finite terms. Some small examples demonstrate the flexibility of this process algebra in the design of distributed reactive systems.

Using binary decision diagrams for representation and analysis of communication protocols

A symbolic representation of a state/transition system based on binary decision diagrams (BDDs) is generally more compact than an explicit representation like a state/transition table. This is due to regular and repetitive patterns occurring in state/transition systems. By exploiting this property, huge state spaces can be represented, and the resulting BDDs can be profitably used for activities such as symbolic model checking and sequential circuit synthesis. This paper shows how such techniques can be applied to communication protocols by presenting a systematic method to build BDD representations from protocol specifications expressed in the ISO standard protocol specification language LOTOS. The method exploits the compositionality of the process algebra of LOTOS to avoid the enumeration of all the states and transitions, takes also data into account, enables building the BDDs in the more convenient disjunctive partitioned form, and can handle any LOTOS specification characterized by a finite LTS. The method consists in partitioning the set of process definitions according to their mutual recursion relationships, building an LTS for each set of mutually recursive process definitions, encoding these LTSs as BDDs which in turn are combined together, according to the process algebraic operators, to obtain the overall BDD representation. An example is used throughout the paper to illustrate the method.

Operational Semantics for MSC'96

Recently, the ITU-standardised specification language Message Sequence Chart has been extended with constructs for more complete and structured specifications. The new version of the language is called MSC'96. Currently, research is performed on the extension of the formal semantics towards a semantics for MSC'96. In this article, we aim at explaining the basic ideas behind the formal semantics. We give formal definitions of parts of the language, but most features are explained by informal examples and drawings. It takes several steps in order to follow the path from an MSC drawing to its formal meaning. First, the drawing must be converted to a concrete textual representation. This conversion is already defined implicitly in Z.120. Next, this syntax is transformed into a formal expression over some process algebra signature. MSC constructs are replaced by appropriate process algebra operators. This transformation is compositional. The operational behaviour of the process algebraic expression can be studied, or the expression can be interpreted into some mathematical model and compared to the interpretation of some other MSC.

The Algebra of Timed Processes, ATP: Theory and Application

The algebra of timed processes, ATP , uses a notion of discrete global time and suggests a conceptual framework for introducing time by extending untimed languages. The action vocabularly of ATP contains a special element representing the progress of time. The algebra has, apart from standard operators of process algebras such as prefixing by an action, alternative choice, and parallel composition, a primitive unit-delay operator. For two arguments, processes P and Q , this operator gives a process which behaves as P before the execution of a time event and behaves as Q afterwards. It is shown that several d -unit delay constructs such as timeouts and watchdogs can be expressed in terms of the unit-delay operator and standard process algebra operators. A sound and complete axiomatization for bisimulation semantics is studied and two examples illustrating the adequacy of the language for the description of timed systems are given. Finally we provide a comparison with existing timed process algebras.