Labelled transition systems as a Stone space

A fully abstract and universal domain model for modal transition systems and refinement is shown to be a maximal-points space model for the bisimulation quotient of labelled transition systems over a finite set of events. In this domain model we prove that this quotient is a Stone space whose compact, zero-dimensional, and ultra-metrizable Hausdorff topology measures the degree of bisimilarity such that image-finite labelled transition systems are dense. Using this compactness we show that the set of labelled transition systems that refine a modal transition system, its ''set of implementations'', is compact and derive a compactness theorem for Hennessy-Milner logic on such implementation sets. These results extend to systems that also have partially specified state propositions, unify existing denotational, operational, and metric semantics on partial processes, render robust consistency measures for modal transition systems, and yield an abstract interpretation of compact sets of labelled transition systems as Scott-closed sets of modal transition systems.Comment: Changes since v2: Metadata updat

Three Simulation Algorithms for Labelled Transition Systems

Algorithms which compute the coarsest simulation preorder are generally designed on Kripke structures. Only in a second time they are extended to labelled transition systems. By doing this, the size of the alphabet appears in general as a multiplicative factor to both time and space complexities. Let $Q$ denotes the state space, $\rightarrow$ the transition relation, $\Sigma$ the alphabet and $P_{sim}$ the partition of $Q$ induced by the coarsest simulation equivalence. In this paper, we propose a base algorithm which minimizes, since the first stages of its design, the incidence of the size of the alphabet in both time and space complexities. This base algorithm, inspired by the one of Paige and Tarjan in 1987 for bisimulation and the one of Ranzato and Tapparo in 2010 for simulation, is then derived in three versions. One of them has the best bit space complexity up to now, $O(|P_{sim}|^2+|{\rightarrow}|.\log|{\rightarrow}|)$, while another one has the best time complexity up to now, $O(|P_{sim}|.|{\rightarrow}|)$. Note the absence of the alphabet in these complexities. A third version happens to be a nice compromise between space and time since it runs in $O(b.|P_{sim}|.|{\rightarrow}|)$ time, with $b$ a branching factor generally far below $|P_{sim}|$, and uses $O(|P_{sim}|^2.\log|P_{sim}|+|{\rightarrow}|.\log|{\rightarrow}|)$ bits

Transforming specifications of observable behaviour into programs

A methodology for deriving programs from specifications of observable behaviour is described. The class of processes to which this methodology is applicable includes those whose state changes are fully definable by labelled transition systems, for example communicating processes without internal state changes. A logic program representation of such labelled transition systems is proposed, interpreters based on path searching techniques are defined, and the use of partial evaluation techniques to derive the executable programs is described

Confluence Detection for Transformations of Labelled Transition Systems

The development of complex component software systems can be made more manageable by first creating an abstract model and then incrementally adding details. Model transformation is an approach to add such details in a controlled way. In order for model transformation systems to be useful, it is crucial that they are confluent, i.e. that when applied on a given model, they will always produce a unique output model, independent of the order in which rules of the system are applied on the input. In this work, we consider Labelled Transition Systems (LTSs) to reason about the semantics of models, and LTS transformation systems to reason about model transformations. In related work, the problem of confluence detection has been investigated for general graph structures. We observe, however, that confluence can be detected more efficiently in special cases where the graphs have particular structural properties. In this paper, we present a number of observations to detect confluence of LTS transformation systems, and propose both a new confluence detection algorithm and a conflict resolution algorithm based on them.Comment: In Proceedings GaM 2015, arXiv:1504.0244

Conformance Testing with Labelled Transition Systems: Implementation Relations and Test Generation

This paper studies testing based on labelled transition systems, presenting two test generation algorithms with their corresponding implementation relations. The first algorithm assumes that implementations communicate with their environment via symmetric, synchronous interactions. It is based on the theory of testing equivalence and preorder, as is most of the testing theory for labelled transition systems, and it is found in the literature in some slightly different variations. The second algorithm is based on the assumption that implementations communicate with their environment via inputs and outputs. Such implementations are formalized by restricting the class of labelled transition systems to those systems that can always accept input actions. For these implementations a testing theory is developed, analogous to the theory of testing equivalence and preorder. It consists of implementation relations formalizing the notion of conformance of these implementations with respect to labelled transition system specifications, test cases and test suites, test execution, the notion of passing a test suite, and the test generation algorithm, which is proved to produce sound test suites for one of the implementation relations

Coalgebraic Weak Bisimulation from Recursive Equations over Monads

Strong bisimulation for labelled transition systems is one of the most fundamental equivalences in process algebra, and has been generalised to numerous classes of systems that exhibit richer transition behaviour. Nearly all of the ensuing notions are instances of the more general notion of coalgebraic bisimulation. Weak bisimulation, however, has so far been much less amenable to a coalgebraic treatment. Here we attempt to close this gap by giving a coalgebraic treatment of (parametrized) weak equivalences, including weak bisimulation. Our analysis requires that the functor defining the transition type of the system is based on a suitable order-enriched monad, which allows us to capture weak equivalences by least fixpoints of recursive equations. Our notion is in agreement with existing notions of weak bisimulations for labelled transition systems, probabilistic and weighted systems, and simple Segala systems.Comment: final versio

Folk Theorems on the Correspondence between State-Based and Event-Based Systems

Kripke Structures and Labelled Transition Systems are the two most prominent semantic models used in concurrency theory. Both models are commonly believed to be equi-expressive. One can find many ad-hoc embeddings of one of these models into the other. We build upon the seminal work of De Nicola and Vaandrager that firmly established the correspondence between stuttering equivalence in Kripke Structures and divergence-sensitive branching bisimulation in Labelled Transition Systems. We show that their embeddings can also be used for a range of other equivalences of interest, such as strong bisimilarity, simulation equivalence, and trace equivalence. Furthermore, we extend the results by De Nicola and Vaandrager by showing that there are additional translations that allow one to use minimisation techniques in one semantic domain to obtain minimal representatives in the other semantic domain for these equivalences.Comment: Full version of SOFSEM 2011 pape

Computation Tree Logic with Deadlock Detection

We study the equivalence relation on states of labelled transition systems of satisfying the same formulas in Computation Tree Logic without the next state modality (CTL-X). This relation is obtained by De Nicola & Vaandrager by translating labelled transition systems to Kripke structures, while lifting the totality restriction on the latter. They characterised it as divergence sensitive branching bisimulation equivalence. We find that this equivalence fails to be a congruence for interleaving parallel composition. The reason is that the proposed application of CTL-X to non-total Kripke structures lacks the expressiveness to cope with deadlock properties that are important in the context of parallel composition. We propose an extension of CTL-X, or an alternative treatment of non-totality, that fills this hiatus. The equivalence induced by our extension is characterised as branching bisimulation equivalence with explicit divergence, which is, moreover, shown to be the coarsest congruence contained in divergence sensitive branching bisimulation equivalence