Characteristic Logics for Behavioural Metrics via Fuzzy Lax Extensions

Behavioural distances provide a fine-grained measure of equivalence in systems involving quantitative data, such as probabilistic, fuzzy, or metric systems. Like in the classical setting of crisp bisimulation-type equivalences, the wide variation found in system types creates a need for generic methods that apply to many system types at once. Approaches of this kind are emerging within the paradigm of universal coalgebra, based either on lifting pseudometrics along set functors or on lifting general real-valued (fuzzy) relations along functors by means of fuzzy lax extensions. An immediate benefit of the latter is that they allow bounding behavioural distance by means of fuzzy bisimulations that need not themselves be (pseudo-)metrics, in analogy to classical bisimulations (which need not be equivalence relations). The known instances of generic pseudometric liftings, specifically the generic Kantorovich and Wasserstein liftings, both can be extended to yield fuzzy lax extensions, using the fact that both are effectively given by a choice of quantitative modalities. Our central result then shows that in fact all fuzzy lax extensions are Kantorovich extensions for a suitable set of quantitative modalities, the so-called Moss modalities. For non-expansive fuzzy lax extensions, this allows for the extraction of quantitative modal logics that characterize behavioural distance, i.e. satisfy a quantitative version of the Hennessy-Milner theorem; equivalently, we obtain expressiveness of a quantitative version of Moss\u27 coalgebraic logic

Characteristic Logics for Behavioural Hemimetrics via Fuzzy Lax Extensions

In systems involving quantitative data, such as probabilistic, fuzzy, or metric systems, behavioural distances provide a more fine-grained comparison of states than two-valued notions of behavioural equivalence or behaviour inclusion. Like in the two-valued case, the wide variation found in system types creates a need for generic methods that apply to many system types at once. Approaches of this kind are emerging within the paradigm of universal coalgebra, based either on lifting pseudometrics along set functors or on lifting general real-valued (fuzzy) relations along functors by means of fuzzy lax extensions. An immediate benefit of the latter is that they allow bounding behavioural distance by means of fuzzy (bi-)simulations that need not themselves be hemi- or pseudometrics; this is analogous to classical simulations and bisimulations, which need not be preorders or equivalence relations, respectively. The known generic pseudometric liftings, specifically the generic Kantorovich and Wasserstein liftings, both can be extended to yield fuzzy lax extensions, using the fact that both are effectively given by a choice of quantitative modalities. Our central result then shows that in fact all fuzzy lax extensions are Kantorovich extensions for a suitable set of quantitative modalities, the so-called Moss modalities. For nonexpansive fuzzy lax extensions, this allows for the extraction of quantitative modal logics that characterize behavioural distance, i.e. satisfy a quantitative version of the Hennessy-Milner theorem; equivalently, we obtain expressiveness of a quantitative version of Moss' coalgebraic logic. All our results explicitly hold also for asymmetric distances (hemimetrics), i.e. notions of quantitative simulation

The foundational legacy of ASL

Abstract. We recall the kernel algebraic specification language ASL and outline its main features in the context of the state of research on algebraic specification at the time it was conceived in the early 1980s. We discuss the most significant new ideas in ASL and the influence they had on subsequent developments in the field and on our own work in particular.

Trees from Functions as Processes

Levy-Longo Trees and Bohm Trees are the best known tree structures on the {\lambda}-calculus. We give general conditions under which an encoding of the {\lambda}-calculus into the {\pi}-calculus is sound and complete with respect to such trees. We apply these conditions to various encodings of the call-by-name {\lambda}-calculus, showing how the two kinds of tree can be obtained by varying the behavioural equivalence adopted in the {\pi}-calculus and/or the encoding

Tau Be or not Tau Be? - A Perspective on Service Compatibility and Substitutability

One of the main open research issues in Service Oriented Computing is to propose automated techniques to analyse service interfaces. A first problem, called compatibility, aims at determining whether a set of services (two in this paper) can be composed together and interact with each other as expected. Another related problem is to check the substitutability of one service with another. These problems are especially difficult when behavioural descriptions (i.e., message calls and their ordering) are taken into account in service interfaces. Interfaces should capture as faithfully as possible the service behaviour to make their automated analysis possible while not exhibiting implementation details. In this position paper, we choose Labelled Transition Systems to specify the behavioural part of service interfaces. In particular, we show that internal behaviours (tau transitions) are necessary in these transition systems in order to detect subtle errors that may occur when composing a set of services together. We also show that tau transitions should be handled differently in the compatibility and substitutability problem: the former problem requires to check if the compatibility is preserved every time a tau transition is traversed in one interface, whereas the latter requires a precise analysis of tau branchings in order to make the substitution preserve the properties (e.g., a compatibility notion) which were ensured before replacement.Comment: In Proceedings WCSI 2010, arXiv:1010.233

Behavioural equivalences for timed systems

Timed transition systems are behavioural models that include an explicit treatment of time flow and are used to formalise the semantics of several foundational process calculi and automata. Despite their relevance, a general mathematical characterisation of timed transition systems and their behavioural theory is still missing. We introduce the first uniform framework for timed behavioural models that encompasses known behavioural equivalences such as timed bisimulations, timed language equivalences as well as their weak and time-abstract counterparts. All these notions of equivalences are naturally organised by their discriminating power in a spectrum. We prove that this result does not depend on the type of the systems under scrutiny: it holds for any generalisation of timed transition system. We instantiate our framework to timed transition systems and their quantitative extensions such as timed probabilistic systems

Consistency in Multi-Viewpoint Architectural Design of Enterprise Information Systems

Different stakeholders in the design of an enterprise information system have their own view on that design. To help produce a coherent design this paper presents a framework that aids in specifying relations between such views. To help produce a consistent design the framework also aids in specifying consistency rules that apply to the view relations and in checking the consistency according to those rules. The framework focuses on the higher levels of abstraction in a design, we refer to design at those levels of abstraction as architectural design. The highest level of abstraction that we consider is that of business process design and the lowest level is that of software component design. The contribution of our framework is that it provides a collection of basic concepts that is common to viewpoints in the area of enterprise information systems. These basic concepts aid in relating viewpoints by providing: (i) a common terminology that helps stakeholders to understand each others concepts; and (ii) a basis for defining re-usable consistency rules. In particular we define re-usable rules to check consistency between behavioural views that overlap or are a refinement of each other. We also present an architecture for a tool suite that supports our framework. We show that our framework can be applied, by performing a case study in which we specify the relations and consistency rules between the RM-ODP enterprise, computational and information viewpoints