Logics for contravariant simulations

Covariant-contravariant simulation and conformance simulation are two generalizations of the simple notion of simulation which aim at capturing the fact that it is not always the case that “the larger the number of behaviors, the better”. Therefore, they can be considered to be more adequate to express the fact that a system is a correct implementation of some specification. We have previously shown that these two more elaborated notions fit well within the categorical framework developed to study the notion of simulation in a generic way. Now we show that their behaviors have also simple and natural logical characterizations, though more elaborated than those for the plain simulation semantics

Graphical representation of covariant-contravariant modal formulae

Covariant-contravariant simulation is a combination of standard (covariant) simulation, its contravariant counterpart and bisimulation. We have previously studied its logical characterization by means of the covariant-contravariant modal logic. Moreover, we have investigated the relationships between this model and that of modal transition systems, where two kinds of transitions (the so-called may and must transitions) were combined in order to obtain a simple framework to express a notion of refinement over state-transition models. In a classic paper, Boudol and Larsen established a precise connection between the graphical approach, by means of modal transition systems, and the logical approach, based on Hennessy-Milner logic without negation, to system specification. They obtained a (graphical) representation theorem proving that a formula can be represented by a term if, and only if, it is consistent and prime. We show in this paper that the formulae from the covariant-contravariant modal logic that admit a "graphical" representation by means of processes, modulo the covariant-contravariant simulation preorder, are also the consistent and prime ones. In order to obtain the desired graphical representation result, we first restrict ourselves to the case of covariant-contravariant systems without bivariant actions. Bivariant actions can be incorporated later by means of an encoding that splits each bivariant action into its covariant and its contravariant parts.Comment: In Proceedings EXPRESS 2011, arXiv:1108.407

Expressive Logics for Coinductive Predicates

The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas in a certain modal logic. In this paper we study this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. We formulate when a logic fully characterises a coinductive predicate on coalgebras, by providing suitable notions of adequacy and expressivity, and give sufficient conditions on the semantics. The approach is illustrated with logics characterising similarity, divergence and a behavioural metric on automata

Equational Characterization of Covariant-Contravariant Simulation and Conformance Simulation Semantics

Covariant-contravariant simulation and conformance simulation generalize plain simulation and try to capture the fact that it is not always the case that "the larger the number of behaviors, the better". We have previously studied their logical characterizations and in this paper we present the axiomatizations of the preorders defined by the new simulation relations and their induced equivalences. The interest of our results lies in the fact that the axiomatizations help us to know the new simulations better, understanding in particular the role of the contravariant characteristics and their interplay with the covariant ones; moreover, the axiomatizations provide us with a powerful tool to (algebraically) prove results of the corresponding semantics. But we also consider our results interesting from a metatheoretical point of view: the fact that the covariant-contravariant simulation equivalence is indeed ground axiomatizable when there is no action that exhibits both a covariant and a contravariant behaviour, but becomes non-axiomatizable whenever we have together actions of that kind and either covariant or contravariant actions, offers us a new subtle example of the narrow border separating axiomatizable and non-axiomatizable semantics. We expect that by studying these examples we will be able to develop a general theory separating axiomatizable and non-axiomatizable semantics.Comment: In Proceedings SOS 2010, arXiv:1008.190

Coalgebraic Geometric Logic: Basic Theory

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor on some full subcategory of the category of topological spaces and continuous functions. We investigate derivation systems, soundness and completeness for such geometric modal logics, and we we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces, again accompanied by a collection of (open) predicate liftings. Furthermore, we compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category

Generic Trace Semantics via Coinduction

Trace semantics has been defined for various kinds of state-based systems, notably with different forms of branching such as non-determinism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these "trace semantics," namely coinduction in a Kleisli category. This claim is based on our technical result that, under a suitably order-enriched setting, a final coalgebra in a Kleisli category is given by an initial algebra in the category Sets. Formerly the theory of coalgebras has been employed mostly in Sets where coinduction yields a finer process semantics of bisimilarity. Therefore this paper extends the application field of coalgebras, providing a new instance of the principle "process semantics via coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page

An ontology for software component matching

The Web is likely to be a central platform for software development in the future. We investigate how Semantic Web technologies, in particular ontologies, can be utilised to support software component development in a Web environment. We use description logics, which underlie Semantic Web ontology languages such as DAML+OIL, to develop an ontology for matching requested and provided components. A link between modal logic and description logics will prove invaluable for the provision of reasoning support for component and service behaviour

Coalgebraic Geometric Logic

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor T on some full subcategory of the category Top of topological spaces and continuous functions. We compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category. Furthermore, we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces