Modal logics are coalgebraic

Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. The coalgebraic approach is generic and compositional: tools and techniques simultaneously apply to a large class of application areas and can moreover be combined in a modular way. In particular, this facilitates a pick-and-choose approach to domain specific formalisms, applicable across the entire scope of application areas, leading to generic software tools that are easier to design, to implement, and to maintain. This paper substantiates the authors' firm belief that the systematic exploitation of the coalgebraic nature of modal logic will not only have impact on the field of modal logic itself but also lead to significant progress in a number of areas within computer science, such as knowledge representation and concurrency/mobility

A Definition Scheme for Quantitative Bisimulation

FuTS, state-to-function transition systems are generalizations of labeled transition systems and of familiar notions of quantitative semantical models as continuous-time Markov chains, interactive Markov chains, and Markov automata. A general scheme for the definition of a notion of strong bisimulation associated with a FuTS is proposed. It is shown that this notion of bisimulation for a FuTS coincides with the coalgebraic notion of behavioral equivalence associated to the functor on Set given by the type of the FuTS. For a series of concrete quantitative semantical models the notion of bisimulation as reported in the literature is proven to coincide with the notion of quantitative bisimulation obtained from the scheme. The comparison includes models with orthogonal behaviour, like interactive Markov chains, and with multiple levels of behavior, like Markov automata. As a consequence of the general result relating FuTS bisimulation and behavioral equivalence we obtain, in a systematic way, a coalgebraic underpinning of all quantitative bisimulations discussed.Comment: In Proceedings QAPL 2015, arXiv:1509.0816

Fuzzy automata as coalgebras

The coalgebraic method is of great significance to research in process algebra, modal logic, object-oriented design and component-based software engineering. In recent years, fuzzy control has been widely used in many fields, such as handwriting recognition and the control of robots or air conditioners. It is then an interesting topic to analyze the behavior of fuzzy automata from a coalgebraic point of view. This paper models different types of fuzzy automata as coalgebras with a monad structure capturing fuzzy behavior. Based on the coalgebraic models, we can define a notion of fuzzy language and consider several versions of bisimulation for fuzzy automata. A group of combinators is defined to compose fuzzy automata of two branches: state transition and output function. A case study illustrates the coalgebraic models proposed and their composition.This work has been supported by the Guangdong Science and Technology Department (Grant No. 2018B010107004) and the National Natural Science Foundation of China under grant No. 61772038, 61532019 and 61272160. L.S.B. was supported by the ERDF—European Regional Development Fund through the Operational Programme for Competitiveness and InternationalisationCOMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT, within project KLEE - POCI-01-0145-FEDER-030947

Named Models in Coalgebraic Hybrid Logic

Hybrid logic extends modal logic with support for reasoning about individual states, designated by so-called nominals. We study hybrid logic in the broad context of coalgebraic semantics, where Kripke frames are replaced with coalgebras for a given functor, thus covering a wide range of reasoning principles including, e.g., probabilistic, graded, default, or coalitional operators. Specifically, we establish generic criteria for a given coalgebraic hybrid logic to admit named canonical models, with ensuing completeness proofs for pure extensions on the one hand, and for an extended hybrid language with local binding on the other. We instantiate our framework with a number of examples. Notably, we prove completeness of graded hybrid logic with local binding

Algebra, coalgebra, and minimization in polynomial differential equations

We consider reasoning and minimization in systems of polynomial ordinary differential equations (ode's). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow this set with a transition system structure based on the concept of Lie-derivative, thus inducing a notion of L-bisimulation. We prove that two states (variables) are L-bisimilar if and only if they correspond to the same solution in the ode's system. We then characterize L-bisimilarity algebraically, in terms of certain ideals in the polynomial ring that are invariant under Lie-derivation. This characterization allows us to develop a complete algorithm, based on building an ascending chain of ideals, for computing the largest L-bisimulation containing all valid identities that are instances of a user-specified template. A specific largest L-bisimulation can be used to build a reduced system of ode's, equivalent to the original one, but minimal among all those obtainable by linear aggregation of the original equations. A computationally less demanding approximate reduction and linearization technique is also proposed.Comment: 27 pages, extended and revised version of FOSSACS 2017 pape

On semantics and refinement of UML statecharts: a coalgebraic view

Statecharts was conceived as a visual formalism for the design of reactive systems. UML statecharts is an object-based variant of classical statecharts, incorporating several concepts different from the classical statecharts. This paper discusses a coalgebraic description of UML statecharts, directly derived from its operational semantics. In particular such an approach induces suitable notions of equivalence and (behavioral) refinement for statecharts. Finally, a few refinement laws are investigated to support verifiable stepwise system development with statecharts.(undefined

An exercise on transition systems

Labelled transition systems admit different but equivalent characterizations either as relational structures or coalgebras for the powerset functor, each of them with their own merits. Notions of simulation and bisimulation, for example, are expressed in the pointfree relational calculus in a very concise and precise way. On the other hand, the coalgebraic perspective regards processes as inhabitants of a final universe and allows for an intuitive definition of the semantics of process’ combinators. This paper is an exercise on such a dual characterisation. In particular, it discusses how a notion of weak bisimilarity can be lifted from the relational to the coalgebraic level, to become an effective reasoning tool on coinductively defined process algebras.(undefined