G-Reactive Systems as Coalgebras

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Modular Construction of Complete Coalgebraic Logics

We present a modular approach to defining logics for a wide variety of state-based systems. The systems are modelled by coalgebras, and we use modal logics to specify their observable properties. We show that the syntax, semantics and proof systems associated to such logics can all be derived in a modular fashion. Moreover, we show that the logics thus obtained inherit soundness, completeness and expressiveness properties from their building blocks. We apply these techniques to derive sound, complete and expressive logics for a wide variety of probabilistic systems, for which no complete axiomatisation has been obtained so far

Interaction and observation: categorical semantics of reactive systems trough dialgebras

We use dialgebras, generalising both algebras and coalgebras, as a complement of the standard coalgebraic framework, aimed at describing the semantics of an interactive system by the means of reaction rules. In this model, interaction is built-in, and semantic equivalence arises from it, instead of being determined by a (possibly difficult) understanding of the side effects of a component in isolation. Behavioural equivalence in dialgebras is determined by how a given process interacts with the others, and the obtained observations. We develop a technique to inter-define categories of dialgebras of different functors, that in particular permits us to compare a standard coalgebraic semantics and its dialgebraic counterpart. We exemplify the framework using the CCS and the pi-calculus. Remarkably, the dialgebra giving semantics to the pi-calculus does not require the use of presheaf categories

GSOS for non-deterministic processes with quantitative aspects

Recently, some general frameworks have been proposed as unifying theories for processes combining non-determinism with quantitative aspects (such as probabilistic or stochastically timed executions), aiming to provide general results and tools. This paper provides two contributions in this respect. First, we present a general GSOS specification format (and a corresponding notion of bisimulation) for non-deterministic processes with quantitative aspects. These specifications define labelled transition systems according to the ULTraS model, an extension of the usual LTSs where the transition relation associates any source state and transition label with state reachability weight functions (like, e.g., probability distributions). This format, hence called Weight Function SOS (WFSOS), covers many known systems and their bisimulations (e.g. PEPA, TIPP, PCSP) and GSOS formats (e.g. GSOS, Weighted GSOS, Segala-GSOS, among others). The second contribution is a characterization of these systems as coalgebras of a class of functors, parametric on the weight structure. This result allows us to prove soundness of the WFSOS specification format, and that bisimilarities induced by these specifications are always congruences.Comment: In Proceedings QAPL 2014, arXiv:1406.156

Towards a Theory of Glue

We propose and study the notions of behaviour type and composition operator making a first step towards the definition of a formal framework for studying behaviour composition in a setting sufficiently general to provide insight into how the component-based systems should be modelled and compared. We illustrate the proposed notions on classical examples (Traces, Labelled Transition Systems and Coalgebras). Finally, the definition of memoryless glue operators, takes us one step closer to a formal understanding of the separation of concerns principle stipulating that computational aspects of a system should be localised within its atomic components, whereas coordination layer responsible for managing concurrency should be realised by memoryless glue operators.Comment: In Proceedings ICE 2012, arXiv:1212.345

Towards a Uniform Theory of Effectful State Machines

Using recent developments in coalgebraic and monad-based semantics, we present a uniform study of various notions of machines, e.g. finite state machines, multi-stack machines, Turing machines, valence automata, and weighted automata. They are instances of Jacobs' notion of a T-automaton, where T is a monad. We show that the generic language semantics for T-automata correctly instantiates the usual language semantics for a number of known classes of machines/languages, including regular, context-free, recursively-enumerable and various subclasses of context free languages (e.g. deterministic and real-time ones). Moreover, our approach provides new generic techniques for studying the expressivity power of various machine-based models.Comment: final version accepted by TOC

A modular approach to defining and characterising notions of simulation

We propose a modular approach to defining notions of simulation, and modal logics which characterise them. We use coalgebras to model state-based systems, relators to define notions of simulation for such systems, and inductive techniques to define the syntax and semantics of modal logics for coalgebras. We show that the expressiveness of an inductively defined logic for coalgebras w.r.t. a notion of simulation follows from an expressivity condition involving one step in the definition of the logic, and the relator inducing that notion of simulation. Moreover, we show that notions of simulation and associated characterising logics for increasingly complex system types can be derived by lifting the operations used to combine system types, to a relational level as well as to a logical level. We use these results to obtain Baltag’s logic for coalgebraic simulation, as well as notions of simulation and associated logics for a large class of non-deterministic and probabilistic systems