Helle Hvid Hansen

A Coalgebraic Perspective on Monotone Modal Logic

The paper has two main parts: First we make the connection between monotone modal logic and the general theory of coalgebras precise by defining functors UpP:Set->Set and UpV:Stone->Stone such that UpP- and UpV-coalgebras correspond to monotone neighbourhood frames and descriptive general monotone frames, respectively. Then we investigate the relationship between the coalgebraic notions of equivalence and monotone bisimulation. In particular, we show that the UpP-functor does not preserve weak pullbacks, and we prove interpolation for a number of monotone modal logics using results on UpP-bisimulations.

Neighbourhood Structures: Bisimilarity and Basic Model Theory

Neighbourhood structures are the standard semantic tool used to reason about non-normal modal logics. The logic of all neighbourhood models is called classical modal logic. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2 . We use this coalgebraic modelling to derive notions of equivalence between neighbourhood structures. 2 2 -bisimilarity and behavioural equivalence are well known coalgebraic concepts, and they are distinct, since 2 2 does not preserve weak pullbacks. We introduce a third, intermediate notion whose wit- nessing relations we call precocongruences (based on pushouts). We give back-and-forth style characterisations for 2 2 -bisimulations and precocongruences, we show that on a single coalgebra, precocongruences capture behavioural equivalence, and that between neighbour- hood structures, precocongruences are a better approximation of behavioural equivalence than 2 2 -bisimulations. We also introduce a notion of modal saturation for neighbourhood models, and investigate its relationship with denability and image-niteness. We prove a Hennessy-Milner theorem for modally saturated and for image-nite neighbourhood mod- els. Our main results are an analogue of Van Benthems characterisation theorem and a model-theoretic proof of Craig interpolation for classical modal logic.

Algebra-coalgebra duality in brzozowski's minimization algorithm

We give a new presentation of Brzozowskis algorithm to minimize finite automata using elementary facts from universal algebra and coalgebra and building on earlier work by Arbib and Manes on a categorical presentation of Kalman duality between reachability and observability. This leads to a simple proof of its correctness and opens the door to further generalizations. Notably, we derive algorithms to obtain minimal language equivalent automata from Moore nondeterministic and weighted automata.

Towards Model Checking Executable UML Specifications in mCRL2

We describe a translation of a subset of executable UML (xUML) into the process algebraic specification language mCRL2. This subset includes class diagrams with class generalisations, and state machines with signal and change events. The choice of these xUML constructs is dictated by their use in the modelling of railway interlocking systems. The long-term goal is to verify safety properties of interlockings modelled in xUML using the mCRL2 and LTSmin toolsets. Initial verification of an interlocking toy example demonstrates that the safety properties of model instances depend crucially on the run-to-completion assumptions.

Synthesis of Mealy machines using derivatives

In Rutten (2005), the theoretical basis was given for the synthesis of binary Mealy machines from specifications in 2-adic arithmetic. This construction is based on the symbolic computation of the coalgebraic notion of stream function derivative, a generalisation of the Brzozowski derivative of regular expressions. In this paper we complete the construction of Mealy machines from specifications in both 2-adic and modulo-2 arithmetic by describing how we decide equivalence of expressions via reduction to normal forms; we present a Haskell implementation of this Mealy synthesis algorithm; and a theoretical result which characterises the (number of) states in Mealy machines constructed from rational 2-adic specifications.

Bisimulation for neighbourhood structures

Neighbourhood structures are the standard semantic tool used to reason about non-normal modal logics. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 22. In our paper, we investigate the coalgebraic equivalence notions of 22-bisimulation, behavioural equivalence and neighbourhood bisimulation (a notion based on pushouts), with the aim of finding the logically correct notion of equivalence on neighbourhood structures. Our results include relational characterisations for 22-bisimulation and neighbourhood bisimulation, and an analogue of Van Benthems characterisation theorem for all three equivalence notions. We also show that behavioural equivalence gives rise to a Hennessy-Milner theorem, and that this is not the case for the other two equivalence notions.

Automated verification of executable UML models

We present a fully automated approach to verifying safety properties of Executable UML models (xUML). Our tool chain consists of a model transformation program which translates xUML models to the process algebra mCRL2, followed by symbolic model checking using LTSmin. If a safety violation is found, an error trace is visualised as a UML sequence diagram. As a novel feature, our approach allows safety properties to be specified as UML state machines.

Axiomatising Nash-Consistent Coalition Logic

We add a rule for Nash-consistency to Coalition Logic, a modal logic for reasoning about the abilities and rights of groups in multi-agent systems. Rights of agents (constitutions) can be formalised using Coalition Logic, and the additional inference rule of Nash-consistency will guarantee that any multi-agent system implementing these rights will be stable, i.e., for any preferences the agents might have, there will be rights they can exercise such that no individual deviation will be profitable. We apply this logic to obtain a formal analysis of Gibbards paradox, and we provide meta-theoretic results, in particular a complete axiomatisation.

Presenting Distributive Laws

Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural transformations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation from a distributive law for the underlying free monad. We apply this result to show the equivalence between two different representations of context-free languages.

A Coinductive Framework for Infinitary Rewriting and Equational Reasoning

We present a coinductive framework for defining infinitary analogues of equational reasoning and rewriting in a uniform way. The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence. This makes the framework especially suitable for formalizations in theorem provers.