Hybrid Automata and Bisimulations

This paper surveys hybrid automata and bisimulation relations. We formally introduce both notions and briefly present the model checking problem over hybrid automata. We show how, in some cases, bisimulations can be used to quotient infinite state systems to finite ones and, hence, we reduce the model checking over hybrid automata to model checking over finite models. Finally, we review some classes of hybrid automata which admit finite bisimulation quotients

Abstract Hidden Markov Models: a monadic account of quantitative information flow

Hidden Markov Models, HMM's, are mathematical models of Markov processes with state that is hidden, but from which information can leak. They are typically represented as 3-way joint-probability distributions. We use HMM's as denotations of probabilistic hidden-state sequential programs: for that, we recast them as `abstract' HMM's, computations in the Giry monad $\mathbb{D}$, and we equip them with a partial order of increasing security. However to encode the monadic type with hiding over some state $\mathcal{X}$ we use $\mathbb{D}\mathcal{X}\to \mathbb{D}^2\mathcal{X}$ rather than the conventional $\mathcal{X}{\to}\mathbb{D}\mathcal{X}$ that suffices for Markov models whose state is not hidden. We illustrate the $\mathbb{D}\mathcal{X}\to \mathbb{D}^2\mathcal{X}$ construction with a small Haskell prototype. We then present uncertainty measures as a generalisation of the extant diversity of probabilistic entropies, with characteristic analytic properties for them, and show how the new entropies interact with the order of increasing security. Furthermore, we give a `backwards' uncertainty-transformer semantics for HMM's that is dual to the `forwards' abstract HMM's - it is an analogue of the duality between forwards, relational semantics and backwards, predicate-transformer semantics for imperative programs with demonic choice. Finally, we argue that, from this new denotational-semantic viewpoint, one can see that the Dalenius desideratum for statistical databases is actually an issue in compositionality. We propose a means for taking it into account

Dagger Categories of Tame Relations

Within the context of an involutive monoidal category the notion of a comparison relation is identified. Instances are equality on sets, inequality on posets, orthogonality on orthomodular lattices, non-empty intersection on powersets, and inner product on vector or Hilbert spaces. Associated with a collection of such (symmetric) comparison relations a dagger category is defined with "tame" relations as morphisms. Examples include familiar categories in the foundations of quantum mechanics, such as sets with partial injections, or with locally bifinite relations, or with formal distributions between them, or Hilbert spaces with bounded (continuous) linear maps. Of one particular example of such a dagger category of tame relations, involving sets and bifinite multirelations between them, the categorical structure is investigated in some detail. It turns out to involve symmetric monoidal dagger structure, with biproducts, and dagger kernels. This category may form an appropriate universe for discrete quantum computations, just like Hilbert spaces form a universe for continuous computation

Kleisli morphisms and randomized congruences for the Giry monad

AbstractStochastic relations are the Kleisli morphisms for the Giry monad. This paper proposes the study of the associated morphisms and congruences. The relationship between kernels of these morphisms and congruences is studied, and a unique factorization of a morphism through this kernel is shown to exist. This study is based on an investigation into countably generated equivalence relations on the space of all subprobabilities. Operations on these relations are investigated quite closely. This utilizes positive convex structures and indicates cross-connections to Eilenberg–Moore algebras for the Giry monad. Hennessy–Milner logic serves as an illustration for randomized morphisms and congruences