Positive Hennessy-Milner Logic for Branching Bisimulation

Labelled transitions systems can be studied in terms of modal logic and in terms of bisimulation. These two notions are connected by Hennessy-Milner theorems, that show that two states are bisimilar precisely when they satisfy the same modal logic formulas. Recently, apartness has been studied as a dual to bisimulation, which also gives rise to a dual version of the Hennessy-Milner theorem: two states are apart precisely when there is a modal formula that distinguishes them. In this paper, we introduce ``directed'' versions of Hennessy-Milner theorems that characterize when the theory of one state is included in the other. For this we introduce ``positive modal logics'' that only allow a limited use of negation. Furthermore, we introduce directed notions of bisimulation and apartness, and then show that, for this positive modal logic, the theory of $s$ is included in the theory of $t$ precisely when $s$ is directed bisimilar to $t$. Or, in terms of apartness, we show that $s$ is directed apart from $t$ precisely when the theory of $s$ is not included in the theory of $t$. From the directed version of the Hennessy-Milner theorem, the original result follows. In particular, we study the case of branching bisimulation and Hennessy-Milner Logic with Until (HMLU) as a modal logic. We introduce ``directed branching bisimulation'' (and directed branching apartness) and ``Positive Hennessy-Milner Logic with Until'' (PHMLU) and we show the directed version of the Hennessy-Milner theorems. In the process, we show that every HMLU formula is equivalent to a Boolean combination of Positive HMLU formulas, which is a very non-trivial result. This gives rise to a sublogic of HMLU that is equally expressive but easier to reason about.Comment: 19 pages + appendices (28 pages total

Fixpoints and relative precompleteness

We study relative precompleteness in the context of the theory of numberings, and relate this to a notion of lowness. We introduce a notion of divisibility for numberings, and use it to show that for the class of divisible numberings, lowness and relative precompleteness coincide with being computable. We also study the complexity of Skolem functions arising from Arslanov's completeness criterion with parameters. We show that for suitably divisible numberings, these Skolem functions have the maximal possible Turing degree. In particular this holds for the standard numberings of the partial computable functions and the c.e. sets.Comment: 12 page

Magnetization switching in bistable nanomagnets by picosecond pulses of surface acoustic waves

International audienceWe perform a theoretical investigation of the magnetization switching in polycrystalline Ni nanoparticles induced by ultrashort pulses of surface acoustic waves via magnetoelastic interactions. In our numerical simulations, a Ni nanoparticle is modeled as an ellipsoidal disk deposited on a dielectric substrate. The in-plane external magnetic field breaks the symmetry and allows us to adjust the height of the energy barrier between two metastable magnetization states of the free-energy density and dramatically lower the amplitude of elastic strain pulses required for magnetization switching. The switching threshold is shown to depend on the duration of an acoustic pulse, the magnetic shape anisotropy of an elliptical nanoparticle, the amplitude of the external magnetic field, and the magnetostriction coefficient. We introduce the magnetoelastic switching diagram, allowing for the simultaneous visualization of the switching threshold and its characteristic timescale as a function of various physical parameters