Héloïse Beaugendre

François Morency

Innovative Model for Flow Governed Solid Motion based on Penalization and Aerodynamic Forces and Moments

Keywords: Penalization, IBM, Aerodynamic forces and moments computation, VIC scheme, Oscillating-airfoil, Fluid-solid interaction

This work proposes a fluid-solid interaction model. It is formulated for a laminar flow modeled by the incompressible Navier-Stokes equations in which we consider the presence of a rigid moving solid. The model is formulated inside an immersed boundary method based on a penalization technique. The penalization technique imposes the body effects on the flow. The velocity field is extended inside the solid region and a penalty term enforces the rigid motion inside the solid region. This technique offers a great flexibility to modify the geometry of the solid. The forces and angular momentum needed for the fluid-solid interaction model are computed with an innovative method suitable for the penalized equations method. The method is validated successfully for an oscillating airfoil in large flapping motion. Effects of different geometries on aerodynamic coefficients can then be explored. The fluid-solid interaction model is also verified for solid motions governed by the flow, such as falling cylinder and plate.

The goal of most external flow analysis in engineering is the evaluation of aerodynamic forces and moments. For complex flows, with massive separations for example, computational fluid dynamics (CFD) is often the only way to reach this goal. CFD simulations are usually performed using two types of grids: body-fitted grids and embedded grids. For body-fitted grids, external mesh faces match up with the body surfaces and external boundary faces. On the contrary, flow simulations by immersed boundary methods (IBM) use a grid that does not fit the body geometry, thus avoiding most of the difficulties associated with grid generation. IBMs can easily model flows around complex geometries in large motion [START_REF] Mittal | Immersed boundary methods[END_REF] and are well suited for simulating complex fluid-solid interactions. The Cartesian grid, most widely used with the IBM, limits the application to Euler or Laminar flow, unless a local grid refinement strategy [START_REF] Rasmussen | A multiresolution remeshed vortex-in-cell algorithm using patches[END_REF] or a specific turbulent wall model is used [START_REF] Capizzano | Turbulent wall model for immersed boundary methods[END_REF]. Methodological advances on IBM allow the simulation of compressible flows at high Reynolds numbers [START_REF] Ghias | A sharp interface immersed boundary method for compressible viscous flows[END_REF] and therefore enable its use for aeronautic applications, such as computation of droplet impingement [START_REF] Capizzano | A eulerian method for water droplet impingement by means of an immersed boundary technique[END_REF].

As discussed by Mittal et al. in [START_REF] Mittal | Immersed boundary methods[END_REF] and most recently by Sotiropoulos et al. in [START_REF] Sotiropoulos | Immersed boundary methods for simulating fluid-structure interaction[END_REF], IBMs can be classified into two categories: sharp interface methods and diffused interface methods. In the first category the presence of boundaries is taken into account at the discrete level. The goal is to insure properties conservation closed to the boundaries and improve the accuracy at the interface. The ghost-cell approach [START_REF] Ghias | A sharp interface immersed boundary method for compressible viscous flows[END_REF][START_REF] Tseng | A ghost cell immersed boundary method for flow in complex geometry[END_REF][START_REF] Gorsse | A simple second order cartesian scheme for compressible euler flows[END_REF] belongs to this category and uses, at ghost points, values that are extrapolated from the fluid to impose the appropriate boundary conditions at the interface. The sub-mesh penalty method [START_REF] Sarthou | The sub-mesh penalty method[END_REF], the immersed interface method [START_REF] Linnick | A high order interface method for simulating unsteady incompressible flows on irregular domains[END_REF] and the cut-cell method [START_REF] Collela | A cartesian grid embedded boundary method for hyperbolic conservation laws[END_REF] are other approaches belonging to this class. In the second category, the diffused interface methods avoid the difficulties associated to the accurate tracking of the fluid-solid interface position. The presence of boundaries is modeled by adding a continuous forcing term directly to the flow equations. Immersed nodes exert a force in the momentum equations. The effect of immersed boundaries is distributed on surrounding nodes using delta or mask functions. Penalization method [START_REF] Angot | A penalization method to take into account obstacles in incompressible viscous flows[END_REF][START_REF] Arquis | Sur les conditions hydrodynamiques au voisinage d'une interface milieu fluide-milieux poreux: application à la convection naturelle[END_REF] and its recent developments [START_REF] Bergmann | Modeling and simulation of fish-like swimming[END_REF][START_REF] Boiron | A high-resolution penalization method for large mach number flows in the presence of obstacles[END_REF][START_REF] Abgrall | An immersed boundary method using unstructured anisotropic mesh adaptation combined with level-sets and penalization techniques[END_REF] belongs to this category. With penalization, solid bodies are represented as porous media with a very small permeability [START_REF] Kadoch | A volume penalization method for incompressible flows and scalar advection-diffusion with moving obstacle[END_REF].

In this paper, we propose to extend the use of an IBM that combines the advantage of the penalization and of the Vortex-in-Cell (VIC) methods [START_REF] Morency | Aerodynamic force evaluation for ice shedding phenomenon using vortex in cell scheme, penalisation and level set approaches[END_REF]. The vorticity field can be seen as the signature of bluff bodies flows. Therefore, a vorticity formulation of the Navier-Stokes equations may appear as the natural framework to study these flows. Since the advection of vorticity is predominant for moderate and high Reynolds number, Lagrangian or semi-Lagrangian schemes are suitable to discretize such equations. Particle methods and VIC methods belong to this kind of approaches. Particle methods have long been used to compute vortex flows [START_REF] Koumoutsakos | High-resolution simulations of the flow around an impulsive started cylinder using vortex methods[END_REF][START_REF] Ploumhans | Vortex methods for direct numerical simulation of three-dimensional bluff body flows: application to the sphere at re = 300, 500, and 1000[END_REF][START_REF] Cottet | Vortex methods: theory and practice[END_REF][START_REF] Cottet | Advances in direct numerical simulation of 3d wall-bounded flows by vortex-in-cell methods[END_REF]. The difficulties of such methods rely on a delicate tuning of the velocity boundary conditions especially for complex geometries. In such sense, immersed boundary methods and especially, penalization, simplify the treatment of boundary conditions [START_REF] Kevlahan | Computation of turbulent flow past an array of cylinders using a spectral method with brinkman penalization[END_REF][START_REF] Coquerelle | A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies[END_REF][START_REF] Mimeau | Passive flow control around a semi-circular cylinder using porous coatings[END_REF][START_REF] Mimeau | Vortex penalization method for bluff body flows[END_REF]. Active research is performed on such methods to improve their accuracy [START_REF] Gazzola | Reinforcement learning and wevelet adapted vortex methods for simualtions of self-propelled swimmers[END_REF][START_REF] Cottet | High order semi-lagrangian particles for transport equations : numerical analysis and implementation issues[END_REF], to increase the range of applications and physical flows modelled [START_REF] Rasmussen | A multiresolution remeshed vortex-in-cell algorithm using patches[END_REF], and to remain competitive in terms of computing performances [START_REF] Rossinelli | Gpu accelerated simulations body flows using vortex particle methods[END_REF].

It is well known that the fluid interacts with the solid through pressure and viscous stress at the wall. However, for vorticity-based numerical simulation, the pressure distribution is not known. The forces can be computed from the velocity and vorticity following the method pro-posed by Noca et al. [START_REF] Noca | A comparison of methods for evaluating time-dependent fluid dynamic forces on bodies, using only velocity fields and their derivatives[END_REF], as done by Ploumhans et al. [START_REF] Ploumhans | Vortex methods for direct numerical simulation of three-dimensional bluff body flows: application to the sphere at re = 300, 500, and 1000[END_REF]. Recently, some authors have expressed some doubt about the validity of the forces predicted by this method for unsteady flows when pressure and vorticity are not uniform at infinity [START_REF] Pocwierz | Calculation of the force on solid body in the unsteady flow of incompressible fluid[END_REF]. Moreover, for vortex flow formulation, the method does not allow the computation of moments on rigid body. Eldredge [START_REF] Eldredge | Dynamically coupled fluid-body interactions in vorticity-based numerical simulations[END_REF] proposes a method to compute both forces and moment on body, based on integration along the surface of the vorticity derivative in the normal direction. For penalization, this method is not easy to apply because the vorticity is not defined at the surface but rather at cartesian grid points near the surface. Instead, we propose an innovative forces and moment computational method based on the global momentum change and angular momentum change inside the body.

The first specific objective of this paper is to propose a mathematical model for the fuid-solid interactions within IBM. The second objective is to validate the forces and momentum calculations within the VIC-IBM scheme using static and moving bodies. The flexibility offered by the use of IBM and the level-set description of the geometries are then demonstrated. IBMs are well suited for parametric study of geometry effects on the flow because no remeshing is needed when the geometry is modified [START_REF] Zhang | Towards simulation of flapping wings using immersed boundary method[END_REF]. The third objective is to verify the fluid-solid interaction model. This is done by using IBMs for the computation of flow induced body motions, such as the free fall of a solid in a fluid [START_REF] Mittal | Flutter, tumble and vortex induced autorotation[END_REF][START_REF] Glowinski | A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: Application to particulate flow[END_REF][START_REF] Bonisch | On the numerical simulation of the unsteady free fall of a solid in a fluid: I. the newtonian case[END_REF]. IBMs are particularly attractive to study motion of non spherical bodies, such as the autorotation of flat plate [START_REF] Andronov | Numerical simulation of plate autorotation in a viscous fluid flow[END_REF][START_REF] Hargreaves | The computational fluid dynamics modelling of the autorotation of square, flat plates[END_REF].

The paper is organized as follows, first, the penalized Navier-Stokes equations are presented in section 2, together with the numerical method based on a VIC scheme. The new model used to compute the forces and moment exerted by a fluid on a solid body is proposed in section 3, along with appropriate governing equations for fluid-solid interactions. In the fourth section, test cases are presented to validate forces and moments computations against literature results and examples of fluid-solid interactions are presented.

2 Penalized Navier-Stokes equations and VIC scheme

Physical model

The fluid-solid interaction flow model proposed in this work is based on an incompressible laminar Newtonian flow around a body considered as rigid (without any deformation) and delimited by level-set functions. The mass and momentum conservation equations are

∇ • u = 0 in Ω (1a) ∂u ∂t + (u • ∇) u -ν∇ 2 u + 1 ρ ∇p = 0 in Ω. ( 1b
)
where u is the velocity vector, ν = µ/ρ is the kinematic viscosity, ρ is the density, and p is the pressure. Now we consider, in Ω, the presence of a rigid moving solid S i . The boundary of S i is computed from a level set function Φ s i . Φ s i is the signed distance function to S i , typically Φ s i will be negative inside the object and positive outside, see figure 1 for an illustration of the signed distance function, embedded into a Cartesian grid, corresponding to a wing and an ice debris.

The penalization technique extends the velocity field inside the solid body, as illustrated in figure [START_REF] Rasmussen | A multiresolution remeshed vortex-in-cell algorithm using patches[END_REF], and solve the flow equations with a penalty term to enforce rigid motion inside the solid as proposed by [START_REF] Coquerelle | A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies[END_REF]. Let u s i be the rigid moving body velocity vector of S i . Inside S i , the momentum equation becomes u = u s i and remains equation (1b) outside S i . This is summarized as follows: given a very large penalization parameter, λ 1, and denoting by χ s i the characteristic function of the solid S i , i.e. χ s i = 1 inside S i and χ s i = 0 outside S i , the penalized Navier-Stokes equations are

Inria

∂u ∂t + (u • ∇) u -ν∇ 2 u + 1 ρ ∇p = λχ s i (u s i -u) for x ∈ Ω and t > 0, (2)
coupled with the incompressible mass conservation (1a). This model can easily be generalized to multiple rigid bodies S i .

To solve our governing equations the following strategies have been chosen:

1. A vortex formulation of our governing equations is used: this formulation is especially well adapted to study oscillatory motions that create large flow separation around aerodynamic bodies.

2. A vortex in cell (VIC) scheme is used to solve the obtained equations: this scheme offers less CFL restrictions than classical schemes.

3. A time splitting algorithm allows to take into account the specific requirement of each equation term, for example the implicit treatment of the penalization term for accuracy purpose.

Let us give more precision on each choice previously described.

Vortex formulation

Let us consider the penalized Navier-Stokes equation in the vorticity formulation by applying the curl operator to equation ( 2), with

ω = ∇ × u in Ω ∂ω ∂t + (u • ∇) ω = (ω • ∇) u + ν∇ 2 ω + λ∇ × [H(Φ s i )(u s i -u)] (3)
with

∇ • u = 0 in Ω . (4)
In equation (3),

χ s i = H(Φ s i ),
where H is the Heaviside function. In this paper, the vorticity field is numerically determined by a particle discretization using the scheme presented in the next section. Because the equations are not written in primitive variables, special treatments are needed to recover the velocity field and to impose the boundary conditions. Since the incompressible velocity field is divergence-free, from the vector field theory, we can define a vector potential Ψ such that

u = ∇ × Ψ . (5)
This potential vector is imposed to be solenoidal, that is ∇ • Ψ = 0, and given ω the updated vorticity field, the stream function field is computed by solving the linear Poisson equation,

∆Ψ = -ω, (6)
on the cartesian grid with boundary conditions on ∂Ω, using a Fast Fourier Transform (FFT) solver.

VIC scheme

The Vortex-In-Cell (VIC) scheme computes the non linear advection by tracking the trajectories of Lagrangian particles through a set of ODEs. An Eulerian grid is adopted to solve the velocity field, the diffusive term, and the penalization term. Given D/Dt(•), the material derivative, equation

becomes

Dω Dt = (ω • ∇) u + ν∇ 2 ω + λ∇ × [H(Φ s i )(u s i -u)] (7)
The domain Ω is meshed using a uniform fixed cartesian grid. We denote the time step ∆t, such that t n = n∆t and Φ n s i , u n , ω n are grid values of the level set functions, velocity, and vorticity. The vorticity field ω is represented by a set of particles

ω(x) = N p=1 v p ω p ζ (x -x p ) , ( 8
)
where N is the number of particles, x p the particle location, v p and ω p are the volume and the strength of a general particle p. ζ is a smooth distribution function, such that ζ(x) dx = 1, which acts on the vortex support. In vortex methods, the rate of change of vorticity is modeled by means of discrete vortex particles, such that the solution of ( 7) is localized only in the rotational regions of the flow field. This is the most important advantage of the vortex methods, that is, the computational efforts are naturally addressed only to specific flow field zones.

Splitting algorithm

A viscous splitting algorithm solves the equation ( 3). Each time step ∆t is solved using three sub-steps as follows.

1 -Advection:

Dω Dt = ∂ω ∂t + (u • ∇) ω = 0. (9a)
2 -Stretching and diffusion:

∂ω ∂t = (ω • ∇) u + ν∇ 2 ω. ( 9b
)
3 -Penalization term:

∂ω ∂t = λ∇ × (H (Φ s i ) (u s i -u)) . ( 9c
)
sub-step 1: advection Starting with ω n , grid vorticity above a certain cut-off value will create particles at grid point locations [START_REF] Cottet | A comparison of spectral and vortex methods in three-dimensional incompressible flows[END_REF], figure 3 a). Then, using equation (9a), particles are displaced with a fourth order Runge-Kutta time-stepping scheme, figure 3 b). From the new vortex particles' location, the vorticity field is remeshed on the grid, ω , by the M 4 third order interpolation kernel introduced by [START_REF] Monaghan | Extrapolating B splines for interpolation[END_REF], figure 3 c).

sub-step 2: stretching and diffusion

The equation to solve for vortex stretching and viscous contribution is given by equation (9b) applied to ω . This equation is approximated onto the grid with an Euler explicit scheme, while the Laplacian is evaluated with a second order accurate standard five points stencil. The results is noted ω .

sub-step 3: penalization

The penalization term is evaluated using equation (9c). In our simulations, λ is fixed to 10 8 /∆t. An implicit Euler time discretization is used to approximate u n+1 in the penalization term:

u n+1 = u + λ∆tH(Φ s i )u n s i 1 + λ∆tH(Φ s i ) . ( 10
)
where u is the velocity computed using ω resulting of sub-step 2 (∆Ψ = -ω and u = ∇ × Ψ ). The vorticity field at t n+1 is then evaluated on the grid by taking the curl of the velocity, ω n+1 = ∇ × u n+1 , and computing the derivative through the second order centered finite differences approximation. This method is unconditionally stable and enables to take a very large penalization parameter to insure accuracy.

Inria 3 Aerodynamic forces and moment computations and fluidsolid interaction model

The penalization term in equation ( 3), the last term on the right hand side, can also be considered as being the rigid body effect on fluid. At each time step, the penalization term forces the velocity inside the rigid body S i to be equal to u s i . In velocity formulation, the local momentum change imposed by the penalization is computed by

∂u ∂t = λH(Φ s i )(u s i -u) (11)
In our formulation, the velocity u is implicitly calculated by [START_REF] Linnick | A high order interface method for simulating unsteady incompressible flows on irregular domains[END_REF]. Thus, the penalization term become

∂u ∂t = λH(Φ s i ) u n s i - u + λ∆tH(Φ s i )u n s i 1 + λ∆tH(Φ s i ) ( 12
)
∂u ∂t = λH(Φ s i ) u n s i (1 + λ∆tH(Φ s i )) -(u + λ∆tH(Φ s i )u n s i ) 1 + λ∆tH(Φ s i ) ( 13
)
∂u ∂t = λH(Φ s i ) u n s i -u 1 + λ∆tH(Φ s i ) . ( 14
)
The global momentum change is obtained by integration over the computational solid domain S i and the forces are defined as

F = S i ρ f λH(Φ s i ) 1 + λ∆tH(Φ s i ) (u n s i -u )dx. (15)
In a similar way, the angular momentum change created by the penalization term requires an integration over the solid domain S i and the instantaneous pitching moment is defined by

T = S i ρ f λH(Φ s i ) 1 + λ∆tH(Φ s i ) r × (u n s i -u )dx. (16)
In 2D flows, the drag, lift and pitching moment coefficients are defined respectively as

C X = F x 1/2ρ f U 2 ∞ c C Y = F y 1/2ρ f U 2 ∞ c C m = T 1/2ρ f U 2 ∞ c 2 . ( 17
)
where the x axis is aligned with the far field velocity vector U ∞ , the y axis is perpendicular to the far field velocity vector and c is the airfoil chord. For an imposed motion of the solid S i , given u s i we compute the aerodynamic coefficients given by equation [START_REF] Kadoch | A volume penalization method for incompressible flows and scalar advection-diffusion with moving obstacle[END_REF].

The model for fluid-solid interaction is different from the one used in the previous paper [START_REF] Beaugendre | Computation of ice shedding trajectories using cartesian grids, penalization, and level sets[END_REF]. For a fluid-solid interaction the forces acting on the solid S i induce its motion, the governing equations are then given by

∂u s i ∂t = F M a + (ρ s i -ρ f ) ρ s i G ( 18
)
∂θ ∂t = T M i (19)
where M a is the mass of the solid S i , M i its moment of inertia, ρ s i its density, ρ f the fluid density and G the gravity vector. The formalism can be easily extended to multiple solids.

Numerical results

In this section, forces and moments computations are validated using a static cylinder and an oscillating flapping wing motion. Examples of fluid-solid interactions are presented to show the capabilities of the IBM.

Static cylinder test case

The computations are done on a square domain of size [-1.85, 6.15] × [-4, 4]. The centre of the cylinder, whose diameter D is 0.3, is at (0, 0). The far field velocity is U ∞ = 1. The fluid viscosity is selected to achieve the desired Reynolds number Re = 150. The potential flow solution around a cylinder defines reliable values of the flow field at domain boundaries. For example, on the top and bottom boundaries a Dirichlet condition is enforced. That is

Ψ = U ∞ y 1 - (D/2) 2 x 2 + y 2 top, bottom. (20)
This is equivalent to impose a symmetry condition on top and bottom boundaries. Neumann boundary conditions are imposed at upstream and downstream locations, that is

∂Ψ ∂x = 2U ∞ (D/2) 2 xy (x 2 + y 2 ) 2 upstream, downstream. (21)
Three meshes with h = 1/128, 1/256 and h = 1/512, where h is the grid spacing, are used for computations. The time step dt for computation is evaluated by

dt = (0.5h) 2 Re ( 22
)
where the Reynolds number is defined as Re = U ∞ D/ν. Table 1 compares some results obtained with the three meshes to evaluate grid sensitivities of the results. The forces F x and F y are obtained with the Noca formulation [START_REF] Noca | A comparison of methods for evaluating time-dependent fluid dynamic forces on bodies, using only velocity fields and their derivatives[END_REF] and Momentum formulation equation [START_REF] Boiron | A high-resolution penalization method for large mach number flows in the presence of obstacles[END_REF]. The mean drag coefficient C D , C X averaged over one period and the Strouhal number St are tabulated. The St value depends on the shedding frequency f or on the dimensionless time period

T p St = f D U ∞ = 1/T p .

Inria

The dimensionless time period is evaluated by looking at the lift coefficient, C Y , evolution in time. For three significant digits, the St values do not depend on the force formulation or on the mesh size. It is expected that for a fine enough mesh, the two formulations should give the same results. The average mean drag coefficient values are between the values of 1.44 from Lai and Peskin [START_REF] Lai | An immersed boundary method with formal second-order accuracy and reduced numerical viscosity[END_REF] and 1.334 from Liu al. [START_REF] Liu | Preconditioned multigrid methods for unsteady incompressible flows[END_REF]. The Strouhal number are close to the value of 0.184 computed in [START_REF] Lai | An immersed boundary method with formal second-order accuracy and reduced numerical viscosity[END_REF].

Validation of forces through an oscillating airfoil

Flapping wing motions are extensively studied for engineering applications in low Reynolds numbers flow where classical fixed wing geometry performance decreases, [START_REF] Gopalan | Lift enhancement of flapping airfoils by generalized pitching[END_REF]. According to previous works, around ten parameters influence the power extraction in flapping wing motions, such as oscillation frequencies and amplitudes (translational and rotational), phase difference between plunge and pitch motion, viscosity, free stream velocity, flapping pattern and airfoil geometry. In this section, we will study the effect of the pitching position and of the geometry's shape, after the validation of unsteady forces calculations.

For validation, an oscillating airfoil experiencing simultaneous pitching θ(t) and heaving h(t) motions is modelled. The infinitely long wing is based on a NACA 0015 airfoil. The pitching axis is located along the airfoil chord at the position (x p , y p ) = (1/3, 0). The airfoil motion, described by Kinsey and Dumas [START_REF] Kinsey | Parametric study of an oscillating airfoil n a power-extraction regime[END_REF], is defined by the heaving h(t) and the pitching angle θ(t) as follows

h(t) = H 0 sin (ωt + Φ) θ(t) = θ 0 sin (ωt) (23)
where H 0 is the heaving amplitude and θ 0 is the pitching amplitude. The angular frequency is defined by ω = 2πf and the phase difference Φ is set to 90 o . The heaving velocity is then given by

V y (t) = H 0 ω cos(ωt + Φ) . (24)
Based on the imposed motion and on the upstream flow conditions, the airfoil experiences an effective angle of attack α(t) and an effective upstream velocity V ef f (t) defined by

α(t) = arctan(-V y (t)/U ∞ ) -θ(t) V ef f (t) = U 2 ∞ + V 2 y (t) , (25)
where the freestream velocity far upstream of the oscillating airfoil is U ∞ = 1.

To validate our simulations, a regime corresponding to the parameters Re = U ∞ c ν = 1100, H 0 /c = 1, f = 0.14, x p /c = 1/3 and θ 0 = 76.33 o has been computed. A view of the motion is sketched in figure 4. Our numerical results, obtained using dx = dy = 1 × 10 -3 , dt = 5 × 10 -4 and λ = 1 × 10 8 , are then compared to the forces predictions presented by Kinsey et al. [START_REF] Kinsey | Parametric study of an oscillating airfoil n a power-extraction regime[END_REF] and by Campobasso et al. [START_REF] Campobasso | Compressible navier-stokes analysis of an oscillating wing in a power-extraction regime using efficient low-speed preconditioning[END_REF]. Results of instantaneous forces, equation ( 17), C X , C Y and pitching moments C m , figure 5, are in good agreements with literature results. Our solution slightly overestimates the drag coefficient amplitude and the minimum/maximum of the lift coefficient but correctly predicts the curve shape of those coefficients. The momentum coefficient is correctly predicted as well, our curve is slightly shifted to the left compared to literature results (corresponding to a time lag), figure (5c). A mesh sensitivity study using different mesh sizes and different domain sizes has been performed to verify the sensibility of the force predictions. All the simulations used the same time step fixed to dt = 5 × 10 -4 . Figures 6 (a), (b) and (c) compare C X , C Y and C m obtained with three different meshes corresponding to a coarse mesh dx = dy = 5 × 10 -3 , a medium mesh dx = dy = 2 × 10 -3 and a fine mesh dx = dy = 1 × 10 -3 and demonstrate the consistency of the method. Each mesh allows to compute a good approximation of the forces and moment evolution through a time period. As the mesh becomes finer the solution converges to the solution obtained on the finest mesh. Simulations using the medium grid scale dx = dy = 2 × 10 -3 and two different domains, one domain of size [-3, 8] × [-4, 4] referenced as dom1 and the other one of size [-3, 8] × [-6, 6] referenced as dom2, have also been performed to investigate boundary conditions effects and more especially blocking effects on the solution. Figure 6 (d) demonstrates that a small blocking effect is observable on this test case. Indeed, reducing slightly the height of the domain leads to an amplitude increase of the C X coefficient. Since the comparisons with the literature results have been done on the smallest domain, the blocking effects are most probably responsible of the overestimation noticed on the C X curve. However, even the coarse mesh on the small domain exhibits the correct behavior of the solution. The following studies are then performed using the smallest domain and the medium grid scale to ensure a good compromise between the quality of the solution and cpu time.

Flexibility offered by penalization

Penalization technique offers a simple way to consider different geometries for the solid S i because the same mesh can be used for all simulations. To illustrate this capability of the method three kind of geometries have been selected. The NACA 0015 profile, a rounded edge rectangular plate with a ratio chord/thickness = 15% and a NACA 0040 profile. Those geometries are sketched on figure (7a). The method enables also to easily modify the position of the pitching axis, thus two different positions have been selected (x p , y p ) = (1/3, 0) and (x p , y p ) = (1/2, 0). Those geometries evolve using the same motion as described in section 4.2.

For the first position of the pitching axis (1/3, 0), the motion of tow of the three geometries is sketched on figures 4 & 7(b). Figure [START_REF] Gorsse | A simple second order cartesian scheme for compressible euler flows[END_REF] compares C X , C Y and C m for the three geometries. As expected for such a low Reynolds number flow the results obtained with the NACA 0015 airfoil and the rectangular plate of the same thickness are very similar. The thicker airfoil on the contrary exhibits a different behavior. As in the work done by Ashraf al. [START_REF] Ashraf | Reynolds number, thickness and camber effects on flapping airfoil propulsion[END_REF], depending on the Reynolds number, flow around thicker flapping airfoils can be radically different than flow around thin geometries. Indeed, results of the mean power, table [START_REF] Rasmussen | A multiresolution remeshed vortex-in-cell algorithm using patches[END_REF], show that thin geometries extract 1.5 to 3 times more power than thick geometry. The instantaneous power results from the sum of the heaving contribution F y V y (t) and the pitching contribution T θ(t), where T is the resulting torque about the pitching axis x p . The mean power extracted over one cycle can thus be computed using equation [START_REF] Mimeau | Vortex penalization method for bluff body flows[END_REF].

C power = 1 0 C Y (t) V y (t) U ∞ + C m (t) θ(t)c U ∞ dt. (26)
Moving the pitching position from the leading edge to the middle of the wing increases the mean power extracted for the thick airfoil and on the contrary decreases the mean power for thin geometries, table 2.

The results related to the motion corresponding to the second pitching axis location (x p , y p ) = (1/2, 0) are presented in figures [START_REF] Linnick | A high order interface method for simulating unsteady incompressible flows on irregular domains[END_REF] and [START_REF] Collela | A cartesian grid embedded boundary method for hyperbolic conservation laws[END_REF]. First, the motion using this pitching axis loaction is sketched in figure [START_REF] Sarthou | The sub-mesh penalty method[END_REF]. Figure [START_REF] Linnick | A high order interface method for simulating unsteady incompressible flows on irregular domains[END_REF] compares the evolution over one cycle of C X , C Y and C m for the three geometries. Again the NACA 0015 airfoil and the rectangular plate exhibit comparable behaviors, the curves evolving with a similar pattern. The thick airfoil results remain different.

The modification of the pitching axis position reduces the C X coefficient for the three geometries, figures (11a-d-g). The pitching moment coefficient become antisymmetric as expected thanks to the symmetry of the motion, figures (11c-f-i). For thin geometries, the modification of the pitching location leads to related modification of the curves. C X , C Y and C m curves from

Fluid-solid interaction

In this sub-section, trajectories of objects are computed using the proposed fluid-solid interaction model. We will first validate our model against a 2D falling cylinder to compare with literature results. Then we will use the rectangular plate described previously to study the effects of density ratios on the trajectories.

Falling cylinder

This test case is representative of the sedimentation of a 2D cylinder on a flat plate. We consider the case of a 2D cylinder in a square cavity, falling under gravity on a flat plane. The dimension of the cavity is [0, 2] × [0, 6]. The viscosity is 0.01. The density inside and outside the cylinder is, respectively, 1.5 and 1. The mesh spacing is dx = dy = 3.9 × 10 -3 .The cylinder has a radius of 0.125, no roughness, and its barycenter is initially located at the point [START_REF] Mittal | Immersed boundary methods[END_REF][START_REF] Ghias | A sharp interface immersed boundary method for compressible viscous flows[END_REF]. To impose wall boundary conditions on each wall of the cavity we use a penalization layer of 10 cells all around the cavity, this is the reason why the domain, in figure (12a) appears slightly larger and longer than the cavity definition. A snapshot of the vorticity field, at time t = 0.3, is depicted in figure (12a). The vorticity along the cavity walls results from blocking effects. The falling cylinder creates an upward flow to satisfy the mass conservation. The cylinder accelerates under gravity, set to g = -980, then stabilizes its velocity around a steady velocity, due to equilibrium between friction forces, buoyancy and gravity. Then it hits the bottom of the cavity and bounces, figure (12b). Information on the collision model used can be found in [START_REF] Morency | Mathematical model for ice wall interactions within a level set method[END_REF], the parameters chosen for the linear spring collision model are the Young modulus set to E = 1.083e9, the Poisson's coefficient set to 0.8235 and the friction coefficient set to 0.1. The evolution of the vertical velocity component v as a function of time is plotted in figure [START_REF] Angot | A penalization method to take into account obstacles in incompressible viscous flows[END_REF] and is in agreement with the solution of Coquerelle et al. [START_REF] Coquerelle | A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies[END_REF]. The influence of the time step on the vertical velocity is also presented in this figure showing the consistency of the model to simulate the physical phenomenon.

Falling rectangular plate

We use the previous rectangular plate, from sub-section 4.3 for which the gravity center of the plate is positionned at (0, 0) with a rotation angle of attack of π/3 with the horizontal axis, the computational domain is set to [START_REF] Ghias | A sharp interface immersed boundary method for compressible viscous flows[END_REF][START_REF] Ghias | A sharp interface immersed boundary method for compressible viscous flows[END_REF]. First we compute the flow motion around the fixed rectangular plate. The Reynolds number is set at Re = 1100 as in the previous study using this solid body, see section 4.3. A snapshot of the vorticity field around the fixed plate is presented in figure [START_REF] Arquis | Sur les conditions hydrodynamiques au voisinage d'une interface milieu fluide-milieux poreux: application à la convection naturelle[END_REF]. At time t = 0.5, the fluid-solid interaction is started. Once the fluid-solid interaction occurs, the aerodynamic forces and moment acting on the solid induce its motion. During these simulations, the pivot axis of the rectangular plate coincides with its barycenter. The first simulations consist in investigating the sensitivity of the trajectory according to the mesh. In those simulations ρ s = ρ f and the time step dt is fixed to 5 × 10 -4 . Figure (14a) presents the position of the plate every 400 steps using 5 meshes. The plate trajectory is very stable according to the grid size. As can be seen through figures (14b-d) the evolution in time of the velocities (u-component, v-component and angular velocity) is similar. The velocities value evolves smoothly from the coarser mesh to the finer mesh. The angle of rotation with the initial position, figure (14d) induce a similar motion of the plate for all the meshes. The range of the trajectory and the reactivity of the rectangular plate to the flow is slightly affected by the grid size.

[-3, 8] × [-
To verify the influence of plate's density on the trajectory, four density ratios have been selected ρ s /ρ f = 2; 1.75; 1.5 and 1.25. The simulations have been performed using the medium mesh dx = dy = 3. × 10 -3 with a time step of dt = 5 × 10 -4 and a domain of [-3, 5] × [-9, 3]. As expected, from previous work in literature [START_REF] Mittal | Flutter, tumble and vortex induced autorotation[END_REF], the plate starts to auto-rotate, figure (15a). The auto-rotation starts sooner for larger density ratio. Again, figure (15a), shows the position of the plate every 400 time steps for each density ratio. The falling velocity, v (figure 15d), of the plate increases with the plate weight. The vertical acceleration of the plate destabilizes its orientation and the plate starts to rotate around its barycenter axis, figure (15e). The modification of the plate orientation with the flow reduces the falling velocity (the gravity force is compensated by buoyancy and aerodynamic forces), figure (15d). Due to its initial orientation and its weight, at the beginning of the fluid-solid interaction, the plate glides to the left (figures 15b & c). As expected, the lighter plate is more affected by the flow and is pushed to the right by the flow. For the cylinder and the rectangular plate test cases the fluid-solid interaction model behaves consistently for varying density ratio, mesh size and time step.

Conclusions

This paper has presented an original way of computing aerodynamic forces and moments along with the governing equations to use them in order to simulate fluid-solid interactions. Forces and moments have been validated within the VIC-IBM scheme using static and dynamic bodies. Predicted forces and moment agree with literature for a static cylinder and a flapping wing motion. The flexibility offered by the use of IBM and the level-set description of the geometries has been demonstrated. The fluid-solid interaction model has been compared to another approach for a falling cylinder test case and the falling velocity agrees with literature results. For varying density ratio, mesh size and time step, the fluid-solid interaction model predicts realistic trajectories for a rectangular plate. Further experimental results are needed to properly validate this model. Since aeronautic applications will be considered with the computation of ice shedding trajectories a specific turbulent wall model is under development.

Ackowledgement

Numerical experiments presented in this paper were carried out using the PLaFRIM experimental testbed and the BOREAS cluster. BOREAS cluster is funded by the Natural Sciences and Engineering Research Council of Canada. PLaFRIM cluster being developed under the Inria PlaFRIM

Figure 1 :

 1 Figure 1: Global level-set function of two rigid bodies (wing and ice debris) computed onto a Cartesian grid.

Figure 2 :

 2 Figure 2: Penalization technique, the flow field is extended inside the bodies, u component of the velocity, v component of the velocity and vorticity.

Figure 3 :

 3 Figure 3: Particles interpolation scheme. The circle's size denotes the strenght of the particle and the solid circles represent the advected particles. a) vortex particles and velocity field; b) advection step; c) remesh-diffusion step.

Figure 4 :

 4 Figure 4: Oscillating airfoil, sketch of the airfoil motion for H 0 /c = 1, f = 0.14, x p /c = 1/3 and θ 0 = 76.33 o .

Figure 5 :

 5 Figure 5: Oscillating airfoil, comparison with the results obtained by Kinsey et al.[46] and Campobasso et al. [47] (a) Drag coefficient, (b) Lift coefficient, (c) Pitching moment.

Figure 6 :

 6 Figure 6: Mesh sensitivity on the drag coefficient (a) on the lift coefficient (b) and on the pitching moment (c). For the medium mesh, dx = dy = 2×10 -3 influence of the domain size on calculated forces and moment, dom1 = [-3, 8] × [-4, 4], dom2=[-3, 8] × [-6, 6].

Figure 7 :

 7 Figure 7: a) Geometries of the NACA 0015 profile, NACA 0040 profile and the rectangular plate. b) Motion of the rectangular plate for a pitching position at x p = 1/3, y p = 0.

Figure 8 :

 8 Figure 8: (a) C X , (b) C Y , (c) Cm corresponding to the 3 geometries for a pitching axis at x p = 1/3, y p = 0.

Inria

Figure 9 :

 9 Figure 9: Motion of the geometries for a pitching position at x p = 1/2, y p = 0.

Figure 10 :

 10 Figure 10: (a) C X , (b) C Y , (c) Cm corresponding to the 3 geometries for a pitching position at x p = 1/2, y p = 0.

Figure 11 :

 11 Figure 11: Pitching axis position comparison (a-b-c) plate, (d-e-f) NACA 0015, (g-h-i) NACA 0040.

Figure 12 :

 12 Figure 12: Sedimentation of a 2D falling cylinder. a) Vorticity field at time t = 0.3. b) Sensitivity of the solution against the time step and comparison against Coquerelle's velocity profile [24] .

Figure 13 :

 13 Figure 13: Solution around the static rectangular plate, vorticity contours (negative positive in red).

Figure 14 :

 14 Figure 14: Grid sensitivity study on the rectangular plate trajectories: (a) position of the plate along the trajectories; (b) evolution of u-velocity component (c) evolution of v-velocity component (d) evolution of the angular velocity and (e) evolution of the rotational angle.

Figure 15 :

 15 Figure 15: Density investigation, the solid-fluid interaction starts at t = 5, mesh size h = 3×10 -3 (a) trajectories; (b) position of barycenter; (c) u-velocity; (d) v-velocity; (e) angular-velocity; (f) angle.

  Modèle novateur pour le mouvement d'un solide entraîné par l'écoulement basé sur la pénalisation et le calcul des forces et moments aérodynamiques Résumé : La méthode d'interaction fluide-solide proposée dans ce rapport est formulée pour un écoulement laminaire. L'écoulement est modélisé par les équations de Navier-Stokes incompressibles modifiées, dans lesquelles nous considérons la présence d'un corps rigide en mouvement. Le modèle est formulé dans le cadre d'une méthode de frontières immergées basée sur une technique de pénalisation pour tenir compte du corps. Le champ de vitesse est étendu dans la région du corps rigide et un terme de pénalisation impose le mouvement rigide à l'intérieur du solide. Cette technique offre une grande souplesse pour modifier la géométrie du corps. Une méthode innovante appropriée pour le calcul des forces et des moments angulaires basée sur les équations pénalisées est proposée et validée pour une aile oscillante dans un grand mouvement de battement. Les effets de différentes géométries peuvent ensuite être explorés. Le modèle d'interaction fluide-solide est aussi vérifié pour des mouvements de solides gouvernés par l'écoulement, comme le cas d'un cylindre ou d'une plaque en chute libre.

Mots-clés : Pénalisation, Méthode des frontières immergées, Forces aérodynamiques, Moment aérodynamique, Schéma VIC, Profil oscillant, Interaction fluide-structure 1 Introduction

Table 1 :

 1 Static cylinder, values of mean C D C D and St obtained at Re=150 with three different

	grids
	h	Methods	C D	St
	1/128	Noca	1.475 0.173
		Momentum 1.523 0.173
	1/256	Noca	1.381 0.173
		Momentum 1.386 0.172
	1/512	Noca	1.347 0.173
		Momentum 1.356 0.173

RR n°8718

Beaugendre & Morency


Emiliano Lorini
email: lorini@irit.fr

Guifei Jiang

Laurent Perrussel

Trust-based belief change

We propose a modal logic that supports reasoning about trust-based belief change. The term trust-based belief change refers to belief change that depends on the degree of trust the receiver has in the source of information.

Introduction

Trust in information sources plays a crucial role in many domains of interaction between agents, in particular when information sources are either human agents or software agents (e.g., banks, companies, consultants, etc.), typical examples are in the field of e-commerce or in the field of stock and bond market. In the latter case, an agent may receive information from a given source about the evolution of a stock's price. In these situations, the agent's trust in the source has an influence on the dynamics of the belief about the evolution of the stock's price. The latter belief is fundamental for the agent to decide whether to buy or sell stocks.

The aim of this paper is to improve understanding of the relationship between belief and trust: we propose a logic for reasoning about trust-based belief change, that is, belief change that depends on the degree of trust the receiver has in the information source. We call this logic DL-BT which stands for Dynamic Logic of graded Belief and Trust. Using this logic, we stress out the interplay between trust and belief change in a modular way. As opposed to numerous approaches such as [START_REF] Barber | Belief revision process based on trust: Agents evaluating reputation of information sources[END_REF] where the interplay is predefined and thus specific, the logic DL-BT allows to implement different trust-based belief change policies.

On the technical level, the logic DL-BT consists of extending Liau's static modal logic of belief and trust [START_REF] Liau | Belief, information acquisition, and trust in multi-agent systems: a modal logic formulation[END_REF] in three different directions: (i) a generalization of Liau's approach to graded trust, (ii) its extension by modal operators of knowledge and by modal operators of graded belief based on Spohn's theory of uncertainty [START_REF] Spohn | Ordinal conditional functions: a dynamic theory of epistemic states[END_REF], and (iii) by a family of dynamic operators in the style of dynamic epistemic logics (DEL) [START_REF] Van Ditmarsch | Dynamic Epistemic Logic[END_REF]. The latter allows for the representation of the consequences of a trust-based belief change operation while the second enables to handle iterated belief change.

Our contribution is twofold. First of all, our concept of trust-based belief change does not presuppose that incoming information is necessarily incorporated in the belief set of the agent. This is a key difference with classical belief revision [START_REF] Alchourrón | On the logic of theory change: partial meet contraction and revision functions[END_REF] whose primary principle of change (or success postulate) leads any agent to accept new information and to revise her beliefs accordingly. This postulate has been widely criticized in the literature and several approaches of nonprioritized belief revision have been proposed [START_REF] Hansson | A survey of non-prioritized belief revision[END_REF]. Credibility-limited revision approach [START_REF] Hansson | Credibility limited revision[END_REF][START_REF] Booth | Credibility-limited revision operators in propositional logic[END_REF] assumes that revision will be successful only if new information is credible, in the sense it does not conflict with the current beliefs of the agent. Differently from this approach, our key criterion for acceptance of new information is not its credibility but trust in the information source.

Secondly, our logic DL-BT provides a solution to the problem of representing the author of a communicative act in the DELframework. Indeed, existing dynamic epistemic logics [START_REF] Plaza | Logics of public communications[END_REF][START_REF] Baltag | The logic of public announcements, common knowledge and private suspicions[END_REF] do not specify the author of the announcement, as they assume that the announcement is performed by some agent outside the system that is not part of the logic's object language.

The paper is organized as follows. We first present the syntax and the semantics of the logic DL-BT and detail two trust-based belief change policies: an additive policy and a compensatory policy. The additive policy cumulates information received by different information sources. In case different sources provide conflicting information, the compensatory policy balances them depending on how much they are trustworthy. We then provide a sound and complete axiomatization for the variant of DL-BT implementing these two policies.

Dynamic logic of graded belief and trust

In the next two sections we present the syntax and semantics of the logic DL-BT that combines modal operators of knowledge, graded belief and trust with dynamic operators of trust-based belief change.

Syntax of DL-BT

Let Atm = {p, q, . . .} be a countable set of propositional atoms and let Agt = {i, j, . . .} be a finite set of agents. Moreover, let Num = {0, . . . , max} be a finite set of natural numbers with max ∈ N \ {0} which represents the scale for trust and belief degrees. For instance, the set Num = {0, 1, 2, 3, 4, 5} can be interpreted as a qualitative scale where 0 stands for 'null' and 5 for 'very high'. Finally, let Plc be a set of trust-based belief change policies.

Let us stress that DL-BT should be conceived as a "family" of logics rather than a single logic, each of which is parameterized by a certain set of trust-based belief change policies Plc. Hereafter, a specific member of the DL-BT family indexed by some set Plc, is denoted DL-BT Plc .

The language L of DL-BT is defined by the following grammar:

ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | Kiϕ | B ≥α i ϕ | T α i,j ϕ | [ * f i ϕ]ϕ
where p ranges over Atm, i and j range over Agt, α ranges over Num\{0}, and f ranges over the set of total functions with domain Agt and codomain Plc.

The other boolean constructions ⊤, ⊥, ∨, →, ↔ are defined in the standard way. Let Obj be the set of all boolean combinations of atoms in Atm.The elements of Obj are called objective formulas.

Ki is the standard S5 epistemic operator [START_REF] Fagin | Reasoning About Knowledge[END_REF]: Kiϕ stands for "agent i knows that ϕ is true".

The formula B ≥α

i ϕ has to be read "agent i believes that ϕ is true with strength at least α". Similar operators of graded belief have been studied in the past by [START_REF] Aucher | A combined system for update logic and belief revision[END_REF][START_REF] Van Ditmarsch | Prolegomena to dynamic logic for belief revision[END_REF][START_REF] Laverny | From knowledge-based programs to graded belief-based programs, part ii: off-line reasoning[END_REF]. The formula [ * f i ϕ]ψ has to be read "after agent i has publicly announced that ϕ is true and each agent j has revised her beliefs according to the trust-based belief change policy f (j), ψ will be true". In other words, the dynamic operator [ * f i ϕ] allows to represent the effect of agent i's announcement of ϕ: each agent revises her beliefs according to the trust-based belief change policy prescribed by the function f . Finally, the formula T α i,j ϕ has to be read as "agent i trusts agent j's judgement on formula ϕ with strength α". Note that, when i = j, the operator T α i,j captures a notion of self-trust (or self-confidence).

We will use the following abbreviations in the rest of the paper. For all i ∈ Agt and for all α ∈ Num \ {0, max} we define:

K i ϕ = def ¬K i ¬ϕ B i ϕ = def B ≥1 i ϕ B i ϕ = def ¬B i ¬ϕ U i ϕ = def ¬B i ϕ ∧ ¬B i ¬ϕ B α i ϕ = def B ≥α i ϕ ∧ ¬B ≥(α+1) i ϕ B max i ϕ = def B ≥max i ϕ B 0 i ϕ = def ¬B i ϕ T i,j ϕ = def α∈Num\{0} T α i,j ϕ
Ki is the dual of Ki and Kiϕ has to be read "ϕ is compatible with agent i's knowledge".

The operator Bi captures the concept of belief and Biϕ has to be read "agent i believes that ϕ is true". Indeed, we assume that "believing that ϕ is true" is the same as "believing that ϕ is true with strength at least 1".

Bi is the dual of Bi and Biϕ has to be read "ϕ is compatible with agent i's beliefs". The operator Ui captures the concept of uncertainty or doubt, and Uiϕ has to be read "agent i is uncertain whether ϕ is true". The operator B α i captures the exact degree of belief. Specifically, B α i ϕ has to be read "agent i believes that ϕ is true with strength equal to α". The special case B max i ϕ needs to be defined independently since B max+1 i ϕ is not a well-formed formula. The abbreviation B 0 i ϕ has to be read "agent i believes that ϕ with strength 0" which is the same thing as saying that agent i does not believe ϕ. Finally, Ti,jϕ has to be read "agent i trusts agent j's judgment on ϕ".

We call L-BT the static fragment of DL-BT, that is, DL-BT formulas with no dynamic operators [ * f i ϕ]. The language L-BT is defined as follows (as previously, p ranges over Atm, i and j range over Agt and α ranges over Num\{0}):

ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | Kiϕ | B ≥α i ϕ | T α i,j ϕ

Semantics for the static fragment

Let us first focus on the semantics of the static L-BT formulas. Semantics is defined in terms of possible worlds with a special function for ranking worlds according to their plausibility degrees, and a family of neighbourhood functions for trust.

Definition 1 (Model).

A model is a tuple M = (W, {Ei}i∈Agt , κ, {Ni,j}i,j∈Agt , V) where:

• W is a nonempty set of possible worlds or states;

• every Ei is an equivalence relation on W with Ei(w) = {v ∈ W :
wEiv} for all w ∈ W ; • κ : W × Agt → Num is a total function mapping each world and each agent to a natural number in Num such that:

(Constr1) for every w ∈ W and for every i ∈ Agt, there is v ∈ W such that wEiv and κ(v, i) = 0;

• Ni,j : W × Num\{0} -→ 2 2 W
is a total function such that for all w ∈ W , for all i, j ∈ Agt, for all X ⊆ W and for all α, β ∈ Num\{0}:

(Constr2) if X ∈ Ni,j(w, α) and α = β then X ∈ Ni,j(w, β); (Constr3) if X ∈ Ni,j(w, α) then X ∈ Ni,j(v, α) for all v ∈ Ei(w); (Constr4) if X ∈ Ni,j(w, α) then X ∩ Ei(w) = ∅; • V : W -→ 2
Atm is a valuation function for propositional atoms.

As usual, p ∈ V(w) means that proposition p is true at world w. The set Ei(w) is agent i's information set at world w: the set of worlds that agent i envisages at world w. As Ei is an equivalence relation, if wEiv then agent i has the same information set at w and v (i.e., agent i has the same knowledge at w and v).

The function κ provides a plausibility grading of the possible worlds for each agent i. κ(w, i) = α means that, according to agent i the world w has a degree of exceptionality α or, alternatively, according to agent i the world w has a degree of plausibility maxα. Indeed, following [START_REF] Spohn | Ordinal conditional functions: a dynamic theory of epistemic states[END_REF], we assume that the degree of plausibility of a world for an agent is the opposite of its exceptionality degree.

(Constr1) is a normality constraint for the plausibility grading which ensures that an agent can always envisage a world with a minimal degree of exceptionality 0. This constraint is important because it ensures that an agent's beliefs are consistent, e.g., an agent cannot believe ϕ and ¬ϕ at the same time (see below for more details).

The neighbourhood function N i,j specifies a trust grading of the subset of possible worlds and is used to interpret the graded trust formulas T α i,j ϕ. Since each set of possible worlds X ⊆ W is the semantic counterpart of a L-BT formula, the meaning of X ∈ Ni,j(w, α) is that, at world w, agent i trusts agent j's judgment on the truth of the formula corresponding to X with strength α. (Constr2)-(Constr4) are natural constraints for trust. Specifically, (Constr2) requires that an agent cannot trust the same agent on the same formula with different strengths. (Constr3) corresponds to a property of positive introspection for trust, i.e., an agent knows how much she trusts someone. It is worth noting (Constr3) and the fact that Ei is an equivalence relation together imply that if X / ∈ Ni,j(w, α) then X / ∈ Ni,j(v, α) for all v ∈ Ei(w). The latter corresponds to negative introspection for trust, i.e., if an agent does not trust someone then she knows this. (Constr4) claims that an agent's trust in someone must be compatible with her knowledge. Specifically, if agent i trusts agent j's judgement on the truth of some formula, then there should be some world that i envisages in which this formula is true.

We use a neighbourhood semantics for interpreting the graded trust operators T α i,j because these modal operators are not normal. We want to allow situations in which, at the same time, agent i trusts agent j's judgement about ϕ with strength α and i trusts agent j's judgement about ¬ϕ with strength α, without inferring that i trusts agent j's judgement about ⊥ with strength α, that is, we want formula T α i,j ϕ ∧ T α i,j ¬ϕ ∧ ¬T α i,j ⊥ to be satisfiable. For example, Bill may trust Mary's judgement about the fact that a certain stock will go upward with strength α (i.e., T α Bill,Mary stockUp) and, at the same time, trust Mary's judgement about the fact that the stock will not go upward with strength α (i.e., T α Bill,Mary ¬stockUp), without trusting Mary's judgement about ⊥ with strength α (i.e., ¬T α Bill,Mary ⊥). 4 Note that Constraint (Constr4) in Definition 1 makes formula ¬T α i,j ⊥ valid for every trust value α. Thus, if T α i,j was a normal modal operator, ¬(T α i,j ϕ ∧ T α i,j ¬ϕ) would have been valid, which is highly counterintuitive.

Before providing truth conditions of L-BT formulas, we follow [START_REF] Spohn | Ordinal conditional functions: a dynamic theory of epistemic states[END_REF] and lift the exceptionality of a possible world to the exceptionality of a formula viewed as a set of worlds.

Definition 2 (Exceptionality of a formula).

Let M = (W, {Ei}i∈Agt , κ, {Ni,j}i,j∈Agt , V) be a model. Moreover, let ϕ w,i = {v ∈ W : v ∈ Ei(w) and M, v |= ϕ} be the set of worlds envisaged by agent i at w in which ϕ is true. The exceptionality degree of formula ϕ for agent i at world w, denoted by κw,i(ϕ), is defined as follows:

κw,i(ϕ) = min v∈ ϕ w,i κ(v, i) if ϕ w,i = ∅ max if ϕ w,i = ∅
The exceptionality degree of a formula ϕ captures the extent to which ϕ is considered to be exceptional by the agent. The value κw,i(¬ϕ) corresponds to the degree of necessity of ϕ according to agent i at w, in the sense of possibility theory [START_REF] Dubois | Possibility theory: qualitative and quantitative aspects[END_REF]. The following definition provides truth conditions for L-BT formulas.

Definition 3 (Truth conditions). Let M = (W, {Ei}i∈Agt , κ, {Ni,j}i,j∈Agt , V) be a model and let w ∈ W . Then: M, w |= p iff p ∈ V(w) M, w |= ¬ϕ iff M, w |= ϕ M, w |= ϕ ∧ ψ iff M, w |= ϕ and M, w |= ψ M, w |= Kiϕ iff ∀v ∈ Ei(w) : M, v |= ϕ M, w |= B ≥α i ϕ iff κw,i(¬ϕ) ≥ α M, w |= T α i,j ϕ iff ϕ M ∈ Ni,j(w, α) with ϕ M = {v ∈ W : M, v |= ϕ}
In the following, we say that the L-BT formula ϕ is valid, denoted by |= ϕ, if for every model M and for every world w in M we have M, w |= ϕ. Moreover, we say that ϕ is satisfiable if ¬ϕ is not valid. The following validity highlights that beliefs are necessarily consistent: |= ¬(Biϕ ∧ Bi¬ϕ). In the next section, we provide truth conditions for DL-BT formulas [ * f i ϕ]ψ, after introducing the concept of trust-based belief change policy.

Trust-based belief change policies

A trust-based belief change policy specifies the way an agent's plausibility ranking of possible worlds should be modified depending on the agent's trust in the information source.

Additive policy

We start by considering an additive trust-based belief change policy and denote it by the symbol add . It is inspired by Darwiche & Pearl's well-known iterated belief revision method [START_REF] Darwiche | On the logic of iterated belief revision[END_REF].

Definition 4 (Additive policy).

Let M = (W, {Ei}i∈Agt , κ, {Ni,j}i,j∈Agt , V) be a model and let f be a function with domain Agt and codomain Plc such that f (j) = add . Then, for all w ∈ W , we define:

κ * f i ϕ (w, j) =                    Case 1. κ(w, j) if M, w |= ¬Tj,iϕ Case 2. κ(w, j) -κw,j(ϕ) if M, w |= ϕ ∧ Tj,iϕ Case 3. Cut(α + κ(w, j)) if M, w |= ¬ϕ ∧ T α j,i ϕ
where:

Cut(x) =      x if 0 ≤ x ≤ max max if x > max 0 if x < 0
Suppose that the information source i publicly announces that ϕ is true. Then, according to Definition 4, the additive rule rigidly boosts the ¬ϕ-worlds up from where they currently are by the degree of trust agent j has in the information source i. We show below that this policy guarantees that information received by different information sources is cumulated, in the sense that agent j 'does not forget' her previous degree of belief about ϕ.

Note that Case 3 is well-defined because of Constraint (Constr2), agent j cannot trust agent i with different strengths. Function Cut is a minor technical device, taken from [START_REF] Aucher | A combined system for update logic and belief revision[END_REF], which ensures that the new plausibility assignment fits into the finite set of natural numbers Num. Moreover, note that the situation in which agent j knows that ϕ is false is a special case of the preceding Case 1. Indeed, because of (Constr4) in Definition 1, formula Kj¬ϕ → ¬Tj,iϕ is valid. Consequently, if i's announcement of ϕ is incompatible with j's knowledge, then i's announcement of ϕ does not have any effect on j's beliefs.

We are in position to give the truth condition of the dynamic operator [ * f i ϕ] for the logic DL-BT {add} . Definition 5 (Truth conditions (cont.)). Let Plc = {add }, let M = (W, {Ei}i∈Agt , κ, {Ni,j}i,j∈Agt , V) be a model and w ∈ W . Then:

M, w |= [ * f i ϕ]ψ iff M * f i ϕ , w |= ψ where M * f i ϕ = (W, {Ei}i∈Agt , κ * f i ϕ , {Ni,j}i,j∈Agt , V
) and function κ * f i ϕ is defined according to the preceding Definition 4.

We generalize the notions of validity and satisfiability for DL-BT {add} formulas from the the notions of validity and satisfiability for L-BT formulas in the obvious way.

Properties of the additive policy

The next proposition highlights that the additive policy defined above is syntax independent, in the sense that two public announcements with logically equivalent formulas produce the same effects. This is a consequence of the fact that the graded trust operator T α i,j is closed under logical equivalence.

Proposition 1. If f (j) = add for all j ∈ Agt and |= ϕ1 ↔ ϕ2 then: |= [ * f i ϕ1]ψ ↔ [ * f i ϕ2]ψ (1)
The following proposition captures two fundamental properties of the additive policy. Proposition 2. For all i, j ∈ Agt and for all α ∈ Num \ {0}, if f (j) = add and ψ ∈ Obj then:

|= (B α j ψ ∧ ¬Tj,iϕ) → [ * f i ϕ]B α j ψ (2) |= Tj,iψ → [ * f i ψ]Bjψ (3)
According to validity (2), if an agent does not trust the information source then her beliefs about objective facts are not affected by what the information source announces. The validity (3) is a weakening of the AGM success postulate: agent j will revise her beliefs with the objective formula ψ only if j trusts the information source's judgment on ψ. Validity (3) can be generalized to a sequence of announcements of any length n. Proposition 3. For all j, i1, . . . , in ∈ Agt and for all n ∈ N, if f (j) = add and ψn ∈ Obj then:

|= Tj,i n ψn → [ * f 1 i 1 ψ1] . . . [ * fn in ψn]Bjψn (4)
Let us consider the special case of the preceding validity with n = 2 and ψ ∈ Obj . We have:

|= Tj,i 2 ψ → [ * f 1 i 1 ¬ψ][ * fn i 2 ψ]Bjψ. (5)
This means that, in the case of the additive policy, if two sources provide contradictory information, then the receiver will give priority to the last information source, if she trusts her. Let us now illustrate the cumulative effect of the additive policy. Proposition 4. For all i, j ∈ Agt and for all α, β ∈ Num \ {0}, if f (j) = add and ϕ ∈ Obj then:

|=(T α j,i ϕ ∧ B β j ϕ) → [ * f i ϕ]B Cut(α+β) j ϕ (6) |=(T α j,i ϕ ∧ ¬Bjϕ) → [ * f i ϕ]B α j ϕ (7)
Validity ( 6) highlights that the additive policy takes into account not only agent j's trust in the source, but also what agent j believed before the source's announcement. In particular, if agent j trusts i's judgment on the objective formula ϕ with degree α and believes ϕ with strength β then, after i's announcement of ϕ, j will believe ϕ with strength Cut(α + β). Validity (7) captures the complementary case in which agent j does not believe ϕ before the announcement. In this case, the strength of j's belief about ϕ is only determined by j's trust in the information source i.

The two validities of Proposition 4 can actually be generalized to a sequence of announcements of any length n as it is highlighted by Proposition 5. In particular, (i) if ϕ is an objective formula and j believes ϕ with a certain degree α, then j's degree of belief about ϕ at the end of a sequence of n announcements of ϕ is equal to the sum of α and j's degrees of trust in the sources of the announcements; (i) if ϕ is an objective formula and j does not believe ϕ, then j's degree of belief about ϕ at the end of a sequence of n announcements of ϕ is equal to the sum of j's degrees of trust in the sources of the announcements. More generally, the additive policy cumulates information about objective facts coming from different sources. Proposition 5. For all j, i 1, . . . , in ∈ Agt, for all α1 . . . , αn, γ ∈ Num \ {0} and for all n ∈ N, if f1(j) = . . . = fn(j) = add and ϕ ∈ Obj then:

|=(T α 1 j,i 1 ϕ ∧ . . . ∧ T αn j,in ϕ ∧ B γ j ϕ) → [ * f 1 i 1 ϕ] . . . [ * fn in ϕ]B Cut(α 1 +...+αn+γ) i ϕ (8) |=(T α 1 j,i 1 ϕ ∧ . . . ∧ T αn j,in ϕ ∧ ¬Bjϕ) → [ * f 1 i 1 ϕ] . . . [ * fn in ϕ]B Cut(α 1 +...+αn) i ϕ (9)
In propositions 2-5, we only consider objective formulas as they do not hold in general. If we drop the restriction to objective formulas, the validity (3) in Proposition 2 does not work anymore. To see this, suppose that ψ is a Moore-like sentence of the form p ∧ ¬Bjp. Then, the formula Tj,i(p ∧ ¬Bjp) → [ * f i (p ∧ ¬Bjp)]Bj(p ∧ ¬Bjp) is clearly not valid. In fact, Bj(p ∧ ¬Bjp) is equivalent to ⊥. Similar observations hold for Propositions 3-5.

Notice that the additive policy satisfies the following commutativity property. Proposition 6. For all i1, i2 ∈ Agt, if f (j) = add for all j ∈ Agt and ϕ ∈ Obj then:

|= [ * f i 1 ϕ][ * f i 2 ϕ]ψ ↔ [ * f i 2 ϕ][ * f i 1 ϕ]ψ (10)
This means that if all agents adopt the additive policy then the order of the announcements of an objective formula ϕ performed by several information sources does not matter. Bill has to decide whether he buys a certain stock. He hesitates because he is uncertain whether the stock will go upward (stockUp). Assume the following initial epistemic state for Bill:

Hyp1 = def U Bill stockUp
Bill asks two stockbrokers their opinions: Mary and Jack. He first asks Mary. Then, he asks Jack. Both Mary and Jack say that the stock will go upward and that it is convenient to buy it. We assume that Bill trusts fairly Mary's judgement on stockUp, and Bill trusts very weakly Jack's judgement on stockUp:

Hyp2 = def T 3 Bill,Mary stockUp ∧ T 1 Bill,Jack stockUp
Suppose Bill uses the additive policy. In this situation, after having received the information from Mary and Jack, Bill will strongly believe that proposition stockUp is true. As Proposition 5 above highlighs, Bill cumulates the information provided by the two information sources. Specifically, if f (Bill ) = f ′ (Bill ) = add then:

|=(Hyp1 ∧ Hyp2 ) → [ * f Mary stockUp][ * f ′ Jack stockUp]B 4 Bill stockUp.
Now, suppose that Mary and Jack provide contradictory information about proposition stockUp. As highlighted by Proposition 3, priority will be given to the last information source. That is, if f (Bill ) = f ′ (Bill ) = add then:

|=(Hyp1 ∧ Hyp2 ) → [ * f Mary ¬stockUp][ * f ′ Jack stockUp]B Bill stockUp.
In the next section we present a new policy, the compensatory policy that does not satisfy the general property given in Proposition 3. In case of two contradictory information provided by two sources, the compensatory policy balances them depending on the degrees of trust in the sources.

Compensatory policy

The compensatory policy, denoted by the symbol comp, is defined as follows.

Definition 6 (Compensatory policy). Let M = (W, {Ei}i∈Agt , κ, {Ni,j}i,j∈Agt , V) be a model and let f be a function with domain Agt and codomain Plc such that f (j) = comp. Then, for all w ∈ W , we define:

κ * f i ϕ (w, j) =                              Case 1. κ(w, j) if M, w |= ¬Tj,iϕ Case 2. Cut(κ(w, j) -α) if M, w |= ϕ ∧ T α j,i ϕ Case 3. Cut(α + κ(w, j)) if M, w |= ¬ϕ ∧ T α j,i ϕ ∧ Bjϕ Case 4. κ(w, j)
if M, w |= ¬ϕ ∧ Tj,iϕ ∧ Bj¬ϕ where Cut(x) is the same as in Definition 4.

Let us focus on Cases 2, 3 and 4, as Case 1 is the same as the one in Definition 4. Case 2 states that, if j trusts i's judgment on ϕ then, after i's announcement of ϕ, the exceptionality degree of a ϕworld for j should be decreased depending on how much j trusts i, in order to decrease the strength of j's belief about ¬ϕ. Cases 3 and 4 distinguish the situation in which ϕ is compatible with j's beliefs from the situation in which it is not. For Case 3, the exceptionality degree of a ¬ϕ-world for j should be increased in order to increase the strength of j's belief about ϕ. For Case 4, agent j should not change her plausibility ordering in order to preserve consistency of beliefs. Case 4 guarantees that (Constr1) in Definition 1 is preserved.

The truth condition of the dynamic operator [ * f i ψ] as well as the notion of validity for the logic DL-BT {add,comp} are defined in a similar way from the ones for the logic DL-BT {add} given above.

It is important to remark that the compensatory policy as well as the additive policy guarantee that the updated model M * f i ϕ is indeed a model in the sense of Definition 1. In particular: Proposition 7. If M is a model in the sense of Definition 1 and f (j) ∈ {add , comp} for all j ∈ Agt, then M * f i ϕ is a model in the sense of Definition 1 too.

Properties of the compensatory policy

Let us consider some basic properties of the compensatory policy. The first point to remark is that, different from the additive policy, the compensatory policy does not satisfy the weakening of the success postulate of Proposition 2. That is, if f (j) = comp and ψ ∈ Obj , the formula Tj,iψ → [ * f i ψ]Bjψ is not valid. The following Proposition 8 provides a list of validities for the compensatory policy. Proposition 8. For all i, j, i1, i2 ∈ Agt, for all α, α1, α2, β ∈ Num \ {0}, if f (j) = f ′ (j) = comp and ϕ ∈ Obj then:

|=(T α j,i ϕ ∧ B β j ϕ) → [ * f i ϕ]B Cut(α+β) j ϕ (11) |=(T α j,i ϕ ∧ Ujϕ) → [ * f i ϕ]B α j ϕ (12)
|=(T α j,i ¬ϕ ∧ B β j ϕ) → [ * f i ¬ϕ]B Cut(β-α) j ϕ (13)
|=(T α 1 j,i 1 ϕ ∧ T α 2 j,i 2 ¬ϕ ∧ B β j ϕ) → [ * f i 1 ϕ][ * f ′ i 2 ¬ϕ]B Cut(Cut(β+α 1 )-α 2 ) j ϕ (14)
|=(T α 1 j,i 1 ϕ ∧ T α 2 j,i 2 ¬ϕ ∧ Ujϕ) → [ * f i 1 ϕ][ * f ′ i 2 ¬ϕ]B Cut(α 1 -α 2 ) j ϕ (15)
According to the validities ( 11) and ( 12), if i announces the objective formula ϕ, ϕ is consistent with j's beliefs and j adopts the compensatory policy, then j will increase the strength of her belief about ϕ w.r.t. her trust's degree in i. The two validities distinguish the situation whether j believes ϕ, or whether j is uncertain about ϕ. The last three key validities ( 13)-( 15) characterize the compensatory aspect. According to the validity [START_REF] Plaza | Logics of public communications[END_REF], if ϕ is an objective formula then, after i announces that ϕ is false, j will decrease the strength of her belief about ϕ depending on how much she trusts i. The validities [START_REF] Spohn | Ordinal conditional functions: a dynamic theory of epistemic states[END_REF] and ( 15) considers information about objective facts coming from two different sources. Suppose that i1 says ϕ, while i2 says ¬ϕ. Then, j should compensate the information received from i1 by decreasing the strength of her belief about ϕ depending on how much she trusts i2. Let us illustrate this by revisiting our previous example.

Example 2. Let us suppose that Mary and Jack provide contradictory information about proposition stockUp. Suppose Bill trusts

Mary's judgment on stockUp with degree 3 and trusts Jack's judgment on ¬stockUp with degree 1:

Hyp2 ′ = def T 3 Bill,Mary stockUp ∧ T 1 Bill,Jack ¬stockUp Now, assume Mary announces that stockUp is true and Jack announces that stockUp is false. If Bill adopts the compensatory policy, he will then believe that stockUp is true with strength

3 -1 = 2. That is, if f (Bill ), f ′ (Bill ) = {comp} : |=(Hyp1 ∧ Hyp2 ′ ) → [ * f Mary stockUp][ * f ′ Jack ¬stockUp]B 2 Bill stockUp.

Axiomatization

In this section, we provide a complete axiomatization for the variant of DL-BT where Plc = {add , comp}, namely DL-BT {add,comp} . This logic has so-called reduction axioms which allow to reduce every formula to an equivalent L-BT formula without dynamic operators [ * f j ψ]. That elimination together with the rule of replacement of equivalent axioms and rules of inference for the static logic L-BT provides an axiomatization.

Proposition 9 provides reduction axioms for boolean formulas, as well as the knowledge and graded trust operators. Proposition 9. The following equivalences are valid:

[ * f j ϕ]p ↔ p [ * f j ϕ]¬ψ ↔ ¬[ * f j ϕ]ψ [ * f j ϕ](ψ1 ∧ ψ2) ↔ ([ * f j ϕ]ψ1 ∧ [ * f j ϕ]ψ2) [ * f j ϕ]Kiψ ↔ Ki[ * f j ϕ]ψ [ * f j ϕ]T α i,k ψ ↔ T α i,k [ * f j ϕ]ψ
The following abbreviation is useful to formulate the reduction axioms for the graded belief operators. For all α > max we give the following abbreviation:

B ≥α i ϕ = def Kiϕ. Proposition 10 provides the reduction axiom for the graded belief operators based on the additive policy. Proposition 10. Let f (i) = add and α ∈ Num \ {0}. Then, the following equivalence is valid:

[ * f j ϕ]B ≥α i ψ ↔ (¬Ti,jϕ → B ≥α i [ * f j ϕ]ψ)∧ β∈Num\{0},γ 1 ∈Num (T β i,j ϕ ∧ B γ 1 i ¬ϕ) → (B ≥α+γ 1 i (ϕ → [ * f j ϕ]ψ)∧ B ≥Cut(α-β) i (¬ϕ → [ * f j ϕ]ψ))
Proposition 11 provides the reduction axiom for the graded belief operators based on the compensatory policy. Proposition 11. Let f (i) = comp and α ∈ Num \ {0}. Then, the following equivalence is valid:

[ * f j ϕ]B ≥α i ψ ↔ (¬Ti,jϕ → B ≥α i [ * f j ϕ]ψ)∧ β∈Num\{0} ((T β i,j ϕ ∧ Biϕ) → (B ≥α+β i (ϕ → [ * f j ϕ]ψ)∧ B ≥Cut(α-β) i (¬ϕ → [ * f j ϕ]ψ)))∧ ((T β i,j ϕ ∧ Bi¬ϕ) → (B ≥α+β i (ϕ → [ * f j ϕ]ψ)∧ B ≥α i (¬ϕ → [ * f j ϕ]ψ)))
These two propositions translate the different cases considered in Definitions 4 and 6. For instance, line 1 of Prop.10 describes Case 1 of Def. 4 (no change if no trust) while lines 2-4 correspond to the two options for changing when trust holds (Cases 2 and 3 of Def. 4).

As the rule of replacement of equivalences ψ 1 ↔ψ 2 ϕ↔ϕ[ψ 1 /ψ 2 ] preserves validity, the equivalences of Propositions 9, 10 and 11 together with this allow to reduce every DL-BT {add,comp} formula to an equivalent L-BT formula. Call τ the mapping which iteratively applies the above equivalences from the left to the right, starting from one of the innermost modal operators. τ pushes the dynamic operators inside the formula, and finally eliminates them when facing an atomic formula. Proposition 12. Let ϕ be a DL-BT {add,comp} formula. Then: (i) τ (ϕ) has no dynamic operators [ * f j ψ], and (ii) τ (ϕ) ↔ ϕ is valid. The axiomatic system of the logic DL-BT {add,comp} consists of the axioms and rules of inference in Figure 1. Notice that the rule of necessitation for graded belief (i.e., from ϕ infer B ≥α i ϕ) does not need to be added, as it is deducible from the rule of necessitation for knowledge, the fifth axiom for graded belief and modus ponens.

(1) Axioms for L-BT :

all tautologies of classical propositional logic axioms K, T, 4 and 5 for knowledge

* K i (ϕ → ψ) → (K i ϕ → K i ψ) * K i ϕ → ϕ * K i ϕ → K i K i ϕ * ¬K i ϕ → K i ¬K i ϕ -axioms for graded belief * B ≥α i (ϕ → ψ) → (B ≥α i ϕ → B ≥α i ψ) * ¬(B ≥1 i ϕ ∧ B ≥1 i ¬ϕ) * B ≥α i ϕ → K i B ≥α i ϕ * ¬B ≥α i ϕ → K i ¬B ≥α i ϕ * K i ϕ → B ≥α i ϕ * B ≥α+1 i ϕ → B ≥α
i ϕ axioms for graded trust * T α i,j ϕ → ¬T β i,j ϕ if α = β * T α i,j ϕ → K i T α i,j ϕ * T α i,j ϕ → K i ϕ

(2) Rules of inference for L-BT:

from ϕ and ϕ → ψ infer ψ from ϕ infer K i ϕ from ϕ ↔ ψ infer T α i,j ϕ ↔ T α i,j ψ

(3) Further axioms and rules of inference for DL-BT:

-from ψ 1 ↔ ψ 2 infer ϕ ↔ ϕ[ψ 1 /ψ 2 ]
reduction axioms of Propositions 9, 10 and 11 (Sketch). Thanks to Prop. 12 and the fact that DL-BT {add,comp} is a conservative extension of L-BT, we only need to prove that L-BT is completely axiomatized by the group of axioms (1) and the group of rules of inference (2) in Figure 1. The proof consists of two steps. First, we provide a relational semantics for L-BT and prove that this semantics is equivalent to the L-BT semantics of Def. 1. Then, we use the canonical model construction in order to show that the group of axioms (1) and the group of rules of inference (2) in Fig. 1 provide a complete axiomatics for L-BT with respect to this semantics.

Conclusion

We have proposed a dynamic logic of graded belief and trust that supports reasoning about trust-based belief change. We have considered two kinds of trust-based belief change policy and studied their logical properties in detail. In addition, we have provided a sound and complete axiomatization for our logic. Following the belief revision tradition, in future work we plan to extend the present work with an axiomatic analysis of the additive and compensatory policies. More concretely, for every policy, we intend to come up with a list of postulates that fully characterize it.

Example 1 .

 1 Let us illustrate the additive policy. Assume that Num = {0, 1, 2, 3, 4, 5} s.t. 0 means 'null', 1 means 'very weak', 2 means 'weak', 3 means 'fair', 4 means 'strong' and 5 means 'very strong'.

Figure 1 .

 1 Figure 1. Axiomatization of DL-BT {add,comp}

ACKNOWLEDGEMENTS

The authors acknowledge the support of the French ANR project EmoTES Emotions in strategic interaction: theory, experiments, logical and computational studies, contract No. 11-EMCO-004-01.