Marco Tognon
email: mtognon@laas.fr

Antonio Franchi
email: antonio.franchi@laas.fr

Nonlinear Observer-based Tracking Control of Link Stress and Elevation for a Tethered Aerial Robot using Inertial-only Measurements

come

I. INTRODUCTION

Aerial robotics is nowadays the focus of many researches in the robotics field, with an enormous increasing interest over the last decade. The recently technological advances, made aerial robots (also referred to as aerial vehicles, UAVs, or UASs in the literature) extremely versatile and usable in many applications, ranging, e.g., from aerial monitoring to search and rescue. Although they were mainly used as remote and moving sensors, nowadays, their use for physically interactive tasks has become a very popular topic.

In addition to the cooperative transportation case [START_REF] Maza | Multi-UAV cooperation and control for load transportation and deployment[END_REF], cables are exploited also in systems where one vehicle is tethered to a ground station. In this case the cable can be used to provide energy and a high-bandwidth communication channel. These systems could find direct application for tasks such as surveillance, border protection, temporary communication relay and so on [START_REF] Pinkney | Unmanned aerial vehicle (UAV) communications relay[END_REF], [START_REF] Muttin | Umbilical deployment modeling for tethered UAV detecting oil pollution from ship[END_REF]. The tethered flight solution could be exploited also to improve the hover stability in the presence of wind or during dangerous maneuvers, e.g., landing and take off from a moving platform as a ship in the presence of rough sea (see [START_REF] Sandino | Advances in modeling and control of tethered unmanned helicopters to enhance hovering performance[END_REF] and references therein).

A controller is presented in [START_REF] Lupashin | Stabilization of a flying vehicle on a taut tether using inertial sensing[END_REF] to stabilize the elevation of a cable-tethered aerial vehicle to a constant value using only inertial onboard sensors. The proposed method is tested on a real quadrotor and relies on a quasi-static assumption. A different controller is presented in [START_REF] Nicotra | Taut cable control of a tethered UAV[END_REF] which is also able to ensure the positivity of the cable tension.

In this work we solve a generalization of the problems considered in [START_REF] Lupashin | Stabilization of a flying vehicle on a taut tether using inertial sensing[END_REF], [START_REF] Nicotra | Taut cable control of a tethered UAV[END_REF]. First of all, we consider the possibility of having a more generic link, either a cable, a strut, or a bar. Second of all we consider a tracking problem for time-varying desired elevation trajectories, instead of just a regulation to a constant desired elevation. Finally, we also consider the problem of simultaneously letting the link stress track any time-varying force trajectory that can assume values both in the positive (tensions) and negative (compressions) domains, instead of only ensuring the positivity of the stress [START_REF] Nicotra | Taut cable control of a tethered UAV[END_REF]. In this way, for example, our control method can be used to follow a certain elevation trajectory while precisely regulating the stress of the link on the basis of its mechanical strength thus avoiding breakages due to excessive tensions or compressions.

The first main contribution of the paper is the design of a nonlinear controller based on dynamic feedback linearization. The second one is the design of a nonlinear observer which uses inertial sensors only. Observation works in dynamics conditions, not only in the quasi-static case [START_REF] Lupashin | Stabilization of a flying vehicle on a taut tether using inertial sensing[END_REF]. The stability of the controlled/observed system is both theoretically proven and validated through extensive simulations.

Finally, another important contribution of this paper is the thorough investigation of the intrinsic properties of the system, such as exact linearizability with dynamic feedback, flatness, state observability, and trajectory feasibility.

The paper is organized as follows. In Sec. II we derive the dynamic model of the system and the explicit characterization of the internal stress of the link. Then, in Sec. III we show dynamic feedback linearizability of the system with respect to elevation and stress of the link and the related nonlinear controller, moreover we ascertain the flatness property of the system. In Sec. IV we develop a nonlinear observer of the state based on onboard inertial-only measurements. Simulation results are presented in Sec. V. Conclusions and future developments are shown in Sec. VI.

II. DYNAMIC MODEL

Similarly to the original problem defined in [START_REF] Lupashin | Stabilization of a flying vehicle on a taut tether using inertial sensing[END_REF], [START_REF] Nicotra | Taut cable control of a tethered UAV[END_REF] we consider a system composed by an underactuated flying vehicle lying on a vertical plane that is connected to the ground through a link, as depicted in Fig. 1 1 . Like [START_REF] Lupashin | Stabilization of a flying vehicle on a taut tether using inertial sensing[END_REF], [START_REF] Nicotra | Taut cable control of a tethered UAV[END_REF] we assume that the link has a negligible mass and inertia, it is not deformable, and its elasticity is minor in the operative conditions. Therefore, the link has a fixed length l ∈ R >0 .

We denote with ϕ ∈ R the elevation of the link, i.e., the angle that it forms with the horizon. With

f L ∈ R we denote f R z W x W x B z B ϑ -m Q gz W -f L d ϕ l O W O B τ R
Fig. 1: Representation of the system and its mainly variables. the link stress, i.e., the internal force that is exerted on the link along its longitudinal line. If f L > 0 then the link is pulled and the stress is a tension, viceversa, if f L < 0 then the link is compressed and the stress is called a compression.

We depart from [START_REF] Lupashin | Stabilization of a flying vehicle on a taut tether using inertial sensing[END_REF], [START_REF] Nicotra | Taut cable control of a tethered UAV[END_REF] by adding three more 'challenges' to the original problem. First of all, we consider the simultaneous control of f L and ϕ instead of only controlling ϕ. Second of all, we consider here a tracking problem for the controlled quantities, i.e., their servoing to a time-varying trajectory, instead of regulation to a constant value. 2 Finally, we consider the more general case of a link that can be either a cable, a strut, or a bar (instead of the cable-only case considered in the original problem). We recall that cables, struts, and bars are structural element designed to resist only to tension, compression, and generic stress, respectively.

The two ends of the link attached to the vehicle center of mass (CoM) and to a fixed point, respectively. Connection is made in such a way that no rotational constraint holds on the two ends. Finally, the aerial vehicle is modeled as a rigid body, and we denote with m R ∈ R >0 and J R ∈ R >0 the mass and the rotational inertia of the vehicle, respectively.

Although we consider here a 2D problem it is still convenient to define our frames in 3D, in order, e.g., to have a properly defined angular velocity vector for the aerial vehicle. Consider in particular two frames: a world frame, F W , with unit vectors along the axes denoted by {x W , y W , z W } and origin O W set on the fixed point, and a body frame, F B , which is rigidly attached to the vehicle, with unit vectors denoted by {x B , y B , z B } and origin O B set on the vehicle CoM, described on F W by p B = [x B y B z B ] T , where y B = 0. The axes y W and y B are parallel to each other and both perpendicular to the vertical plane where the system evolves, see Fig. 1. The vehicle motion on the {x W , z W } plane can be controlled acting on two inputs: the intensity f R ∈ R of the thrust force f R = -f R z B ∈ R 3 ; and the torque τ R ∈ R acting about the y B axis.

Since the motion of the vehicle CoM is constrained on a circle of radius l, the system is completely described by the generalized coordinates q = (ϕ, ϑ ), where ϕ and ϑ are the elevation of the link (defined before) and the attitude of the vehicle, respectively. Deriving kinematic and potential energy, omitted for the limited space, and applying the Euler-Lagrange equation we obtain the system model:

m R l φ = -m R g cos ϕ + f R cos(ϕ + ϑ ) (1)
J R θ = τ R . (2)
Considering the motion system constraint, the acceleration of the CoM of the vehicle, with respect

F W , is pB = -ld φ2 + ld ⊥ φ, (3)
where d = [cos ϕ 0 sin ϕ] T and d ⊥ = [-sin ϕ 0 cos ϕ] T are unit vectors, in R 3 , parallel and perpendicular to the link, respectively. The balance equation of all the forces is

m R pB = -f L d -f R z B -m R gz W . (4)
The stress f L can be retrieved by projecting the equation ( 4) along the link axis

f L = -m R g sin ϕ + f R sin (ϕ + ϑ ) + m R l φ2 . (5)
The problem that we are facing in this paper is that of designing a control law for the inputs ( f R , τ R ) that is able to asymptotically steer (ϕ, f L ) along any smooth desired trajectory (ϕ d , f d L ) making use only of an accelerometer and a gyroscope mounted on-board. Moreover, regardless the proposed solution, we aim at investigating the intrinsic characteristics of the system, as dynamic feedback linearizability, flatness and its observability.

III. DYNAMIC DECOUPLING CONTROL

In order to study the system and design the controller, we define

x = ϕ φ ϑ θ T = [x 1 x 2 x 3 x 4 ] T and u = [ f R τ R ] T = [u 1 u 2 ]
T as the state and input vectors of the system, respectively. Rewriting ( 1) and ( 2) in the state-space form we have

ẋ1 = x 2 (6a) ẋ2 = a 1 cos x 1 + a 2 cos (x 1 + x 3 ) u 1 (6b) ẋ3 = x 4 (6c) ẋ4 = a 3 u 2 , (6d)
where the constant model parameters are

a 1 = -g/l, a 2 = 1/(m R l), a 3 = 1/J R .
To control the elevation and the stress of the link, we consider as outputs of the system the variables y

= [ϕ f L ] T = [y 1 y 2 ] T .
Using a feedback linearization approach, we show now how the nonlinear system can be completely transformed in an equivalent controllable and decoupled linear system. First of all, we differentiate the outputs until the inputs appear. From ( 5) and ( 6), the outputs can be written as

y 1 = x 1 (7a) y 2 = -m R g sin x 1 + m R lx 2 2 + sin (x 1 + x 3 ) u 1 . (7b)
We can see that y 2 already contains the input u 1 , while y 1 has to be differentiated twice, obtaining

y (2) 1 y 2 = a 1 cos x 1 m R lx 2 2 -m R g sin x 1 + a 2 cos (x 1 + x 3 ) 0 sin (x 1 + x 3 ) 0 E(x) u, (8)
where

• (n) = d n •
dt n indicates the n-th time-derivative, for n ≥ 1. We can observe that the decoupling matrix E(x) is always singular which means that the system is not linearizable with a static feedback. In these cases one can delay the appearance of the input u 1 in y 2 (i.e., increasing the relative degree of y 2 ) by adding one or more integrators in the input channel u 1 thus resulting in a dynamic compensator. To this aim, we redefine the input as ū = [ ü1 u 2 ] T = [ ū1 ū2 ] T , considering the acceleration of the thrust intensity as new controllable input, ü1 = fR . The system is now described by the extended state x = ϕ φ ϑ θ u 1 u1

T , that contains also the thrust intensity and its derivative. Considering the extended system and the new input, y 1 and y 2 have to be differentiated four and two times, respectively, in order to see the new input ū appear:

y (4) 1 y (2) 2 = b(x) + a 2 cos (x 1 + x 3 ) -a 2 a 3 sin (x 1 + x 3 ) u 1 sin (x 1 + x 3 ) a 3 cos (x 1 + x 3 ) u 1 Ē(x)
ū, [START_REF] De Luca | Trajectory planning and control for planar robots with passive last joint[END_REF] where b(x) is simply derived differentiating equations [START_REF] Khalil | Nonlinear Systems[END_REF].

The new decoupling matrix Ē(x) is now always invertible except for the zero measure case u 1 = 0,3 indeed det Ē(x) = a 2 a 3 u 1 . Furthermore the total relative degree4 r = r 1 + r 2 = 4 + 2 = 6 is equal to the dimension of the extended state. Therefore the system does not have an internal dynamics [START_REF] Khalil | Nonlinear Systems[END_REF]. Thus designing the control input as

ū = E -1 (x) [-b(x) + v] , (10)
where

v = [v 1 v 2 ]
T are virtual inputs, we obtain y (4)

1 = v 1 and y (2) 2 = v 2 i.e.
, a linear decoupled model equivalent to the original nonlinear system through the state feedback transformation [START_REF] Marino | Nonlinear Control Design: Geometric, Adaptive and Robust[END_REF].

Given any desired trajectory y d 1 (t) of class C 3 for the link elevation y 1 and any desired trajectory y d 2 (t) of class C 1 for the link stress y 2 , one can set the virtual inputs as

v 1 = y d 1 (4) + k 11 e 1 + k 12 e (1)
1 + k 13 e (2)
1 + k 14 e (3) 1 (11a) v 2 = y d 2 (2) + k 21 e 2 + k 22 e (1) 2 , (11b)
where e 1 = y d 1 -y 1 and e 2 = y d 2 -y 2 . Choosing the gains

k 1 = [k 11 k 12 k 13 k 14 ] ∈ R 4 >0 and k 2 = [k 21 k 22 ] ∈ R 2 >0
, one can arbitrarily assign the poles of the error dynamics in order to guaranties an arbitrarily fast exponential tracking of (y d 1 (t), y d 2 (t)) for (y 1 (t), y 2 (t)). The block diagram of the controller is depicted in Fig. 2. We summarize the obtained results in the following: Proposition 1. Consider the system composed by a link connected to the ground and an aerial vehicle with passive joints, whose dynamic model is described by [START_REF] Nicotra | Taut cable control of a tethered UAV[END_REF]. Consider as outputs the elevation and the stress of the link, y = [ϕ f L ] T . Then the system is fully linearizable via dynamic feedback for every state configuration, iff the thrust intensity f R = 0.

Furthermore, considering as input the second derivative of the thrust and the torque provided by the aerial vehicle, ū = [ fR τ R ] T , the control law described by [START_REF] Marino | Nonlinear Control Design: Geometric, Adaptive and Robust[END_REF] and [START_REF] Tognon | Extended simulations for the link stress and elevation control of a tethered aerial robot[END_REF] globally exponentially steers y along any desired trajectory Remark III.1. Thanks to the exactness of the feedback linearization (i.e., the absence of the internal dynamics) the dynamics of the pitch of the aerial vehicle is automatically stabilized while tracking the desired output.

y d = [ϕ d f d L ] T with
f R τ R + + - - x f R ḟR ḟ d L .... ϕ d
Remark III.2. If the link is a bar it is completely feasible to pass from compression to tension, and viceversa.

Remark III.3. In the case that the link is a tie (like, e.g., a cable) a preliminary phase has to bring the link in a taut condition. As shown in [START_REF] Gioioso | Turning a near-hovering controlled quadrotor into a 3D force effector[END_REF], using a near hovering controller, is possible to reach the initial position and to provide a sufficient force to make the tie taut. Then the control action [START_REF] Marino | Nonlinear Control Design: Geometric, Adaptive and Robust[END_REF], [START_REF] Tognon | Extended simulations for the link stress and elevation control of a tethered aerial robot[END_REF] can be switched on and used to control the stress in the positive domain.

For the equivalence between dynamic feedback linearizability and differential flatness on an open and dense set of the state space [START_REF] De Luca | Trajectory planning and control for planar robots with passive last joint[END_REF], the following proposition is valid, whose detailed proof is omitted for space limitation.

Proposition 2. Consider the model (1) (2), then, the system is flat with respect the flat outputs: elevation and stress of the link, y = (ϕ, f L ). In other words the state (ϕ, φ, ϑ , θ ) and the inputs ( f R , τ R ) can be written as algebraic function of (ϕ, f L ) and their time derivatives (up to ϕ (4) and f

(2) L ).

IV. STATE OBSERVER

Implementation of the control laws [START_REF] Marino | Nonlinear Control Design: Geometric, Adaptive and Robust[END_REF], [START_REF] Tognon | Extended simulations for the link stress and elevation control of a tethered aerial robot[END_REF] requires the knowledge (measurement) of the state x, the output y and its derivatives (up to the third-order for y 1 and firstorder for y 2 ). Nevertheless, y and all its needed derivatives can be computed from x and ū as done, e.g., in [START_REF] Khalil | Nonlinear Systems[END_REF], ( 8) and ( 9) for some of the derivatives. Furthermore, u 1 and u1 are internal states of the controller and therefore they are known. Ultimately the knowledge of x is sufficient in order to implement the presented controller. In this section we demonstrate how the needed x can be observed based only on the measurements from the onboard IMU (accelerometer plus gyroscope). Then we prove that the observed state can be used in the controller in place of the real state while preserving the closed-loop system stability.

Assume to have an IMU mounted at O B and oriented according to F B . The gyroscope provides the angular velocity along y B , representable in 2D by the scalar ω = θ . The accelerometer gives the specific acceleration in F B , i.e.,

a = R B W ( pB -gz W ) = [a x , 0, a z ] T , (12)
where R B W ∈ R 3 is the rotation matrix from F W to F B . Putting (1) and ( 3) in (12), we obtain:

a x = cos (ϕ + ϑ ) l φ2 -g sin ϕ + f R m sin (ϕ + ϑ ) (13a) a z = sin (ϕ + ϑ ) l φ2 -g sin ϕ + f R m sin (ϕ + ϑ ) - f R m . (13b)

A. State/Output Transformations and Observability

First of all, since x 4 = ω is directly measured, we can restrict the observability study and the observer design to the reduced system (6a)-(6c). Then, applying the state transformation z = T (x) : R 3 → R 3 defined by

z 1 = x 1 , z 2 = x 2 , z 3 = x 1 + x 3 , (14)
and defining the new input u 3 = x 4 = ω, and the new measurements w 1 = a x /l and w 1 = a z /l + f R /(ml), the system (6a)-( 6c) is transformed into

     ż1 = z 2 ż2 = a 1 cos z 1 + a 2 cos z 3 u 1 ż3 = z 2 + u 3 ( 15
)
w 1 = cos z 3 [z 2 2 + a 1 sin z 1 + a 2 sin z 3 u 1 ] w 2 = sin z 3 [z 2 2 + a 1 sin z 1 + a 2 sin z 3 u 1 ].
Clearly, if we are able to estimate z 1 , z 2 , z 3 then we can easily recover the original state using

x 1 = z 1 , x 2 = z 2 , x 3 = z 3 -z 1 .
We then apply another output transformation

w 1 = w 2 w 1 = tan z 3 if | w 2 w 1 | ≤ 1 w 1 w 2 = cot z 3 if | w 1 w 2 | < 1 w 2 = w 2 1 + w 2 2 = |z 2 2 + a 1 sin z 1 + a 2 sin z 3 u 1 |. (16)
Note that this transformation is not allowed when w 2 = 0. Indeed in this case, both w 1 and w 2 are zero and w 1 cannot be evaluated. From [START_REF] Lupashin | Stabilization of a flying vehicle on a taut tether using inertial sensing[END_REF], one can notice that w 2 = | f L /(ml)|. Therefore, w 2 is zero when the stress on the link is equal to zero. In the observer design, we shall take into account this singularity, particularly important in the case that the desired stress trajectory passes through zero.

We can now invert the measurement map (16) in order to obtain a direct measure of z 3 . However, since the sign of z 2 2 + a 1 sin z 1 + a 2 sin z 3 u 1 is unknown, one obtains two solutions

w1 (k) =    tan -1 ( w 2 w 1 ) [-π 2 , π 2 ] + kπ if | w 2 w 1 | ≤ 1 cot -1 ( w 1 w 2 ) [0,π] + kπ if | w 1 w 2 | < 1 w2 (k) = cos( w1 (k))w 1 + sin( w1 (k))w 2 ,
where k ∈ {0, 1}. Depending on k we have then two distinct pairs of measures: ( w1 (0), w2 (0)) and ( w1 (1), w2 (1)). We know that, at each time t one of the two pairs of measurements is correct, i.e., it is equal to (z 3 , f L ml ), while the second one is wrong, i.e., it is equal to (z 3 + π, -f L ml ). As a matter of fact, it is not possible to discriminate whether the correct one corresponds to k = 0 or to k = 1 by only considering a single measurement. For the analysis of the observability and design of the observer, we consider from now on the correct measurement case. We shall show in Sec. IV-E a discrimination method that exploits the filtering error dynamics. We can then write

w1 = z 3 , w2 = z 2 2 + a 1 sin z 1 + a 2 sin z 3 u 1 = f L ml . ( 17
)
In a compact form we can write ( 15), ( 17) as

ż = f(z, u), w = h(z, u), where z = [z 1 z 2 z 3 ] T , u = [u 1 u 3 ]
T and w = [ w1 w2 ] T . The system is observable if the nonlinear observability matrix O(z, u) has full rank [START_REF] Marino | Nonlinear Control Design: Geometric, Adaptive and Robust[END_REF]. After some algebra (not reported for reason of space) it is possible to see that two minors of O(z, u) are equal to a 1 cos z 1 and a 1 sin z 1 , respectively, which are never contemporarily zero. Therefore we can conclude that O(z, u) is full rank for every z ∈ R 3 and u ∈ R 3 , i.e.:

Proposition 3. The full state of the system (6) is observable with accelerometer and gyroscope measurements only, except along the trajectories that keep the link stress constantly zero.

We divide the observer in two parts, as depicted in the block diagram of Fig. 3. The first part makes use of a High Gain Observer (HGO) approach [START_REF] Khalil | Nonlinear Systems[END_REF], in order to estimate the quantities z 3 , z 2 , ż2 . The second part applies a nonlinear transformation in order to retrieve z 1 . We describe these two parts separately in the next sections.

B. High Gain Observer for z 2 and z

3 Let us define ζ 1 = z 3 , ζ 2 = z 2 , ζ 3 = ż2 and ζ ζ ζ = [ζ 1 ζ 2 ζ 3 ] T and µ µ µ = (u 1 , u1 , u 3 ).
Taking the derivative of ż2 in (15) we can hide the dependency on z 1 , in fact:

z2 = -[a 1 sin z 1 + a 2 sin z 3 u 1 ]z 2 -a 2 sin z 3 u 3 u 1 + a 2 cos z 3 u1 = [z 2 2 -w2 ]z 2 -a 2 sin z 3 u 3 u 1 + a 2 cos z 3 u1 σ (ζ ζ ζ , w2 ,µ µ µ)
.

Therefore, we, can write the dynamics of

ζ ζ ζ as ζ ζ ζ = 0 1 0 0 0 1 0 0 0 A ζ ζ ζ + u 3 0 σ (ζ ζ ζ , w2 ,µ µ µ) σ σ σ (ζ ζ ζ , w2 ,µ µ µ) , w1 = [ 1 0 0 ] C ζ ζ ζ . (18)
Thanks to the previous transformations we have been able to put the system in the triangular form (18) for which is possible to use the following high gain observer

ζ ζ ζ = A ζ ζ ζ + σ σ σ ( ζ ζ ζ , w2 , µ µ µ) + H( w1 -C ζ ζ ζ ), (19)
where

H = α 1 ε α 2 ε 2 α 3 ε 3
T , ε ∈ R >0 , and α i ∈ R >0 are set such that the roots of s 3 +α 1 s 2 +α 2 s+α 3 have negative real part. 5C. Observation of z 1 and Final x

From the expressions of ż2 in (15) and w2 in (17) we have:

cos z 1 sin z 1 = 1 a 1 ż2 -a 2 cos z 3 u 1 w2 -z 2 2 -a 2 sin z 3 u 1 = h(ζ ζ ζ , u 1 , w2 ).
The state z 1 can then be interpreted as the phase of the unit-

vector h(ζ ζ ζ , u 1 , w2 ), which we denote with ∠ h(ζ ζ ζ , u 1 , w2 ).
We can then obtain a direct estimate of z 1 as Finally, using the first state transformation (14), we get the estimation of the original state x = ( x1 , x2 , x3 , x4 ) needed by the controller shown in Sec. III, i.e,:

ẑ1 = ∠ h( ζ ζ ζ , u 1 , w2 ). (20)
x1 = ẑ1 , x2 = ẑ2 , x3 = ẑ3 -ẑ1 , x4 = u 3 . (21)

D. Closed-loop System Stability with State Observation

Due to space limitations we provide here only the main steps of the convergence proof for the controlled closed loop system with observer. From (20) and (21) we can notice that the estimated state, x, is a function of ζ ζ ζ , w2 and

u 3 x = T ( ζ ζ ζ , w2 , u 3 ). (22)
Therefore, writing the output controller [START_REF] Marino | Nonlinear Control Design: Geometric, Adaptive and Robust[END_REF] as function of the estimated state, and replacing (22) we obtain

ū = Γ(x, u 1 , u1 ) = Γ( ζ ζ ζ , w2 , u 3 , u 1 , u1 ).
The controller depends only on the estimation of the HGO, the measures, and the states of the dynamic compensator. Then, since the system is observable, it can be transformed in triangular form, and the controller stabilizes the nominal system, there exists ε * > 0 such that, for every 0 < ε < ε * , the error dynamic of the observer and the closed-loop system are asymptotically stable [START_REF] Khalil | Nonlinear Systems[END_REF].

The observer presented here is globally convergent. However, to minimize the transient phase it is wise to use some method to initialize the estimate as close as possible to the actual state. For example one can use for this purpose the method proposed in [START_REF] Lupashin | Stabilization of a flying vehicle on a taut tether using inertial sensing[END_REF], which produces an acceptable estimate for quasi-static motions and a distorted estimate for more general motions.

E. Disambiguation of w Using the gyroscope measure, u 3 = ω, and the knowledge of the control inputs, the observer Σ k , described by (19) (20) (21), based on the measures w(k), provides an estimation of the state xk = ( x1k , x2k , x3k , x4k ), depending on k = {0, 1}. Implementing this observer for both the k values, we get two estimations of the same state, x0 and x1 . Thus the problem arises of which observer should be considered as the correct one. We propose here a selection method based on the prediction error. We assign to each observer a prediction error ẽk , smoothed with an exponential discount factor:

ėk = λ ( w -ŵk -ẽk ) , (23)
where λ ∈ R >0 sets the discount rate, w = [w 1 w 2 ] T = a x l a x l + f R ml T , and ŵk = [ ŵ1k ŵ2k ] T is defined as

ŵ1k = cos ( x1k + x3k ) [ x2 2k + a 1 sin x1k + a 2 sin ( x1k + x3k ) u 1 ] ŵ2k = sin ( x1k + x3k ) [ x2 2k + a 1 sin x1k + a 2 sin ( x1k + x3k ) u 1 ]. (24)
Then we select the estimation provided by the observer with the minimum prediction error

x = x0 if ẽ0 ≤ ẽ1 x1 if ẽ1 < ẽ0 .
If the system starts with zero elevation rate (i.e., x 2 = 0), after a transient the two estimations will converge to

x1k = x 1 + kπ x2k = 0 x3k = x 3 x4 = x 4 ,
where (x 1 , x 2 , x 3 , x 4 ) are the real state. In this condition, from equation (24) it is easy to verify that the prediction errors of the two observers converge both to zero. Thus the proposed method could select the incorrect observer. Nevertheless, this is not a problem in practice. Because, by computing the control action based on the wrong estimation, the controller will react forcing x 2 = 0 and, in turn, the predictions errors to be different from each other.

V. SIMULATIONS We conducted several simulations using Simulink and modeling the system with the toolbox SimMechanics, thus obtaining a more realistic validation independently from the model used for the control and observer design. In all the simulations we consider an aerial robot of mass m R = 1 [Kg] and moment of inertia J R = 0.15 [Kg m 2 ], connected to the ground with a generic link of length l = 2 [m]. We assume the vehicle is able to provide both positive and negative thrust. The gains k 1 and k 2 in [START_REF] Tognon | Extended simulations for the link stress and elevation control of a tethered aerial robot[END_REF] are chosen such that the poles of the error dynamics e 1 and e 2 are placed in (-3, -6, -12, -24) and in (-6, -12) respectively. We set ε = 0.1 and (α 1 , α 2 , α 3 ) in (19) in order to obtain an error dynamics with negative real poles. The discount rate in (23) is set to λ = 20.

Each simulation considers different initial state and pair of desired trajectories for the stress and elevation. The complete set of simulation results is provided in [START_REF] Tognon | Extended simulations for the link stress and elevation control of a tethered aerial robot[END_REF]. We show in Fig. 4 a representative simulation chosen from that set. Each simulation result is composed by three figures showing: a stroboscopic trajectory, the controller plots and the observer plots, respectively.

In the stroboscopic trajectories (as the one in Fig 4a ) we show the trajectory evolution as a sequence of state images discriminated by the color that represents the time. In order to show the stress variation, the width of the link is thinner when the tension is higher, and wider when the compression is higher. Moreover to better distinguish the positive from the negative stresses, we draw the link dashed or solid, respectively. This recalls the usual mechanical behavior of a bar under stress. Similarly, the arrow representing the thrust vector is wider and longer when the thrust intensity is higher.

In the controller and observer plot figures (like the ones in Figs. 4b, and4c) it is possible to see that, after a transient due to the state estimation convergence, the controller is able to obtain an exact tracking of all the desired trajectories for both the elevation and stress of the link. This is also obtained for desired stress trajectories passing from tension to compression and viceversa. One can also notice that the transition from tension to compression is also stable and that compression is automatically achieved with a facing down attitude and positive thrust.

A video of the simulations is available in the supplementary material. Additional simulations are provided in [START_REF] Tognon | Extended simulations for the link stress and elevation control of a tethered aerial robot[END_REF].

VI. CONCLUSIONS

We presented a solution to a new tethered aerial vehicle problem that goes beyond the quasi static case considered in the literature. We showed the possibility to precisely and independently control tension and compression of a generic link only resorting to onboard inertial measurements.

In the future we plan to experimentally validate the proposed approach and to design a trajectory planner based on the proven differential flatness of the tethered vehicle. . The complete simulation results can be found in [START_REF] Tognon | Extended simulations for the link stress and elevation control of a tethered aerial robot[END_REF].

Fig. 2 :

 2 Fig. 2: Graphic representation of the controller.

  Observer Σ k : The first block is the high gain observer, the next is the linear observer for z 1 , the third is the state transformation and in the end the prediction error dynamics.

Fig. 3 :

 3 Fig. 3: Graphic representation of the observer.

  It is easy to see that the convergence of ζ ζ ζ to ζ ζ ζ automatically implies the convergence of ẑ1 to z 1 .

  Trajectory from time Begin = 2 [s] to End = 4.5 [s]. Controller Results: thrust stays positive even when the strees becomes negative and compression is obtained turning the robot. Observer Results: zoom on the first 0.7 seconds of simulation. The remaining part is not displayed because the observer follows the state with high fidelity. θ and θ are always equal because the estimated value comes from the direct measure of the gyroscope. The prediction error does not increase even when the tension goes to zero for an instant.

Fig. 4 :

 4 Fig. 4: The desired smooth elevation trajectory of class C 3 goes from the initial value ϕ d 0 = π/4 [rad] to the final ϕ d f = 3π/4 [rad]. The desired smooth stress trajectory of class C 1 goes from the initial tension f d L 0 = 20 [N] to the final compression f d L f = -20 [N]. The complete simulation results can be found in[START_REF] Tognon | Extended simulations for the link stress and elevation control of a tethered aerial robot[END_REF].

  ϕ d of class C 3 and f d L of class C 1 . The behavior of the convergence can be arbitrarily assigned by suitably choosing k 1 ∈ R 4 >0 , and k 2 ∈ R 2 >0 .

	f d L , ḟ d L ϕ d , . . . ,	... ϕ d	e ϕ e fL	Virtual Inputs	v 1 v 2	Feedback
	x	f R ḟR	Output Deriv. and	f L , ḟL ϕ, . . . , ... ϕ		Lineariz. fR 1 s	ḟR 1 s

Consideration of the

2D model has been done often in the related literature and does not limit the range of the results. In fact, the 2D case captures most of the nonlinear and challenging features of the problem.

In[START_REF] Nicotra | Taut cable control of a tethered UAV[END_REF] it is guaranteed only the positivity of the stress, while here we allow exact tracking of any stress trajectory.

Preprint version, final version at http://ieeexplore.ieee.org/ 2

In practice, if a trajectory requires zero thrust for a short time, one can switch off the controller when |u 1 | < ε for a sufficiently small ε ∈ R.

The sum, for each entry of the output, of the number of times that each output entry has to be differentiated in order to see the input appear.

Preprint version, final version at http://ieeexplore.ieee.org/ 3

Preprint version, final version at http://ieeexplore.ieee.org/ 4

The difference ( w1 -C ζ ζ ζ ) stands here for the unique angle β ∈ (-π, π] such that β +C ζ ζ ζ = w1 + k2π for a certain k ∈ Z.

Preprint version, final version at http://ieeexplore.ieee.org/ 5

Preprint version, final version at http://ieeexplore.ieee.org/ 6 2015 IEEE ICRA

This work has been partially funded by the European Community under Contract ICT 287617 ARCAS, and through a resourcing action of the LAAS CNRS Carnot Institute, namely, MDrones project.


Jonathan Ginzburg
email: yonatan.ginzburg@univ-paris-diderot.fr

Robin Cooper

Quotation via Dialogical Interaction

Keywords:

Quotation has been much studied in philosophy. Given that quotation allows one to diagonalize out of any grammar, there have been comparatively few attempts within the linguistic literature to develop an account within a formal linguistic theory. Nonetheless, given the ubiquity of quotation in natural language, linguists need to explicate the formal mechanisms it employs. The central claim of this paper is that once one assumes a dialogical perspective on language such as provided by the KoS 1 framework, formalized in a rich type theory like Type Theory with Records (TTR), much of the mystery evaporates. In particular, one can utilize as denotations for quotative constructions entities that are independently motivated for dialogue processing-utterance types and locutionary propositions, austinian propositions about speech events.

Introduction

Quotation has been much studied in philosophy (see [START_REF] Cappelen | Quotation. Stanford Encyclopedia of Philosophy Clark H (1996) Using Language[END_REF]; De [START_REF] Brabanter | The semantics and pragmatics of hybrid quotations[END_REF] for some recent surveys). There have been comparatively few attempts within the linguistic literature to develop an account within a linguistic theory.

This research was supported in part by VR project 2009-1569, Semantic analysis of interaction and coordination in dialogue (SAICD), by the Lab(oratory of )Ex(cellence)-EFL (ANR/CGI), and by the Disfluency, Exclamations, and Laughter in Dialogue (DUEL) project within the projets franco-allemand en sciences humaines et sociales funded by the ANR and the DFG. [START_REF] Partee | The syntax and semantics of quotation[END_REF] is a classic paper in this area. More recent proposals include [START_REF] Geurts | Quotation in context[END_REF]; [START_REF] Potts | The dimensions of quotation[END_REF]; [START_REF] Bonami | On the syntax of direct quotation in French[END_REF]. This apparent lack of interest in the phenomenon on the part of linguists might be attributed to the fact that quotation is a means of diagonalization (in the Cantorian sense) out of any grammar-for any expression e deemed ungrammatical by grammar Γ , one can produce via quotation a well formed string that includes e, hence undermining Γ . Thus, we can quote something that is ungrammatical in our own language as in (1a) or something that is in a different language to the one we are speaking (1b) or even something that doesn't seem to involve speech sounds that occur in any natural language (1c) or sounds made by inanimate objects (1d).

(1) a. Damien, who's only four years old, said 'I go'ed to Grandma's' b. Pelle, whose native language is Swedish, said 'Jag har varit hos mormor' (meaning "I've been at Grandma's") c. Morry aimed the toothbrush at David and went '[ aeP aeP aeP aeP]' [START_REF] Partee | The syntax and semantics of quotation[END_REF] d. The blender went 'plplplpl' Linguists might be forgiven if they do not regard it as being part of the job of somebody writing a formal grammar of English to include everything that can occur between quotation marks in sentences like those in (1). From a linguistic perspective, one could easily form the opinion that quotation is on the margins of linguistic data, something that a formal grammarian could, and possibly should, ignore. In a similar vein, a wide range of philosophical accounts (e.g. [START_REF] Davidson | Quotation[END_REF]; [START_REF] Werning | Making quotation transparent[END_REF], among many others) assume that what occurs between the quotation marks in the surface syntax is not part of the sentence in which those quotation marks occur. This, however, presents challenges for dealing with examples where the form or the content of the quotation is referred to from outside the quotation as in (2).

(2) a. 'I talk better English than the both of youse!' shouted Charles, thereby convincing me that he didn't. (Partee, 1973, ex. 20) b. The sign says 'George Washington slept here', but I don't believe he really did. (Partee, 1973, ex. 26) c. What he actually said was, 'It's clear that you've given this problem a great deal of thought,' but he meant quite the opposite. (Partee, 1973, ex. 32) And indeed there is no little amount of evidence that quotation is subject to general grammatical principles governing word order, ability to be embedded and psuedo-clefted, and semantic selection (Postal, 2004;[START_REF] Bonami | On the syntax of direct quotation in French[END_REF].

Moreover, all natural languages seem to have quotation of some kind. Children use direct quotation from their earliest utterances [START_REF] Ginzburg | The earliest utterances in dialogue: towards a formal theory of parent/child talk in interaction[END_REF]. Indirect quotation is acquired substantially later, but is nevertheless standardly present in the adult language. 2,3 Given the ubiquity of quotation in natural language, linguists need to explicate the mechanisms it employs. Indeed, one is obligated to do so in a way that offers an answer to the question: why, rather than being a heterodox linguistic process, is in fact quotation so straightforward?

The short answer, we suggest, is that this is because quotation involves entities and mechanisms utilized ubiquitously during dialogue processing. In slightly more detail-dialogue processing involves a periodic interaction concerning whether an utterance has been understood (grounding [START_REF] Brennan | Conceptual pacts and lexical choice in conversation[END_REF]) and if not the need to engage in clarification. On one model of this process developed in the KoS framework (see [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF]) this involves assessing the truth of a locutionary proposition-a proposition constructed from the utterance event and a linguistic sign made available by grammatical resources available to the dialogue participants-assessing whether the utterance can be fully characterized or not by the sign and if not inferring a clarification question. On this view, locutionary propositions, of which signs (speech event types) are fundamental components, are the essential means for representing utterances in an interlocuter's view of the context-their dialogue gameboard. We will propose, in section 4, that signs and locutionary propositions offer simple and semantically parsimonious denotations for (pure/direct) quotation, thereby directly tying them to utterance representations used ubiquitously in dialogue.

The central claim of this paper, then, is that once one adopts a semantic perspective rooted in dialogue and utilizing a rich type theory many of the recaltricant issues associated with a semantic account of quotation dissolve. This perspective incorporates four fundamental assumptions:

1. Language as a system in flux. [START_REF] Cooper | Natural Languages as Collections of Resources[END_REF]; [START_REF] Cooper | Type theory and semantics in flux[END_REF] argue that natural language grammar should be regarded in terms of a collection of resources that a linguistic agent has available in order to build local (possibly formal in the sense of Montague, 1974) languages on the fly appropriate to a particular situation. This is meant to provide a basis for the fact that speakers coordinate both the form and meaning of their language with their interlocutors [START_REF] Cooper | Compositional and ontological semantics in learning from corrective feedback and explicit definition[END_REF][START_REF] Larsson | Towards a formal view of corrective feedback[END_REF], a fact which is normally abstracted away from in formal treatments of grammar but which has attracted a good deal of attention in the psychological literature on language (e.g. [START_REF] Brennan | Conceptual pacts and lexical choice in conversation[END_REF][START_REF] Healey | Expertise or expertese?: The emergence of task-oriented sublanguages[END_REF][START_REF] Healey | Graphical Language Games: Interactional Constraints on Representational Form[END_REF]. The constant micro-adjustment of language as we talk with others (particularly, but not solely, on the part of children acquiring the language) is the driving mechanism in linguistic change.

For quotation this means that it is no longer surprising that a speaker should create an ad hoc language blending resources from different languages or dialects which the speaker has available and even including non-linguistic resources. Since we are not in the business of building one monolithic formal grammar for English but rather explaining what resources a linguistic agent has to have available to construct a local language for a particular purpose, the formal problems associated with these kinds of quotations do not arise. 2. Linguistic events. In our previous work (most recently [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF][START_REF] Cooper | Type theory and semantics in flux[END_REF] we take seriously the fact that natural language grammar concerns linguistic events and that this should be reflected in grammatical theory. Our inspiration for this comes originally from early work in situation semantics [START_REF] Barwise | Situations and Attitudes[END_REF]) which showed us how to make grammatical sense out of speech act theory [START_REF] Austin | How to Do Things with Words[END_REF]Searle, 1969, and, of course, much subsequent literature).

For quotation this means that we analyze direct quotation as referring to a particular linguistic (speech, reading or writing) event. This situates our approach within the family of demonstrative theories [START_REF] De Cornulier | L'incise, la classe des verbes parenthétiques et le signe mimique[END_REF][START_REF] Clark | Quotations as demonstrations[END_REF]. 3. Event types as central to linguistic processing and understanding. On our event based approach to language it is important that linguistic generalizations and procedures be expressed not in terms of the events themselves but in terms of types of events (or situations). This use of event types enables us to express rules which predict possible future events. It also allows us to characterize certain aspects of events and underspecify the nature of an event by using a type to which the event in question belongs but to which other events could also belong. The formalism we will use for this is Type Theory with Records (Cooper, 2012, TTR) which incorporates ideas from situation theory and Martin-Löf Type Theory (Martin-Löf, 1984;Nordström et al, 1990). The relevance of this for quotation has to do with the fact that while we are referring to a previous speech event with a direct quotation we cannot reproduce that event directly (except perhaps by playing a recording of it, which would not count as a quotation). The direct quotation of the previous speech event is a new event which shares a type with the original event. Which type we choose to illustrate with the quotation varies depending on the context and the resources we have available.

Types of linguistic events also play a role in our analysis of pure quotation. Words (like pieces of music, Cooper, 2013b) can be thought of as types of events, either types of utterances or types of written occurrences. Thus when we say that "Anna" begins with "a" we mean that any written occurrence of the type "Anna" begins with an instance of the letter "a", itself a type of written occurrence of a letter. Alternatively, we mean that any utterance event of the type "Anna" begins with an event of an utterance of the type which is phoneme /ae/, itself a type of utterance event. 4. Language as a sign and feature based system. It will be important for our approach to quotation to adopt the notion of linguistic sign [START_REF] De Saussure | Cours de linguistique générale[END_REF], that is a pairing of linguistic form and content. For us the linguistic form will be an event and we will be interested in types of signs, that is types of formcontent pairs where the form and the content are required to be of certain types. Our approach to this will be based largely on a type theoretical version of Head-Driven Phrase Structure Grammar (HPSG, Sag et al, 2003). The adaptation of de Saussure's notion of sign in HPSG uses feature structures. We will model feature structures as record types.

Using such structures gives us a simple minded but powerful notion of similarity between types where similarity can be construed as sharing some features but not necessarily all. Such a notion of similarity is central to a theory of quotation. For example, a quotation in a different language from the original may share the content of the original but not the form. Furthermore, the content may not be exactly the content of the original but similar enough for the purposes of the dialogue. In exploiting the notion of linguistic sign in quotation we are following the lead of [START_REF] Bonami | On the syntax of direct quotation in French[END_REF] and also [START_REF] Potts | The dimensions of quotation[END_REF] who essentially uses a sign-based (though not feature-based) approach although he does not refer to it explicitly as such.

The rest of the paper is structured as following: Section 2 provides a brief survey of relevant data and relates them to these four aspects. Section 3 introduces TTR and show how it enables the modelling of dialogical interaction. Section 4 applies these tools to develop analyses of pure and direct quotation.

Some data

In this section we will present some data on quotation that motivates our account: on pure quotation it justifies our decision to postulate a sign-denoting construction; for direct quotation we provide data suggesting that a similarity measure and that grammatical resources are contextual parameters of such uses; we demonstrate that direct quotation is selected by certain predicates, raising the issue of what the semantic type of such uses is; finally, we demonstrate the close relationship between direct and indirect quotation.

Pure Quotation

Pure quotation involves general, rule-like statements about utterance types. As example (3) shows, these can target various dimensions of the linguistic sign, but they do not directly concern any utterance tokens as such:

(3) a. syntax: To capture the data in (3), one needs to assume that quotation has a use in which it refers to an entity that incorporates information about the variety of linguistic dimensions (phonological, morphosyntactic, semantic, . . . ) relative to a particular language-it is crucial, e.g., that neither intra-linguistic, nor cross-linguistic homophones (English 'rapport' (relationship) and French 'rapport' (report); Hebrew 'xatuna' (wedding) and Georgian 'xatuna' (a proper name)) be conflated. Hence, breezy assumptions about the pure quotation as 'denoting the expression' cannot be maintained. 4By shifting the content of a proper name to a type of occurrence of a proper name we can explain apparent failures of compositionality and existential generalization which have been discussed in the literature such as in (4).

'
(4) a. Karl was looking for "Mark Twain" in the phone book.

→ Karl was looking for "Samuel Clemens" in the phone book. b. "Anne" begins with a vowel.

→ ∃x "x" begins with a vowel [START_REF] Werning | Making quotation transparent[END_REF] We conceive of the use of quotation marks in written text as a symbolic way of representing that the content of the word or phrase within the quotation marks is to be shifted to such a type. So, for example, if we were to treat proper names following the classical Montague (1974) treatment as denoting the set of properties of some object, the quoted version of the proper name "Anne" could be the set of properties of the type of strings of phoneme utterances that consist of an utterance whose phonological dimension has the phoneme /ae/ followed by the phoneme /n/. This could be derived from the sign for Anne by replacing the content of the sign with a noun-phrase denotation derived from the original sign. Note that on this conception, while (4b) indeed represents a failure of existential generalization, if we allow quantification over types and predicates of types in the way we have suggested the existential generalization in (5) holds.

(5) a. "Anne" begins with a vowel. b. → ∃x x begins with a vowel

The point, of course, of (4b) is that we cannot consider there to be a quotation function, ".", which takes the content of an expression a and yields the content of "a". Thus in a strict sense quotation is not compositional. (For a recent discussion of compositionality and quotation see [START_REF] Pagin | Pure quotation and general compositionality[END_REF] In the case where the quotation refers to a type of utterance event, we might say that "." is defined not on the content but on the whole sign, picking out the phonology of the sign as the content of the quoted phrase. (This is an essential part of the proposal by Potts, 2007.)

Direct quotation

Unlike pure quotation, direct quotation involves reference to a particular linguistic event or situation (either a speech act or a textual situation of a given text containing a particular subtext). 5 We consider two contextual parameters that seem crucial for an account and have not been explicitly considered in previous work-similarity measures and grammatical resources. For a start, there is a question of whether quotation involves reference to an event or, following [START_REF] Clark | Quotations as demonstrations[END_REF]; [START_REF] Brennan | Conceptual pacts and lexical choice in conversation[END_REF], a demonstration of an event. For and [START_REF] Geurts | Quotation in context[END_REF] say: 'If the quotation is a 'pure' one, it denotes a linguistic expression and its semantic type may differ from that of the quoted expression. For example, in [the example 'Monosyllabic' is not monosyllabic.] quotation maps the semantic type of the adjective 'monosyllabic' to that of singular terms, and the resulting expression denotes the word 'monosyllabic'. 5 We will ignore the possibility that texts can be regarded abstractly as types of physical manifestations, although we find this plausible.

an illuminating discussion of this issue see [START_REF] Recanati | Truth-Conditional Pragmatics[END_REF], Chap. 7. Our discussion of direct quotation here is closely allied with Recanati's discussion. We will follow Recanati in talking of reference in cases of what he calls closed quotation, that is, when the quoted material plays a grammatical function in an utterance. The speaker who is quoting a previous linguistic event cannot, of course, reproduce the exact previous event. It is in the nature of events in general that they happen only once. The closest we can come to repeating an event is to create a new event which is similar to the previous one. This is the notion of demonstration introduced by Clark and Gerrig. The kind of reference that Recanati ascribes to what he calls closed quotation is achieved by demonstrating the event which is referred to. On our type theoretic approach there are two ways in which one can analyze similarity between two events. One way is to say that there is a type which the two linguistic events (that is, the one being quoted and the demonstration) share. Recanati discusses ( 6). ( 6) . . . And then Greta Garbo said, 'I want to be alone!'

In demonstrating Garbo's utterance the speaker produces an utterance which shares a type with the original utterance being referred to. In terms of a type of sign which specifies both the form and the content of the utterance we can say that the demonstration and the original share a sign type which specifies the phonological form of the utterance to be of the phonological type represented by "I want to be alone" and the content type corresponding to Greta Garbo wants to be alone. Note that in order for this to be a quotation of Greta Garbo's utterance, more is required than that the two speech events share just any type. After all, there are much more general types to which both belong such as Event, the type of events or SpeechEvent, the type of speech events. One cannot be said to be quoting somebody by producing some random speech event simply because the original and the quotation are both speech events. The type which the two events share has to be specific enough for it to count as similarity.

We want to claim that this need for specificity leads us ultimately not to look at similarity in terms of sharing types but rather to develop similarity measures on types themselves. Let us consider a non-linguistic example first. Suppose that I want to demonstrate the way that John Cleese walks in his Minister of Funny Walks sketch. I am not able to do the walks so they are exactly like those of John Cleese but I might be able to produce walks of a similar type. There might well be types to which both Cleese's performance and my demonstration belong but there is not obviously a guarantee that these types will be specific enough to count as guaranteeing similarity. For language, Recanati discusses Sperber and Wilson's (1986) example ( 7). ( 7)

He said 'We're going to close early tonight' which, in Recanati's words, "can be used to report an utterance in Italian, provided the Italian sentence had a meaning sufficiently similar to that of the English sentence 'We're going to close early tonight' ". In terms of signs this means that the phonological form of the sentence does not have to be the same and the content only has to be "sufficiently similar". For us the content of a declarative sentence like we're going to close early tonight will be defined in terms of a type of situation which could be described by the sentence. Thus it will be expedient for us to consider similarity mea-sures on the types themselves rather than characterizing similarity in terms of shared types.

A question that arises on this kind of view is whether there is a single similarity measure which can be used in all cases of quotation. It seems that this is not the case. The fact that we can quote utterances that were made in a different language, as in the discussion of ( 7), suggests that it can be sufficient for the content of the quoting utterance to be the same. Here there is no requirement that the form of the utterance be similar to the original utterance. Even if the quotation is in the same language as the original it normally cleans up disfluencies such as hesitations or mispronunciations unless these are a relevant part of the demonstration involved in the quotation. This point is made clearly in the quotation in (8 6 See also the following from the economist Brad DeLong:

It is conventional-at least in the circles in which I travel-to clean-up transcripts. Raw human speech looks silly in print. You eliminate the "ummms" and the "you knows" and the false starts. You collapse into one grammatical sentence those times when people start a sentence, go back and start it again, and then go back and start it again and finally finish it. The convention is to edit the transcription of what they said into something different-into what they would have written if they had been not speaking but writing, sentence by sentence and paragraph by paragraph, and editing as they go. But this seems to go rather beyond that, so much so as to convey a substantially distorted impression of how the symposium actually went . . . http://delong.typepad.com/sdj/2013/12/thursday-idiocy-weblogging.html d. Jean said 'Je le ferai' which I think means that he will do it.

These differences in the similarity measures used in different contexts are not surprising if the whole linguistic system follows the principle of flux. We would suggest that it is not the job of a linguistic theory to specify exactly which similarity measures are used but rather to exhibit a linguistic theory which enables the definition of relevant similarity measures.

Agents who know a language have grammatical resources which will allow them to assign grammatical types, such as sign types, to speech events they observe. In computational terms we can think of the types assigned to a speech event (utterance) as the result of parsing the utterance. It is these types, not types in general, which have to be similar between the original event and the quotation event. We assume that agents have a collection of grammatical resources and that they draw on different subcollections of these resources on different occasions, depending, for example, on which language or dialect they are speaking, whether they are speaking about theoretical physics or the family outing at the weekend or are involved in a technical discussion about the offside rule.

In many cases of quotation the resources applied to the original utterance-in terms of notation, Γ 1 -will be the same as the resources applied to the quotation utterance--in terms of notation, Γ 2 . However, this will not be the case when the original utterance was in a different language to the quoting utterance. In this case it may even be that the agent of the quoting utterance does not have access to resources Γ 1 needed to process the original utterance -they may have benefitted from a trans- lation at the time. Similarly the addressee of the quoting utterance may also not have access to Γ 1 . Note also, that Γ 2 need not be the same resources that are needed to process the rest of the sentence Anna said. . . if the actual quotation is in a different language or dialect. We can switch resources in the middle of a sentence in a similar way that speakers of more than one language can in code-switching.

The choice of Γ 1 and Γ 2 has consequences for the kind of similarity measure, µ, that can be used. When Γ 1 = Γ 2 we can choose µ to be a strict measure which requires that the resources assign exactly the same sign type to both the original and the quoting utterance. But even here we will often want a looser notion of similarity. For example, a demonstration of an utterance editing out disfluencies we may want to count as a quotation, at least some of the time. Different choices of synonymous words may for some purposes count as quotation for others not. µ may be more or less strict depending on the purposes at hand, an extreme case being (12): (12) [An article about an orphaned walrus arriving in a new zoo:] During [the orphan walrus'] first look at a walrus, he was like, 'What's that?' (New York Times, 22/01/2014) When Γ 1 and Γ 2 are resources for different languages the most we can hope for µ only requires that the contents required by the two sign types are identical. However, as is well known, identity of content between expressions in two languages can be too much to hope for. Suppose the original utterance quoted in (13a) had been the Swedish utterance (13b).

(13) a. Max said 'Bo departed'. b. Bo har åkt ("Bo has gone/went (by vehicle)") There are three things which could be different about the content of (13b) which could make it different from the content of the quotation in (13a). Firstly, it is in the perfect tense. There appear to be cases where a Swedish perfect is appropriately rendered in English by a simple past, but this is not always true. Secondly, Swedish like many other languages makes a distinction between travelling by vehicle and walking. Finally, the verb which serves for leave here has the broader meeting went and could be used to describe a situation where Bo has arrived. All of these subtle differences are familiar to anybody who has been involved in translation. This has as a consequence that µ must say something about similarity of non-identical contents. Our aim here is not to go into detail characterizing similarity measures that can be used, but rather to make the point that there cannot be a single similarity measure for all instances of quotation. The similarity measures, like the grammatical resources involved, will vary from one quotation event to another.

Finally, we need to allow for the possibility that the types compared by the similarity measure µ are not necessarily the types assigned by the resources Γ 1 and Γ 2 but subtypes of those types. This is to allow for cases where the demonstration in the quoting event is meant to show a non-linguistic aspect of the original event in addition to the linguistic type, for example, that the speaker of the original event had a cold, was drunk or had a particular expression on their face while making the utterance. This information could be added to the grammatical types to form subtypes and similarity computed between these subtypes. Note that it is important here that we construe linguistic types as types of events so that we can, for example, not only talk of types of events which are utterances of the phoneme /ae/ but also types of events which are utterances of the phoneme /ae/ with a blocked nose. While the blocked nose may not be "linguistically significant" in the standard sense, it can be significant for the content of the quotation.

Quotative Operators

There are a number of words that require direct quotation as their complements. In English the marker like and the verb go have a certain usage which requires a direct quotation as in (14a) and (14b) and does not allow an indirect quotation, as exemplified in (14c) and (14d). Such constructions do not even require a speech act to be demonstrated as Partee's example (1c) shows and also (14e). Such constructions exist in many, if not all, languages although they tend to be restricted to an informal spoken register. One might cite French faire and Swedish typ/ba as in (14f). ( 14) a. I asked her if she wanted to read my paper and she was like "Are you crazy?" b. I asked her if she wanted to read my paper and she went "Yuck!" c. *I asked her if she wanted to read my paper and she was like whether I was crazy d. *I asked her if she wanted to read my paper and she went that she didn't, in no uncertain terms e. I asked her if she wanted to read my paper and she was like [expression of disgust, slowly shaking head] f. först sa han typ, ("first he said like") hej, hur gammal är du ? ("hi, how old are you?") sen är du såblond ? ("then, are you so blond?") ja ba, nae inte precis, ("I just, no not exactly") han typ är du singel ? ("he like, are you single?") http://nellierosengren.bloggsida.se/okategoriserad/13 (accessed 6th Dec. 2013)

Given the possibility of non-speech act demonstrations in these examples one might come to the conclusion that direct quotations simply refer to the demonstration and lose the content of the original event, if it was a speech act. There is much discussion in the literature which shows this not to be true, including the examples (2) discussed by Partee. These examples involve predicates that can combine with direct quotation, as well as indirect quotation, as in (15). Indeed, there is a clear relationship between the indirect and direct quotation complements in (15a,b)-both support the inference (15d):

(15) a. Zohar asked whether she snored. b. Zohar asked 'naxarti?'. c. Zohar asked 'naxarti?', a Hebrew sentence often uttered by people who have just woken up. d. Zohar asked a question, a question about herself.

The examples in (2) and in (15) indicate that, in one way or another, the complement of a direct quotation behaves as a bona fide semantic entity, giving rise to nominal and verbal anaphora, as well as to propositional ('the opposite') or interrogative inference. These facts seem to hold even of constructions which require a direct quotation as in ( 16). ( 16) I asked her if she wanted to read my paper and she was like "Are you crazy?" But I'm not. I'm perfectly sane.

Our earlier claim that language is sign-based and feature-based, will play an important role in identifying different aspects of an utterance and enabling an utterance to be a demonstration of a prior utterance while at the same time making available the normal content of the utterance.

A Dialogical Perspective on Semantics

In this section we introduce the tools underlying our view of dialogical perspective on grammar and semantics. We introduce Type Theory with Records (TTR), which enables us both to develop a grammatical ontology, a semantic ontology, and a theory of interaction.

Grounding and Clarification Interaction Conditions

In dialogue all words and phrases are, in principle, subject to clarification interaction (see [START_REF] Ginzburg | Clarification, ellipsis, and the nature of contextual updates[END_REF][START_REF] Purver | Clarie: Handling clarification requests in a dialogue system. Research on Language and Computation[END_REF][START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF]Cooper, 2013a) This is a conventionalized process: as corpus studies have shown, there are a very small number of possible construals of CRs-as request for confirmation, repetition, intended content resolution (see [START_REF] Purver | On the means for clarification in dialogue[END_REF]; [START_REF] Rodriguez | Form, intonation and function of clarification requests in german task-oriented spoken dialogues[END_REF]). There are thus in the aftermath of an utterance (and indeed while it is ongoing) two essential branches:

-Grounding: the utterance is understood and signalled as such [START_REF] Brennan | Conceptual pacts and lexical choice in conversation[END_REF], its content added to the common ground, uptake occurs. Clarification Interaction: some aspect of the utterance causes a problem; this triggers an exchange to repair the problem.

Hence, we need an entity off of which both 'shared understanding' update AND clarification potential (Grounding and Clarification Interaction conditions) can be read. In order to understand how this can be formalized we turn to sketch the framework that provides the logical underpinnings of KoS, Type Theory with Records (TTR).

Developing a semantic and grammatical ontology in Type Theory with Records (TTR)

An utterance of a proper name like Bo is a sign, that is, it is an event of a certain type relating a certain kind of utterance to a content. We can think of a sign essentially as a pair of an event e and a content c. (We will argue later that more things are involved.) One way of thinking of this pair is as the set {e, c}. This set, however, does not label its elements in any way. Given an arbitrary sign modelled in this way it may not be obvious which element is playing which role. For this reason we use records, which include not only the objects in a set but also labels for those objects. For now, we will use the labels 's-event' (for "speech event") and 'cont'. (for "content"). Each label can label only one object in a record. Formally, we can model records as sets of ordered pairs consisting of a label and an object as in (18a). We will represent this as in (18b) in order to emphasize the relationship with feature structures in linguistics and to make larger structures more readable.

(18) a. { s-event, e , cont, c } b. s-event = e cont = c
Part of what makes TTR a type theory is that each object a belongs to a type T , in symbols, a : T . In fact in TTR objects may belong to many types, even infinitely many types. Thus, for example, we may say that e belongs to the type SEvent (for "speech event") and c belongs to the type Cont (for "content"). Types are regarded as mathematical entities in their own right within the semantic universe. They are not just collections of objects. ':', the "of-type relation", represents a relation which holds between objects and types. This means that types are intensional in the sense that two distinct types can be related to the same set of objects by the of-type relation. We also have record types which are labelled sets of types, that is, sets of ordered pairs of labels and types such that each label in the record type is associated with exactly one type. The type of signs as we have so far discussed them is thus (19a). As with records we represent this in the, for linguists, more convenient form (19b).

(

) a. { s-event, SEvent , cont, Cont } b. s-event : SEvent cont : Cont 19
The of-type relation between records and record types is defined such that a record, r is of a record type T just in case for each labelled type in T r contains a labelled object of that type:7

(20)

The record

    l 1 = a 1 l 2 = a 2 . . . l n = a n     is of type:     l 1 : T 1 l 2 : T 2 . . . l n : T n     iff a 1 : T 1 , a 2 : T 2 , . . . , a n : T n
The type SEvent may be considered to be a record type. Its exact nature is a matter of the analysis you wish to propose. One candidate is ( 21), although one can argue that there are speech events which do not belong to this type.

         
This means that a speech event would be modelled as a record of this type, that is, a record containing a spatial location labelled 'e-loc' ("event location"), two individuals ("speaker" and "audience") and an utterance event (labelled 'e') which is a phonological event, an event of type Phon(ology). The remaining fields in the type place constraints on the relations that hold between these objects. Predicates like 'loc', 'speaker' and 'audience' are type constructors. When supplied with arguments they yield a type (normally a type of situation) which intuitively is to be thought of a providing a proof of a given fact. Thus, if e is an event and l is a location then loc(e, l) is intuitively to be construed as a type of situation such that any situation of the type shows that e is located at l. In other words s : loc(e, l) if and only if s constitutes a proof that the e's location is l. Before describing what a sign is and how utterances enter into context, we briefly describe the view of propositions we assume. Building on a conception articulated 30 years earlier by [START_REF] Urmson Austin | Truth[END_REF]), [START_REF] Barwise | The Liar[END_REF] developed a theory of propositions in which a proposition is a structured object prop(s, σ ), individuated in terms of a situation s and a situation type σ .

TTR offers a straightforward way for us to model propositions using records. A proposition is a record of the form in (22a). The type of propositions is the record type (22b) and their truth can be defined, as in ( 22c In ( 23) we exemplify a sign or utterance event type in the format of the grammatical framework HPSG: this has fields for phon, (syntactic) category, contextual parameters, and content-the contextual parameters include minimally those discussed earlier in our explication of the type SEvent above:8 ( 23)

                           PHON :
            CONT = Ask(spkr,addr, ?   sit = s0 sit-type = c : In(l,g)   ) : IllocProp                           

Grounding and Clarification Interaction conditions in KoS

The original conception of Austinian propositions was that s is a situation deictically indicated by a speaker making an assertion. [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF] extends this idea to the notion of locutionary propositions to characterize the communicative process, here the propositions individuated in terms of an utterance event u 0 as well as its grammatical type T u 0 . In terms of an interlocuter's dialogue gameboard, the issue can be formulated as follows: what information needs to be associated with Pending, that component of the dialogue gameboard which keeps track of utterances undergoing integration into the dialogue gameboard, to enable the formulation of grounding conditions/clarification interaction potential? The requisite information needs to be such that it enables the original speaker to interpret and recognize the coherence of the range of possible clarification queries that the original addressee might make. [START_REF] Ginzburg | The Interactive Stance: Meaning for Conversation[END_REF] offers detailed arguments on this issue, including considerations of the phonological/syntactic parallelism exhibited between clarification requests and their antecedents and the existence of clarification requests whose function is to request repetition of (parts of) an utterance. Taken together with the obvious need for Pending to include values for the contextual parameters specified by the utterance type, Ginzburg concludes that the type of Pending combines tokens of the utterance, its parts, and of the constituents of the content with the utterance type associated with the utterance. An entity that fits this specification is the locutionary proposition defined by the utterance. In ( 24) we exemplify a locutionary proposition, whose utterance type we introduced earlier. The content of the utterance is given by the path sit.cont:

(24)                                                             sit =                            PHON = izjorjhiya CAT = V[+fin,+root] CONSTITS = u1(iz),u2(jorj), u3(hiya), u4( izjorjhiya) C-PARAMS =           spkr = A addr = B s0 = sit1 l = l0 g = g0 c3 = pr1           cont = Ask(A,B, ?   sit = sit1 sit-type = c : In(l0,g0)   )                            sit-type =                           
PHON : is georges here CAT = V[+fin,+root] : syncat CONSTITS = is, georges, here, is georges here : set(sign)

C-PARAMS :

            spkr: IND addr: IND c1 : address(s,a) s0: SIT l: LOC g: IND c3: Named(g,'georges')             CONT = Ask(spkr,addr, ?   sit = s0 sit-type = c : In(l,g)   ) : IllocProp                                                                                       
In the immediate aftermath of a speech event u, Pending gets updated with a record of type locutionary proposition (LocProp). Here T u is a grammatical type for classifying u that emerges during the process of parsing u. Here we identify this with a sign in the sense of Head Driven Phrase Structure Grammar (HPSG), as we exemplified above in (23). The relationship between u and T u -describable in terms of the proposition p u given in (25a)-can be utilized in providing an analysis of grounding/CRification conditions, as shown in 25b:

(25) a. p u = sit = u sit-type = T u b. Grounding/clarification interaction conditions:

1. Grounding: p u is true: the utterance type fully classifies the utterance token.

2. clarification interaction: p u is false, either because T u is weak (e.g., incomplete word recognition) or because u is incompletely specified (e.g., incomplete contextual resolution-problems with reference resolution or sense disambiguation).

It is useful to conceive of the integration of an utterance in an information state as a potentially cyclic process. Instantiation of some, perhaps all, contextual parameters will occur as soon as an utterance has taken place, assuming T u is uniquely specified; if this is not the case, then clarification interaction can occur on that level. Parameter instantiation can also take place subsequently, as when more information is provided as a consequence of clarification interaction. Given this, utterance integration can be broken into three components:9 (26) a. Pending update: in the immediate aftermath of a speech event u, Pending gets updated with a record of the form sit = u sit-type = T u b. Contextual extension: If T u is uniquely specified, try to instantiate the contextual parameters of T u relative to the context provided by the dialogue gameboard: find a record w that extends u and such that w contains a subrecord of the c-param anchoring intended by u's speaker; integrate w into MaxPending: MaxPending := sit = w sit-type = T u c. Move update/Pending downdate: if MaxPending is true, update Moves, so that LatestMove:= MaxPending, downdate MaxPending from Pending.

Back to Quotation

The ontology we have developed for dialogue processing can now be utilized for analyzing the quotation data from earlier sections, with pure quotation denoting signs and direct quotations denoting locutionary propositions. Quotative predicates select for clauses restricted to denote either abstract entities simpliciter or locutionary propositions. We could formulate our formal account in a number of ways. One would involve type shifting-postulating mappings between utterance types triggered by quotation. Such an approach is highly consonant with the view of grammar developed as a dynamic resource available to an agent to create a language on the fly [START_REF] Cooper | Natural Languages as Collections of Resources[END_REF]; [START_REF] Cooper | Type theory and semantics in flux[END_REF]). Another possibility, somewhat more conservative in a way, and in line with [START_REF] Bonami | On the syntax of direct quotation in French[END_REF], is to appeal to one or more quotative constructions. This approach, emphasizing that quotation fits into a traditional view of grammar, is the format we adopt here, though it would be straightforward to adopt the former approach.

Pure Quotation

The simplest construction we could associate with pure quotation is one in which it denotes the phonological type of its unquoted correlate, as in ( 27 Note that the pure quotation type from (27) will lose the information about the original content of the sign. Moreover, we argued in section 2 that, in fact, a pure quotation utterance concerns potentially the entire information about the sign, not merely its phonological form. This can be easily fixed by using construction type in (28), wherein the content of the quoted phrase is the whole sign-type, rather than that in (27). One additional aspect of this construction concerns its contextual parameters. We specify that these all get absorbed and the construction, hence, has no contextual parameters: Recall that we proposed that in dialogue an utterance u0 is represented in an interlocutor's DGB by means of a locutionary proposition, a proposition whose situational component is the utterance event u0 and whose type component is a grammatical type T u0,Γ 0 assigned to the utterance by a given grammatical resource Γ 0 . Not surprisingly, our analysis of direct quotation assigns to such clauses as their content a locutionary proposition, with the situational component being the (original) utterance event and the type component a grammatical type T u0,Γ 0 assigned to the utterance by a grammatical resource associated with the person reporting the speech.

Whereas with pure quotation we associated solely the grammatical type, we include here also the utterance event which, in conjunction with the grammatical type, enables the utterance's content to be read off, as we explained in section 3.3. This is key both in capturing the similarity with indirect quotation (see data above examples (15)) and Partee-inspired data such as examples (2) above. Beyond this, though we emphasize that by postulating locutionary propositions as the denotations of direct quotation, we are offering a direct answer to the question we had posed in section 1, namely Why is direct quotation so easy? The answer is that this is because it exploits mechanisms in place for dialogue processing. 10By analyzing direct quotation complements as denoting locutionary propositions, which include as a component a sign we can capture simultaneously the type and token aspects of such complements. Indeed the fact that a single predication can address both type and token aspects, as in ( 29):

(29) a. 'Was I snoring' was asked by Bill and is a frequently used interrogative clause.

b. 'Am I snoring?' asked Bill, a sentence frequently uttered by men who don't think they snore. It is usually answered by 'You were before you woke up.'

The construction we associate with this type is a fairly minor variant on the one we posited for pure quotation. The crucial difference is that although the contextual parameters of the original type have been projected away, a new one is projected to include the utterance event. To this we need to add constraints that relate the utterance event and the grammatical type appearing in the construction. As we discussed in section 2, what we need to enforce is similarity between the original utterance u and the type T used to quote it-we capture this in terms of a similarity measure µ and a set of grammatical resources Γ , the relationships between which we explicate below: 11,12 (30) a. direct-quot-ph

=                 PHON = T.phon : phoncat CAT = T.cat : syncat C-PARAMS :      u : Rec µ : SimilarityMeasure Γ : set(GrammType) c1 : Similar(T,u, µ,Γ )      CONT= sit = u sit-type = T : LocProp                
T : GrammType HEAD-DTR : T

We turn to explicate briefly the relationship among the contextual parameters of a direct quotation:

(31) a. Let Γ be a set of grammatical resources. We will not specify this formally here. If u is a speech event and Γ a set of grammatical resources we use Γ (u) to represent the sign type assigned to u by the Γ (ignoring the possibility that several alternative sign types may be provided for u). b. A similarity measure on types is a pair of a threshold θ and a binary function µ 0 on types which returns a real number for any pair of types and whose maximum value is returned for any pair of identical types and for which any value greater than θ indicates similarity.13 c. If e is a speech event, T a grammatical type, µ a similarity measure and Γ a set of grammatical resources, the type Similar(T, u, µ,Γ ) is non-empty iff there exists a grammatical type T 1 such that T 1 ⊑ Γ 1 (e 1 ) e : T , and µ 0 (T 1 , T ) ≥ θ

Quotative Predicates

One crucial point the data we discussed earlier establishes is that there are predicates ('go', 'be like', 'faire', 'typ') that select only for direct quotation complements. Another is that there are predicates that select simultaneously for direct and indirect quotation complements, with mutual entailments between corresponding com-plements (see (15 above). 14 We specify direct quotation selection via the semantic type we postulated for them-LocProp, whereas predicates selecting for both types will be specified via the abstract entity associated with an utterance of the direct complement-question ('ask'), proposition ('claim'), outcome ('order') etc.

We illustrate this with three lexical entries: one for a quotative predicate like 'be like', one for 'ask' taking a direct quotation complement, and one for an indirect quotation complement of 'ask'. In the two first cases the complement selected is a locutionary proposition, representing an utterance whose speaker is identical to the entity associated with the subject of the predicate. The cont value of the predicate is identified with the value of the utterance component of the locutionary proposition. For 'be like' this can be any kind of illocutionary act,15 whereas for direct 'ask' this has to be a query. In the final indirect case of 'ask', one simply unifies in the question-denoting complement daughter and the referent of the subject daughter:

(32) a.               PHON : be like CAT = v[+fin] : PoS SPR-DTR : np ∧ content = x : Ind COMP-DTR :    CONT= p : LocProp p.sit-type.ctxt-params.spkr = x : IND i = p.sit.cont : IllocProp    CONT = i : IllocProp               b.                 phon : ask CAT = v[
i = Ask(x,q) : IllocProp      CONT = i : IllocProp                
14 And to this one should add, as noted by [START_REF] Bonami | On the syntax of direct quotation in French[END_REF], the existence of predicates incompatible with direct quotation complements. But we will not attempt to capture this pattern here, which extends to verbs of cognition-an analogy much emphasized by Recanati. In this section we discuss some recent formal grammatical accounts of quotation whose insights we draw on.

Bonami and Godard

We have already mentioned Bonami and Godard's paper [START_REF] Bonami | On the syntax of direct quotation in French[END_REF]) several times. This paper makes several important contributions. Descriptively, it highlights the canonical nature of quotative complements, thereby providing arguments for their within-grammar nature; it points out the need for the grammar to accommodate 'agrammatical' quotative complements. Formally, it offers an account for a quotative complement as a sign-denoting construction. And it points to intrinsic difficulties that standard typed feature structure based grammar, of which their account is self-consciously one, has with quotation. One fundamental problem faced by Bonami and Godard's account involves direct quotation, their intended quarry: whereas a sign-based construction is desirable for dealing with pure quotation, it cannot work for direct quotation since what one needs there in order to evaluate similarity is, in addition, the (source) utterance token. But this latter is not made available in standard typed feature structure based grammar.

Potts

In the concluding section of his paper, [START_REF] Potts | The dimensions of quotation[END_REF] writes (a propos of a remark by Bart Geurts) that 'Quotation is a hugely important matter for linguistic theory. It forces us to enrich our stock of basic entities.' We agree with the first sentence, but disagree with the second sentence, albeit recognizing its subjectivity. Potts introduces a standard grammar which he enriches with a type for expressions whose denotations he identifies with triples of phonological types, syntactic types, and logical expressions. These ad hoc entities are used as denotations for pure quotation. In contrast, our approach simply reuses entities that are assumed to exist for independent reasons, namely signs (utterance types). For dealing with direct quotation Potts introduces a two dimensional semantics involving the regular denotations and an additional dimension dubbed the 'speech report dimension'. Our account of direct quotation utilizes locutionary propositions, independently introduced as the contextual representation of utterances.

Geurts and Maier

The general framework in which Geurts and Maier(GM) work in is the 'binding theory' of presupposition (BTP, see e.g., [START_REF] Geurts | Presuppositions and Pronouns[END_REF]), an offshoot of Discourse Representation Theory [START_REF] Kamp | From Discourse to Logic: Introduction to Modeltheoretic Semantics of Natural Language, Formal Logic and Discourse Representation Theory[END_REF]). 16 The general intention is to bring to bear BTP tools to capture the context dependency of a quotation analogously to anaphoric pronouns, definite noun phrases, and other presuppositioninducing expressions. On GM's account, the semantic effect of quotation is to shift the ordinary meaning of an expression α to a new meaning that involves mentioning α. If the quotation is a 'pure' one, it denotes a linguistic expression and its semantic type may differ from that of the quoted expression. For example, in (33) quotation maps the semantic type of the adjective 'monosyllabic' to that of singular terms, and the resulting expression denotes the word 'monosyllabic':

(33) 'Monosyllabic' is not monosyllabic. (GM's (31))

In other cases, e.g. mixed and direct quotation, the semantic type of the quotation is identical to that of the quoted expression. For example, on the GM analysis, (34a) denotes the DRS in (34b), with the crucial innovation the postulation of the discourse referent Q which denotes (or purports to denote) the same kind of object the word 'police' would have denoted. The semantic content expressed by the quotation involves reference to the expression 'POlice'; it might be, e.g., 'whatever it is Tony meant when he used the expression 'POlice". In this sense 'POlice' is used and mentioned at the same time: the mentioning becomes part of the expression's content; On GM's account there are at least two ways in which mixed quotations depend on the context. A mixed quotation always has a source, which has two coordinates: a speaker and a speech event. The speaker coordinate may be a concrete individual or it may be institutional or generic. Similarly, the speech-event coordinate may be a concrete event or it may be habitual or generic. (34b) illustrates this in that x, e ′ are discourse referents that need to be bound via presuppositional binding-in this case (34c) emerges with x identified with Tony and e ′ with the saying event. Hence, (34a) gets a reading paraphrasable as there is an occasion e on which Tony used the expression 'POlice' to express the property Q, and that on the same occasion Tony said that the Q are on their way. But for a variety of other cases, discussed by GM, different binding possibilities emerge, mediated by binding operators such as quantification, negation, and focus. In contrast to Potts' two dimensional account, which GM critique, their own account is one dimensional and captures elegantly and precisely various interactions between variable/presupposition binding and quotation, which in a larger study we would try to integrate into our account.

Nonetheless, there are a number of intrinsic shortcomings. For a start, given the unrelated ways of dealing with pure and direct quotation, it is not clear how copredication cases like (29) can be handled. An additional problem, relating to the framework in which the account is couched concerns selection-there is no obvious way in DRT and its close variants to specify the type which a direct quotation predicate such as 'be like' selects for.17 . Similarly, the account of pure quotation appeals to denotations as expressions, whereas in fact we have argued that the requisite denotations need to be signs. With respect to direct quotation, given the lack of a rich type theory, it cannot formulate similarity as a condition relating source and quotation, as we do here. Nor can the account deal with diagonalization, as we do here by postulating grammatical resources as contextual parameters.

Conclusions

Direct and pure quotation have often been viewed as mysterious, outside the pale in most formal grammars or requiring unorthodox semantic/logical treatment, despite the fact that direct quotation is typologically universal and ontogenetically prior to indirect quotation. We have suggested that once one assumes a dialogical perspective on language such as provided by KoS, formalized in a rich type theory like TTR, much of the mystery evaporates. In particular, one can utilize as denotations for quotative constructions entities that are independently motivated for dialogue processing-utterance types and locutionary propositions, austinian propositions about speech events. The "diagonalizing property" of quotation, its ability to accommodate the unwashed and the alien, viz. ungrammatical and foreign utterances, is unproblematic on a view of language in flux, where there is no overarching notion of grammar, but recourse is made to multiple grammatical resources, depending on the purpose and setting. Our account of direct quotation invoking similarity and grammatical resources as contextual parameters provides the basis for an account of fluctuating standards of faithfulness between quotative and original utterance.

What we have provided here is of course a sketch: a detailed grammar needs to be developed and extended to mixed and scare quotation. We hope we have at least motivated the desirability of a more detailed picture in these terms.

  = sit : Record sit-type : RecType c. A proposition sit = r 0 sit-type = p 0 is true iff r 0 : p 0

  -type.ctxt-params.spkr = x : IND q : Question

  (i) Bill thought to himself 'What a guy I am'. (ii) #Bill knew 'What a guy I am'. (cf. We all know what an impressive candidate Bo is.)

(

  34) a. Tony says the 'POlice' are on their way. (GM's (42)) b. [y: Tony(y), e: say e (y,[x e' Q: E e ′ (x,Q,POlice), z: Q(z), e": onTheirWay e ′′ (z)]) ] (GM's (43)) c. [y: Tony(y), e Q: E e ′ (x,Q,POlice), z: Q(z), say e (y,[x [e": onTheirWay e ′′ (z)]]) ] (GM's (44))

KoS is not an acronym, despite emphasizing a Konversationally Oriented Semantics.

[START_REF] Everett | Quotatives in wari' and piraha: different strategies for reporting the contents of other minds[END_REF] claims that Pira ha lacks indirect quotation.

We limit our discussion and treatment here to 'pure', 'direct' and 'indirect' quotation, putting aside the recently much discussed 'mixed' and 'scare' quotation. We believe the account we develop can be extended to these, but that remains a future project.

The locus classicus for this view is Quine: 'Take quotation marks: applied to any sort of expression, what they produce is a singular term (naming, as it happens, the expression inside).[START_REF] Quine | Word and object[END_REF]. Geurts

Note that r may contain more objects with labels not mentioned in the type.

The convention is that a field ℓ=a:T is used as a convenient notation for ℓ:T a where T a is the type T restricted to a singleton type whose only witness is a. This is why the apparent asymmetry between the 'phon' and 'c-params' fields on the one hand, and the 'cat' and 'cont' fields is merely notational.

Moves is the component of the dialogue gameboard where grounded utterances are represented.

Of course there is a bit of a chicken/egg issue here, given that direct quotation is one of the mechanisms that gets language off the ground.

In order to accommodate quotations of non-linguistic events, we would simply need to modify our characterization of the type of the head daughter as being grammatical to a somewhat wider class of types. Such types would certainly have a field for phon and might also have a field for cont.

An anonymous reviewer for the Journal of Logic, Language, and Information asks whether by invoking similarity measures we are not rendering our proposal to be unfalsifiable. S/he writes 'Suppose that John gave a lecture, and somebody asks Peter about the lecture, how it was. Peter: John said: 'blablabla'. Can 'blablabla' stand for the content of John's lecture? If not, how is it blocked in the model?' On our account the truth of Peter's report requires an utterance of 'blablabla' to be similar to the sequence of utterances that made up John's lecture, relative to a contextually given similarity measure. A number of similarity measures we have mentioned above would not accommodate such similarity, e.g., one based on identity of content and a fortiori on identity of form. In fact, a measure of similarity that would make Peter's report true would need to be very coarse grained, essentially one that made all utterances of a given language similar, as for instance when one hears a lecture in a language one does not understand or when one wishes to convey that, for all intents and purposes what someone has said is contentless.

Additional axioms need to be met but they need not concern us here.

And we could extend this somewhat to allow for speech events like, say, grunts that might not be illocutionary in a strict sense.

For a more detailed discussion of GM's approach, also developed in[START_REF] Maier | Mixed quotation: between use and mention[END_REF] and subsequently, see[START_REF] Brabanter | The semantics and pragmatics of hybrid quotations[END_REF].

See Asher (2011) for such a type theoretic framework, though it is not integrated with the discourse semantics, in contrast to TTR/KoS.

Acknowledgements Earlier versions of this work have been presented at the workshop Quotation: Perspectives from Philosophy & Linguistics, held in October 2012 in Bochum, and at UNILOG 2013, held in April 2013 in Rio de Janeiro. We would like to thank audiences there for their comments, in particular Yuri Gurevich, Stefano Predelli, Franc ¸ois Recanati, and Markus Werning. For comments on this paper we would like to thank Olivier Bonami and two anonymous reviewers for the Journal of Logic, Language, and Information. We also wish to thank Marcos Lopes and Urszula Wybraniec-Skardowska for much help and encouragement.