Johan Markdahl

Jorge Goncalves

Towards Intrinsic Almost Global Consensus on the n-Sphere

This paper concerns the global level convergence properties of a continuous consensus protocol for multi-agent systems that evolve in continuous time on the n-sphere. The feedback is intrinsic to the n-sphere since it does not rely on the use of local coordinates obtained through a parametrization. It is shown that, for any connected undirected graph topology and all n ∈ N\{1}, the protocol yields convergence that is akin to almost global consensus in a weak sense. Simulation results suggest that almost global consensus hold. This results is of interest in the context of consensus on Riemannian manifolds since it differs from what is known with regard to 1-sphere and on SO(3) where more advanced intrinsic consensus protocols are required in order to generate equivalent results.

I. INTRODUCTION

Consider a system consisting of N agents, each of which is equipped with some limited communication and sensing capabilities. The goal is for all agents to converge to the same state, i.e., to reach a consensus. This type of problem is widely studied in the literature, see e.g., [START_REF] Mesbahi | Graph Theoretic Methods in Multi-Agent Networks[END_REF] and the references therein. As the field of networked and multiagent system has matured, research focus shifts from linear dynamics to more realistic models such as switched and highly nonlinear systems including those featured in the attitude synchronization problem [START_REF] Sarlette | Autonomous rigid body attitude synchronization[END_REF]- [START_REF] Thunberg | Distributed attitude synchronization control of multi-agent systems with switching topologies[END_REF]. Research on attitude synchronization is motivated by applications such as satellite formation flying [START_REF] Beard | A coordination architecture for spacecraft formation control[END_REF], [START_REF] Lawton | Synchronized multiple spacecraft rotations[END_REF], cooperative robotic manipulation [START_REF] Rodriguez-Angeles | Mutual synchronization of robots via estimated state feedback: a cooperative approach[END_REF], multi-camera networks [START_REF] Tron | Distributed 3-D localization of camera sensor networks from 2-D image measurements[END_REF], and distributed rotation averaging [START_REF] Hartley | Rotation averaging[END_REF], [START_REF] Matni | A convex approach to consensus on SO(n)[END_REF].

The reduced attitude is a property of objects that for various reasons, such as task redundancy, cylindrical symmetry, or actuator failure, lack one degree of rotational freedom in three-dimensional space and whose orientation corresponds to a pointing direction with no regard for the rotation about said axis of pointing [START_REF] Chaturvedi | Rigid-body attitude control: Using rotation matrices for continuous singularityfree control laws[END_REF]. The reduced attitude synchronization problem is equivalent to the consensus problem on the sphere. The problem of cooperative control on the n-sphere, S n , has received some attention in the literature [START_REF] Olfati-Saber | Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks[END_REF]- [START_REF] Li | Unified cooperative control of multiple agents on a sphere for different spherical patterns[END_REF], but comparatively less than the full attitude synchronization problem. Applications aside from reduced attitude synchronization include planetary scale mobile sensing networks [START_REF] Paley | Stabilization of collective motion on a sphere[END_REF] and systems consisting of periodic oscillators described by so-called Kuramoto models [START_REF] Kuramoto | Self-entrainment of a population of coupled non-linear oscillators[END_REF].

The problem of almost global consensus has been studied on S 1 [START_REF] Sarlette | Geometry and symmetries in coordination control[END_REF], on SO(3) [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF], and on S n in the special case of a complete graph [START_REF] Olfati-Saber | Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks[END_REF], [START_REF] Li | Unified cooperative control of multiple agents on a sphere for different spherical patterns[END_REF]. The work [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF] apply an optimization based method to characterize the stability of all equilibria for a particular discrete-time consensus protocol. Their result is akin to almost global consensus over any connected graph topology. The algorithm uses a particular reshaping function which depends on a constant parameter that must be bounded below. However, the bound depends on the graph topology which is unknown to the agents. Moreover, the overall convergence speed of the algorithm decreases with increasing values of the parameter. By shifting consideration from SO(3) to the n-sphere, this paper uses the direct method of Lyapunov to establish stability results of the same type as those in [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF] for a basic consensus protocol that does not require the use of any reshaping function.

The work [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF] divides the literature on attitude consensus into two categories: extrinsic and intrinsic algorithms. An algorithm is said to be extrinsic if it makes use of a parametrization that embeds SO(3) in some Euclidean space. There are algorithms in this class that provide consensus on a global level. For the second category of algorithms that work with SO(3) directly, global level results had not been obtained prior to the publication of [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF]. Much of previous work on concerning intrinsic cooperative control on the nsphere only regards the case of a complete graph [START_REF] Olfati-Saber | Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks[END_REF], [START_REF] Li | Unified cooperative control of multiple agents on a sphere for different spherical patterns[END_REF]. Some preliminary results regarding a class of of non-trivial topologies, essentially the case of cycle graphs, is found in the main authors previous work [START_REF] Markdahl | Rigid-body attitude control and related topics[END_REF]- [START_REF] Markdahl | Global and invariant aspects of consensus on the n-sphere[END_REF]. The gossip algorithm of [START_REF] Sarlette | Geometry and symmetries in coordination control[END_REF] achieves consensus on S n with probability 1 for a class of digraphs, but does not apply to mechanical systems.

The 2-sphere is akin to a subset of SO(3), and as such many results obtained for SO(3) also applies to S 2 . Special cases sometimes allow for stronger results. The findings of this paper indicates that the conditions for achieving almost global consensus are, in a certain sense, more favorable on the n-sphere for n ≥ 2 than the particularization of previously known results concerning the 1-sphere and SO(3) would imply. A basic intrinsic consensus protocol over connected, undirected graph topologies render all equilibria except the consensus point unstable on S 2 whereas simulation results indicate that some of the corresponding equilibria for certain graph topologies are asymptotically stable on the 1-sphere [START_REF] Sarlette | Geometry and symmetries in coordination control[END_REF] and SO(3) [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF].

The main contribution of this paper is to provide a convergence results on a global level for all connected, undirected graph topologies, i.e., for a larger class of topologies than what has previously been found on the n-sphere [START_REF] Tron | Riemannian consensus for manifolds with bounded curvature[END_REF], [START_REF] Olfati-Saber | Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks[END_REF], [START_REF] Li | Unified cooperative control of multiple agents on a sphere for different spherical patterns[END_REF], [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF], [START_REF] Markdahl | Rigid-body attitude control and related topics[END_REF], [START_REF] Markdahl | Global and invariant aspects of consensus on the n-sphere[END_REF]. Unlike [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF], the control law does not require the use of a reshaping function that depends on the graph topology, i.e., on unavailable information. This result is conjectured in [START_REF] Olfati-Saber | Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks[END_REF], [START_REF] Markdahl | Rigid-body attitude control and related topics[END_REF], but may be considered unexpected since it is qualitatively different from what is known to hold with regard to discrete-time consensus on Riemannian manifolds for the Lie groups S 1 and SO(3) of which S 2 may be considered a subset [START_REF] Sarlette | Geometry and symmetries in coordination control[END_REF], [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF].

II. PROBLEM DESCRIPTION

The following notation is used in this paper. The inner product and outer product of x, y ∈ R n are denoted by x, y and x ⊗ y respectively. The Euclidean norm is used for vectors and the Frobenius norm is used for matrices. The special orthogonal group is

SO(n) = {R ∈ R n×n | R -1 = R ⊤ , det R = 1}. The Lie algebra of SO(n) is so(n) = {S ∈ R n×n | S ⊤ = -S}.
The n-sphere is denoted by

S n = {x ∈ R n+1 | x = 1}
, where n ∈ N. We do not consider the trivial 0-sphere S 0 = {-1, 1}. A graph is a pair G = (V, E) where V is the node set and E ⊂ V × V is the edge set.

An equilibrium manifold is a set of equilibria of a system that also constitutes a manifold. The concepts of stability of an equilibrium can be extended to sets and hence to manifolds [START_REF] Hinrichsen | Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness[END_REF]- [START_REF] Shub | Global stability of dynamical systems[END_REF]. This paper uses the following terminology to describe stability properties of manifolds. Definition 1. An equilibrium manifold is said to be maximal if it is connected and not a strict subset of any other connected equilibrium manifold. Definition 2. A maximal equilibrium manifold is said to be uniquely stable if it is stable and no other maximal equilibrium manifold is stable, uniquely attractive if it is attractive and no other maximal equilibrium manifold is attractive, and uniquely asymptotically stable if it is both uniquely stable and uniquely attractive.

A. Distributed Control Design on the n-Sphere

Consider a multi-agent system where each agent corresponds to an index i ∈ V. The agents are capable of limited pairwise communication and local sensing. The topology of the communication network is described by an undirected connected graph G = (V, E), where V = {1, 2, . . . , N }, and (i, j) ∈ E implies that agent i and j can communicate, or equivalently that agent i and j can sense the so-called local or relative information regarding the displacement of their states.

Control is assumed to be based on relative information and to be carried out on a kinematic level. The information I ij that agent i has access to regarding its neighbor agent j includes

span{(I -X i )(x j -x i )} = span{x j -x j , x i x i }, (1)
where X i = x i ⊗ x i and I -X i : R n+1 → T x i S n is a projection. The set of neighbors of agent i is denoted N i = {j ∈ V | (i, j) ∈ E}. The space of relative information known to any agent i ∈ V is ∪ j∈N i I ij . The subset of I ij given by (1) corresponds to the customary relative information in linear spaces I ij ⊃ span{x jx i } projected on T x i S n . An agent can calculate this aspect of I ij based on local sensing since all it needs to discern is the direction towards its neighbor along its tangent space. System 3. The system is given by N agents, an undirected and connected graph G = (V, E), agent states x i ∈ S n , where n ∈ N\{1}, and dynamics

ẋi = u i -u i , x i x i = (I -X i )u i , (2)
where u i :

j∈N i I ij → R n+1 is the input signal of agent i and X i = x i ⊗ x i for all i ∈ V.
Note that the dynamics (2) projects the input u i on the space of relative information. While some agent i ∈ V may not be able to calculate some u i ∈ I ij based on the information (1) obtained from all its neighbors, that agent may still be able to calculate an input v i whose projection on T x i S n is identical to that of u i . This holds for inputs that belongs to span{x j | j ∈ N i }, and in particular for elements of the positive cone pos{x j | j ∈ N i }. Intuitively speaking, it is reasonable to assume that agent i should be able to move towards any point in

pos{x j | j ∈ N i }, i.e., that pos{x j | j ∈ N i } ⊂ I ij .
All vectors in this section are defined in the world frame W. To implement the control law in a distributed fashion, u i must be transfered to the body frame B i of agent i. Let [x] F denote that the coordinates of vector x are given with respect to the F . Suppose

B i is related to W by means of a rotation R i : [v] W → [v] B i . The control law in W is given by [u i ] W = j∈N i w ij [x j ] W . Hence [u i ] B i = j∈N i w ij R i [x j ] W = j∈N i w ij [x j ] B i . Moreover, [ ẋi ] B i = R i [u i ] W + [u i ] W , R ⊤ i R i [x i ] W R i [x i ] W = [u i ] B i + [u i ] B i , [x i ] B i [x i ] B i , (3)
due to inner products being invariant under orthonormal changes of coordinates. From the perspective of stability analysis, (3) is the same as equation ( 2). The problem of multi-agent consensus on the n-sphere concerns the design of distributed control protocols {u i } N i=1 based on relative information, as discussed in the above paragraphs, that stabilize the consensus manifold

C = {{x i } N i=1 ∈ (S n ) N | x i = x j , ∀ i, j ∈ V} (4)
of System 3. If all agents converge to one point on the n-sphere, then they are said to reach consensus. For all connected graphs, it can easily be shown to suffice that the states of any pair of neighboring agents are equal.

B. Problem Statement

This paper concerns the following aspects of control design and stability analysis for a basic consensus protocol on the n-sphere. By basic we mean that the algorithm should be as simple as possible.

Problem 4. Find a basic, intrinsic consensus protocol, i.e., input signals u i :

j∈N i I ij → R n+1
for all i ∈ V, for System 3 such that the consensus manifold is uniquely asymptotically stable.

Problem 4 concerns the global behavior of the system. Under certain assumptions regarding the connectivity of G, local consensus on SO(3) can be established with the region of attraction being the largest geodesically convex sets on S n , i.e., open hemispheres. See for example [START_REF] Tron | Riemannian consensus for manifolds with bounded curvature[END_REF] in the case of an undirected graph and [START_REF] Thunberg | Distributed attitude synchronization control of multi-agent systems with switching topologies[END_REF] in the case of a directed and time-varying graph. A global stability result for discretetime consensus on SO(3) is provided in [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF]. Almost global asymptotical stability of the consensus manifold on the nsphere is known to hold when the graph is a tree [START_REF] Tron | Riemannian consensus for manifolds with bounded curvature[END_REF] or is complete in the case of first-and second-order models [START_REF] Olfati-Saber | Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks[END_REF], [START_REF] Li | Unified cooperative control of multiple agents on a sphere for different spherical patterns[END_REF]. The author of [START_REF] Olfati-Saber | Swarms on sphere: A programmable swarm with synchronous behaviors like oscillator networks[END_REF] conjectures that global stability also holds for a larger class of topologies whereas [START_REF] Sarlette | Geometry and symmetries in coordination control[END_REF] provides counter-examples of basic consensus protocols that fail to generate consensus on S 1 . [START_REF] Bhat | A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon[END_REF]. It is however possible to achieve almost global asymptotical stability [START_REF] Sarlette | Geometry and symmetries in coordination control[END_REF]. The paper [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF] argues that adding a small perturbation to the feedback of a uniquely asymptotically stable system yields almost global consensus. However, that is not true in general. To prove almost global consensus is difficult since basic tools such as the Hartman-Grobman theorem or stable-unstable manifold theorems are unavailable due to the equilibria being nonhyperbolic [START_REF] Sastry | Nonlinear systems: analysis, stability, and control[END_REF]. Instead, center manifold theory for equilibrium manifolds may be required. That is the topic of future work whereas this paper deals with unique asymptotical stability.

Remark 5. It is not possible to achieve global consensus on S n by means of a continuous feedback due to topological constraints

III. STABILITY OF DESIRED EQUILIBRIA

This section concerns System 3 governed by Algorithm 6 below which provides a continuous protocol for continuoustime consensus on the n-sphere. It could be argued that Algorithm 6 is the most basic conceivable feedback for this problem, and actually corresponds to the weighted linear consensus protocols given by u i = i∈N i w ij (x jx i ) where w ij ∈ (0, ∞), w ij = w ji for all (i, j) ∈ E since the dynamics on the n-sphere project an input to the orthogonal complement of the state of each agent. Algorithm 6. The feedback is given by u i = j∈N i w ij x j , where w ij ∈ (0, ∞) and w ij = w ji for all (i, j) ∈ E.

Algorithm 6 can be derived by taking the gradient of the potential function

V = 1 2 (i,j)∈E w ij x i -x j 2 = (i,j)∈E w ij (1 -cos ϑ ij ), (5)
where ϑ ij is the angle between x i and x j . It is possible to work with more general gains than the constant weights of Algorithm 6, but we prefer constants for ease of notation. Proposition 7 implies that there are no limits cycles in System 3 under Algorithm 6. If the graph in System 3 under Algorithm 6 were directed, then there would exist examples of topologies that result in limit cycles.

Proposition 7. System 3 converges to an equilibrium. The following equilibrium configurations exist:

(x i , u i ) ∈ - u i u i , u i , u i u i , u i , (x i , 0) ,
where u i = j∈N i w ij x j for all i ∈ V.

Proof. Consider the potential function [START_REF] Tron | Riemannian consensus for manifolds with bounded curvature[END_REF]. It holds that

V = (i,j)∈E w ij x i -x j , ẋi -ẋj = (i,j)∈E w ij x i -x j , (I -X i )u i -(I -X j )u j = - (i,j)∈E w ij [ x i , (I -X j )u j + x j , (I -X i )u i ] = - i∈V j∈N i w ij x j , (I -X i )u i = - i∈V j∈N i w ij x j , (I -X i )u i = - i∈V u i , (I -X i )u i = - i∈V u i 2 -u i , x i 2 .
System 3 converges to the set {{x i } N i=1 | u i x i } by LaSalle's theorem, i.e., the input and state of each agent align up to sign. This implies ẋi = 0 for all i ∈ V, i.e., that the system is at an equilibrium by inspection of (2).

The following result, Theorem 8, may be considered as one of the many known facts concerning consensus on convex subsets of manifolds [START_REF] Tron | Riemannian consensus for manifolds with bounded curvature[END_REF]. To solve Problem 4 this paper provides a companion to Theorem 8 that regards all equilibrium manifolds of System 3 under Algorithm 6.

Theorem 8. Consider System 3 under Algorithm 6. The system reaches consensus asymptotically if and only if there is some finite time such that all agents belong to an open hemisphere.

Remark 9.

A proof of Theorem 8-or generalizations of this result such as local consensus over switched directed graph topologies-can be obtained by following lines of reasoning that is found in many works within the consensus literature, and is therefore omitted here.

IV. INSTABILITY OF UNDESIRED EQUILIBRIA

The global behavior of the system is determined by the stability of the equilibrium configurations given in Proposition 7. Proposition 11 below establishes that any equilibrium where x i = -u i/ u i or u i = 0 for some i is unstable. Set all weights in Algorithm 6 to be equal. An example of x i = -u i/ u i is given by a tetrahedron formation with a tetrahedral graph, i.e., the complete graph on four nodes. An example of x i = 0 for all i ∈ V is provided by the octahedral graph, see Figure 1. The case of x i = u i u i for all i ∈ V poses a more difficult challenge since it includes the consensus manifold as a special case. The difficulty is overcome by the indirect method of Lyapunov. As such, we study the signs of the real part of the linearization of System 3 under Algorithm 6 in order to establish the instability of all undesired equilibria.

A ij = -(x i ⊗ u i + u i , x i I)(I -X i ) if j = i, w ij (I -X i )(I -X j ) if j = i,
for (i, j) ∈ E and A ij = 0 otherwise. The matrix A is symmetric.

Proof. For systems evolving on manifolds, a perturbation technique is used to obtain the linearized dynamics. Let x i for all i ∈ V be a solution to [START_REF] Sarlette | Autonomous rigid body attitude synchronization[END_REF]. Consider a perturbed solution x i (ε, v i ) given by

x i (ε, v i ) = x i + εv i x i + εv i ,
where v i is a smooth function. The perturbed solution satisfies the differential equation

ẋi (ε, v i ) = u i (ε, v i ) -u i (ε, v i ), x i (ε, v i ) x i (ε, v i ).
The linearized dynamics on S n can be derived by studying the linear effect of v i on ẋi (ε, x i ). Define

w i = d dε x i (ε, v i )| ε=0 = v i x i + εv i ε=0 - x i + εv i x i + εv i 3 x i , v i ε=0 = v i -x i ⊗ x i v i = (I -X i )v i , (6)
where X i = x i ⊗ x i . The matrix I -X i projects onto the tangent space T x i S n where w i lives. Note that

d dε X i (ε, v i )| ε=0 = w i ⊗ x i + x i ⊗ w i by the product rule. Then ẇi = d 2 dtdε x i (ε, v i )| ε=0 = d dε ẋi (ε, v i )| ε=0 = d dε (I -X i (ε, v i )) j∈N i w ij x j (ε, v j ) = -d dε X i (ε, v i ) j∈N i x j (ε, v j ) ε=0 + (I -X i (ε, v i )) j∈N i d dε x j (ε, v j ) ε=0 = -(w i ⊗ x i + x i ⊗ w i ) j∈N i x j + (I -X i ) j∈N i w j = -(w i ⊗ x i + x i ⊗ w i )u i + (I -X i ) j∈N i w j = -( u i , x i I + x i ⊗ u i )w i + (I -X i ) j∈N i w j
where the relation x, y z = (z ⊗ x) y = (z ⊗ y)x for all x, y, z ∈ R n+1 and ( 6) are used.

The vector w = [w ⊤ 1 . . . w ⊤ N ] ⊤ has N (n + 1) components whereas the linearized system actually evolves on an N ndimensional space that lies embedded in R N (n+1) . The dimension reduction is given implicitly by the definition of w i which requires w i ∈ T x i S n-1 . This constraint can be removed by using variables that are premultiplied by the projection matrices I -X i : R n+1 → T x i S n-1 , i.e., the variables v i , whereby the matrix A is obtained.

It remains to show that A is symmetric. Write

A ii = -(x i ⊗ u i + u i ⊗ x i + u i , x i I)(I -X i )
which is clearly symmetric. Moreover,

A ⊤ ji -A ij = w ji [(I -X j )(I -X i )] ⊤ - w ij (I -X i )(I -X j ) = 0,
since w ij = w ji for all (i, j) ∈ E.

Proposition 11. Any equilibrium {x i } N i=1 / ∈ C of System 3 under Algorithm 6 where n ∈ N and either x i = -u i/ u i or u i = 0 for some i ∈ V is unstable.

Proof. The proof makes use of the linearization provided by Proposition 10. The Courant-Fischer-Weyl min-max principle bounds the range of the Rayleigh quotient of a a symmetric matrix by its minimum and maximum eigenvalues [START_REF] Horn | Matrix analysis[END_REF]. Recall that if A has a positive eigenvalue at an equilibrium, then that equilibrium is unstable by the indirect method of Lyapunov [START_REF] Khalil | Nonlinear systems[END_REF].

In the case of x i = -u i / u i for some i ∈ V, note that the diagonal block of the linearization matrix A is given by A ii = u i (I -X i ), which is a positive semidefinite, nonzero matrix. There is hence an eigenpair (λ, u) of A ii with λ ∈ (0, ∞), u ∈ S n . Let v ∈ R N (n+1) denote a vector composed of the blocks v 1 , . . . , v n ∈ R n+1 . Set v i = u and v j = 0 for all j / ∈ V/{i} whereby v, Av = λ. The largest eigenvalue of A is hence bounded below by a positive number.

In the case of u i = 0 and n ≥ 2, take a j ∈ N i and set v j = εu for a u ∈ S n such that u, x i = 0, u, x j = 0 and an ε ∈ (0, ∞). Set v i = u. Then v, Av = u, A jj u ε 2 + 2w ij ε, which is positive given that ε is sufficiently small. In the case of u i = 0 and n = 1, if possible, take a j ∈ N i and set v j = εu for a u ∈ S n such that u, x i = 0, u, x j ∈ (-1, 1) and an ε ∈ (0, ∞). Then v, Av = u, A jj u ε 2 + 2w ij (1u, x j 2 )ε, which is positive given that ε is sufficiently small. If this is not possible, it is because

x i , x j = 0, ∀ j ∈ N i . Then x j ∈ {-Rx i , Rx i } where R is a rotation of π /2 radians. It follows that X j = X i whereby A ij = w ij (I -X i ). Take a u ∈ S n such that u, x i ∈ (-1, 1) and set v j = εu, v i = u for some ε ∈ (0, ∞). Then v, Av = u, A jj u ε 2 + 2w ij (1 -u, x i 2 )ε, which
is positive given that ε is sufficiently small.

The following basic idea from [START_REF] Markdahl | Rigid-body attitude control and related topics[END_REF], [START_REF] Markdahl | Global and invariant aspects of consensus on the n-sphere[END_REF] forms the intuition behind our next result. Consider an equilibrium such that all agents belong to the intersection of S n and a hyperplane in R n . Perturb all agents into an open hemisphere by an arbitrarily small movement along a direction orthogonal to the hyperplane. Theorem 8 then yields asymptotical consensus. The idea of perturbing all agents in one direction is the basis of the next result. The effect of such a perturbation depends of the position of an agent, e.g., if all agents are perturbed along the direction x i , then agent i is invariant whereas the other agents move towards it.

Proposition 12. Any equilibrium {x

i } N i=1 / ∈ C of System 3 where n ∈ N\{1} under Algorithm 6 is unstable.
Proof. The proof continues where that of Proposition 11 left off. It remains to consider the case of x i = u i/ u i . Then u i = u i x i and hence

A ij = -u i (I -X i ) if j = i, w ij (I -X i )(I -X j ) if j = i,
for (i, j) ∈ E and A ij = 0 otherwise. The matrix A is symmetric at all equilibria by Proposition 10.

Let z = [y ⊤ . . . y ⊤ ] ⊤ and consider

z, Az = i∈V y, A ii y + j∈N i y, A ij y = y, i∈V A ii + j∈N i A ij   y Denote B = i∈V A ii + j∈N i A ij . The matrix B is symmetric due to A being symmetric whereby σ(B) ⊂ R by the spectral theorem. If (λ, v) is an eigenpair of B, then (λ, [v ⊤ . . . v ⊤ ] ⊤ ) is an eigenpair of A.
Let us prove that B has a positive eigenvalue, which in turn will imply that A has a positive eigenvalue. Consider

tr B = - i∈V u i (n + 1 -tr X i )+ j∈N i w ij (n + 1 -tr X i -tr X j + tr X i X j ) = - i∈V n u i + j∈N i w ij (n -1 + x i , x j 2 ) = n   i∈V -u i + j∈N i w ij   - i∈V j∈N i w ij (1 -cos 2 ϑ ij ) = n   i∈V j∈N i w ij (1 -cos ϑ ij )   - i∈V j∈N i w ij (1 + cos ϑ ij )(1 -cos ϑ ij ) = i∈V j∈N i w ij (n -1 -cos θ ij )(1 -cos ϑ ij )
where we used that tr X i = x i 2 = 1 and u i =

x i , u i = j∈N i x i , w ij x j = j∈N i w ij cos ϑ ij .
Since tr B ≥ 0 for all n ∈ N\{1} with strict inequality unless θ ij = 0 for all (i, j) ∈ E, i.e., unless {x i } N i=1 ∈ C, it follows that B has an eigenvalue λ ∈ (0, ∞).

Remark 13. The case when the angle ϑ i between at least one agent and the average of its neighbors, defined by ϑ i = arccos u i ,x i / u i , belongs to [ π /2, π] turns out to be, roughly speaking, more unstable than the case of ϑ i ∈ [0, π /2) for all (i, j) ∈ E since instability holds for n ∈ N rather than for n ∈ N\{1}, see also [START_REF] Sarlette | Geometry and symmetries in coordination control[END_REF]. Proposition 14. Any connected equilibrium manifold M ⊂ C of System 3 where n ∈ N\{1} under Algorithm 6 is unattractive.

Proof. Consider an equilibrium {x

i } N i=1 ∈ M. Since {x i } N
i=1 is unstable by Proposition 12, there must be a smooth trajectory T : (0, ∞) → (S n ) N with T (0) / ∈ M such that, for some ε ∈ (0, ∞), there is no δ ∈ (0, ∞) that would make d(T (0),

{x i } N i=1 ) < δ imply sup t∈(0,∞) d(T (t), {x i } N i=1 ) < ε.
Suppose that the trajectory returns to the manifold, i.e., lim t→∞ T (t) ∈ M. The potential function V given by ( 5) decreases along every trajectory of the system but is constant over each connected equilibrium manifold. If the value of V is lower on some point on T than it is on M, then T cannot return to M. If V is higher on some point on T than on M, then it increases even higher backwards in time and by moving T (0) arbitrarily close to M, the continuity of V is contradicted. Finally, if V is constant on T and lim t→∞ T (t) ∈ M, then T (0) ∈ T ⊂ M, a contradiction. Proposition 15. Any maximal equilibrium manifold M ⊂ C of System 3 where n ∈ N\{1} under Algorithm 6 is unstable.

Proof. Reasoning as in the proof of Proposition 14 we find there is a smooth trajectory T : (0, ∞) → (S n ) N where T (0) can be chosen arbitrarily close to an unstable equilibrium {x i } N i=1 ∈ M that leaves an ε-neighborhood of {x i } N i=1 for some fixed ε ∈ (0, ∞). Suppose that this trajectory stays in an ε-neighborhood of M. It must converge to an equilibrium that does not belong to M by Proposition 7 and 14. Hence there is a second equilibrium manifold N at a distance at most ε from M. Since ε is arbitrary, there must be some point where d(N , M) = 0. Since M is closed, this contradicts the assumption of M being maximal.

Remark 16. The reasoning of Proposition 14 and 15 can be generalized to establish that stability and attractiveness are equivalent properties of maximal equilibrium manifolds for gradient systems.

V. MAIN RESULT Theorem 17. The consensus manifold C of System 3 where n ∈ N\{1} under Algorithm 6 is uniquely asymptotically stable.

Proof. The proof follows directly by verifying that the requirements of Definition 2 are satisfied as is evident by Theorem 8, Proposition 14, and Proposition 15.

Remark 18. It is shown in [START_REF] Sarlette | Geometry and symmetries in coordination control[END_REF] that certain graph topologies generate stable equilibria on S 1 . The circle is a subset of any n-Sphere yet this result does not contradict Theorem 17 since any configuration that is confined to a circle can be perturbed into a hemisphere when n ∈ N\{1} whereby Theorem 8 guarantees a consensus. It would seem the additional degrees of freedom is the source of qualitatively different stability properties on S 1 and the higher dimensional spheres. This suggests that the result of [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF] concerning the failure of basic consensus protocols to yield almost global consensus on SO(3) may not apply on SO(n) for n ∈ N\{2, 3}.

VI. SIMULATION

The question of whether Theorem 17 can be extended to the case of almost global consensus remains unanswered. This section provides simulations that argue in favor of such a result. It also compares the global performance of two consensus protocols on S 2 and SO(3) respectively in simulation.

A. A Case for Almost Global Consensus on S 2

Let R denote the union of the regions of attraction over all equilibrium manifolds except the consensus manifold. Suppose the probability measure of R on (S 2 ) N is ε ∈ [0, 1]. Consider the statement 'R has measure zero' as a null hypothesis. Let the random variable X denote the number of draws from R after m draws from U(S 2 ). If X > 0 then the null hypothesis is rejected. The probability of the null hypothesis not being rejected is P (X = 0) = (1p) m , where p is the probability of an outcome belonging to R, i.e., p = ε /(4πN). If m is large and the null hypothesis is not rejected, then it is very likely that p is small. Of course, the expression for P (X = 0) does not account for the effect of numerical errors, but convergence to the desired manifold in the presence of errors is anyhow a positive quality.

Good candidates of graphs that results in stable nonconsensus equilibria on the 2-sphere are the Platonic solids and the fullerene graphs. The Platonic solids are symmetric three-dimensional bodies whose corners lie on the surface of a sphere. There are a total of five platonic bodies, the tetrahedron, hexahedron, octahedron, icosahedron and dodecahedron, see Figure 1 and2. A planar depiction of all graphs is given in Table I. If the number of agents equals the number of nodes of a Platonic graph and are positioned as on the corners of the corresponding Platonic solid, then the multi-agent systems is at an equilibrium since the contributions of all neighboring agents on each tangent space cancel for reasons of symmetry, see Figure 1.

Table I displays the outcome of running 10 6 trials of Algorithm 6 with w ij = 5 for all (i, j) ∈ E on System 3 for the five platonic graphs. The initial conditions are drawn uniformly from the sphere using that x ∈ N (0, I) implies that x / x ∈ U(S n ), see [START_REF] Muller | A note on a method for generating points uniformly on n-dimensional spheres[END_REF]. By inspection of Table I there were no failures to reach consensus. There is hence no reason to reject the null hypothesis. It can be shown, as a consequence of Proposition 11, that the platonic body configurations in the case of the tetrahedron and octahedron are unstable equilibria on S 2 , see also Figure 1. It appears that almost global consensus holds for all the platonic graphs. System 19. The system is given by N agents, an undirected graph G = (V, E), agent states R i ∈ SO(3), and dynamics Ṙi = Ω i R i where Ω i ∈ so(3) for all i ∈ V. It is assumed that G is connected and that the system can be actuated on a kinematic level, i.e., Ω i is the input signal of agent i.

Recall that Algorithm 6 is derived by taking the gradient of the potential function in [START_REF] Tron | Riemannian consensus for manifolds with bounded curvature[END_REF]. A related consensus protocol on SO(3) can be derived by taking the gradient of the corresponding potential function, V : SO(3) × SO(3) → [0, ∞), given by

V (R i , R j ) = 1 2 (i,j)∈E w ij R i -R j 2 F ,
where the Euclidean metric x ix j on S 2 has been replaced by the Frobenius metric d(R i , R j ) = R i -R j F on SO(3). As such, Algorithm 6 is rather similar to the following algorithm on System 19.

Algorithm 20. The feedback is given by

Ω i = j∈N i w ij (R ⊤ i R j -R ⊤ j R i )
, where w ij ∈ (0, ∞) and w ij = w ji for all (i, j) ∈ E.

Table II displays the outcome of running 10 6 trials of Algorithm 6 with w ij = 5 for all (i, j) ∈ E on System 3 and Algorithm 20 with w ij = 5 for all (i, j) ∈ E on System 19 for the five platonic graphs. The method is also used to draw uniform samples from S 2 is also used to draw from U(SO(3)) by first generating a uniform distribution on the unit sphere in quaternion space, i.e., drawing from U(S 3 ), and then mapping the sample to SO [START_REF] Sarlette | Coordinated motion design on Lie groups[END_REF]. II, note that almost global consensus does not hold for Algorithm 20 on System 19 over SO [START_REF] Sarlette | Coordinated motion design on Lie groups[END_REF]. The findings of our simulations agree with those of [START_REF] Tron | Intrinsic consensus on SO(3) with almost-global convergence[END_REF]. However, there were no failures to reach consensus on the graphs in on S 2 despite a factor of 10 2 more trials. Neither Table I nor II give us reason to reject the null hypothesis formulated in the previous section. It is hence not unreasonable to conjecture that Algorithm 6 may give almost global consensus on the 2-sphere for all graph topologies in the case of all weights being equal.

By inspection of Table

VII. FUTURE WORK

This paper defines and establishes the uniquely asymptotical stability of the consensus manifold for of a basic consensus protocol for systems that evolve on the n-sphere in the case of n ∈ N\{1} and for all connected, undirected graph topologies. Future work plans include the consideration of a larger class of feedback laws and the case of undirected graph topologies. It would also be interesting to consider the case of SO(n) when n ∈ N\{1, 2, 3}.

Fig. 1 .Proposition 10 .

 110 Fig. 1. An unstable equilibrium of a system on S 2 (left) with an octahedral graph (right).

Fig. 2 .

 2 Fig. 2. The icosahedral (left) and dodecahedral (right) graph depicted in three dimensions.

  Comparison of Consensus on S 2 and SO(3)Consider the following multi-agent system on the special orthogonal group SO(3).

TABLE I .

 I Number of failures to reach consensus on S 2 over 10

6

trials for the five platonic graphs.

TABLE II .

 II Number of failures to reach consensus on S 2 , and SO(3) over 10 6 and 10 4 trials respectively.

	S 2	0	0	0
	SO(3) 695	54	82

*This work was not supported by any organization.[START_REF] Mesbahi | Graph Theoretic Methods in Multi-Agent Networks[END_REF] Luxembourg Centre for Systems Biomedicine (LCSB), University of Luxembourg, Belval, Luxembourg. Email: johan.markdahl@uni.lu, jorge.goncalves@uni.lu


Annie Montaut
email: annie.montaut@inalco.fr

The discourse particle to and word ordering in Hindi: From grammar to discourse

Word reordering is the most common way in Hindi for focalizing and thematizing a constituent, without morphological restructuration. A theme is thus fronted and a focus is preverbal. However, the language also displays other devices, namely particles, for this purpose, particularly but not only, for constituents already respectively fronted and preverbal in the unmarked order (SOV). The paper bears on the most frequent of such particles, the thematic enclitic particle to. Previous analyses have pointed to the extreme diversity of its meanings --up to four "homonyms": intensive, contrastive, assertive, request particle. The present study explains this diversity in terms of a common operation which involves intersubjectivity. As a thematic particle, to requalifies the constituent as contrasting either with other elements present in the dialogue or with a view point on the constituent which has been previously expressed or attributed the other. Being highly sensitive to the exchange of distinct viewpoints, the particle may also cliticize to non-initial constituents, in which case it marks the whole statement as dismissing of opposing a previous proposal. The behaviour of the particle to may ultimately be traced to the other and older functions of the word used as a correlative pronoun and as a conjunction, which also presuppose an initial content as the starting point for a distinct predication or discourse sequence.

Introduction

The Hindi word to is both a conjunction ('so', 'then'), and a discourse clitic particle, both of which have usually been considered as different (homonymous) units. The discourse particle itself, described as intensive, vaguely contrastive (McGegor 1972: 141), or emphatic [START_REF] Kellogg | A Grammar of the Hindi Language[END_REF](Kellogg [ ] 1938 : 490) : 490), covers such a collection of highly distinct meanings ('sure, well, at least, finally, will you, but', etc.) that it too has been assumed to represent homonymous words (Shapiro 1999;[START_REF] Bai | Syntax and Semantics of the Particle To in Hindi[END_REF]. This study aims to show that the diversity of the surface meanings and functions of the latter (clitic particle) may be accounted for by a common abstract operation, implemented according to the various specific contexts of occurrence, both syntactic and enunciative (discursive). After a brief presentation of the grammatical to, always clause initial, in section 1, I will attempt in the following sections to disentangle the various factors (position and scope, intonation) at play in the scope of the discourse particle to, and show that the operation underlying all surface meanings, as a topic marker (section 2) and as the so-called contrastive marker (section 3), involves intersubjectivity. By connecting this abstract operation with the meaning of conjunctive to, we assume polysemy rather than homonymy as the more effective explanation of the various readings of to.

1. From the conjunction to the theme marker 1.1. The conjunction to: Discourse paragraph, sentence or clause initial

The non-clitic to, strongly stressed, essentially behaves as a coordinative conjunction ('so', 'then'), and as a correlative, particularly in conditionals.

In the first case, to mostly operates as a linker between two discourse paragraphs, the second one still remaining in the logical sequence of the first one but adding to it a new element instead of a simple relation of cause-effect or of temporal sequence 1 : it introduces a new turn or phase of the story in narrative contexts. It may also introduce a new idea or argument in an argumentative sequence, in the same way as its English equivalent 'then/so'. Given the fact that to occurs at cardinal moments in a narration, for chaining a distinct episode or for shifting the frame or scenario or viewpoint , when a story teller interrupts his tale after a non-final event, the hearer who wishes to know what happens next (and is not predictable) usually maps his expectation by somewhat providing the empty frame for further instantiation, and he does so by using to (1a), a to which may even behave as sentential (1b):

(1) a to kyā huā? b to ? to interr be.PFV to 'And then what happened?' 'And then ?'

Using this to-sentence in interrogative contexts which do not convey interrogation regarding some following event as in (1b) is potentially polemical (cf English "so what?") with underlying aggressive denegation ("what does it prove? what does it amount to?"): such rhetorical interrogations indeed question the potential of the previous statement to provide a step for further innovative developments in a logical argument. Hence the polemical meaning, since it dismisses the relevance of what has just been said: the speaker A who uses this rhetorical to means that there is nothing to make out of what has been said, and that the information or argument given by the speaker B is nil. With to, A pretends to provide a frame to further instantiate, in order to mean that it has not been instantiated yet and cannot be:

( The other main use of the non-clitic to is to introduce the apodosis after a conditional clause (agar/yadi: 'if'), in a correlative system typical of Indo-Aryan languages. 2 (2) a agar tum merī madad karte to main saphal ho-jātā if 2 my help do.CTF to 1SG successful become.CFT

1 Two relations which can also be lexically expressed respectively by islie 'therefore' and phir 'after that', 'then'.

2 Cf. Montaut (1999Montaut ( , 2012)). The correlative to can also correlate temporal dependent clauses (protasis introduced by jab 'when'), adding a nuance of cause-effect in contrast with the simply temporal tab. It is also more frequent in iterative contexts, where a whole series is occurrence is scanned. The term introducing the first element of the correlative dyptich is often omitted in Hindi but never is to, required in the hypothetic system even when the protasis is left unexpressed:3

(2) b yah pahle hī jāgī huī to…? this.one already just awake be.PFV.PL to 'And if she is already awake?'

The correlative to in the conditional system is then a marker of entailment which chains the protasis p to the apodosis q in the way p > q (you help me > I succeed), but in such a situation where two divergent paths are available: p' (you do not help me) would entail q' (I do not succeed). The speaker presents an alternative, and the apodosis is constructed in leaving aside one of the possible scenarii. To corresponds to the selection of one of these two scenarii in contrast with the other one in a given situation where the speaker has constructed what Culioli (1999) calls a fictive landmark, that is, a point of localisation for the predication q which is not real (not asserted) but virtual ("if": if we admit, let us imagine). 4 It behaves as an indicator of contrastive selection triggering q by contrast with q'. 1.2. A continuity from coordination and correlative to thematic particle? Whereas, as a correlative in conditional constructions, to involves the selection of one path in contrast with another one, as a coordinative conjunction, it opens a new sequence which resets the narrative flow on the basis of the previous statement, by shifting the scenery or the viewpoint, which makes the sequence following to distinctively salient.

The word etymologically stems from an ancient pronominal basis (Sanskrit ta-) referring to third person ('that', 'he'), which is still used as such in certain Indo-Aryan languages such as Marathi (to 'he') although not in Hindi. 5 This basis is typically used throughout Sanskrit, Prakrits and modern Indo-Aryan in the correlative system, as a linker between a relative clause (Sanskrit ya-'wh-') and its correlated clause. It is well known that anaphoric devices have often provided for coordinators in many languages (Brill & Rebuschi 2006, Haspelmath 2004: 3-39), and their use as correlators results from the same logic.

As for the shift from correlative to discourse particle, the evolution from conditional markers into theme markers has also been well described [START_REF] Haiman | Conditional are Topics[END_REF], Haspelmath 2008: 1005 sq): the conditional clause (protasis) behaves as the frame for the following predication in the apodosis. Since in Hindi the correlator which introduces the apodosis, represents the end part (right position) of the protasis in truncated systems (cf. ex 2d), it is all the more understandable that it came to be used as a topic marker, cliticized at the right of the topicalized term (Montaut 2012). The non-thematizing functions of the clitic, depending on its scope in the sentence, will be described in section 3.

To as a thematic particle

When the particle to is a clitic which forms a single accentual word with the term at its left, a word which can always be followed by a brief pause, it behaves mostly as a topic marker.

(3) ājkal to āp akele hain nowadays to 2H alone be.PRES.PL 'These days, you are alone' In (3), the intonation shows an ascending tone on to which bears the accent in the "word" ājkal to which is the topic, while the remaining of the clause (comment) is intoned in lower pitch 2.1. Theme, rheme and focus in Hindi Whether according to Chafe (1976)'s more cognitive definition based on the "given" and "known", then [START_REF] Gundel | Cognitive Status and the Form of Referring Expressions in Discourse[END_REF]'s finer definition of "givenness", including "semiactivated" and "inferred", or to [START_REF] Bonnot | La Particule to et la polémique cachée en russe moderne. A propos du statut énonciatif du thème[END_REF][START_REF] Bonnot | Lorsque la focalisation porte sur l'ensemble de la relation prédicative : les énoncés à accent non final en russe moderne[END_REF] and Guimier (1999)'s more formal definition based on the position and intonation, the topic or theme comes first in the sequence. Although significant research has been carried out during the last decades on focus and focalization in Hindi, mainly within the generative frame, practically none is available on the theme. I am using the term "focus" in its restricted meaning, and not in reference to the part of the sentence which comments the theme, which I will label rheme or rhematic content. A focalized constituent within the rheme is marked by its linguistic prominence (emphasis) by means of intonational or accentual marking, specific word ordering or morpho-syntactic devices such as cleft sentence or focus particles (Wlodarczik 2006[START_REF] François | Fonction et moyens de la focalisation à travers les langues[END_REF]. 6A statement may have no theme nor focus, and be an entirely rhematic content, what is currently designed as a thetic sentence ("it is two o' clock"). 7 In Hindi, an ergative head final language usually considered with SOV order,8 the equivalent entirely rhematic statement (do baj gae, two strike went) shows no pause and no accentual pitch. Similarly in (4a), uttered in answer to a medical inquiry, is entirely rhematic:

(4) a mainne bacpan se ainak rakhī hai 1SG.ERG childhood from glasses put PFT 'I wear glasses since my childhood' Although the theme is essentially marked by its position, always initial, and by the possibility of marking a pause after the thematized expression, not all fronted constituents are thematic, since a subject placed between the object and the verb is in the focus position without necessarily the first constituent being a theme. In (4b), uttered as a polemical answer to the previous statement "I don't see any around", the object ainak "glasses" is not a theme although first in the sequence, with the subject in the second place as a contrastive focus. But it is in (4c), in answer to "where are my glasses, everything gets lost in this house", the object ainak, with the same position, is a theme:

( Regarding the intonation pattern, sentences with a focalized constituent does not make a significant difference, as opposed to better studied languages such as English or French. The constituents in Hindi or Bengali are in a strict down step relationship regardless of word order [START_REF] Lahiri | Emphatic clitics and focus intonation in Bengali[END_REF], probably because of the prominence of lexical stress (Ohala 1986). The only difference observed by Patil et al. ( 2008) is that pitch excursion and duration are higher in OSV sentences where the subject is focused compared to SOV sentences, and focus constituents show greater pitch excursion and longer duration. But no acoustic study is available for OSV sentences where the object is thematized.

It is often considered that "scrambling" results in adding definiteness to all constituents in Hindi,9 but the reason why the object is always definite in (5b) is that a term shifted to the initial position is potentially a theme and a theme is necessarily definite (on the other hand, a term shifted to the preverbal position is focalized and potentially definite):

(

Constructing a theme as contrastive

Practically all lexical categories (noun, adjective, adverb, verb) may be thematized by the particle to in all syntactic functions (subject, object, beneficiary, oblique, attribute, predicate). Since a topic does not require to for behaving as such -position and accent are sufficient indicators of thematization -, it must be hypothesized that to constructs a particular kind of theme, intuitively perceived as "contrastive" or "emphatic" (McGregor 1972: 141). This section shows that this operation involves either a contrast with another term belonging to the same paradigm, or a contrast in judgements on the same term.

Contrast with other elements belonging to the same paradigm

Example (3) above makes the temporal frame (ājkal 'nowadays') a contrastive theme, implicitly suggesting that before or after, the situation was/will be different: in the narrative, it corresponds to the moment when the addressee gets up to take leave and the speaker gives the answer in (3), suggesting that he should stay some more since "he is alone", and implicitly there is nobody home waiting for him these days, as opposed to last week or next week for instance. To makes the adverb a theme by actualizing the relation which the other terms of the same paradigm could have had with the implicit meaning: usually you are not free, the fact that you are alone nowadays is a good occasion not to miss. The answer given ("I am alone/alone, sure, but I still must absolutely be back home" in example (13) below) makes this intended meaning ("you should stay longer as usual") clear. In ( 6), the elements of the paradigm contrasting with the to-topic are explicit in the previous context: "give me your school books for a few hours, I forgot mine". The particle to here builds the item as contrasting with the other items (available now) of the paradigm of school books:

(6) ye lo! hindī kī kitāb to tumhen kal dūngī these take Hindi of book to 2.DAT tomorrow give.FUT.1SG 'Take these! The Hindi book, I will give it to you tomorrow' In ( 7), the possible substitutable terms in the paradigm out of which mujhe "to me" is contrastively selected are implicit: Example (7) occurs as an answer to the description of X as extremely generous. The speaker reacts by representing himself (theme), as somebody who, contrary to the others, has never benefited from it. The operation consists in constructing the topic in contrast with the implicit paradigm of the other terms which could illustrate the generosity of X, and thus amounts to contest the previous representation of a supposedly universal giver. Its argumentative meaning derives from its basic function, which consists in making its scope as a contrastive theme.

An even more polemical meaning appears in (8) where the thematized numeral tīn "three" is followed by to. The speaker B, a Muslim, in a train, answers a question from A, a Hindu, which does not really aim at getting an information, since A only seeks to blame his own wife for her stupidity in not going and sit in the "ladies compartment". A's rhetorical question aims at having his judgment on his wife cautioned by the other travellers. The term tīn 'three' is constructed by to as contrasting with the number presupposed by A's rhetoric, zero (none would be so stupid as not to travel in the ladies compartment). When A answers that he sees three of them, he emphasizes the dishonesty of B in the count (you want us to believe that there are none but there are three I can see), and so dismisses his rhetoric: hence the polemical interpretation of to here, followed by a hot discussion between both A and B. If A had answered by a thematic "I"-to, contrasting then his own vision to B's vision, he would simply have opposed his own count to B's count, with a milder polemical impact: And if he had simply uttered (8c) without the particle to, the statement would have meant that B interprets A's question as a real question, devoid of polemic undertones, to which B could give a factual answer. Such an answer, at least strange in the above context, is natural after a real question like "how many of them do you see?" In the context of a quite aggressive discussion between Hindu and Muslim travellers just before the Partition of India, (8a) has the expected impact of a very polemic statement: it discards the Hindu husband's rhetoric about the stupidity of his wife. 11 And indeed the further context shows that B has only wished to contest the Hindu husband's tyranny over his wife, a tyranny he wants to emphasize because he is a Muslim and the Muslims are generally the ones blamed for depriving their women of freedom. This polemical use of to, whose efficiency is immediately perceptible in the dialogue following this sequence (violent quarrel about women's freedom in both communities), is in conformity with the logic of the particle's meaning, always more or less argumentative.

To as a re-qualification of an already mentioned term

When to has scope over a term which repeats an already mentioned term (either or not a previous theme), it re-qualifies it. Shapiro (1999: 182-3) interprets this "reiteration" as "sustaining" a previous interpretation, or "in contradistinction to a parallel form that is operative in the discourse at hand". Yet this "sustaining", echoing of McGregor "emphasis", always implies a re-qualification. In (9) for instance, the attribute "shameless", part of the previous statement by A in a non topical position (attribute), is repeated in B's statement: 'Shamelessness is rather yours!' 11 The wife (C) consequently gets over her own shyness to protest and then the husband (A) overtly insults her, finding no longer escape in pseudo-rhetorical arguments since they have been torn out by (8a):

C -zanāne Dabbe men koī bandā na pardā! main kaise baiTh jātī? A -tū chup karegī ki nahīn? Bevakūf ! 'C-There is not a soul in the ladies compartment! How could I sit there? A-Will you shut up (or not)? Idiot!'

The term besharam "shameless", formerly part of the comment, is promoted by B in the theme position, but a theme re-qualified in its relation with its referent since the referent is now the opposite group (it is you, not us, who are shameless): the already given term, when thematized with to, is redefined as a different type of shamelessness in relation to the new subject.

In (10), to occurs three times, and is used each time in order to re-qualify a term which has been previously introduced by another speaker with a different viewpoint. Two parents are complaining about the Hindi teacher of their daughter and the stupid homework she has required, an essay on hunger: In the first sequence, the requalification of the same referent (vah to thematize a previous subject "my Hindi teacher" which itself requalifies a previous object "her Hindi Madam") amounts to construct the theme as contrastive in relation to its occurrence in the mother's statement. The occurrence of the referent in the daughter's talk without to ("my Hindi teacher"), already associated with a positive evaluation contrary to the mother's viewpoint, only serves as an argument given by the daughter for not firing her Hindi teacher. The next sentence with to presents the positive qualification as relevant per se, in total opposition with the other speaker's viewpoint. To expresses the discrepancy between an initial viewpoint on a class (the type: Hindi teacher, generally considered as dull) and an individual (who gives too much work) as well as a different viewpoint on a particular individual belonging to the class but not fitting the usual qualification.

(10) MOTHER --itnā
The second sequence of the dialogue opens with a proposal by the father to substitute a tutor for school classes in Hindi: what he emphasizes is the competence of the tutor in helping for homework, the idea of having a tutor is itself taken for granted. And it is this very idea (presupposition) which the mother opposes, by shifting the debate to tutors in general and then to one in particular, her former tutor. Attached to "tutor" in the mother's discourse, to makes it a theme which negatively contrasts with the implicitly positive opinion on tutors expressed by the other speaker, the father. Without to, the statement would have only expressed a mere comparison between various school props. To gets the class of tutors questioned, hence the following shift towards personal (bad) memories.

And when the father tries to come back to the initial question, it is again to (third occurrence) that brings back to the foreground the main theme ("our daughter"). Thematizing "daughter" is a device used for going on with the initial problem -how to help the girlwhile contrasting the relation girl-tutor with the relation mother-tutor previously commented by the mother: contrary to the previous relation, bad because of male tutors, father's statement qualifies positively the relation daughter-tutor (lady tutor): to does not contrast a distinct viewpoint on the daughter, but the relation 'daughter-tutor' with the parallel relation 'pupiltutor' represented in the viewpoint of previous speaker (mother).

Example ( 11), also exhibits a conflict in viewpoints, on the predicative notion this time: B and C have just expressed that they do not understand what C means (hence asking for more information), whereas C questions his own statement as potentially dubious (regarding his own understanding). ( 11 But it frequently happens that the particle, although placed after the first segment of the statement, has scope on the whole sentence: in this case to re-qualifies the statement as a whole, most of the time with polemic intentions.

.

The argumentative particle

The formal difference between thematizing and non thematizing uses of discourse particle to relies not so much on the position of to than on the unavailability of any pause after the term on which the particle cliticizes. The position however may be relevant in certain cases: when to is clause final, it is always strongly argumentative.

Denying or shifting the relevance of a previous argument

The particle to, with scope on the whole statement, usually consists in limiting the relevance of an argument previously proposed by the other speaker, while pretending to confirm it. To rephrase the argument with to amounts to giving this argument a limited or factual confirmation, in order to better deny its wider or real relevance. In (12) for instance, a statement by B, to has scope over a clause P which rephrases the clause P' previously stated by speaker A, "speak the truth": no segmentation is possible and to receives a light stress (stronger stress on hī), the meaning intended by B amounts to agree with A but only in order to relativize the relevance of A's argument (sure you are right, but it is not the point).

(12) A -shaharī log baRe beīmān hote hain. Marad kyā aur aurat kyā. Hindū kyā aur musalmān kyā. Sāre fitne-fasād shaharon se hī shurū hote hain. kyon bābājîī main jhūTh bol rahā hūn yā sac ? 'The city folk is all very dishonest. Men and women alike. Hindus and Muslims alike. All communal riots start in cities. Right, Baba-ji ? Do I speak lie or truth ?' B -bol to tū sac hī rahā hai, speak to you true just PROG PRES 'You (indeed) speak true (Sure you are right), par isse bhī baRā sac yah hai ki insāf na shahar men na gānv men. Insāf to12 tum dekh lenā Pakistān men bhī nahīn hogā. but there is a bigger truth than that, and this is that justice is neither in cities or villages. Justice, you will see, you won't find it either in Pakistan'

The fronted thematized verb (bol 'tell'), in the strongly marked order V-to-S-O-aux, is not the simple quote of A's utterance, which is unmarked in its ordering (SOV) with the same object (sac 'truth'). The new mapping within to… hī does not simply discard the alternative ("or wrong") nor does it bring a different viewpoint on the predicate or the clause P, whose validity is not questioned. But if B agrees with A on the initial location of communal violence, he does so in order to signal that this "truth" has little relevance (to) in the present discussion. B implicitly disqualifies A's interpretation of facts, while granting him the confirmation he rhetorically asked for ("right or wrong": you can't say I am not right). But he grants him such a confirmation ("sure you are right") only for the sake of shifting from P1 to P2: the real problem is not your being right on the initiation of violence (P1) but to deal with justice (P2). B does not enter in A's rhetoric and to marks this refusal to accept A's premises while accepting only the conclusions.

Similarly, in ( 13), the continuation of (3) supra, B in the first clause, with the same structure as (12), qualifies A's argument as weakly relevant, and in the second adversative clause provides another argument as crucially relevant, in contradiction to A's request:

(13) A-ājkal to āp akele hain… B -akelā to hūn, lekin ghar to jānā hī hai nowadaysto 2H alone are alone to am but house to go hi is '-These days you are alone… --Sure I am alone, but I really must go home'

In the first (underlined) clause of B's answer, the absence of to would result in a rather strange meaning in this context, since A was not asking for a confirmation but was using the argument to convince B to stay longer. In B's second sentence, to… hī in an adversative clause makes the clause a crucially relevant argument in countering A's offer.

Making the clause a crucial discursive argument

Discursive cruciality is involved in the last clause of last example (13) ghar to jânâ hî hai 'I really have to go home'. Although apparently opposed to the meaning of to in the first clause of the same example (denial of relevance), it nonetheless participates in a similar operation: speaker A provides a new element in the discussion, something that the addressee had either not thought of or had not wished to consider. What is at stake is the promotion of the propositional content within a perspective opposed to the addressee's view, for which the proposal is either not relevant or not conceivable. Such an operation is then symmetrical to the denial of relevance, where P was relevant for speaker A and not for speaker B. When this operation occurs in non adversative contexts (of the type 'sure, but'), the salience intuitively perceived in to-statements is due to the fact that such statements are constructed in opposition to a previous implicit assumption of speaker A. Hence their objection-like character instead of simple assertions. The series of answers to the suggestion 'let us go and have some tea' given by Lakshmi [START_REF] Bai | Syntax and Semantics of the Particle To in Hindi[END_REF] as examples of "assertive particle" can be explained in this way: Each of the answers (14a-d) represents according to Lakshmi Bai the choice of the speaker among various alternatives, and to asserts this choice. However, a single speaker does not really select one by dismissing the other alternatives, and the 'assertive force' is the result of the inter-subjective relation involved in the whole series as an answer to a proposal which the speaker wishes to discard. Speaker A in (14a) signals to speaker B (let us have some tea) something of which B was not aware (not enough money, closing time, unfinished work), which makes it difficult to obey the request and therefore acts as an indirect objection or refusal. To here behaves as a request for taking into account what the speaker valuates as a crucial argument that has been neglected by the addressee. Hence the possible translations opening with "but you know", "the problem is", "but"… When the discourse particle to has scope on a proposal already stated by a previous speaker, it denies it full (or any) relevance. When it has scope on a proposal which is new, it builds the relevance of this new P in contradiction with what the speaker thinks is the viewpoint of the other speaker.

(
3.3 Final to: winning over the other's reluctance I have shown elsewhere that preverbal to in the last syntagm of questions amounts to checking if the proposition has been successfully realized (in conformity with the speaker's expectations and against the obvious difficulties barring access to that realization ("let us hope that P"). 13 In a postverbal position, mostly after a verb in the imperative, the particle to acquires such a specific meaning that it is often deemed a quasi homonym of the "assertive" particle (Lakshmi Bai 1977: 73;Shapiro 1999: 186).

(15) batāo to ! 13 Montaut (2002). Such a use is idiomatized in the expression āp acche to hain? 'How are you/hope you are fine?' [you well to are] but still differs from the same statement uttered without to by its contextual constraints: it is natural only when the speaker has not seen the person he greets since a long time, or has got bad news, or thinks he may not be so well… example of the meaning "insistence" in McGregor. tell to 'But tell it (now)! Come on, why don't you tell? Will you finally tell it?' Statement (15) strongly differs from its counterpart without to and requires a specific context: it echoes a strong reluctance on the part of the addressee, or at least the speaker's anticipation of such reluctance. 14 Example ( 16) shows concrete evidence of B's reluctance: When A utters the last sentence (with to), he has already ordered B twice to come, without to, and this order has not yet been fulfilled. Only in the third utterance of the same order, once fully aware of B's reluctant behaviour, A uses to after the order verb. The intuitive perception of insistence results in reality from the feature "unwillingness" imputed by A to B.

Far from a mere emphasis, stylistically optional, such examples show that the particle to operates systematically in constructing P as countering a non-P imputed to the addressee. With to, an order then conjures up the risk of not reaching its goal, and at the same time emphasizes the addressee's reluctant behaviour. Here again, the meaning of to stems from contrastive and even conflicting attitudes of both partners. Similarly, phrasal nahīn ('no') in answers to P, when followed by to (nahīn to) acquires its "emphasis" from the fact that the speaker refutes an assumption he imputes to the addressee.

C o n c l u s i o n s

The very high frequency of the occurrences of to, particularly remarkable in oral interaction, as well as the variety of its meanings and functions, may be correlated to the limited number of words behaving as discourse particles in Hindi. 15 This contrasts with the situation in Germanic languages which display far more numerous discourse particles, a dozen in German (Weydt 1969) with more than one equivalent for each in Serbo-croatian (Diewald and Krescic 2010), in which each particle has consequently a more restricted usage.

In spite of the apparent heterogeneity of the meanings involved in the data studied, one can argue that the multiple surface meanings and functions of the discourse particle to obey a single basic meaning, although not concretely semantic but rather pertaining to an abstract operation. This basic meaning consists in triggering an operation which deals with alterity ("altérité": 'otherness'), at various levels: to seems to convey the speaker's judgment on a 14 A weaker (because idiomatized) expression of the same meaning is still associated to the common polite way of welcoming a visitor baiThie to (sit-down to) 'why don't you sit down '. 15 As evidenced by the various examples: (13), ( 20), ( 21), ( 22), ( 27) are from Bhūkh āg hai (KB Vaid, Hunger is fire), contemporary theatre, or (11), ( 14), ( 15), ( 16), from Guzrā huā zamānā (KB Vaid, The broken Mirror [lit. Time past]), a contemporary Partition novel, or from recorded conversations (Standard Hindi, middle class, 40-60 years old informants, Delhi). The other two commonly used discourse particles are hī (focus particle, particularly in restricted focus) and bhī 'even'. term or sequence on which it has scope, in such a way that to-P triggers the implicit or explicit representation of P' (P': non P or other than P).

When to is a thematic particle, P is the thematized term and P' either refers to other members of the same paradigm where P belongs, or it constructs a different relation between P and its rheme, or a viewpoint on P differing from the viewpoint previously expressed by (or attributed to) the other speaker (re-qualification). This thematic function has no parallel in the Germanic discourse particles (more often called modal particles or Abtönungspartikel), since their common characteristic is to have for "focus the relational category of discourse rheme in a sentence", and to have "scope over the whole VP" (Abraham 1991: 356). In Hindi the nonthematic to only has scope over the whole sentence.

In such cases, that is, in Hindi, in sentences entirely rhematic, structured as thetic or focus sentences, to presents the whole statement on which it has scope as a correction of another distinct viewpoint: either a new argument is presented as countering an initial proposal, or a proposal already voiced is presented with weaker relevance or no relevance, or (in pre-or postverbal positions) to aims at rejecting an opposite eventuality (conjuring a fear, dismissing a fictive assumption, winning over an anticipated reluctance from the other speaker). The operation triggered by the discourse particle to always involves inter-subjectivity, since it negotiates with the other's viewpoint, distinct from and often conflicting with the utterer's viewpoint. 16 This prerequisite of inter-subjectivity is paralleled by the behaviour of Germanic modal particles as described in terms of dialogic interaction in Weydt (2006) or grounding and common grounds in [START_REF] Fischer | Grounding and common ground: modal particles and their translation equivalents[END_REF], Diewald and Krescic (2010) or [START_REF] Bross | German modal particles and the common ground[END_REF]: the speaker acknowledges what the hearer thinks as a common ground for anchoring his/her proposal. The specificity of Hindi to is that this inter-subjective interaction rarely fosters conviviality or connivance since it basically involves conflicting or at least diverging viewpoints.

This orientation is compatible with the origin of the word, initially a pronominal basis related to the sphere of non-ego. As for the conjunction to, when it is a correlative, to, which is directly related to the anaphoric pronoun, correlates the apodosis to a conditional, acting both as an anaphoric device and as a selecting a given path among others; when it coordinates two discourse sequences ('then'), it opens a new scenario resulting yet distinct from the previous discourse sequence.

The behaviour and meaning of to as a discourse particle certainly differ from its syntactic meanings: they are highly diversified and often involve polemical values. Yet they form a constellation of meanings and uses whose consistency becomes clear as soon as the operation at stake is analyzed on the basis of interactive contexts; besides, they are not totally disconnected from the grammatical word. 17 But the discourse particle always involves intersubjectivity, whereas the conjunction is not sensitive to it. The above study is of course only a fist attempt to disentangle the meanings and functions of to in interactive situations, and finer grain studies are still awaited, particularly regarding the exact intonational contours in the various scopes (thematizing vs non-thematizing) of the particle. Abraham, Werner. 1991. The Grammaticization of the German Modal Particles. In Edith 16 As opposed to the uncontroversial meaning of the focus marker derived from the tag question marker in Even (Matic,this volume,example 26b). There is probably a logical correlation between the fact that the focus marking element produces uncontroversial common ground in the inter-subjective relation of speaker A and B, whereas the topic marking element emphasizes diverging view points in the interaction. 17 Which is also true of the Russian particle to, analyzed by [START_REF] Bonnot | La Particule to et la polémique cachée en russe moderne. A propos du statut énonciatif du thème[END_REF], equally issued from the same Indo-European pronominal basis. Russian to has numerous uses comparable to those of Hindi to, but not the correlative one nor the meanings observed in section 3.2. l 'INALCO 4 : 111-134. Montaut, Annie. 2004. Hindi Grammar. Munchen : Lincom Europa. Montaut, Annie. 2012

Abstract

The Hindi particle to may be both a conjunction, which occurs in the initial position with the meaning "so, then", and a discourse particle, usually described as an intensive, or vaguely contrastive or emphatic device. Both have been treated as different units, and the discourse particle as several distinct (homonymous) words. The aim of the paper is to show that the diversity of the meanings and functions of the particle may be explained by a common abstract operation, realized in various ways according to the various specific contexts of occurrence, both syntactic and discursive. The first section briefly summarizes and interprets the main functions of grammatical to, as a coordinator and as a correlative, before I analyze the discourse particle, either with restricted scope and fixed position, as a topic marker (section 2), or with wider scope and no fixed position, as an argumentative particle (section 3).

Key words Discourse particle, Hindi to, argumentation, thematization, inter-subjectivity, correlation La particule hindi to peut être une conjonction (de coordination et de subordination), qui se place en première position et correspond aux sens « alors, donc », et un mot du discours, décrit d'ordinaire comme un intensif, ou un dispositif d' « emphase » vaguement contrastif. Les deux sont généralement considérés comme deux mots distincts, homonymes, tout comme le mot du discours est considéré comme recouvrant (au moins) deux particules distinctes, homonymes. En fait, les divers effets de sens attachés à la particule s'expliquent par une opération commune, qui se réalise différemment en fonction du contexte (position, portée). Par ailleurs les fonctions du mot grammatical montrent en quoi il a pu se spécialiser parallèlement comme marqueur de thème.

  ) to me (I, as far as I am concerned) she has not given me anything'

  (8) a A--koī aur aurat bhī hai baiThī huī mardāne Dabbe men INDEF other woman even is seated being male compartment in A-'Is there any another woman sitting in the men compartment ?' B --tīn to mujhe nazar ā rahī hain three to 1SG.DAT look come PROG PRES B-'There are three (you can't deny) I can see'

  to three look come PROG PRES 'As for me I (But if you ask me) I can see three of them'

  (8) c mujhe tīn nazar ā rahī hain 1SG.DAT three look come PROG PRES 'I can see three of them'

(

  26) A -ā ! B --hān... A --idhar ā! B -ū hūn… come yes here come ok ok C --is buddhū ko ek bār men bāt mān lene kī ādat nahīn. this idiot DAT one time in thing accept take of habit NEG A -are! ā to! hey come to 'A -Come! B -Ya… A -Come here ! B -Ok ok…C -This idiot is not used to obey at the first time. A -Hey you! you do come! [what are you waiting for!/ will you move your ass?]'

  If you helped me [had helped me] I would succeed [would have succeeded]'

	jab	āp	log	cale gae	to /tab mainne usko	bulā liyā
	when 2H	people leave go.PFV

to /tab 1S.ERG 3S.ACC call take.PFV 'When you left I called him' Abbreviations : ERG (ergative), ACC (accusative), DAT (dative), PFV (perfective: simple anterior), PFT (perfect), PRES (present), CTF (counterfactual), H (honorific), PPFT (pluperfect). '

  Detached after the verb, the same object behaves as a delayed theme or post rheme ("antitopic" in Chafe 1976, "mneme" in Fernandez-Vest 2004 and in this volume) in the same context as (4c): You had put them on the table yesterday night, the glasses'

	4) b	ainak tūne lagā rakhī hai	yā	mainne?
		glasses 2S.ERG place put PFT		or	1SG.ERG
		'Is it you who is wearing glasses or I?'
	(4) c	ainak kal	rāt	mez par	rakhī thī
		glasses yesterday	night table on	put PPFT
		'The glasses, you had put them on the table yesterday night'
	(4) d	kal rāt	tumne xud	mez par	rakhī thī	ainak
		yesterday night	you.ERG REFL table on	put PPFT	glasses
		'

Apnī biTiyā ke lie to ham

  sārā homework, vah bhī hindī men ! Ab hindī bhī koī sabjekt rah gayā hai, ājkal ke zamāne men ! islie main kahtī hūn iskī hindī kī Madam ko haTvā do ! 'So much of homework! And in Hindi on top of that! As if Hindi was still a real subject nowadays! That's why I am asking you to have this Hindi Madam (teacher) fired!' DAUGHTER --nahīn Pāpā, please aisā mat karnā ! merī hindī kī Madam bahut kyūT hai. 'No, daddy, please, don't do that! My Hindi teacher is very cute' Vah to hindī kī lagtī hī nahīn, bahut smart hai, ekdam ! 3SG to hindi of seem just NEG much smart be.PRES.3SG totally She (you know) just does not look like a Hindi (teacher) at all, she is very smart, absolutely! [Both parents discuss the question of homework between themselves] FATHER -ham apnī biTiyā ke lie ek aisā tutor rakh lenge jo sārā homwork phaTā-phaT kar diyā karegā Well, we will find a tutor for our daughter who will get the homework done in a minute. MOTHER -Tutor se to computer behter. (…) Computer yes, tutor no. Never ! Tutor badmāsh hote hain. Always! Merā ek tutor hotā thā, very bad. A tutor [do you realize what a tutor means ?], a computer would do better (…).Computor yes, tutor no. Never ! All tutors are wrecks. I used to have a tutor, very bad . FATHER -(…) khair, is māmle ke bāre men main pūchtāch phir kabhī karūngâ. lady tutor hī rakhenge (…) Well, I will look in that matter some other time. [But] For our daughter [at any rate] we will find a lady tutor [not a man])

discourse particle to and word ordering in Hindi: From grammar to discourse Annie

  . De l'anaphore à la subordination en passant par la corrélation, In Olga Inkova & Pascale Haderman (eds.). La Corrélation, 193-213. Paris : Champion. Ohala, Manjari. 1986. A Search for Acoustic Correlates in Hindi Stress. In Bh. Krishnamurti (ed.), South Asian Languages: Structure, Convergence and Diglossia, 81-90. Delhi: Motilal Banarsidass. Patil, Umesh, Gerrit Kentner, Anja Gollrad, Frank Kügler, Caroline Féry & Shravan Vasishth. 2008. Focus, Word Order and Intonation in Hindi. JSAL 1: 55-72. Shapiro, Michael. 1999. Hindi to as discourse marker. In P. J. Mistry and Bharati Modi (eds.). Vidyopaasanaa: Studies in Honor of Harivallabh C. Bhayani, 179-89. Mumbai and Ahmedabad: Image Publications Pvt. Ltd. Weydt, Harald. 1969. Abtönungspartikel. Die deutschen Modalwörter und ihre französischen Entsprechungen. Berlin West & Zürich : Bad Homburg. Weydt, Harald. 2006. What are particles good for? In Kirsten Fischer (ed.). Approaches to Discourse Particles. Amsterdam: Elsevier. Wlodarczik, Hélène & André Wlodarczik (eds.). 2006. La focalisation dans les langues. Paris : L'Harmattan. The

Montaut Inalco USPC, SeDyL UMR 8202

Whether to belongs to the protasis or to the apodosis is a real question: it is traditionally considered to belong to the protasis (as correlatives in general), yet it cannot be omitted if the protasis is suspended.

The notions of fictive landmark ("repère fictive"), entailment ("entraînement"), scanning ("parcours") and access path ("chemin d'accès") are borrowed from[START_REF] Culioli | Pour une linguistique de l'énonciation 2. Formalisation et opérations de repérage[END_REF]. Conditionals are considered in Culioli's model as involving "bifurcation" from one possible path to the other.

The Sanskrit pronominal basis ta-displays in certain flexions and genders a form in sa-(sah 'he', tat 'it'), a form (Hindi so) which is still used in Hindi as an adverbial coordinator ('then', 'so') and has long been used as a correlator for relative clauses(Montaut 2012.)

This restricted meaning is distinguished from the "general" or psychological meaning of focus of attention byGundel & al. (1993: 279, note 1, after Hajičova). Focus in the psychological/cognitive meaning (focus of attention) is equivalent to the rhematic content, but not in its restricted meaning (linguistic focus).

As clearly stated byGutierrez-Alvares (2003: 17): "Rheme is present in all statements. Focus is not, and it requires a specific marking: it is always a marked function".

It has postpositions and no prepositions, goal complements always precede the verb, and only the SOV order allows all syntactic transformations are available. Some scholars[START_REF] Mohanan | Issues in Word Order[END_REF]) consider Hindi as a free order language because all constituents can shift position without morpho-syntactic restructuration. but they mention "scrambling", which implicitly refers to some basic or dominant order.

[START_REF] Dvivedi | Topicalization in Hindi and the Correlative Construction[END_REF] however shows that contrastive focuses may, exceptionally, remain indefinite.

Example fromMohanan & Mohanan (1994 : 169). In fact, hār 'necklace' is thematized in (5b) with a possible pause after hār, and the literal translation would rather be 'the necklace, the jeweller sent it to the girl' ('to a girl' would correspond to Hindi kisī laRkī ko, INDEF girl DAT).

To here is a topic particle which backs the topic of justice (in contrast to previously mentioned: justice in God's realm =Pakistan) after the digression about cities/villages.