Victor Boone
email: victor.boone@univ-grenoble-alpes.fr

When do discounted-optimal policies also optimize the gain?

In this technical note, we establish an upper-bound on the threshold on the discount factor starting from which all discounted-optimal deterministic policies are gain-optimal, that we prove to be tight on an example. To address computability issues of that theoretical threshold, we provide a weaker bound which is tractable on ergodic MDPs in polynomial time.

Concepts and main result

We consider Markov decision processes (MDPs) with finitely many states and actions, whose sets are respectively denoted X and A = x∈X A(x). A MDP is given by a tuple M = ⟨X , A, p, q⟩ where p and q are respectively the transition kernel and reward distributions. The mean reward associated to (x, a) is denoted r(x, a), viz., r(x, a) := E R∼q(x,a) [R].

A policy π ∈ Π is any deterministic stationary decision rule. Upon iterating a policy π on M starting from x ∈ X , we obtain a sequence of states, actions, and rewards {(X t , A t , R t ) : t ≥ 0} whose probability measure will be denoted P π x (-) and expectation operator E π

x [-]. We will write Z t for the pair (X t , A t ). The iterates of a policy also define a Markov reward process (MRP) {(X t , R t )} whose transition kernel will be denoted P π and mean reward vector r π , i.e., r π (x) := r(x, π(x)). To each policy are associated various notions of scores:

• the finite-horizon score J π T (x) := E π x [ T -1 t=0 r(Z t )]; • the β-discounted score V π β (x) := E π x [ ∞ t=0 r(Z t )β t ] for β ∈ [0, 1); • the gain g π (x) := lim T →∞ E π x [ 1 T T -1 t=0 r(Z t )]; • the bias h π (x) := lim T →∞ E π x [ ∞ t=0 (r(Z t )-g π (X t ))
], or the Cesáro-limit when the limit doesn't exist. 1 Note that those quantities depend of the underlying MDP M -this M will sometimes be added to notations to avoid ambiguities.

A policy that achieves maximal β-discounted score from all state is said βdiscounted optimal and we write π ∈ Π * β . A policy that achieves maximal gain from all state is said gain-optimal and we write π ∈ Π * -1 ; if in addition it achieves maximal bias from all state, it is said bias-optimal and we write π ∈ Π * 0 . All these classes are non-empty [Puterman, 1994]. It is known [START_REF] Blackwell | Discrete dynamic programming[END_REF] that when β ↑ 1, Π * β eventually converges to a single class of policies known as Blackwell-optimal policies, which is a non-trivial subset of bias-optimal policies [Puterman, 1994].

There are few explicit bounds on how large β needs to be so that all βdiscounted optimal policies are Blackwell-optimal [Grand-Clément and Petrik, 2023]. For a less demanding problem, and to the best of our knowledge, the question of finding a threshold on the discount factor β -1 that guarantees that every βdiscounted optimal policy is gain-optimal (i.e. in Π * -1 ) for all β > β -1 has not been addressed so far. This threshold is formally given by:

β -1 := inf β * ≥ 1 : ∀β ∈ (β * , 1], Π * β ⊆ Π * -1 . (1)
Theorem 1 establishes an upper-bound on β -1 . If u is a vector (e.g., g π , h π , . . .), its span is sp(u) := max x u(x) -min x u(x). We denote g * and h * the respective optimal gain and bias vectors, equal to g π * and h π * respectively, where π * is any bias-optimal policy. Theorem 1. For all MDP with finitely many states X and actions,

β -1 ≤ 1 -inf g * (x) -g π (x) sp(h * ) + sp(h π ) : x ∈ X and π ∈ Π s.t. g π (x) < g * (x) . (2)
This result underlines a trade-off to be found between how close to gain-optimal π is and how large its bias tends to be. This result follows from relatively folklore identities linking the gain and the discounted score. A complete proof is provided later in this note.

The bound (2) is tight. Consider the MDP with deterministic transitions pictured in Figure 1. There is a single action from states 2, 3 and two from 1 (left or right). The associated mean rewards are represented as arc weights.

0 1 2 1 1 + ϵ h -ϵ g 1 -ϵ g 1 Figure 1: A deterministic MDP achieving the bound of Theorem 1 (ϵ g , ϵ h > 0).
The policy going to i ∈ {1, 2} from 0 is denoted π i . Only π 1 is gain-optimal.

π 1 : V π 1 β (0) = (1 -β) -1 , g π 1 (0) = 1, h π 1 = (0, 0, 0); π 2 : V π 2 β (0) = (1 -ϵ g )(1 -β) -1 + ϵ h , g π 1 (0) = 1 -ϵ g , h π 2 = (0, ϵ h , 0).
From the values above, we derive that

V π 1 β (0) > V π 2 β (0) ⇐⇒ β > 1 - ϵ g ϵ h = 1 - g π 1 (0) -g π 2 (0) sp(h π 1 ) + sp(h π 2 )
.

A tractable lower bound for ergodic MDPs

The inf-bound in Theorem 1 is hard to compute because one needs to check all suboptimal policies. Our second result is a weaker version of (2) for ergodic MDPs that can be computed in polynomial time. Recall that a MDP is ergodic if P π is ergodic for all π -this condition is not easy to check in general, and is usually decided by the problem that the MDP models.

Theorem 2. For all ergodic MDP with finitely many states X and actions,

β -1 ≤ 1 - ∆ g 2sp(r)D (3)
where ∆ g := inf {g * (x) -g π (x) : x ∈ X and g π (x) < g * (x)} > 0 is the gain-gap of the MDP and D := max π max x̸ =y E π x [τ y ]1 < ∞ is the worst diameter. We later show that ∆ g and D are both computable in polynomial time.

Proof. We only have to show that for all policy π, sp(h π ) ≤ Dsp(r). This result is variation on [Bartlett and Tewari, 2009, Theorem 4] that we prove using a different technique. Consider the Markov chain {X t : t ≥ 1} induced by the iterations of π. Let x, y ∈ X such that sp(h π ) = h π (x)-h π (y). Because r π -g π = (I -P π )h π , the quantity

(r π (X t ) -g π (X t )) -(h π (X t ) -h π (X t+1 ))
is a martingale difference sequence. Its differences are a.s. bounded by sp(r) + sp(h π ) < ∞ and τ y is an a.s. finite stopping time. Hence:

h(x) -h(y) = E π x h π (X 0 ) -h π (X τy ) = E π x τy-1 t=0 (r π (X t ) -g π (X t )) ≤ E π x [τ y ]sp(r) ≤ Dsp(r).
Remark 1. We see from the definition that D < ∞ if and only if M is ergodic. If M is not ergodic, then the bound of Theorem 2 is not informative.

Computation of ∆ g

Define, for (x, a) a state-action pair,

∆ * (x, a) := h * (x) -[r(x, a) -g * (x) + ⟨p(x, a), h * ⟩] (4)
the suboptimality gap of (x, a), which is non-negative. For π ∈ Π, µ π x denotes the (empirical) invariant measure of π achieved by iterating π starting from x ∈ X .

Lemma 1. For all policy π and all x ∈ S, g π (x) ≤ g * (x) -y∈X µ π x (y)∆ * (y, π(y)).

Proof. This result can either be established algebraically or using the martingale technique used in Theorem 2. We go for the algebraic proof here. Denote ∆ π (x) := ∆ * (x, π(x)). By definition of ∆ * , we have r π = g * + (I -P π )h * + ∆ π . Multiplying by (P π ) t and summing over t, we obtain:

T -1 t=0 (P π ) t r π = T -1 t=0 (P π ) t g * + I -(P π ) T h * + T -1 t=0 (P π ) t ∆ π ≤ T g * + I -(P π ) T h * + T -1 t=0 (P π ) t ∆ π .
Dividing by T and making T go to infinity, we obtain:

g π ≤ g * + lim T →∞ 1 T T -1 t=0 (P π ) t ∆ π
whose x-th line readily provides the result.

Following Lemma 1, we deduce that for any policy π such that g π (x) < g * (x), there must be y ∈ X such that µ π x (y)∆ * (y, π(y)) > 0. More precisely, since µ π

x (y) > 0 for all π and all y in the ergodic case, a policy is gain-suboptimal if and only if it uses a suboptimal action at some point, i.e., an action such that ∆ * (x, a) > 0. Because, given a MDP M , the computation of g * (M ) is polynomial time, we deduce that ∆ g is also computed in polynomial time with the following procedure:

Algorithm 1 Computation of ∆ g for a MDP M

Computation of D

The computation of D follows the same idea than ∆ g . Denote M y the copy of M where (1) y is zero-reward absorbing state and (2) all rewards, except from y, are set to 1. Because y is recurrent under every policy π (on M ), the iterates of π are eventually stationary to y on M y . In particular, g π (M y ) = 0 for all π and y. We deduce that h

* (x; M y ) = max π E π,M x [τ y ], hence: max x h * (x; M y ) = max π max x E π,M x [τ y ].
But is the computation of h * (M y ) polytime? In general, Bellman's equations are not enough to compute h * . We show that they are sufficient for M y .

Lemma 2. Write M y = S, A, p My , q My . Let π any policy that satisfies the Bellman equation:

∀x ∈ X , g π (x; M y ) + h π (x; M y ) = max a∈A(x) r(x, a; M y ) + p My (x, a), h π (M y ) .
Then π is bias-optimal on M y .

Proof. This result is a special case of a much more general result that says that, if all policies have the same recurrent states, Bellman equations automatically guarantee bias-optimality. We provide an ad-hoc proof for the special case of M y for self-containedness. For short, denote g ′π , h ′π , r ′π and P ′π the quantities related to π on M y . We know that g ′π = 0 automatically (all policies are gain-optimal on M y ). Let π * bias-optimal on M y . By applying Bellman's equation iteratively, we get:

h ′π ≥ r ′π * + P ′π * h ′π ≥ r ′π * + P ′π * r ′π * + P ′π * h ′π . . . ≥ T -1 t=0 P ′π * r ′π * + (P ′π * ) T h ′π .
We know that the only recurrent state of π * is y, so when T → ∞, (P ′π * ) T h ′π → h ′π (y) = 0. So overall, when T → ∞, we get h ′π ≥ h * . So h ′π = h * .

In the end, D is computed in polynomial type as follows.

Algorithm 2 Computation of D for ergodic MDPs.

1: Construct {M y : y ∈ S} where M y is the y-absorbing copy of M with 0 reward on y and 1 reward everywhere else; 2: For each y, compute π y a policy satisfying the Bellman's equations on M y ; 3: return max y max x h πy (x; M y ).

Proof of Theorem 1

Lemma 3. Every policy π satisfies:

g π (x) -1 T sp(h π ) ≤ 1 T J π T (x) ≤ g π (x) + 1 T sp(h π
). Proof of Lemma 3. Bias, gain and reward vectors are linked by the following Poisson equation: r π = g π + (I -P π )h π . Multiplying by (P π ) t and summing up, we obtain:

J π T := T -1 t=0 (P π ) t r π = T -1 t=0 (P π ) t g π + T -1 t=0 (P π ) t -(P π ) t+1 h π = T g π + (I -(P π ) T )h π
where the last equality is obtained using P π g π = g π . Now, because I -(P π ) T is a difference of line-stochastic matrices, all entries of (I -(P π )) T h π are upper bounded by sp(h π ) in absolute value. If e denotes the vector whose entries are all 1s, we thus obtain T g π -sp(h π )e ≤ J π T ≤ T g π + sp(h π )e which is the claimed result.

Lemma 4. For all policy π and discount factor β ∈

[0, 1), V π β (x) -g π (x) 1-β ≤ sp(h π ). Proof of Lemma 4. Denote r t := E π x [r(Z t )] the t-th expected reward. Let F π β (x) := (1 -β) ∞ t=0 β t r t the normalized β-discounted reward of π starting from x. Then for |β| < 1, (1 -β) -2 F π β (x) = (1 -β) -1 ∞ t=0 r t β t = ∞ t=0 β t ∞ t=0 r t β t = ∞ t=0 J π t+1 (x)β t . So F π β (x) = (1 -β) 2 ∞ t=0 J π t+1 (x)β t .
Using 1 = (1 -β) 2 ∞ t=0 (t + 1)β t , we also have g π = (1 -β) 2 ∞ t=0 g π (t + 1)β t , and

F π β (x) -g π (x) ≤ (1 -β) 2 ∞ t=0 1 t + 1 J π t+1 (x) -g π (x) (t + 1)β t ≤ (1 -β) 2 ∞ t=0 sp(h π )β t = (1 -β)sp(h π ),
where the second inequality is obtained by applying Lemma 3 for J π t+1 . Proof of Theorem 1. Let β * the right-hand side of (2) and let β > β * . Let π * a bias-optimal policy and π such that g π (x) < g * (x) for some x ∈ X . From Lemma 4 follows that

V π * β (x) ≥ V π β (x) + g * (x) -g π (x) 1 -β -sp(h * ) -sp(h π ).
A sufficient condition for V π * β (x) > V π β (x) is thus g * (x) -g π (x) 1 -β -sp(h * ) -sp(h π ) > 0.

Solving the above inequality in β, this sufficient condition is equivalent to:

β > 1 -g * (x) -g π (x) sp(h * ) + sp(h π ) .

This holds by choice of β * . So V π * β (x) > V π β (x) and π is not β-discounted optimal.

1:

  Construct {M xa : (x, a) ∈ S × A} where M xa is the copy of M whose only available action from x is a; 2: Compute {g * (M xa ) : (x, a) ∈ S × A} and g * (M ); 3: return min {g * (M ) -g * (M xa ) : g * (M xa ) < g * (M )}.

τ y is the reaching time of y, starting from the initial state: τ y = inf{t ≥ 0 : X t = y}.


domain: math phys

text:
Gourab Bhattacharya

A METRIC THAT SATISFIES THE GRAVITATIONAL MONOPOLE EQUATIONS-I

We proved the existence of a Gravitational Monopole using Theorem 0.1. Let (M, g, J, ω) be a compact almost Kähler surface with a negative definite signature σ < 0 and c 1 ̸ = 0, c 1 (L) 2 > 0, and (M, J) admits no holomorphic vector fields. Then on these surfaces, we have

and if the inequality is strict, there exist deformations which are strictly almost-Kähler.

The condition ensures that the following theorem holds good Theorem 0.2. The obstruction to the existence of the self-dual Gravitational Monopole metric lies in H 2 (Z, Θ Z,D ). Therefore the vanishing condition H 2 (Z, Θ Z,D ) = 0 and s < 0 will guarantee the existence of the Gravitational Monopole metric.

Introduction

We chose a self-dual metric in 4 dimensions and up to a conformal equivalence we find an obstruction to the deformation of a self-dual metric to a Gravitational Monopole. We find a condition on that guarantees the existence of the Gravitational Monopole metric.

The Gravitational Monopole Equations

In [cf.1] the Gravitational Monopole equations were introduced in the following sense. Let (X 4 , g) be a Riemannian spin 4-manifold. Then the Clifford algebra bundle Cl(X 4 ) is a vector bundle over X 4 with fibre at x ∈ X 4 is the Clifford algebra Cl(T x X). With respect to the metric g, one identifies (isomprphism) Cl(T x X) with Cl(T * x X). Therefore, as a vector space, this is isomorphic to ∧T * x X. Let us also assume E → X is a Clifford module bundle with a covariant derivative ∇ E . Then for each x ∈ X there is a Clifford action c : T * x X ⊗ E x → E x via c(α ⊗ s) = c(α)s. Definition 2.1. The twisted Dirac operator associated to (E, ∇ E ) is the operator, (2.1)

/ ∇ := c • ∇ E : C ∞ (X, E) → C ∞ (X, E).
The equations we wish to consider are (sometimes we omit the mapping c and the dimension 4 for the convenience of computations),

/ ∇ψ = (d + d * )ψ = 0, c(W + g ) = 1 4 ⟨e i • e j ψ, ψ⟩e i ∧ e j ,
or, c (W + g ) ijkl e i ∧ e j = 1 4 ⟨e k • e l ψ, ψ⟩.

(2.

2)

The Weitzenböck's formula [START_REF] Bavnbek | Elliptic Boundary Problems for Dirac Operators[END_REF]: the decomposition of the Laplace-Beltami operator as a generalized Laplacian is, with the left-Clifford multiplication ϵ l = ext l -int l , and corresponding right Clifford action ϵ r , (2.3)

(d + d * ) 2 = ∆ ∧T * X - ijkl R ijkl ϵ i l ϵ j l ϵ k r ϵ l r + 1 4 s.
Proposition 1. For k ̸ = l, ⟨e k e l ψ, ψ⟩ is purely imaginary.

1 Proposition 2.

(2.4) |⟨e k e l ψ, ψ⟩| 2 = 2|ψ| 4 .

We keep in mind the Seiberg-Witten analysis, and analogously define and get the following definition of Gravitational-Monopole functional, Definition 2.2. The Gravitational-Monopole functional of a pair (ψ, g) is given by,

S(g, ψ) = X | / ∇ψ| 2 + |W + - 1 4 ⟨e i • e j ψ, ψ⟩e i ∧ e j | 2 d(vol) g i.e., S(g, ψ) = X |(d + d * )ψ| 2 + |W + - 1 4 ⟨e i • e j ψ, ψ⟩e i ∧ e j | 2 d(vol) g .
(2.5)

Proposition 3. (2.6) S(g, ψ) = X |∇ψ| 2 + |W + | 2 + s 4 |ψ| 2 + 1 8 |ψ| 4 d(vol) g Proposition 4.
As a direct consequence of (3), if the scalar curvature of X is non-negative, all solutions of (2.2) have ψ ≡ 0.

2.1. Action of Special Orthogonal Group. Let X be an oriented Riemannian manifold of even dimension 2l and we also assume X is a spin manifold, that is the first and second Stiefel-Whitney classes vanish. We denote by ∧ p the bundle of exterior p-forms with A p = Γ(∧ p ) its space of smooth sections. The Hodge star operator * ∧ p → ∧ 2l-p is defined by,

(2.7) α ∧ * β = (α, β)ω ∈ ∧ 2l
where α, β ∈ ∧ p , (α, β) is the induced inner product on p-forms and ω is the volume form.

From now everything will be 4-dimensional unless otherwise stated. We start with the symmetry of the equations, namely the Lie algebras. The Lie algebra so(4) of the special orthogonal group SO(4) is not simple. It can be decomposed into the direct sum of two copies of the Lie algebra so(3) of the group SO(3):

(2.8) so(4) ∼ = so(3) ⊕ so(3).

In terms of the group theory, one understands the above decomposition corresponds to the fact that the universal covering group of SO( 4) is the product of the two copies of SU (2). This fact in quantum mechanics corresponds to ± 1 2 spins of an electron for each factor SU (2). In terms of the geometry of the vector bundles, the decomposition so(4) ∼ = so(3) ⊕ so(3) induces the following decomposition (for a choice of g on X 4 ) for the vector bundle 2 T * X → X,

(2.9) ∧ 2 T * X ∼ = ∧ + ⊕ ∧ -,
as a Whitney sum of two oriented 3-plane bundles. One can choose an oriented orthonormal frame for T * U X for an open set U ⊂ X. One therefore has, (2.10) ∧ ± = Span (e 1 ∧ e 2 ± e 3 ∧ e 4 ), (e 2 ∧ e 3 ± e 1 ∧ e 4 ), (e 3 ∧ e 1 ± e 2 ∧ e 4 ) .

We now use the unique Levi-Civita connection ∇ on ∧ 2 T * X to find a suitable decomposition of the curvature tensor under the action of O(4). The first step towards it is to note that ∇g = 0, this however means that ∇ is covariantly constant, that is ∇ maps sections of ∧ ± into ∧ ± ⊗ T * X; there is no mixed term mapping ∧ + into ∧ -⊗ T * X. The curvature of the Levi-Civita connection defines a section of ∧ 2 T * X ⊗ 2 T * X, correspondingly a decomposition of ∧ 2 T * X ⊗ 2 T * X into four matrix-blocks of size 3 × 3, more precisely, the Riemann curvature tensor defines, in general, a self-adjoint linear transformation R :

∧ 2 → ∧ 2 such that, (2.11) R(e i ∧ e j ) = 1 2 k,l
R ijkl e k ∧ e l , relative to the decomposition ∧ 2 = ∧ + ⊗ ∧ -, the operator R has the following form, (2.12)

R = A B B t C
where, B ∈ Hom(∧ -, ∧ + ) (is the traceless Ricci curvature) 0 Ric, and A ∈ End(∧ + ), that is A is symmetric about its diagonal, that is A t = A, similarly for C ∈ End(∧ -) we have C t = C. This representation of the curvature tensor R gives us a complete decomposition of it into irreducible components, namely

(2.13) R → (Tr A, B, A - 1 3 Tr A W + , C - 1 3 Tr C W - ) T rA = Tr C = 1 4
s where s is the scalar curvature. More elaborately, if the basis of ∧ ± in (2.10) is denoted by {x i ± } 3 i=1 , then the curvature tensor R has the following expansion

(2.14) R = W + ij x i + ⊗ x j + + W - ij x i -⊗ x j -+ B ij x i + ⊗ x j -+ B t ij x i -⊗ x j + - s 12 (x i + ⊗ x j -+ x i -⊗ x j + ).
If we denote the projection operator by, (2.15)

P ± := 1 2 (1 ± * ) : ∧ → ∧ ± , then, (2.16
)

W ± = P ± • Rm • P ± - s 12 Id ± 2.
2. The construction of the Elliptic complex. To get an elliptic complex corresponding to the data of the Gravitational monopole equations, we modify the data a little bit. So, we assume, A is a spin connection on the smooth 4-manifold X. ψ is a section of S + , so we relace the Dirac data ψ ∈ ker(d + d * ) to ψ ∈ ker D A .

In this way, the Dirac operator becomes dependent on A and we get a more general context. Let A be the induced connection on the line bundle L → X. We denote by F A the curvature corresponding to the connection A. We put no restriction on F A . All the restrictions are on the self-dual part of the Weyl tensor of X.

In the case of a line bundle, the gauge group G = M(X, U (1)) as a space of maps is well-defined as it is only dependent on the transition functions. The action of G on the pair (A, ψ) is given in the following way:

(2.17)

λ : (A, ψ) → (A -λ -1 dλ, λψ),
and on A by

(2.18) A -2iλ -1 dλ.
We verify the following:

(2.19) D A-λ -1 dλ (λψ) = λD A ψ + dλ • ψ -dλ • ψ.
The self-dual part F + A of the curvature tensor F A also remains invariant (2.20)

F + A-2λ -1 dλ = F + A -2d + (λ -1 dλ) = F + A . Since the metric g ij → g AA ′ BB ′ = ε AB ε A ′ B ′ , corresponding U (1)
-action on the metric is just multiplication by |λ| 2 = 1, therefore W + remains invariant, on the other hand (2.21) ⟨e i e j λψ, λψ⟩ = |λ| 2 ⟨e i e j ψ, ψ⟩ = ⟨e i e j ψ, ψ⟩, as |λ| = 1. Now we study the kernel of the linearized operator T g W + : T g M → Γ(Sym 2 0 ∧ 2 + ). The deformations we shall consider will be represented by the first cohomology of the complex

(2.22) Γ(T X) ⊕ Γ(R) τ * -→ Γ(Sym 2 0 ∧ 2 + ) T W + ---→ Γ(Sym 2 0 ∧ 2 + ), where R represents trivial R-bundle over X. Γ(Sym 2 T * X) = T g M = Hom(∧ 2 + , ∧ 2 -)⊕R. But Hom(∧ 2 + , ∧ 2 -) ∼ = ∧ 2 + ⊗ ∧ 2 -.
If we want to mod out the trivial line bundle R then we must work with M/C ∞ + , and the section space Γ(R) is replaced by the orbit space C ∞ + (g). The problem is not underdetermined as the Gravitational monopole equation (2.2) gives additional restriction on the scalar curvature, that is s = s g < 0 [cf.1], hence a linearization at g gives the following complex

(2.23) Γ(T X) δ * -→ Γ(Sym 2 T * X) T W+⊕Tg ------→ Γ(Sym 2 0 ∧ 2 + ) ⊕ Γ(R). Theorem 2.3. The complex (2.23) is elliptic with index equals to (2.24) 1 2 (29|σ(X)| -15χ(X)),
where χ(X) is the Euler characteristic of X and σ(X) are the signature of X.

By the index theorem for the twisted Dirac operator

(2.25) Index(D A ) = - X ch( √ L) A(X).
The Chern character is known to be

(2.26) ch( √ L) = 2(1 + c 1 (L) 2 + • • • ) the A-genus is (2.27) A(X) = 1 - 1 24 p 1 (X) + • • • , where p 1 (X) is the first-Pontrjagin class. The top degree form of ch( √ L) A(X) is 1 12 p 1 (X) + c 1 (L) 2 , but 1
3 p 1 (X) = σ, therefore we have the result (2.28)

Index(D A ) = c 1 ( √ L) 2 - σ 4 .
We find the following [cf.2])

Theorem 2.4. The moduli space of solutions of the Gravitational monopole equation has dimension (assuming the index σ(X) ≥ 0)

d := 1 2 (29σ(X) -15χ(X)) + c 1 ( √ L) 2 - σ 4 = (c 1 ( √ L)) 2 + 57σ(X) -30χ(X) 4
(2.29)

If √ L is globally defined, then, d Global √ L := (c 1 ( √ L)) 2 + 57σ(X) -30χ(X) 4 = 1 4 (c 1 (L)) 2 + 57σ(X) -30χ(X) 4 = 1 4 (c 1 (L)) 2 + 57σ(X) -30χ(X)
(2.30)

Almost Kähler Manifolds

Proposition 5. Let (M, g, ω, J) be an almost-Kähler structure. Then ω is an anti-self-dual harmonic 2-form ω with

(3.1) |ω| = √ 2.
Proof. As the almost complex structure J is orthogonal with respect to g, there exists an orthonormal basis of the form {e 1 , e 2 = J(e 1 ), e 3 , e 4 = J(e 3 )}. Then the corresponding anti-self-dual 2-form ω is

(3.2) ω := e 1 ∧ e 2 -e 3 ∧ e 4 , |ω| = √ 2.
ω is closed, therefore dω = 0, anti-self-dual, * ω = -ω, hence

(3.3) d * ω = * d * ω = * dω = 0 =⇒ ∆ω = (dd * + d * d)ω = 0. □ Proposition 6.
The scalar curvature of an almost-Kähler self-dual metric is non-positive. Moreover, the scalar curvature is identically zero if and only if the metric is Kähler.

Proof. For an anti-self-dual 2-form ω on an oriented, smooth, compact Riemannian 4-manifold Weitzenböck formula (2.3) with respect to the decomposition of the curvature tensor under the action of SO(4) as in (2.12), (2.13), (2.14) imply (as

W + (ω, •) = 0 for ω anti-self-dual) (3.4) ∆ω = ∇ * ∇ω -2W -(ω, •) + s 3 ω
One takes an inner product with (3.4) and gets

(3.5) ⟨∆ω, ω⟩ = ⟨∇ * ∇ω, ω⟩ -2W -(ω, ω) + s 3 |ω| 2 .
If g is an almost-Kähler, then the corresponding symplectic 2-form ω is harmonic and as shown in the Proposition ( 5

□

Similarly, in case of the Gravitational Monopoles (2.2), the Proposition (4) implies the following statement:

Proposition 7. The scalar curvature of a Gravitational Monopole is strictly negative.

The following theorem is due to Salamon and can be found in [cf.6].

Theorem 3.1. Let (M, g, J, ω) be an almost-Kähler structure. Then the zero-set of the Nijenhuis tensor N is equal to the zero-set of ∇ω.

Therefore when dω = 0, the zero-set of Nijenhuis tensor N is equal to the zero-set of ∇ω. The following theorem is proved in [cf.7]. Theorem 3.2 (Armstrong). Let (M, J) be a compact smooth 4-manifold with an almost complex structure J. If the Nijenhuis tensor N is nowhere vanishing, then 5χ + 6τ = 0. Theorem 3.3. A compact almost Kähler Gravitational monopole with a nowhere vanishing Nijenhuis tensor can never be Einstein.

Proof. If possible then 5χ + 6σ = 0 =⇒ σ = - 5 6 χ but Hitchin-Thorpe inequality says σ ≤ -2 3 χ = -4 6 χ, so σ can not be simultaneously less or equal than - 4 6 χ and equal to -5 6 χ hence χ = 0 =⇒ W + = s = 0. A contradiction to the definition of Gravitational monopole. □ Proposition 8. Let (M, g, ω, J) be an almost-Hermitian 4-manifold. Then we have

∧ + C = Cω ⊕ ∧ 2,0 ⊕ ∧ 2,0 , ∧ - C = ∧ 1,1 0 (3.8)
where ∧ 1,1 0 is the orthogonal complement of ω in ∧ 1,1 . Proof. For, smooth 4-manifold M admits an almost-complex structure J and a compatible metric g. Then the complex-valued 2-forms decompose as (3.9)

∧ + C T * M = ∧ 2,0 ⊕ ∧ 2,0 ⊕ ∧ 1,1
. Also, on an oriented, smooth 4-manifold, we have the following decomposition holds (3.10)

∧ + C T * M = ∧ 2,+ C ⊕ ∧ 2,- C .
We define an associated 2-form ω by (3.11) ω(v, w) := g(Jv, w).

When dω ̸ = 0 then (M, g, ω, J) is called an almost-Hermitian manifold. Hence the proposition.

□ It is proved in [cf.8]
Theorem 3.4. Let g be a Kähler metric on a complex surface. Then g is self-dual (anti-self-dual) if and only if its scalar curvature s = 0.

The Ricci form ϱ is defined by

(3.12) ϱ(X, Y ) := Ric(JX, Y )
ϱ is a closed real (1, 1)-form. Let K M be the canonical line bundle of M and K -1 M be the dual of K M . The first Chern class is given by

(3.13) c 1 (M ) := c 1 (K -1 M ) = ρ 2π
The volume form dµ is determined by ω,

namely dµ = 1 2 ω ∧ ω. Let φ ∈ ∧ 1,1 then (8) implies (3.14) φ = 1 2 ⟨φ, ω⟩ω + φ 0 , φ 0 ∈ ∧ 1,1 0 .
Therefore as the Ricci form ρ is a (1, 1)-form, we can write it as

(3.15) ρ = 1 4 sω + ρ 0
Therefore using (3.15) and dµ = 1 2 ω ∧ ω

(3.16) 2π M c 1 ∧ ω = M ρ ∧ ω = 1 4 M sω ∧ ω = 1 2 M sdµ.
Since, for a Gravitational Monopole s < 0 by Propositions ( 7) and ( 4), we get Proposition 9. If g is a strictly almost-Kähler metric that is also a Gravitational monopole, then

(3.17) c 1 • [ω] < 0.
One can compare the above result i.e., Proposition [START_REF] Pontecorvo | On twistor spaces of anti-self-dual hermitian surfaces[END_REF] with the following proposition Proposition 10. Let g be an almost-Kähler self-dual metric on a compact 4-manifold. If g is a strictly almost-Kähler self-dual metric, then

(3.18) c 1 • [ω] < 0.
3.1. Twistor correspondence for the Gravitational Monopoles. Let t : Z → M denote the twistor fibration and suppose M is hermitian and self-dual. One concludes from the definition of the almost complex structure of Z : that t is not a holomorphic map and the complex structure J of M defines a cross section J : M → Z. Let us denote the image of J by Σ. One can use the integrability of J, and can show Σ is a complex hypersurface of Z biholomorphic to M . Similarly -J : M → Z defines hypersurface Σ. The "real structure" σ of Z switches the two hypersurfaces identifying one with the other in an antiholomorphic fashion [cf.9]. If D denotes the divisor Σ + Σ in Z , we can consider the corresponding holomorphic line bundle

L = [D] ; since σ(D) = σ(Σ + Σ) = Σ + Σ = D, so L is called a "real" bundle.
The following theorem is due to Penrose Theorem 3.5.

(3.19) L ∼ = K -1/2 Z
, if and only if the metric h is conformai to a Kähler metric.

Theorem 3.6. If M is not Ricci flat then,

(3.20) H 0 (Z, Θ) ∼ = H 0 (M, Θ).
Proof. See [START_REF] Pontecorvo | On twistor spaces of anti-self-dual hermitian surfaces[END_REF]. □ Proposition 11. H 0 (Z, Θ) is the complexification of the Lie algebra of parallel vector fields, and

(3.21) h 0 (M, Θ) = h 0 (M, Ω 1 ) = 1 2 b 1 (M ).
Proof. See [START_REF] Pontecorvo | On twistor spaces of anti-self-dual hermitian surfaces[END_REF]. □ Theorem 3.7. Let (M, ω) be a compact almost Kähler surface with a positive definite signature σ > 0 and c 1 ̸ = 0, c 1 (L) 2 > 0, we also assume M is a Gravitational Monopole and let Z be its twistor space, we also assume

H 2 (Z, Θ ⊗ I D ) = 0 then H 0 (M, Θ M ) = H 0 (Z, Θ Z ) = H 2 (Z, Θ Z,D ) = H 2 (Z, Θ Z ) = 0.
Proof. The condition H 0 (M, Θ M ) = H 0 (Z, Θ Z ) = 0 follows from the non-existent of Killing vector fields, showed in [cf.Equation 3.17 of 3].

To prove the second part we use [cf.8]. Consider the following exact sequence of sheaves

0 → Θ Z,D → Θ Z → N D → 0, (3.22) and, (3.23
) 0 → Θ Z ⊗ I D → Θ Z,D → Θ D → 0.
From the short exact sequence (3.23), we get the long exact sequence Remark 3.9. One can compare with [cf.9], it was proved there that when M is a scalar-flat Kähler surface but not Ricci-flat

(3.24) • • • → H 2 (Z, Θ Z ⊗ I D ) → H 2 (Z, Θ Z,D ) → H 2 (D, Θ D ) → • • • , since (3.
(3.29) H 0 (M, Θ M ) = H 0 (Z, Θ Z ) = 0
holds true in contrast to our strictly negative scalar curvature treatment of Equation 3.17 in [cf.3] and corresponding vanishing of Killing fields.

The following theorem can be found in [cf.10], [cf.9],

Theorem 3.10. Let M be an oriented, compact, smooth 4-manifold with an SD metric g and assume the first Betti number b 1 (M ) is even. Suppose there is a complex structure J such that g(v, w) = g(Jv, Jw).

Then the conformal class of g contains a unique scalar-flat Kähler metric.

One therefore concludes that deforming scalar-flat Kähler metrics is equivalent to deforming self-dual hermitian conformal structures, this however means the deformation of the pair (Z, D) preserving the real structure of the twistor space. Therefore, when obstruction vanishes, the moduli space of the self-dual hermitian conformal structures is the real slice of H 1 (Z, Θ Z,D ) [cf.8]. In [cf.8] they proved the following theorem Theorem 3.11. When obstruction vanishes, the deformation of scalar-flat Kähler metrics corresponds to the real slice of H 1 (Z, Θ Z,D ), where Θ Z,D is the sheaf of holomorphic vector fields on Z which are tangent to D; and its obstruction lies in H 2 (Z, Θ Z,D ).

The deformation of scalar-flat Kähler metrics with a fixed complex structure corresponds to the sheaf Θ Z ⊗ I D with the ideal sheaf of D is denoted by I D . Theorem 3.12. The obstruction to the existence of the self-dual Gravitational Monopole metric lies in H 2 (Z, Θ Z,D ). Therefore the vanishing condition H 2 (Z, Θ Z,D ) = 0 and s < 0 will guarantee the existence of the Gravitational Monopole metric. Theorem 3.13. Let M be a compact, smooth four-dimensional manifold which admits a scalar-flat Kähler metric g and suppose c 2 1 > 0. Assume obstruction for deforming g as a scalar-flat Kähler metric, (and similar statement for a self-dual metric) vanishes. Let g ′ be a self-dual metric which is obtained by deforming g. Then there is a unique almost-Kähler self-dual metric in the conformal class of [g ′ ] that satisfies the Gravitational Monopole equation. Theorem 3.14. Let (M, g, J, ω) be a complex surface with a scalar-flat Kähler metric. Suppose c 1 ̸ = 0 and assume (M, J) admits no holomorphic vector fields.Then on these surfaces, we have -9χ ≥ 13σ and if the inequality is strict, there exist deformations which are strictly almost-Kähler. In particular for L = K -1 the following holds

2 +s 3 |ω| 2 is 3 |ω| 2 .( 3 . 6 )

 22236 |∇ω| 2 -2W -(ω, ω) + .e., 0 = |∇ω| 2 -2W -(ω, ω) + Since g is self-dual, we get by using the fact W -(ω, ω) = 0 the following relationship (3.7) 0 = |∇ω| 2 + 2s 3 .

  25) H 2 (Z, Θ Z ⊗ I D ) = 0 we have (3.26) H 2 (Z, Θ Z,D ) = 0. Now one uses the short exact sequence (3.22) to get the long exact sequence(3.27) • • • → H 2 (Z, Θ Z,D ) → H 2 (Z, Θ Z ) → H 2 (D, N D ) → • • •Now equation (3.26) says H 2 (Z, Θ Z,D ) = 0, therefore from the long exact sequence we get(3.28) H 2 (Z, Θ Z ) = 0.This proves the theorem. □ Remark 3.8. We can compare with [cf.8] to see the non-existence of holomorphic vector fields on Z also imply H 2 (Z, Θ Z ⊗ I D ) = 0.

Proof.

  In the following we use the same technique as employed in [cf.8],[cf.11]. Since,(3.30) 0 → Θ Z ⊗ I D → Θ Z,D → Θ D → 0 one gets (3.31) χ(Z, Θ Z,D ) = χ(Z, Θ Z ⊗ I D ) + 2χ(M, Θ M ).If the obstruction vanishes, we can think of -χ as the dimension of the moduli space [cf.8], and it is shown in[cf.8] that (3.32) χ(Z, Θ Z ⊗ I D ) = σThe dimension the moduli space of scalar-flat Kähler metrics is given by-χ(Z, Θ Z,D ) = -σ -2χ(M, Θ M ) = -σ -(3χ + 7σ) = -3χ -8σ. (3.33) The Atiyah-Singer index theorem gives [cf.11] χ(M, Θ M ) = of the moduli space of scalar We have N D = K -1 D [cf.9] and short exact sequence (3.22) (3.35) 0 → Θ Z,D → Θ Z → N D → 0 to give (3.36) χ(Z, Θ Z ) = χ(Z, Θ Z,D ) + 2χ(M, K -1 M ). But by the Riemann-Roch theorem, we get for any holomorphic line bundle L on M , (3.37) χ(L) = χ(O M ) + M c 1 (L)(c 1 (L) + c 1 (K -1 )) 2 .

( 3 . 1 ( 3

 313 38) χ(K -1 ) = χ(O M ) + c 1 (K -1 ) • c 1 (K -1 ) = χ(O M ) + c 2 ) = 2χ + 3σ, c 2 (M ) = χ we get, , Θ Z ) = χ(Z, Θ Z,D ) + 2χ(M,of the moduli space M of self-dual metrics are given by (using Theorem (3.13), we conclude if-1 2 (15χ + 29σ) > -(3χ + 8σ)i.e., -9χ > 13σ(3.45) holds true then there exists strictly almost Kähler self-dual metrics.

□

Now, if these self-dual metrics are also Gravitational monopoles, then as σ < 0, the dimension of the moduli space of solution of the Gravitational Monopole equation becomes

(3.47) Theorem 3.15. Let (M, g, J, ω) be a compact almost Kähler surface with a negative definite signature σ < 0 and c 1 ̸ = 0, c 1 (L) 2 > 0, and (M, J) admits no holomorphic vector fields. Then on these surfaces, we have

and if the inequality is strict, there exist deformations which are strictly almost-Kähler.

Proof. An application of the Theorem (3.13) applies whenever the condition

holds true. Therefore by Theorem (3.12), we prove the existence of the Gravitational Monopole □


A healthcare software company is developing a patient triage tool to assess the level of risk of stroke based on a machine learning algorithm. The AI system was trained on data sold by a provider and included only stroke cases in patients >70 years old, mostly men.

A 35-year-old woman lost consciousness while jogging, was brought to hospital by a friend and the hospital prioritised other patients who arrived after him. The medical diagnosis then identifies a stroke for which earlier management would have limited the long-term consequences. The hospital used the AI system to prioritise patients in emergency.

A healthcare software company is developing a patient triage tool to assess the level of risk of stroke based on a machine learning algorithm. The AI system was trained on data sold by a provider and included only stroke cases in patients >70 years old, mostly men.

A 35-year-old woman lost consciousness while jogging, was brought to hospital by a friend and the hospital prioritised other patients who arrived after him. The medical diagnosis then identifies a stroke for which earlier management would have limited the long-term consequences. The hospital used the AI system to prioritise patients in emergency.

-What are the ethical issues involved in this situation?

-What questions for the law?

-What are the specificities of using an AI system?

→ in the context of development of AI as an epidemiological studies! 4/54 AI Systems --UNESCO definition AI systems are information-processing technologies that integrate models and algorithms that produce a capacity to learn and to perform cognitive tasks leading to outcomes such as prediction and decision-making in material and virtual environments. AI systems are designed to operate with varying degrees of autonomy by means of knowledge modelling and representation and by exploiting data and calculating correlations. AI systems may include several methods, such as but not limited to:

machine learning, including deep learning and reinforcement learning;

machine reasoning, including planning, scheduling, knowledge representation and reasoning, search, and optimization. AI systems with learning capabilities concentrate ethical and legal issues -Learning AI system = Decision-making system

• Representable as any computer system (Turing machine)

-The design of such a system changes

• It is no longer a programmer who describes how to make the decision -Techno-optimism is the only good solution to our current problems (...) We can build AGI. We can colonize space. We can get fusion to work and solar to mass scale. We can cure all human disease. We can build new realities.

•

9/54

Risks and fears of AI in health

There are many situations where one can imagine "damaging facts".

for patients

• misdiagnosis, diagnostic delays, intervention error

• limitation of possible remedies for caregivers

• loss of jobs

• reduced freedom of prescription and practice

• loss of technical skills for citizens -Axis 3 -Construction and integration of predictive tools useful for clinical practice involving the conduct of supervised analyses based in particular on machine learning approaches Study example: OPTISOINS, understanding and improving the care pathways of patients hospitalised for lung resection. -80-90's: Return of connectionism and the subsymbolic movement

• Multi-layer perceptron (1985)

• Distributed AI (Genetic Algorithms, Artificial Life, ...)

-90-00's: revival of machine learning -Several times predicted ...

-OpenAI: Planning for AGI and beyond

• Our mission is to ensure that artificial general intelligence-AI systems that are generally smarter than humans-benefits all of humanity.Sam Altman: Future of AI will "create new knowledge "AI does not exist" L. Julia The contribution of epistemology to thinking about AI Is AI a science? A technique?

-AI system as a tool

• Performs a function

• Its design is put at a distance (unknown)

• Do I need to understand the technique to use it?

-AI as a technique for science (AI is seen in its full cycle?)

• In health, more than anywhere else: the discovery of new techniques is governed by the scientific approach

• AI: empowerment tool for epidemiologists (analyse more data, more accurately)

• Essentially this tool in many other sciences

• Application of ethical/legal principles of research?

Action 21/54

An AI system for patient triage The contribution of epistemology to AI design What would be new?

1) AI: data-driven approach (Elodie Giroux)

-Pure" discovery science from data (knowledge building without theory)

• Data neutrality

• Broadening the concept of data

• The truth would be in the mass of data (either hidden or debiased)

-Relativisation of causality in favour of correlation

• Sufficiency of the empirical result

• No need to research causes -Overcoming traditional logics

• Idea of better exploitation of digital data by digital

• Questioning the explanation of decision logic

23/54

The contribution of epistemology to AI design What would be new?

1) AI: data-driven approach (Elodie Giroux)

Pure" discovery science from data (knowledge building without theory)

• Data neutrality

• Broadening the concept of data

• The truth would be in the mass of data (either hidden or debiased)

• There is no such thing as "pure data" (S. Leonelli): assumptions implicit in the data (+PIML)

Relativisation of causality in favour of correlation

• Sufficiency of the empirical result

• No need to research causes

• Explicability of models, identification of causes required in health research

Going beyond traditional logics

• Idea of better exploitation of digital data by digital

• Questioning the explanation of decision logic 24/54

The contribution of epistemology to AI design What would be new?

2) AI: towards autonomous systems • On ease of access to use These changes, particularly in their application in health, require the promotion of ethics and regulation -Seven key requirements for Trustworthy AI (Ethics Guidelines for Trustworthy AI, 2019)

• human agency and oversight,

• technical robustness and safety,

• privacy and data governance,

• transparency,

• diversity, non-discrimination and fairness,

• environmental and societal well-being and • accountability -In fact ... there are many documents that set out ethical principles for AI design, from :

• scientific societies • private companies • NGOs

Limits

For the question of the effective application of the principles in the framework of companies: move from principles to implementation! -Goes through regulation (see part three)

-Information (fair, understandable and undistorted) -Device/system certification Limits of principalism -Limits of ethics that are reduced to compliance with guidelines -Who defines the principles to be respected?

Excellence and trust in artificial intelligence

Trustworthy AI can bring many benefits, such as better healthcare, safer and cleaner transport, more efficient manufacturing, and cheaper and more sustainable energy.

Artificial intelligence (AI) can help find solutions to many of society's problems. This can only be achieved if the technology is of high quality, and developed and used in ways that earns peoples' trust. Therefore, an EU strategic framework based on EU values will give citizens the confidence to accept AI-based solutions, while encouraging businesses to develop and deploy them. The framework of accountability in law A prejudice is analysed as a consequence of a damage that the law will apprehend. It is a subjective notion and is likely to give rise to a right to compensation. Accountability must then be imputed.

A Damage is an impairment of a situation or a person (e.g. privacy issue, personal injury). Damage is an objective concept which is subject to a finding. Damage is the potential source of the loss.

E.g.: personal injury (paralysis)

The « fait générateur »: For a person to be liable, the event that gives rise to liability must be the direct cause of the damage. It may be at fault or not (no-fault liability). For our example ...

A healthcare software company is developing a patient triage tool to assess the level of risk of stroke based on a machine learning algorithm. The AI system was trained on data sold by a provider and included only stroke cases in patients >70 years old, mostly men.

A 35-year-old woman lost consciousness while jogging, was brought to hospital by a friend and the hospital prioritised other patients who arrived after him. The medical diagnosis then identifies a stroke for which earlier management would have limited the long-term consequences. The hospital used the AI system to prioritise patients in emergency room.

-Is there a fault? -Who should take accountability? The AI tool? The regulating doctor? -Is the AI tool accountable? Is it the AI as a person? Its designer(s)? ... and how to identify the responsibilities of each? -How can the patient make a claim against the AI tool if he/she does not know that it was used?

39/54

Our research questions Questions Does the current legal framework allow these questions to be answered?

What are the new legal issues raised by AI?

What are the current avenues of reflection?

What AI research challenges do these questions raise? 40/54

Evolution and innovation in law

What exists in law?

-product liability

• This covers certain situations related to a device defect with an AI

• Nevertheless, the notion of defect defined in the context of products may need to be revised (does an AI learned on ill-fitting examples have a defect?).

-civil liability

• Allows to deal with the case of malice (e.g. modification of an AI)

-the bioethics laws (+ health code) revised in 2021

-the GDPR An AI is a system that will evolve -A challenge in relation to the function of medical devices based on certification (at a time t)

An AI is a complex system -Difficulty in proving fault, identifying the cause and tracing responsibility

Giving an AI a legal personality?

=> The problems are not in the identification of new losses or new events, but rather in the modalities defining the distribution of accountabilities in the case of damage and in the qualification as a harmful event (direct link).

42/54

Questions addressed in the DRIAS project -I. Le professionnel de santé qui décide d'utiliser, pour un acte de prévention, de diagnostic ou de soin, un dispositif médical comportant un traitement de données algorithmique dont l'apprentissage a été réalisé à partir de données massives s'assure que la personne concernée en a été informée et qu'elle est, le cas échéant, avertie de l'interprétation qui en résulte.

• Dans notre exemple : utilisation d'une IA dans le cadre du triage (acte diagnostic ?) ?

• Pas de mention d'opposition, juste une information : « La personne concernée a le droit de ne pas faire l'objet d'une décision fondée exclusivement sur un traitement automatisé, y compris le profilage, produisant des effets juridiques la concernant ou l'affectant de manière significative de façon similaire » (GDPR, Art 22.1)

-L'interprétation actuelle est qu'une intervention humaine « mineure » n'est pas suffisante.

• Capacité d'explication par le médecin ?

-III. Les concepteurs d'un traitement algorithmique mentionné au I s'assurent de l'explicabilité de son fonctionnement pour les utilisateurs. The regulatory framework is under construction: between legislation and technique (CNIL/LNE/AFFNOR)

Three specific examples :

• Existing contradictory injunction in these studies due to the RGPD • What anonymisation/pseudonymisation methods can provide practical solutions to these health data issues adaptation of the rules on extra-contractual civil liability to the field of artificial intelligence -Developments still to be expected in 2023

Definition of an AI system in November 2022 "'Artificial intelligence system' (AI system) means a system that is designed to operate with elements of autonomy and that, based on machine and/or human-provided data and inputs, infers how to achieve a given set of objectives using machine learning and/or logic-and knowledge based approaches, and produces system-generated outputs such as content (generative AI systems), predictions, recommendations or decisions, influencing the environments with which the AI system interacts" • AI systems intended to be used as safety components of products with third party ex-ante conformity assessment; or Other autonomous AI systems raising fundamental rights issues (list in Annex III: 8 AI systems)

• 2020 text "Health is an area where a high risk ranking can be expected".

-Other systems

• Fault-based liability regime : burden of proof and of causality to the claimant

• Less user protective than the proposal in 2020 A private insurance system

Raises questions about the principles of assessing contributions (risk assessment)

48/54

The issue of interpretability of AIs for accountability Explicability/interpretability of AI tools Many AI researches programs aim at bringing explicability to

• of the models or

• decisions made on the basis of these models

The distinction between classes of methods is also reflected in the legal issues raised by AI systems 49/54

The issue of interpretability of AIs for accountability Explicability of models in case of injury : A crucial issue for lawyers in the redress of damages related to the use of AI -Question of the causal link: need for elements that relativise or remove the causality between the event and the damage -Question of imputability: How to apportion responsibility in case of injury?

=> the explicability of the models must allow the reasons for a decision to be explained posthoc

H. Muscat and C. Paillard 50/54
The issue of interpretability of AIs for accountability

Explicability of decisions to users

A strong expectation from health service users,

• Principle 7 France Asso Santé: "Tout système algorithmique utilisé dans le cadre d'une aide à la décision médicale est explicable à un usager sous une forme intelligible".

RGPD obligations on automated decisions: tension on the role of explicability between dis-accountability and informed information

Conditions of (moral) accountability: control and epistemics [Habli, 2020].

• Does the use of AI (and the inexplicability of some decisions) mitigate the accountability of decision-makers?

• Does this shift the accountability to the prescriber of the tool?

• What training obligations can be imposed on AI users? What are the consequences for the attribution of their liabilities? An example ...

A healthcare software company is developing a patient triage tool to assess the level of risk of stroke based on a machine learning algorithm. The AI system was trained on data sold by a provider and included only stroke cases in patients >70 years old, mostly men.

A 35-year-old woman lost consciousness while jogging, was brought to hospital by a friend and the hospital prioritised other patients who arrived after him. The medical diagnosis then identifies a stroke for which earlier management would have limited the long-term consequences. The hospital used the AI system to prioritise patients in emergency.

-What are the ethical issues involved in this situation?

-What questions for the law?

-What are the specificities of using an AI system? More and more open source implementations are available to the machine learning world:

-Access to data via perturbed query results (Laplace, Gaussian mechanisms, etc.)

-Provision of summary data

The guarantee of non-re-identification reduced to limited situations (easily overcome in practice):

-Which model for certification?

-What is the liability in case of re-identification? "Uncertainty about risks should not lead to prohibitively high insurance premiums, which would act as a brake on research and innovation".

A private insurance system

• "the Commission should work closely with the insurance industry to consider how to use data and innovative models

-It is a question of facilitating access to data by insurers: what are the risks with regard to personal data? How to ensure the separation between personal data (usable by the insurer) and the need to quantify risks?

• Should (wish) to be complemented by a special compensation fund (for specific needs)

-What is the relationship with ONIAM (Office National d'Indemnisation des Accidents Médicaux) in the case of health damage?

In the Artificial Intelligence Act, 2022

-No compulsory insurance mentioned in AI legislation -The appropriateness of this issue is subject to a future report (Directive on liability in the field of AI)

-

  The program is built indirectly from examples • Abandonment of full control over the decision-making process 6/54 Quelques acteurs (personne morale ou physique) autour des systèmes d'IA en Santé (cycle de vie) Inserm, https://www.inserm.fr/dossier/intelligence-artificielle-et-sante/ AI intervention in health care systems ... -prescription (e.g. medication) -diagnosis (radiological image analysis more efficient than humans) -surgery assistance (Da Vinci robot) -the organisation of care (prediction of hospitalisation times, organisation of cares) -the organisation of the healthcare system (epidemiology based on massive health data) The use of AI technologies could improve decision making e.g. in radiology/dermatology today Improving efficiency -Improvement of the care system -Focusing caregivers' time on useful tasks -Speed/timeliness of diagnosis and intervention Care personalisation -Tailoring care to patients -Prevention through monitoring More objective decisions -Rationalisation of the decision At the extreme, Sam Altman (Twitter, 4th Feb 2022)

-

  of data from the APHP data warehouse -Secondary data (idem SNDS): Data collected in the context of care for medical or administrative reasons Heterogeneous but structured data -Available data: no delay in data collection -Data representative of the information available in the care setting Exploitation of these data by machine learning methods -Identification of care trajectories (phenotyping) -Prognostic algorithms -Deep learning: non-selective exploitation of data -Framework with similar objective as in our example Action 12/54 Some ethical and legal questions? Development of AI systems in health -Strong expectations to improve our care systems and care -Potential risks: personal and social How to address the many questions that arise about the opportunities and limitations of developing AI systems in health Need for an inter-disciplinary approach between -Philosophy/epistemology: thinking about the evolution of technology and science induced by AI -Law/Ethics: defining a collective and responsible framework for the development of AI systems -Computer science: identification of real risks linked to the use of AI (today and tomorrow) -Health: a source of practical cases and questions 13/54

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  10-20's: Era of deep learning (AI = ML) • Increasing the size of the architectures • Wide dissemination of technologies -20's: Generative models • ChatGPT / MidJourney • Content (text, image, sound) generation 15/54 Emergence and transformation of AI A little history -Pascal ... Babbage ... -50's : Early work on cybernetics • Neural networks McCulloch & Pitts, Hebb) • Macy conferences -70-80's: expert systems • Knowledge representation and modelling of reasoning • Diagnostic support (MYCIN) -80-90's: Return of connectionism and the subsymbolic movement • Multi-layer perceptron (1985) • Distributed AI (Genetic Algorithms, Artificial Life, ...) -90-00's: revival of machine learning • Preferences for Big Data approaches • 1997 Deep blue vs Kasparov -00-10's: emergence of Big Data • Automates the identification of features useful to a model • Spectacular advances in image classification -10-20's: Era of deep learning (AI = ML) • Increasing the size of the architectures • Wide dissemination of technologies -20's: Generative models • ChatGPT / MidJourney • Content (text, image, sound) generation multilevel hype cycles 16/54 Emergence and transformation of AI Rising of new machine learning thanks to the convervence of Increasingly abstract automation of cognitive tasks -An unstoppable and limitless progression towards Artificial General Intelligence -Transition to a "strong" AI that would • Autonomous • Able to deal with many problems • Even with a 'conscience'.

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  In the sense of AGI (Artificial General Inteligence) -Rethinking human/machine comparisons in terms of energy https://openai.com/blog/planning-for-agi-and-beyond 18/54 Role of the term 'Artificial Intelligence' in the myth The term helps to convey an imaginary -Today, no AI system, no matter how complex our computational capabilities, can be called intelligent -Wide visibility in media -Dump AI ? A marketing buzzword -Term carrying the promise of innovation -AudikaFull "all the sounds that are important to you are played back thanks to the power of artificial intelligence Should a more neutral term be preferred for these technologies? Experiment by Pascal Hitzler with chatGPT 19/54 The contribution of epistemology to thinking about AI Is AI a scientific 'revolution'? -Specific concepts in epistemology (T. Kuhn) -"paradigm shift" = change of world view / relationship to the world -The new paradigm has to solve more puzzles than the old one • What was the old paradigm 20/54

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  (...) The hospital used the AI system to prioritise patients in emergency.. (...) -What characterises an AI system? -Three 'systems' of stroke stratification • NIHSS score (Brott et al, 1989) -reference score -Evaluation of 14 criteria (Level of consciousness, Simple order, Facial paralysis, ...) -Severity classification » 1-4 points : Minor stroke » 5-15 points : Moderate stroke » 15-20 points : Severe stroke » > 20 points : Severe stroke • Quick-NIHSS (Geiser, 2019) -5 binary variables + global score from 0 to 5 Constructed by ANOVA on a mono-centric database, 3995 patients • Machine learning model (Yu et al., 2019) Learning a C4.5 model from 227 patients [1] Brott T, Adams HP; Olinger CP, Marler JR, Barsan WG, Biller J, Spilker J, Holleran R, Eberne R, Hertzberg V, Rorick M, Moomaw CJ, Walker M. Measurements of acute cerebral infarction: a clinical examination scale. Stroke 1989;20:864-70 [2] Geiser A. Elaboration of the Quick-NIHSS: A simple and rapid score for prehospital triage of suspected stroke, PhD Thesis, 2019 [3] Yu J. et al, Semantic Analysis of NIH Stroke Scale using Machine Learning Techniques, International Conference on Platform Technology and Service (PlatCon), 2019 22/54

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  Continuous learning: giving up some control by the designer • Raises many new questions -Issue of the emergence of functional systems without control • Treated by Cybernetics (1950s) -Cybernetics: science of control (Norbert Wiener) -Second order cybernetics: theory of self-organising systems (von Foerster) • Autopoietic systems (H. Maturana) Connected watches that measure physiological constants and make personalised recommendations → online learning of user habits 25/54 Take home message Does AI introduce revolutions in technology? -Little change in the nature of the technology • Developments to be expected as a result of technological change • The usual battle between technophobes and technolaters • Epistemological developments that need attention -Data-driven epistemology -Cybernetics epistemology -But changes in magnitude • On the number of people affected by a single technology • On the speed of their dissemination

•-

  e.g. AI design principle in China: in line with the Party's ambitions Ethical design of AI systems vs. Design of ethical AI systems e.g. Bloom: his "ethical" conception does not guarantee his behaviour 32/54 On going work Pro-ethical design of an artificial intelligence systems (ED-AIM, E. Pardoux) -Reflecting on and proposing a framework for the ethical design of AI systems in health -How to promote and encourage the implementation of good practices • Throughout the AI life cycle • Throughout the AI design process (from training to validation) -Example of ongoing work on the design of epidemiological studies • Towards the formalisation of epidemiological studies leading to the design of AI systems (models) from data • According to the FAIR principles -FAIR: Findable, Acceccible, Interoperable and Reusable -Transposed to the notion of study • Desired inputs -Objectivise the studies to facilitate the identification of potential problematic steps -Accompanying (and tracing) the design of an AI system 33What (re)evolution does AI bring? -Is there a real need to treat AI differently from any technical object? AI development: which ethical principles? -Ethical principles of AI (in health) -A critical look at principalism and epidemiology AI development: what regulatory framework (in Europe)? -Motivation of the European regulatory framework -Use of the data -Responsibility framework Work carried out in the framework of the CNRS/DRIAS project http://drias.irisa.fr/ P. Saurel, C. Paillard, H. Muscat, D. Bourcier, N. Le Meur, E. Oger, T. Alalrd 34/54 General context Development of AI systems in health -Strong expectations to improve our care systems and care -Potential risks: personal and social What guides the (European) regulator: excellence and trust strategy -Providing protection for people -Promote technological innovation -Ensuring free competition

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  Strong expectations to improve our care systems and care -Potential risks: personal and social A central question: That of responsibility in a context of medical use and involving an AI? Need for an inter-disciplinary approach between -Law: the concept of liability is governed in part by law -Computer science: identification of risks related to the use of AI (today and tomorrow) Strong expectations to improve our care systems and care -Potential risks: personal and social A central question: That of responsibility in a context of medical use and involving an AI? Need for an inter-disciplinary approach between -Law: AI raises very topical questions (law under construction, eg by the EU) Computer science: liability drives current AI research + legal design -Health: how AI methods should look in future medical practice 37/54

E

  .g. misuse of an AI technology (faulty), overloading of emergencies requiring prioritisation (therapeutic hazard, non-faulty, Kouchner Law 2002) 38/54

1

  ) liability for the use of personal data framework of the new bioethics laws (2021) regulatory difficulties anonymisation and pseudonymisation of data 2) liability and automated decision support 43/54 Regulatory framework for the use of AI in healthcare Modification of the Health code (Art. L. 4001-3)

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  Accountability: according to the proportionality of the risks assumed by the actors -Raises the issue of traceability and interpretability of decisions Separation of AI systems into two classes -High risk systems • AIs identified and listed by a commission • Strict liability regime

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  Supplement the RGPD (for data risk management) -Legal framework in Europe • Mainly for high-risk AIS • AIS for Health are not considered as high risk systems • Does not constraint so much the AIS designers -To be further investigated … Close connections between computer science and -Personal data risk management Explicability What (re)evolution does AI bring? -Is there a real need to treat AI differently from any technical object? -How can we distinguish between myths, fantasies and reality? AI development: which ethical principles? Ethical principles of AI (in health) -A critical look at principalism and epidemiology -A framwork for pro-ethical design of artificial intelligence systems AI development: what regulatory framework (in Europe)? -Motivation of the European regulatory framework -Use of the data -Responsibility framework Action 53/54

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  Access to the learned model (on nonanonymised data) with privacy guarantees Do models learned from personal data contain personal information?(CNIL, F. Vallet)Startup XXX: request for advice from a company marketing a tool implementing automatic learning methods for the coding of medical acts (PMSI coding)-OK to deploy to an X centre if the learning and model remain on the centre's premises.-Can the startup transfer the model learned in centre X to centre Y and adapt it there? -If possible, what measures and practices should be recommended to minimise the risks? Development of learning methods (on non-anonymised data) with guarantees : DP-SGD, DP-GAN, ... DP-* 60/54 Type de mécanisme -Responsabilité objective pour défaut Source de responsabilité -Art. 6 Caractère défectueux du produit Charge de la preuve -Art. 9.1 et 9.2 Charge à la victime de produire tous les éléments au soutien de la plausibilité de l'action et de prouver le défaut du produit et le dommage subi (Art. 4.1.a Manquement à un devoir de vigilance Charge de la preuve -Art. 4.1, 4.2 et 4.3 Charge à la victime de de prouver la faute et du lien de causalité avec le dommage (lien de causalité présumé sous conditions, notamment pour les SIA à haut risque, et réfragable) https://www.actu-juridique.fr/ntic-medias-presse/les-nuances-des-nouveaux-regimes-deresponsabilite-des-ia-en-europe/ 61/54 Defective Products / certification: what risk management for AIs? Look at the possible damaging facts -Injury caused by a surgical robot -Disease caused by AI-assisted diagnostic error -Disclosure of sensitive data -Failure to inform the patient • Information on the use of AI -Respecting the wish not to use an AI -Equity of AI based on machine learning -Unequal access to technology -Decision bias on a model learned from unbalanced data Complex burden of the proof (EP Directive adapting liability of defective products) Burden of the proof belongs to the claimant in general -Defectiveness and causal link presumption in case of « technical or scientific complexity » Very different time frames for decisions : no distinction for AI -How important is it to qualify responsibilities? 62/54 Product defects / certification: what regime for AIs? French policy guided by the Innovation Council -Grand Challenge "Securing, making reliable and certifying systems that use Artificial Intelligence -White Paper (Oct 2022): https://www.confiance.ai/ AI certification (G. Avrin, ex-LNE) -AI compliance assessment is a major issue in the practical implementation of a liability regime -A company developing a service seeks to assess the legal risks of its products -How to 'evaluate' AI: many questions still open Certification issues for evolving systems (by learning/customisation) -FDA experience with evolving DMs • Classification of 4 families of modifications + methodology for evaluating Software as a Medical Device 63

  The challenge of ensuring the development of efficient/accurate AI methods while preserving personal rightstension between privacy and the need for doctors to be state of the art User data protection strategies of the HAS (P.-A. Jachiet)

	Questions for AI research
	44/54

• Explicabilité du fonctionnement vs explication d'une décision ! -Limited dissemination / statistical aggregation / remote access / pseudonymisation -Need to adapt the strategy to the use of data

  Practical implications for this multicentre study -To have an individual means of informing the patient about the study -Allocation of a specific number from the NIR in each university hospital• Risk of patients known to several institutions => deduplication to be carried out• Could be avoided, but CNIL recommends an "RGPD compatible" data model for the warehouse making it more difficult to access personally identifiable information. But the controller must be able to escalate to the patient concerned to exercise his or her right of withdrawal « si les finalités pour lesquelles des données à caractère personnel sont traitées n'imposent pas ou n'imposent plus au responsable du traitement d'identifier une personne concernée, celui-ci n'est pas tenu de conserver, d'obtenir ou de traiter des informations supplémentaires pour identifier la personne concernée à la seule fin de respecter le présent règlement.

	Principled artificial intelligence Example 1: Regulatory difficulties (REPERES) An example of a regulatory difficulty (REPERES) Example 2: Access to data with privacy guarantees
	54/54 55/54 Epidemiological study conducted on several hospitals (multi-centric) requiring access Thank you Privacy: Privacy Control over Use of Data Transparency and Explainability: Explainability Transparency Open Source Data and Algorithms to health data What anonymisation/pseudonymisation methods can provide practical solutions to these health data issues (S. Gambs) Consent Privacy by Design Recommendation for Data Protection Laws Ability to Restrict Processing Right to Rectification Right to Erasure Accountability: Accountability Recommendation for New Regulations Impact Assessment Evaluation and Auditing Requirement Verifiability and Replicability Liability and Legal Responsibility Ability to Appeal Environmental Responsibility Creation of a Monitoring Body Remedy for Automated Decision Safety and Security: Security Safety and Reliability Predictability Security by Design Notification when Interacting with an AI Notification when AI Makes a Decision about an Individual Regulatory framework for access and processing authorisations First issue of privacy qualification: a very active IT issue Regular Reporting Requirement Right to Information Open Procurement (for Government) Fairness and Non-discrimination: Non-discrimination and the Prevention of Bias Fairness Inclusiveness in Impact Equality Human Control of Technology Human Review of Automated Decision Multistakeholder Collaboration Responsible Design Consideration of Long Term Effects Accuracy Scientific Integrity Human Values and Human Flourishing Access to Technology 56/54 57/54 58/54 Leveraged to Benefit Society Promotion of Human Values: individuelle prévue par les dispositions de l'article 58 de la loi Informatique et Libertés et qui de la santé. Elle rappelle que cette information générale ne peut se substituer à l'information informés de l'éventualité de l'utilisation des données à des fins de recherche dans le domaine -Extract from the CNIL deliberation: La Commission relève par ailleurs que les patients sont Professional Responsibility: Ability to Opt out of Automated Decision warehouse • for the re-use of data from the warehouse for research, studies or evaluations based on data held in the Human Control of Technology: • for the creation of health warehouses Representative and High Quality Data -Two distinct legal regimes Inclusiveness in Design -CNIL authorisations by each university hospital of its warehouse -k-anonymity -CNIL authorisation for the project including the multicentre study between the university hospitals + matching with the historical SNDS -Differential Privacy (DP) → being adopted by industry

devra être réalisée pour chaque traitement de données réalisé à partir des données de l'entrepôt, en application des dispositions du chapitre IX de la loi informatique et libertés source: CNIL Deliberation 2018-295 of 19 July 2018 https://www.legifrance.gouv.fr/cnil/id/CNILTEXT000037509951/ Contradictory injunction in these studies due to the RGPD -The data must be pseudonymised (impossible to trace back to individuals)

-Lorsque, dans les cas visés au paragraphe 1 du présent article, le responsable du traitement est à même de démontrer qu'il n'est pas en mesure d'identifier la personne concernée, il en informe la personne concernée, si possible. En pareils cas, les articles 15 à 20 ne sont pas applicables, sauf lorsque la personne concernée fournit, aux fins d'exercer les droits que lui confèrent ces articles, des informations complémentaires qui permettent de l'identifier. » (RGPD, Art 11)

Implementing ethical principles

The case of digital epidemiology In this context of medical research, reasonable application of standard and adapted ethical principles How can similar practices be encouraged for products resulting from industrial developments?

A healthcare software company is developing a patient triage tool to assess the level of risk of stroke based on a machine learning algorithm. The AI system was trained on data sold by a provider and included only stroke cases in patients >70 years old, mostly men.


• Questions de recherche :

• Les langues kiranti sont-elles véritablement apparentées ?

• Propositions de correspondances régulières et de reconstruction de la proto-langue, mais surtout les initiales • Certains spécialistes se demandent s'il s'agit d'un sous-groupe : « It has never been shown that Kiranti... is a valid genetic unit. » Ebert 2003

• Peut-on utiliser des matériaux autochtones, plutôt qu'issus de stimuli non adaptés, pour en savoir plus ?

• Cycle mythologique partagé par les langues kiranti, décrit (entre autres) par Ebert et Gaenszle 2008 [Ebert, K. 2003. Kiranti languages: an overview. In G. Thurgood and R. LaPolla (eds)

The Sino-Tibetan languages. • Avantages :

• le concordancier permet une comparaison lexicale rapide, qui peut ensuite être contextualisée par l'accès aux phrases • intérêt éthnographique d'avoir des histoires « alignées », puisque le processus permet de réfléchir aux éléments constituants de la proto-histoire (à moins que ces histoires ne soient empruntées ...?!) • projet attractif et valorisant pour les communautés locales Objectif principal : base de données relationnelle nous permettant de répondre de manière empirique à un certain nombre de questions de recherche 1) établir une taxinomie des schèmes structurels ou organisationnels de grammaires descriptives 2) corréler ces schèmes avec des critères biographiques et ou bibliographiques 3) déterminer des principes d'appartenance à des écoles descriptives sur la base de structuration et de terminologie 4) établir une base de données terminologiques 'alignée' à Mieux comprendre les choix que font les linguistes écrivant des grammaires et l'impact de ces choix sur l'utilisation faite par les typologues de ces données • basés sur l'idée de comparaison de données semblables alignées :

-dans un cas, les expressions linguistiques codant (dans différentes langues) des évènements communs dans le cadre d'un narratif ; -dans l'autre, les tables de matières des grammaires descriptives • visée typologique :

-dans le cas du CCK, pouvoir comparer des matériaux natifs et naturels pour découvrir les différences lexicales, morphosyntaxiques, constructionnelles entre les langues du corpus... ; -dans le cas de Taxogram, pouvoir comprendre les influences sur les choix structurants et terminologiques sur les grammairiens, et les prendre en compte lorsqu'on utilise ces grammaires pour faire de la typologie linguistique

FichierFichier

  Objet d'étude = les grammaires (descriptives) Pour chaque grammaire que nos 'codons' nous collectons : a) métadonnées biographiques (linguiste) et bibliographiques (grammaire) b) Table des matières c) Liste d'abréviations utilisées Fichier métadonnées Codes couleur : orange : données à trouver dans la grammaire orange pale : un peu de recherche nécessaire pour trouver les données orange foncé : décisions subjectives Métadonnées portant sur le grammairien et sur la grammaire (biographiques et bibliographiques) corrélations entre métalangue et organisation/terminologie (par nationalité ? mais plus difficile d'obtenir l'information) 39 : gabarits imposés ou transmis ? Nombre de chapitres Nombre de pages à corrélations avec des types d'organisation du contenu ? (Pawley 2014, dans Nakayama & Rice eds : thèses de plus en plus longueshtl.cnrs.fr/ Liste d'abréviations + termes correspondants aperçu rapide de ce qui est grammaticalisé dans la langue Parfois les abréviations ne sont par fournies : Ex. Grammaires missionnaires Présupposition que les lecteurs et grammairiens partagent le même cadre grammaticographique/terminologique (cadre gréco-latin) Parallèle contemporain : articles de description/typologie qui ne listent que les abréviations non incluses dans les gloses de Leipzig à cadre partagé Connexion entre interlinéarisation et utilisation d'abréviations 43 Fichier métadonnées : abréviations • Comparaison de tables de matières et de structuration de grammaires sur un sous-corpus (selon critères sélectionnées dans les métadonnées) • Recherche terminologique : première apparition de termes, distribution géographique de termes.... • Visualisation de la proportion de la grammaire accordée à différents sous-domaines ou thématiques : évolution du contenu des grammaires (apparition de sections sur la syntaxe, sur la sémantique, sur les propositions relatives, sur le mouvement associé....) à tendances en typologie • Evolution de l'appareil descriptif : glosage interlinéaire, abréviations, collections de textes, de glossaires.... 49 Que peut-on faire avec cette base ? htl.cnrs.fr/ Deux projets très différents, mais avec quelques points communs :

  • les idiosyncrasies dans les versions des histoires sont individuelles et non pas dialectales/linguistiques • les constructions ne sont pas glosées (ex. pas moyen de faire une recherche « propositions relatives » ou « verbes dérivés » ), et les éléments qui permettraient vraiment de comparer les langues sur le plan morphologique/syntaxique ne sont pas interrogeables facilement

	Deux problèmes (parmi tant d'autres) dans l'utilisation des grammaires descriptives
	pour la typologie :
	Toba 1979 Khaling 40 pages 0. Introduction 1. Phonology 1) Structuration des tables de matières très hétérogène Michailovsky 1988 Hayu 234 pages 1. Les hayu et leur langue 2. Phonologie Questions chronologiques Ouvrages Date d'apparition : • Felix Ameka, Alan Dench & Nicholas Evans (eds.), 2006, Catching Language: de tables de matières The standing challenge of grammar writing, 41-68. Berlin & New York: de gloses interlinéaires Mouton de Gruyter. de différentes catégories grammaticales/termes
	2. Word order	3 Morphologie (1): structure
	3. Nouns 4. Pronouns Quelle influence de la langue native du linguiste ou métalangue du mot verbal 4. Morphologie (2) : structure
	5. Numerals choisie sur la description ?	du mot non verbal ; parties du
	6. Verbs	discours
	7. Adjectives	5. Syntaxe
	8. Adverbs	Texte 1
	9. Conjunctions	Texte 2
	10. Articles
	11. Interjections

• Difficultés : • alignement non automatisable (réduisant la taille maximale possible du corpus) 27 [Carte basée sur Michailovsky 1975] Ex. idéophones « open lexical class of marked words that depict sensory imagery » (Dingemanse 2019) 2) Différences terminologiques Université Paris Cité UMR 7597 HTL, Case 7034 5 rue Thomas Mann 75205 Paris cedex 13 https://htl.cnrs.fr/ Métagrammaticographie ? méta + grammatico + graphie = rédaction de grammaires + méta = regard évaluatif/critique sur la rédaction de grammaires "Grammaticography is an activity related to grammar 2 […]. At the object level, grammaticography is (the practice -experience or art -of) grammar writing. At the meta-level, it is the investigation of methodological principles that reconcile this practice with linguistic theorizing. [...]" [Lehmann & Maslova 2004, in Morphologie: Ein Handbuch zur Flexion und Wortbildung, vol. 2] Université Paris Cité UMR 7597 HTL, Case 7034 5 rue Thomas Mann 75205 Paris cedex 13 https://htl.cnrs.fr/ Quelques travaux (+/-descriptifs) sur la métagrammaticographie • Thomas Payne & David Weber (eds.), 2007, Perspectives on grammar writing (Benjamins Current Topics 11). Amsterdam & Philadelphia: John Benjamins. • Toshihide Nakayama & Keren Rice (eds.), 2014, The art and practice of grammar writing (Language Documentation and Conservation Special Publication 8). Honolulu, HI: University of Hawai'i Press. [open access] Questions de recherche Quelles écoles de description ? Comment les reconnaitre ? Quels schèmes organisationnels pour les grammaires ? Quelle approche est utilisée par le grammairien ? Quel cadre théorique de description ? Comment le reconnaitre ? Quels types de grammaires ? Quels types de publics pour des grammaires ? Peut-on les reconnaitre à l'organisation de la grammaire ? Quelle est la place des glossaires/textes dans les grammaires ? Les réponses à ces questions sont souvent basées sur une expérience de lecture de grammaires qui est biaisée par les intérêts et expériences du linguiste 32 Questions de recherche Université Paris Cité UMR 7597 HTL, Case 7034 5 rue Thomas Mann 75205 Paris cedex 13 https://htl.cnrs.fr/ Projet Taxogram, https://taxogram.huma-num.fr/home.html 33 Base de données Taxogram Université Paris Cité UMR 7597 HTL, Case 7034 5 rue Thomas Mann 75205 Paris cedex 13 https://htl.cnrs.fr/

rue Thomas Mann 75205 Paris cedex 13 https://htl.cnrs.fr/

rue Thomas Mann 75205 Paris cedex 13 https://htl.cnrs.fr/ Sélection du sous-corpus pour visualisation :

https://htl.cnrs.fr/

rue Thomas Mann 75205 Paris cedex 13 https://htl.cnrs.fr/ http://himalco.huma-num.fr

rue Thomas Mann 75205 Paris cedex 13 https://htl.cnrs.fr/ Base de données métagrammaticographiques

rue Thomas Mann 75205 Paris cedex 13 https://htl.cnrs.fr/ 36 Fichier métadonnées

rue Thomas Mann 75205 Paris cedex 13 https://htl.cnrs.fr/ MERCI DE VOTRE ATTENTION