Fatima-Zahra Bensidhoum
email: fzbensidhoum@yahoo.fr

Hacen Dib
email: hdib@mail.univ-tlemen.dz

On some Regular Fractional Sturm-Liouville Problems with Generalized Dirichlet Conditions

Keywords: MSC 2010 Primary 26A33, 34A08: Secondary 34B24, 34B27, 34L05, 34L10, 34L15, 47G10 -sided Riemann-Liouville fractional derivatives, fractional Sturm-Liouville problem, fractional Green's function, Hilbert-Schmidt operators, min-max principle

The present work deal with some spectral properties of the problem (P)

D α b,-and D α a,+ are the right-sided and left-sided Riemann-Liouville fractional derivatives of order α ∈ (0, 1), respectively. λ is a scalar parameter.

First, we prove, using the spectral theory of linear compact operators, that this problem has an infinite sequence of real eigenvalues and the corresponding eigenfunctions form a complete orthonormal system in the Hilbert space L 2 q [a, b]. Then, we investigate some asymptotic properties of the spectrum as α -→ < 1. We give, in particular, the asymptotic expansion of the first eigenvalue.

Introduction

The Sturm-Liouville theory has an important role in mathematics, physics and engineering. During the last few years, fractional Sturm-Liouville problems appear in the literature. They were introduced by M.Klimek and O.P.Agrawal in [START_REF] Klimek | Fractional Sturm-Liouville problem[END_REF][START_REF] Klimek | On a regular fractional Sturm-Liouville problem with derivatives of order in (0, 1)[END_REF]. Both Riemann-Liouville and Caputo derivatives were used. These authors proved that the eigenvalues are real and the eigenfunctions are orthogonal by using a Green-type formula as in the classical case. In [START_REF] Rivero | A fractional approach to the Sturm-Liouville problem[END_REF], more problems have been presented.

The purpose of this paper is to investigate some basic spectral properties of the fractional Sturm-Liouville problem (P) with Generalized Dirichlet conditions. By Generalized Dirichlet conditions, we mean lim

x-→a > (x -a) 1-α y(x) = 0 = y(b).

Note that this kind of conditions involves continuous functions at the point a, as well as noncontinuous ones. For example, lim

x-→a > ln (x -a) = -∞ but lim

x-→a > (x -a) 1-α ln (x -a) = 0.

Obviously, Dirichlet conditions are particular case of the Generalized ones and the limit in [START_REF] Dib | Generalized Variational Problems and Euler-Lagrange equations[END_REF] implies that y is in L 1 . We start by considering the problem (P) on the domain Dom where y ∈ Dom if and only if y verifies the following conditions with the aim of transform (P) into an integral equation problem, which can be analyzed by classical tools of operator theory. The condition (C1) is nothing but the generalized Dirichlet condition at x = a. The second one (C2) will appear naturally further in the calculation of the integral operator.

The number λ is said to be an eigenvalue for the problem (P) if there exists a function in Dom, not identically zero, which satisfies the fractional regular Sturm-Liouville equation and the Generalized Dirichlet conditions.

In what follows, we prove that the set of eigenvalues of (P) is real and infinite. This result answers partially a question in [START_REF] Rivero | A fractional approach to the Sturm-Liouville problem[END_REF] (Remark 1, p.9) where the situation is quite similar. However the simplicity of the eigenvalues is still an open problem.

The corresponding eigenfunctions form a complete orthonormal system in the Hilbert space L 2 q [a, b] endowed with the usual scalar product and the associated norm

u, v q = b a u(x) v(x) q(x) dx , u q = u, u q
This result follows from the theory of linear compact operators and the fractional Green's function properties following so, the approach in [START_REF] Klimek | Regular fractional Sturm-Liouville problem with discrete spectrum: Solutions and applications[END_REF] adapted to our generalized boundary conditions. The second result investigate the asymptotic behaviour of the first eigenvalue of (P) when α -→ < 1. This can be the starting point to put the classical and the fractional cases in the same theory.

Basic Concepts

In this section, we recall some definitions and properties of Riemann-Liouville fractional integrals and fractional derivatives used below. For more details see [START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF][START_REF] Miller | An Introduction to the Fractional Calculus and Fractional Differential Equations[END_REF][START_REF] Podlubny | Fractional Differential Equations[END_REF][START_REF] Samko | Fractional Integrals and Derivatives -Theory and Applications[END_REF].

Definition 2.1 [9, 13] Let f ∈ C([a, b]
). The right-sided and left-sided Riemann-Liouville fractional integrals of order α > 0 are defined, respectively, by

(I α b,-f )(x) = 1 Γ(α) b x (t -x) α-1 f (t) dt (2)
and

(I α a,+ f )(x) = 1 Γ(α) x a (x -t) α-1 f (t) dt ( 3
)
where Γ is the Euler Gamma function.

Definition 2.2 [START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF][START_REF] Podlubny | Fractional Differential Equations[END_REF] The right-sided and left-sided Riemann-Liouville fractional derivatives of order α ∈ (0, 1) are defined by

(D α b,-f )(x) = - d dx (I 1-α b,-f )(x) (4)
and

(D α a,+ f )(x) = d dx (I 1-α a,+ f )(x) (5)
respectively.

Proposition 2.1 [START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF][START_REF] Podlubny | Fractional Differential Equations[END_REF][START_REF] Samko | Fractional Integrals and Derivatives -Theory and Applications[END_REF] Let

0 < α < 1 and f ∈ C((a, b)). 1. [(D α b,-o I α b,-) f ](x) = f (x). 2. If I 1-α b,-f ∈ AC([a, b]
), then we have

[(I α b,-o D α b,-) f ](x) = f (x) - (b -x) α-1 Γ(α) lim x-→b < (I 1-α b,-f )(x). 3. [(D α a,+ o I α a,+ ) f ](x) = f (x). 4. If I 1-α a,+ f ∈ AC([a, b]
), then we have (1) If there exists a limit lim

[(I α a,+ o D α a,+ ) f ](x) = f (x) - (x -a) α-1 Γ(α) lim x-→a > (I 1-α a,+ f )(x).
x-→a > (x -a) 1-α y(x) = c, c ∈ C then also the following limit exists

I 1-α a,+ y (a+) := lim x-→a > I 1-α a,+ y (x) = c Γ(α).
(2) Symmetrically, if there exists a limit

lim x-→b < (b -x) 1-α y(x) = d, d ∈ C
then also the following limit exists

I 1-α b,-y (b-) := lim x-→b < I 1-α b,-y (x) = d Γ(α).
Finally, we recall the following fractional integration by parts formula, proved in [START_REF] Samko | Fractional Integrals and Derivatives -Theory and Applications[END_REF] (Corollary 2 -p.46 with simple sufficient conditions)

Proposition 2.2 Let f, g, (D α a,+ f ), (D α b,-g) ∈ C([a, b]) then b a (D α a,+ f )(x) g(x) dx = b a f (x) (D α b,-g)(x) dx. ( 6
)
3 Main Results

Spectral properties

In this subsection, we prove that the problem (P) admits an infinite sequence of eigenvalues, which are real and negative. Also, the normalized eigenfunctions form an orthonormal basis in L 2 q [a, b].

Proposition 3.1 Let α ∈ 1 2 , 1 then
1. ∀y ∈ Dom, the problem (P) is equivalent to

y(x) = λ (T α y)(x)
where T α is the linear operator defined by

T α : Dom -→ Dom y(•) -→ (T α y)(x) = b a G α (x, τ ) q(τ ) y(τ ) dτ G α (x, τ ) is the Fractional Green's Function defined by G α (x, τ ) =            φ(b, x) φ(b, τ ) φ(b, b) Γ(α) - φ(x, τ ) Γ(α) , if a ≤ τ ≤ x φ(b, x) φ(b, τ ) φ(b, b) Γ(α) - φ(τ, x) Γ(α) , if x ≤ τ ≤ b (7)
and

φ(u, v) = I α a,+ (u -v) α-1 p(v) (v) = 1 Γ(α) v a (u -τ ) α-1 (v -τ ) α-1 dτ p(τ ) . ( 8
)
2. T α extends to a compact, self-adjoint operator on L 2 q [a, b] and Ker(T α ) = {0} 3. All the eigenvalues of the problem (P) are negative.

Proof.

By applying

I α b,-to the equation D α b,-(p(x) D α a,+ y)(x) + λ q(x) y(x) = 0 we obtain (D α a,+ y)(x) = (b -x) α-1 Γ(α) p(x) lim x-→b < (I 1-α b,-p(x) D α a,+ y)(x) - λ p(x) (I α b,-q(x) y(x))(x).
Now, we apply the operator I α a,+ . Then

y(x) = (x -a) α-1 Γ(α) lim x-→a > (I 1-α a,+ y)(x) + C φ(b, x) Γ(α) -λ I α a,+ 1 p(x) (I α b,-q(x) y(x)) (x) (9)
where, according to Proposition 2.1 (property 2), we have

C = lim x-→b < (I 1-α b,-p(x) D α a,+ y)(x)
and φ(b, x) is defined in [START_REF] Klimek | Regular fractional Sturm-Liouville problem with discrete spectrum: Solutions and applications[END_REF].

• Note that the generalized Dirichlet condition lim

x-→a > (x -a) 1-α y(x) = 0 implies that lim x-→a > (I 1-α a,+ y)(x) = 0.
according to lemma 2.1.

• The application of the second boundary value condition y(b) = 0, gives

C = λ Γ(α) φ(b, b) lim x-→b < I α a,+ 1 p(x) (I α b,-q(x) y(x)) (x) where φ(b, b) = lim x-→b < φ(b, x) exists for 1 2 < α ≤ 1 since the integral b a (b -x) 2 α-2 dx p(x) (10)
converges if and only if 2 α -1 > 0.

Let us compute the constant C. By using Fubini's theorem, we obtain

I α a,+ 1 p(x) (I α b,-q(x) y(x)) (x) = 1 Γ(α) x a φ(x, τ ) q(τ ) y(τ ) dτ + 1 Γ(α) b x φ(τ, x) q(τ ) y(τ ) dτ (11)
Then we have

C = λ Γ(α) φ(b, b) lim x-→b < I α a,+ 1 p(x) (I α b,-q(x) y(x)) (x) = λ φ(b, b) b a φ(b, τ ) q(τ ) y(τ ) dτ .
So, by substituting the first boundary value condition, the constant C and the integral [START_REF] Mckiernan | On the nth derivative of composite functions[END_REF] in the equation ( 9) we get the first implication.

We can easily verify that ∀ y ∈ Dom, T α y ∈ Dom.

Then, the equivalence stated above follows immediately.

2.

We can get the results from the properties of the fractional Green's function. The operator T α defined on Dom can be continuously extended to L 2 q [a, b] because Dom is dense, since it contains the space of infinitely differentiable functions with compact support. The compactness of T α follows from the properties below. In fact,

i. ∀(x, τ ) ∈ [a, b] × [a, b], G α (x, τ ) is a continuous function because φ(x, τ ) is a continuous function in the domain a ≤ τ ≤ x ≤ b and lim τ -→x < G α (x, τ ) = lim τ -→x > G α (x, τ ), then the kernel G α ∈ L 2 q⊗q ([a, b] × [a, b]) therefore T α is a Hilbert-Schmidt operator, so compact. (see [3], [16]) ii. ∀(x, τ ) ∈ [a, b]×[a, b], G α (x, τ ) = G α (τ, x) then G is a symmetric function on [a, b]×[a, b] so the operator T α is self-adjoint in L 2 q [a, b].
iii. To prove that the kernel of T α is reduced to {0}, it suffice to solve the equation

(T α y)(x) = 0, ∀x ∈ [a, b]
By application of left-sided and right-sided Riemann-Liouville fractional derivatives, respectively, we get the result after some simple calculations.

3.

Let us consider the Fractional Sturm-Liouville equation λ n = -∞, which can be ordered as

D α b,-(p(x) D α a,+ y)(x) + λ q(x) y(x) = 0.
• • • ≤ λ n ≤ • • • ≤ λ 3 ≤ λ 2 ≤ λ 1 ≤ 0
ii) The normalized eigenfunctions {φ n } n corresponding to the eigenvalues

{λ n } n are orthogonal in L 2 q [a, b] that is ∀ i = j, b a φ i (x) φ j (x) q(x) dx = 0
iii) Let {φ n,j } dn j=1 an orthonormal basis of E n , the eigenspace corresponding to λ n (d n = dim E n ). The eigenfunctions {φ n,j } n,j form an orthonormal basis of the Hilbert space

L 2 q [a, b] that is ∀ y ∈ L 2 q [a, b], y = lim N →∞ N n=1 dn j=1
< y, φ n,j > q φ n,j .

Proof.

Let λ be a non-zero eigenvalue of (P) and µ be a non-zero eigenvalue of T α , then according to Proposition 3.1, we have

(P) ⇐⇒ 1 λ y = T α y and µ y = T α y ⇐⇒          D α b,-(p(x) D α a,+ y)(x) + 1 µ q(x) y(x) = 0 lim x-→a > (x -a) 1-α y(x) = 0 = y(b)
The homogeneous problem has not a non-zero solution hence zero is neither eigenvalue of (P) nor eigenvalue of T α .

Then, the non-zero eigenvalues of (P) are exactly the inverses of those of T α . These last are real countable, negative, tend to zero when n -→ ∞ and they can be ordered as

|µ 1 | ≥ |µ 2 | ≥ |µ 3 | ≥ • • • ≥ |µ k | ≥ • • • → 0 because T α is self-adjoint and compact in L 2 q [a, b].
We can also conclude the properties of the eigenfunctions. Now, we prove the infinity of eigenvalues by absurd. Let us assume that T α has a finite sequence of non-zero eigenvalues. The spectral theorem of compact self-adjoint operators gives

L 2 q [a, b] = (Ker T α ) ⊕ N k=1 E n .
Since the corresponding eigenspaces E 1 , • • • , E n are finite dimensional and according to Proposition 3.1 we have Ker T α = {0}, then L 2 q [a, b] must be finite dimensional which is a contradiction.

Remark 3.1 The integral [START_REF] Lebedev | Special Functions and their applications[END_REF] converges if and only if α > 1 2

. So, the operator T α remains compact and self-adjoint even if α ≥ 1. But the corresponding fractional Sturm-Liouville problem must contain much more boundary conditions.

Recall the expression of the Hilbert-Schmidt norm of a compact integral operator on L 2 q [a, b], [START_REF] Weidmann | Linear Operators in Hilbert Spaces[END_REF].

If (Sy)(x) = b a K(x, t) y(t) q(t)dt then S 2 HS = b a b a |K(x, t)| 2 q(t)q(x) dt dx (12)
Proposition 3.2 All the eigenvalues of the problem (P) verify

|λ| ≥ 1 T α HS
Proof. If λ is an eigenvalue then there exists y = 0 such that y = λ T α y so y q = |λ| . T α y q ≤ |λ| . T α . y q ≤ |λ| . T α HS . y q then |λ| ≥ 1 T α HS .

Remark 3.2

The optimal disc free from eigenvalues has 1 T α as radius. But the operator norm is more difficult to compute than the Hilbert-Schmidt one. In our case

T α 2 HS = b a b a
|G α (x, t)| 2 q(t)q(x) dt dx can be computed for explicit functions p and q.

Asymptotic behaviour

In this section, we prove that the first eigenvalue of T α is close to the first eigenvalue of the following problem

(P 1 )    -(p(x) y ) (x) + λ q(x) y(x) = 0, a < x < b y(a) = 0 = y(b) when α -→ < 1.
To prove this result we need some lemmas and propositions.

Lemma 3.1 The n th derivative of 1 Γ(α) and Γ(α) have the following properties

∀ n ≥ 0, ∃ M n > 0, ∀α ∈ 1 2 , 1 , 1 Γ(α) (n) ≤ M n and (Γ(α)) (n) ≤ M n (13)
Proof.The functions 1 Γ(•)

, Γ(•) ∈ C ∞ (]0, +∞[), then they are bounded on a compact sets, as well as all their derivatives. Lemma 3.2 For a ≤ t ≤ x ≤ b, we have the following property

∀ ε > 0, ∀ 0 ≤ j ≤ k, ∃ C k,j (ε) > 0 1 2 + ε < α ≤ 1, t a (x -τ ) α-1 ln j (x -τ ) (t -τ ) α-1 ln k-j (t -τ ) dτ ≤ C k,j (ε). (14)
Proof.

Let us consider the function

f m (ξ, β) = ξ β ln m (ξ) with m ∈ N and 0 < β < α - 1 2 . It is not hard to see that f m (ξ, β) is uniformly bounded in [0, c] × 0, α - 1 2 . t a (x -τ ) α-1 ln j (x -τ ) (t -τ ) α-1 ln k-j (t -τ ) dτ = t a (x -τ ) α-β-1 f j ((x -τ ), β) (t -τ ) α-β-1 f k-j ((t -τ ), β) dτ ≤ A k,j (ε) t a (x -τ ) α-β-1 (t -τ ) α-β-1 dτ ≤ A k,j (ε) (2α -2β -1) (x -a) 2α-2β-1 -(x -t) 2α-2β-1 . (t -a) 2α-2β-1 ≤ A k,j (ε) (2α -2β -1) (b -a) 2α-2β-1
where we used the Cauchy-Schwartz inequality. By choosing 2β = α -

1 2 , we get t a (x -τ ) α-1 ln j (x -τ ) (t -τ ) α-1 ln k-j (t -τ ) dτ ≤ A k,j (ε) ε d ε 1 2 = C k,j (ε)
where

d ε (γ) = (b -a) γ if b ≥ a + 1 (b -a) 2 γ ε if b < a + 1
Lemma 3.3 Let g be the function defined by

g(α) = b a (b -τ ) 2 α-2 dτ p(τ )
then g has the following properties

1. ∀ ε > 0, ∃ K min , K max > 0 1 2 + ε < α ≤ 1 ⇒ K min ≤ |g(α)| ≤ K max . 2. ∀ ε > 0, ∀ n ≥ 0, ∃ η n > 0 1 2 + ε < α ≤ 1 ⇒ g (n) (α) ≤ η n . 3. ∀ ε > 0, ∀ n ≥ 0, ∃ δ n > 0 1 2 + ε < α ≤ 1 ⇒ 1 g(α) (n) ≤ δ n .

Proof.

Let

g(α) = b a (b -τ ) 2 α-2 dτ p(τ ) .
Since p ∈ C([a, b]) and p(t) > 0, then

1 M = 1 max τ ∈[a,b] p(τ ) ≤ 1 p(τ ) ≤ 1 min τ ∈[a,b] p(τ ) = 1 m hence (b -a) 2 α-1 M (2 α -1) ≤ |g(α)| ≤ (b -a) 2 α-1 m (2 α -1) . Let ε > 0 1 2 + ε < α ≤ 1 , so we conclude that c ε M ≤ |g(α)| ≤ d ε (1) m (2 ε) where c ε = (b -a) 2 ε if b ≥ a + 1 b -a if b < a + 1
which gives the first property.

For g (n) (α), we have

g (n) (α) = b a 2 n ln n (b -τ ) (b -τ ) 2 α-2 dτ p(τ ) = b a 2 n f n (b -τ, β) (b -τ ) 2α-β-2 dτ p(τ ) and then g (n) (α) ≤ 2 n m A n (ε) (b -a) 2α-β-1 2α -β -1 Now, we choose β = α - 1 2 to get g (n) (α) ≤ 2 n m ε A n (ε) d ε 1 2 = η n (15)
Finally, we can compute 1 g(α)

(n) by application of the n th derivative of composite functions (see [START_REF] Mckiernan | On the nth derivative of composite functions[END_REF]) to the following functions

f (α) = 1 α and g(α) = b a (b -τ ) 2 α-2 dτ p(τ ) . So (f • g) (n) (α) = n k=1 f (k) (g(α)) k l=0 (-1) k-l l! (k -l)! g k-l (α) g l (α) (n) (16)
where

∀ k / 1 ≤ k ≤ n , f (k) (g(α)) = (-1) k k! g k+1 (α) (17)
and ∀ l / 0 ≤ l ≤ n , j 0 = 0 and j l+1 = n we have

g l (α) (n) = j 1 j 0 =0 j 2 j 1 =0 j 3 j 2 =0 • • • j l+1 j l =0 l i=1 j i+1 j i (g(α)) (j i+1 -j i ) (18)
which can be proved easily by induction. By substituting (17) and ( 18) in ( 16) we get

(f • g) (n) (α) = n k=1 k l=0 k l (-1) l g l+1 (α) j 1 j 0 =0 • • • j l+1 j l =0 l i=1 j i+1 j i (g(α)) (j i+1 -j i )
where ∀l/ 0 ≤ l ≤ n, j l+1 = n.

Using triangular inequality, the first property of g(α) and the property (15), we conclude that

∀ ε > 0, ∀ n ≥ 0, ∃ δ n > 0 1 2 + ε < α ≤ 1 =⇒ (f • g) (n) (α) ≤ δ n . ( 19
) Proposition 3.3 For a ≤ t ≤ x ≤ b and φ(x, t) = 1 Γ(α) t a (t -τ ) α-1 (x -τ ) α-1 dτ p(τ )
we have

∀ ε > 0, ∀ n ≥ 0, ∃ K n > 0 1 2 + ε < α ≤ 1 ⇒ ∂ n ∂ α n φ(x, t) ≤ K n . ∀ ε > 0, ∀ n ≥ 0, ∃ B n > 0 1 2 + ε < α ≤ 1 ⇒ ∂ n ∂ α n 1 φ(b, b) ≤ B n .
Proof.

• ∂ n ∂ α n φ(x, t) = ∂ n ∂ α n 1 Γ(α) t a (t -τ ) α-1 (x -τ ) α-1 dτ
p(τ ) by application twice the Leibniz's formula, we obtain

= n k=0 n k 1 Γ(α) (n-k) t a (t -τ ) α-1 (x -τ ) α-1 dτ p(τ ) (k) = n k=0 k j=0 n k k j 1 Γ(α) (n-k) t a (x -τ ) α-1 ln j (x -τ ) (t -τ ) α-1 ln k-j (t -τ ) dτ p(τ ) .
Then, according to the triangular inequality we obtain

∂ n ∂ α n φ(x, t) ≤ n k=0 k j=0 n k k j 1 Γ(α) (n-k) t a (x -τ ) α-1 ln j (x -τ ) (t -τ ) α-1 ln k-j (t -τ ) dτ p(τ ) the fact that p ∈ C([a, b]) p(τ ) > 0 give 1 p(τ ) ≤ 1 min τ ∈[a,b] p(τ ) = 1 m .
So by Lemma 3.1 and Lemma 3.2, we obtain

∀ ε > 0, ∀ n ≥ 0, ∃ K n = 1 m n k=0 k j=0 n k k j M n-k C k,j > 0 1 2 + ε < α ≤ 1 =⇒ ∂ n ∂ α n φ(x, t) ≤ K n . • We have φ(b, b) = 1 Γ(α) b a (b -τ ) 2 α-2 dτ p(τ ) ⇐⇒ 1 φ(b, b) = Γ(α) g(α)
then by application of Leibniz's formula, we obtain

∂ n ∂ α n 1 φ(b, b) = n k=0 n k (Γ(α)) (n-k) 1 g(α) (k) . ( 20
)
Hence, from the properties ( 13) and ( 19) we get the desired result.

Proposition 3.4 Consider once more the operator T α

(T α y)(x) = b a G α (x, τ ) q(τ ) y(τ ) dτ
where G α is defined in [START_REF] Klimek | Regular Sturm-Liouville Problem with Riemann-Liouville Derivatives of Order in (1,2)[END_REF].

Then T

(n) α = ∂ n ∂ α n T α is uniformly bounded for 1 2 + ε ≤ α ≤ 1, that is ∀ ε > 0, ∀ n ≥ 0, ∃ θ n > 0 1 2 + ε ≤ α ≤ 1 =⇒ T (n) α ≤ θ n . (21)
Proof.We have

(T (n) α y)(x) = b a G (n) α (x, τ ) q(τ ) y(τ ) dτ and • for a ≤ τ ≤ x then G (n) α (x, τ ) = n m=0 n m 1 Γ(α) (n-m) . m l=0 m l φ (m-l) (b, x) l k=0 l k φ (l-k) (b, τ ) 1 φ(b, b) (k) -φ (m) (x, τ ) • and for x ≤ τ ≤ b then G (n) α (x, τ ) = n m=0 n m 1 Γ(α) (n-m) . m l=0 m l φ (m-l) (b, τ ) l k=0 l k φ (l-k) (b, x) 1 φ(b, b) (k) -φ (m) (τ, x) (22)
obtained by Leibniz's formula.

Then by the Cauchy-Schwartz inequality, the fact that q ∈ C([a, b]) and q(τ ) > 0, we can easily prove that

T (n) α ≤ T (n) α H.S. ≤ sup τ ∈[a,b] q(τ ) b a b a G (n) α (x, τ ) 2 dτ dx.
So by of Lemma 3.1, the Proposition 3.3 and the formula (22) of G

(n) α , we obtain the result (21). Now, we can state the second main result.

Theorem 3.2 i) Let m ≥ 0 be a fixed integer. Then, T α is (m + 1) th differentiable operator with respect to α at α = 1 and we have the following expansion

T α y = m k=0 T (k) α y α=1 k! (α -1) k + R m (α)y (23)
where the Lagrange reminder is expressed by

R m (α)y = (α -1) m+1 (m + 1)! T (m+1) α y α=ξ (α < ξ < 1) (24) and R m (α) = o(|α -1| m ), α -→ < 1. ii) If µ - 1 (α) is the first negative eigenvalue of the operator T α , then lim α-→ < 1 µ - 1 (α) = µ - 1 (1). ( 25
)
where µ - 1 (1) is the first negative eigenvalue of the operator T 1 . Proof.

i) From the expression (7) of the kernel G α (x, τ ), one can see that G α (x, τ ) is C ∞ in any small neighbourhood of α = 1. So we can write the Taylor formula with Lagrange reminder at this point

(T α y) (x) = m k=0 T (k) α y (x) α=1 k! (α -1) k + R m (α)
where

T (k) α y (x) α=1 = b a G (k) α (x, τ ) α=1 q(τ ) y(τ ) dτ
and R m (α) is the Lagrange reminder defined in (24). Then,

R m (α) ≤ T (m+1) ξ . |α -1| m+1 (m + 1)! , α < ξ < 1.
By applying the property (21), we obtain

∃ θ m+1 > 0/ R m (α) ≤ θ m+1 . |α -1| m+1 (m + 1)! which gives R m (α) = o(|α -1| m ), α -→ < 1.
ii) Let us take the previous expansion of T α (23) with m = 0, then

(T α y) (x) = (T 1 y) (x) + (α -1). T ξ y (x)
⇐⇒ T α y, y q -T 1 y, y q = |α -1| . T ξ y, y q .

However, T ξ y, y q ≤ T ξ . y q ≤ T ξ , ∀y/ y q = 1.

Hence,

T α y, y q -T 1 y, y q ≤ |α -1| . T ξ ⇒ -|α -1| . T ξ ≤ T α y, y q -T 1 y, y q ≤ |α -1| . T ξ ⇒ -|α -1| . T ξ + T 1 y, y q ≤ T α y, y q ≤ T 1 y, y q + |α -1| . T ξ ⇒ -|α -1| . T ξ + inf

y q =1 T 1 y, y q ≤ inf y q =1 T α y, y q ≤ inf y q =1 T 1 y, y q + |α -1| . T ξ ⇒ -|α -1| . T ξ + µ - 1 (1) ≤ µ - 1 (α) ≤ µ - 1 (1) + |α -1| . T ξ (26)
from this and the fact that T ξ verifies (21), we get lim

α-→ < 1 µ - 1 (α) = µ - 1 (1)
.

and consequently, lim

α-→ < 1 λ - 1 (α) = λ - 1 (1)
Remark 3.3 We think that the previous expansion of T α can be of great importance in the expression of the next terms in asymptotic development of µ - 1 (α).

Illustrative example

Let us see a simple example with p ≡ 1, q ≡ 1 and where the fractional Green's function is

G α (x, τ ) =    (2 α -1) ϕ(1, x) ϕ(1, τ ) -ϕ(x, τ ), if 0 ≤ τ ≤ x (2 α -1) ϕ(1, x) ϕ(1, τ ) -ϕ(τ, x), if x ≤ τ ≤ 1 with ϕ(τ, x) = 1 Γ(α) x 0 (x -t) α-1 (τ -t) α-1 dt = x α τ α-1 Γ(1 + α) 2 F 1    1 -α , 1 ; x τ 1 + α   
where 2 F 1 is the Gauss-hypergeometric function which can be defined by its Euler representation (see [[10]]) where u(x) = x ln x and γ is the Euler constant. We remark that the linear operator associated to this Green's function which contains Logarithmic functions is not connected to any classical differential operator. Finally, we have lim

α-→ < 1 λ - 1 (α) = -π 2 .

(

  C1) y ∈ C(]a, b]) and lim x-→a > (x -a) 1-α y(x) = 0 (C2) D α a,+ y ∈ C([a, b[) and lim x-→b < (b -x) 1-α D α a,+ y(x) exists

Lemma 2 . 1 [ 9 ]

 219 (Lemma 3.2-p.151) Let 0 < α < 1 and y(x) be a Lebesgue measurable function on [a,b].

  By multiplying this equation by y(x) and integrate it on [a, b] we get b a D α b,-(p(x) D α a,+ y)(x) y(x) dx + λ b a q(x) y 2 (x) dx = 0 Now, by applying the property (6) of proposition 2.2, which is also satisfied with the generalized Dirichlet conditions (1), we obtain b a p(x) D α a,+ y 2 (x) dx + λ b a q(x) y 2 (x) dx = 0 hence λ must be negative. Theorem 3.1 Let us consider the Fractional Regular Sturm-Liouville problem (P) for 1 2 < α < 1, then i) The problem (P) has an infinite countable set {λ n } n≥1 of real negative eigenvalues such that lim n-→∞

1 0G

 1 [a, b] = [0, 1]. Then, our R.F.S.α y(x) = 0 = y(1)and the "inverse" compact operator defined inL 2 ([0, 1]) is (T α y)(x) = α (x, τ ) y(τ ) dτ

2 τ

 2 1 (1 -s) c-b-1 (1 -z s) -a ds.So, this problem has an infinite sequence of real negative eigenvalues. The Taylor expansion up to order 1 gives(T α y) (x) = (T α y) (x)| α=1 + T (1) α y (x) α=1 (α -1) + R 1 (α)where the kernel Gα (x, τ ) α=1 = x τ -min(x, τ ) is the classical Green's function of the (P 1 ) problem corresponding to our data. G[START_REF] Dib | Generalized Variational Problems and Euler-Lagrange equations[END_REF] α (x, τ ) α=1 can be computed by some integrations by partsG(1) α (x, τ ) + 2 (γ -1) x τ -u(x) -u(τ ) + u(x -τ ) +u(x τ ) -x u(1 -τ ) -τ u(1 -x), if 0 ≤ τ ≤ x 2 x + 2 (γ -1) τ x -u(τ ) -u(x) + u(τ -x) +u(τ x) -τ u(1 -x) -x u(1 -τ ), if x ≤ τ ≤ 1

Acknowledgements We are grateful to the referees for their careful reading of the manuscript. Their comments and questions were of great importance for us to improve the quality of the paper. We thank them warmly.


Bruno Mery

Christian Retoré

Are Books Events? Ontological Inclusions as Coercive Sub-Typing, Lexical Transfers as Entailment

Keywords: Lexical Semantics, Type Theory, Coercive Sub-Typing, Composition, Ontological Inclusion

We present a way to combine coercive sub-typing relations, used to model ontological inclusions, and entailment using polymorphic terms provided by the lexicon in order to provide a coherent calculus integrated in L TY n . The resulting system is able to parse different types of lexical phenomena. We also illustrate differences of assumptions between related theories of lexical meaning, using the different ways to define coercive sub-typing and type systems as examples.

Introduction

Human language is highly polysemic, and present many challenges to the compositional semantics in the tradition of [Montague, 1974]: ambiguous words, vagueness, context-dependent usages, creative use of words, hypostasis. . . In publications such as [Moravcsik, 1982] and [Cruse, 1986], many of these issues have been spelt out, and the Generative Lexicon Theory have been proposed as a guideline of integration of lexical data into compositional semantics in [Pustejovsky, 1995]. More recently, several systems using different type theories for the purpose of modelling lexical semantics have been proposed. Those include, among others: Asher's Type Composition Logic presented in [Asher, 2011], Bekki's Dependent Type Semantics presented in [Bekki, 2014], Cooper's Type Theory with Records presented in [Cooper, 2007], Luo's Unified Type Theory presented in [Luo, 2011], as well our framework L TY n , developed in [Mery andRetoré, 2013,Retoré, 2014].

Over the last several years, the specific issue of sub-typing has been examined in several of these settings, as in [START_REF] Bekki | Logical polysemy and subtyping[END_REF] and [START_REF] Luo | Coercive subtyping: Theory and implementation[END_REF], for instance.

In the present paper, we present a way to combine simple mechanisms, using subtyping to represent ontological inclusions only, while other lexical phenomena are modelled using optional terms derived from the lexicon and entailment. We present rules (originally provided for the simply-typed l -calculus) for establishing coercive sub-typing relations in L TY n , a system able to derive formulae of predicate calculus with the desired formal results. We precise the combination of sub-typing and standard entailment, forming a coherent calculus able to articulate the composition of meaning from syntax to semantics and logical representation. Thereafter, we survey the other common uses of "coercive sub-typing" in different formal frameworks for lexical semantics, clarifying what is meant by coercion. This allows us to illustrate some of the differences between two classes of type-theoretic systems for lexical semantics. While we do not claim that the choices presented here are the only possible ones, we want to argue that they are adequate for our purposes.

1 Coercive Sub-Typing and Entailment in Lambda TYn

Formal and Applicative Context

The goal of our L TY n is to integrate lexical phenomena such as restriction of selection and polysemy, presented in theories such as the Generative Lexicon (see [Pustejovsky, 1995]), into Montague-style compositional analysis. As in that tradition, we analyse utterances made in natural language starting from the syntax and semantics to produce a logical representation in the form of a formula of (higher-order) predicate calculus, suitable for interpretation in game semantics, model-theoretic semantics and many other theories. The integration of lexical phenomena is largely made by the distinction of many sorts or base types, in contrast with the standard single-sort e in Montague; having a collection of sorts e 1 , e 2 .. .e n is a way to distinguish selection classes as might appear in a dictionary. We have noted that, in Le Petit Robert (in French) and the online wiktionary (in English), the entry for the verb to bark is noted to be said of animals, especially dogs2 . We need to be able to distinguish such selection classes in order to decide if a restriction of selection is respected (as in the dog barked), violated (as in *the chair barked), or indicates that some lexical phenomena is in play (as in the sergeant barked, which indicates a certain judgement from the speaker).

In our framework, and in others for lexical semantics that rely on a many-sorted type theory, sub-typing is of particular interest. Relations between types can allow the system to accommodate some properties of selection classes. Specifically, we argue that it is quite suited to the represent ontological inclusions ("is-a" relations, named "hyponymy" in the linguistic literature when denoting relations between words).

For instance, if we define Apple < Fruit and Fruit < Food as types with a simple hierarchy, if the predicate to eat is defined to apply to patients of type Food, it will apply to objects that are of type Apple as well.

The foundations of sub-typing as a tool for lexical semantics have been under discussion since [Pustejovsky, 1995] and coercive sub-typing rules have been proposed for Martin-Löf Type Theories as recently as [START_REF] Luo | Coercive subtyping: Theory and implementation[END_REF]. For L TY n , we rely on System-F, the l -calculus with polymorphic types. Attempts have been made since the late 80's to equip System-F with a sub-typing relation, in order to present a formal system of computation equivalent to most current programming languages, culminating in [START_REF] Cardelli | An extension of System-F with subtyping[END_REF]; such systems remain complex (and not really satisfactory) in nature. Subsequent efforts have relied on extending System-F and detailing how a sub-typing relation defines a logic system akin to entailment, as in [START_REF] Longo | Coherence and transitivity of subtyping as entailment[END_REF]]. In the present article, we will not attempt to provide sub-typing rules for higher-order types, but will argue that a simple version of the sub-typing relation (i.e., coercive subtyping as an entailment judgement for simply-typed terms) is sufficient for our purposes.

Coercive sub-typing is a relatively new means to formalise sub-typing relations ( [START_REF] Luo | Coercive subtyping: Theory and implementation[END_REF] uses coercive sub-typing). Building a coercive sub-typing relation requires a collection of base types (the sorts, e i ) and of coercions -given as l -terms -between some of those sorts, reflecting ontological inclusions, written c i j : e i ! e j . The collection of coercions should form a partial order on sorts, such as there is at most one coercion for every (ordered) pair of sorts (if the coercion c i j is defined, it is noted that e i < e j ) and that the graph of the relation forms, at most, a directed acyclic graph. Note that "diamond formations" in which are defined coercions c in , c im , c n j , c m j (i.e., it is true that e n < e i , e m < e i , e j < e n and e j < e m with four different types) are only allowed when c in c n j = c im c m j ; i.e., when all possible compositions of coercions yielding the same type are equal. In that case, there is one single well-defined composed coercion, c i j , and it can been unambiguously written that e j < e i (Soloviev, pc.). It is of course possible to have a formulation that does not contain such formations; that issue is, of course, very similar to allowing or not multiple inheritance in object-oriented languages. Observe that ontological inclusions do respect that unicity requirement, as they represent "is-a" relations (if an object is an Apple, it is also a Fruit -the coercion involved is object identity.

Given suitable rules, this axiomatic partial order is extended to complex types (e.g. functional types). The result is a coercive sub-typing when:

1. between any two (base or complex) types X,Y such as X < Y , there is an unique coercion3 that turns inhabitants of X into objects of type Y , and this coercion can be reconstructed from the coercions given between base types, and 2. the extended relation remains acyclic.

Definition and Propagation of Sub-Typing

Coercive sub-typing can be viewed as an abbreviation: when having an object of x : X applied to f : Y ! U for some result type U, if X < Y , then there is an unique coercion c that can be used in order to have ( f (c x)) : U, a well-formed term. In such cases, the (unambiguous) coercion can be omitted and the term written as ( f x).

Formally, Coercive application is properly defined as:

`x : A `f : B ! U A < B `f (x) : U
As seen above, coercive sub-typing also requires a set of axioms in the form e i < e j for suitable sorts; each sub-typing axiom is an abbreviation for a coercion c i j : e i ! e j , respecting the properties of coherence (at most one coercion from a sort to anotherincluding composed coercions) and acyclicity.

The following rules are used to extend the sub-typing relation < to complex types:

Sub-typing transitivity A<B B<C A<C Co-variance and contra-variance of implication

A<B T !A<T !B Simple co-variance A<B B!T <A!T Simple contra-variance A<B C<D D!A<C!B
These rules are sufficient, and correspond to the rules of sub-typing for simplytyped l -calculus. It was demonstrated in [Lang, 2014] that, for a larger set of rules (including the propagation of sub-typing over some form of type quantification), the < relation is coercive, that is, that coherence and acyclicity hold; they therefore also hold for the present set of rules. Accordingly, if there is a coercion c X!Y that can be constructed using these rules, then there is an unique canonical way to build it from axiomatic coercions c i! j , which is detailed in [Retoré, 2014].

Combining Sub-Typing and Entailment in a Coherent Calculus

The relation <, defined above for the simply-typed l -calculus, is a special case of the usual entailment `. Specifically, the "coercive application" `x:A `f :B!U A<B f (x):U can be simulated by the following:

`x : A `f : B ! U `c : A ! B f (c(x)) : U
In other words, the relation of coercive sub-typing is a sub-calculus of System-F, the syntax of which is used for L TY n . A deduction in our system with coercive subtyping can thus combine < and `proof-trees seamlessly in order to from the proof corresponding to the logical representation of a sentence.

A Purely Logical Syntax and Semantics Analysis

We will be using the example sentence John began a book. This sentence, while simple in itself, illustrates the following phenomena: -Epsilon quantification (with the article a); choice functions are also suitable candidates for this form of selection. The article a is polymorphic and, when applied to a sort X and a predicate P X!t , yields a (new) object satisfying P; see [START_REF] Mery | Typed Hilbert Operators for the Lexical Semantics of Singular and Plural Determiner Phrases[END_REF] for details. -Qualia adaptation with the telic of book; the telic quale is a classic use of lexical items in the Generative Lexicon in which the most common usage for an object is substituted for it (here, the event of reading the book in question). -Ontological inclusion, the topic of this article (reading events are a sub-type of events).

We will use coercive sub-typing solely for ontological inclusions, and entailment with terms provided by the lexicon for the other phenomena. The semantic sorts used here are: H for human beings, RM for reading materials (books, magazines. . . ), Evt for events, REvt for reading events. The base type t for propositions is also used.

Syntax analysis: using a Lambek-style grammar and lexicon, the sentence is parsed as in the Montagovian tradition: John : NP began : (NP/S)\NP a : NP/N book : N a book : NP began a book : NP/S John began a book : S Syntactic to semantic terms: the syntactic terms (categories of the grammar) are translated systematically into semantic terms (l -terms) using a complex lexicon. As in most accounts of lexical semantics with many sorts, type mismatch indicates a violation of restriction class. We provide two mechanisms for the resolution of such mismatches: using coercive sub-typing, or using explicit terms introduced by the lexicon. Accordingly, additional terms might be provided (in this case, associated to the lexical entry for book, the term f Telic : RM ! REvt is provided as a lexical transformation giving access to the event of reading a book) in order to account for semantic phenomena in a co-compositional way; the general hierarchy of types also provides relevant sub-typing judgements (here, REvt < Evt). Like in standard Montague semantics, the syntactic proof is transformed into a semantic proof. However, complying with the additional information from the lexicon, the resulting proof might be more complicated and is not always valid; we do not want to give an interpretation to infelicitous sentences. The syntax-semantics correspondence is as follows:

Lexemes, Syntax Semantics Remarks John, NP j : H began, (NP/S)\NP be :

P X. X ! (X ! Evt) ! H ! t
The first syntactic argument can be anything, given a suitable lexical transformation to an event.

a, NP/N e : P X.(X ! t) ! X

A is a polymorphic Hilbert operator as in [START_REF] Mery | Typed Hilbert Operators for the Lexical Semantics of Singular and Plural Determiner Phrases[END_REF].

book, N b : RM ! t

Common nouns are predicates.

Meaning assembly: the standard process of natural deduction combines proofs including `and, if needed, <, in order to derive a term of type t that corresponds to the sentence. In the exemple above, this will yield the following semantic proof: A : (Specialisation of began for the type Reading Material:) In summary, the resulting calculus is fully integrated in L TY n . Being a sub-calculus of System-F, it is well-founded and strongly normalising; thus, if the final semantic term is of type t, it also constitutes a valid formula of predicate calculus. Everything is compliant with our guiding principle: to be able to establish the semantic representation of utterances from their syntax giving a formula of predicate calculus as a result, detailing every step of the way. At the end, the result is, as usual, a formula of (potentially higher order) predicate calculus that can be interpreted easily in any suitable theory, including model-theoretic semantics.

`be : P X.X ! (X ! Evt) ! H ! t `(be {RM}) : RM ! (RM ! Evt) ! H ! t B : (

Differences between Type-Theories of Lexical Semantics

We want to illustrate some differences between two classes of systems: our proposal, L TY n , and MTT-based frameworks for type-theoretic lexical semantics. There are, of course, many similarities between our system and the others: all aspire to be able to model the lexical phenomena detailed in [Pustejovsky, 1995] via type-driven mechanisms.

The goal of all frameworks representing lexical semantics in type theory is to be able to derive a representation of the meaning(s) of the felicitous utterances, and to the correct rejection of the infelicitous ones. This is why every system makes some assumptions as to the selection classes of lexical items, that are absent from classical compositional semantics (in [Montague, 1974], every syntactically correct utterances has a semantic representation). There are many ways to characterise the selection classes; as we will discuss in Section 2.3, dictionaries are can provide some insight on this point. But listing restrictions of selection can only be the start, as they are routinely softened, adapted, accommodated using lexical mechanisms (predicates that are listed as applying to Human beings, such as to be joyful, can also be applied to cities); those meaning shifts are what concern all frameworks of lexical semantics, and are modelled via type operations in all of these systems (some semantic terms will have their type changed, coerced, or made compatible with others). Co-predications have proven a difficult issue, as it is both difficult to have terms being of different types at once (all of the systems involved have specific solutions), and to get information on whether a co-predication is valid or not; such informations are not available in dictionaries.

Our system differs from MTT-based frameworks in several ways. There are several systems that are inspired by Martin-Löf's Type Theory and derivatives (sometimes named "Modern" Type Theories, or Type Theories with Canonical Objects). They include Asher's TCL presented in [Asher, 2011] (though TCL presents many aspects that are closer to Montague than many other formalisms), Bekki's DTS presented in [Bekki, 2014], Cooper's TTR presented in [Cooper, 2007], and Luo's UTT presented in [Luo, 2011]. They incorporate complex typing information, at least on the form of Dependent Types, and common nouns are typically considered to be atomic types, with a high or even unbounded number of lexical sorts. Each lexical entry is given as a single type and (sometimes) term. Composition rules are quite complex, but effective; MTTbased systems have proven very apt at tasks such as reasoning, predicting or rejecting inferences, and learning, and there are several working implementations. However, there are no direct link to an analysis of syntax, and syntactic information is vital for establishing the boundaries of local phenomena.

L TY n has a different approach as to the organisation of the lexicon. Each lexical entry is a main l -term with a type, associated with some (or none) optional terms that provide all lexical information. We have a limited (and finite) number of sorts, and the rules of composition are abstraction and application (of terms and types). Note that we have described L TY n using the syntax of System-F, but that is not a formal requirement; we would be able to give the same system using MTT, DTS, UTT, or even the simply-typed l -calculus (the enumeration of optional terms would be burdensome in the latter case, but manageable, as every use of polymorphic terms is made over a finite family of types). Our main goal was to keep close to Montague, as by analysing the syntax of the utterances using categorial grammars, we are able to derive semantic representations seamlessly. We do not claim that this is the only way, or even that we are able to represent all possible phenomena in our systems, but we think that the approach is sound.

In the following, we discuss the specific differences as to the use of coercive subtyping in MTT-based systems and as in presented in this article.

Different "Coercions" in Lexical Semantics

On the one hand, the examples used above illustrate an intuitive notion to link to sub-typing, namely ontological inclusion. This is the simplest linguistic phenomenon covered by coercive sub-typing: the coercion is that of "asymmetrical identity" or "isa" relation, familiar to any database modeler, forming a directed acyclic graph (more precisely a rooted tree: the taxonomy). Coherence and acyclicity are thus guaranteed; when there is a coercion between any two types, it is unique (modulo equivalence) and canonically defined from axiomatic coercions.

On the other hand, the word "coercion", applied to lexical semantics, is often used in a much broader sense: it is the use of a word that is associated with a primary type in the lexicon, that the context of enunciation "coerces" into another type. That notion of "linguistic coercion" subsumes ontological inclusions ("accommodation" according to Pustejovsky), but also qualia adaptation, grinding, and many other lexical phenomena.

An extreme example from [Nunberg, 1993] is the ham sandwich left without paying, a meaning shift that coerces Food items into the Human beings that have ordered them in conversations among restaurant staff. We differentiate the two notions and argue that using coercive sub-typing for ontological inclusions, but not for other forms of linguistic "coercions", allows us to have a complete model of lexical phenomena that is coherent with our assumptions. Other systems for lexical semantics do differently.

Martin-Löf-inspired Systems

In MTT-based systems, a lot of the lexical information is encoded in the types themselves, with complex relations indicated in the dependencies between terms and types. In some cases, the interpretation of such terms is much less straightforward than with predicate calculus. Systems such as TTR that encode all information as typing judgements, for example, have issues with logical connectives such as negation (negation in predicate calculus is simply a predicate over propositions, while negating a type will result in a type that has no inhabitants in the usual interpretation); these issues sometimes require a very different interpretation than usual. In most of those systems, coercive sub-typing becomes a vital component, as it could be used to reflect all the (general) linguistic coercions.

Of course, doing this without would result in very a disruptive notion of sub-typing; none of the systems above do this. Some issues are as follow: recall that a well-behaved (normal and complete) system of coercive sub-typing is such that, between two given types, if one is a sub-type of the other, there is a single canonical coercion between the two, and that the graph made by elementary coercions does not contains cycles. With general linguistic coercions, there not only several ways to coerce one type into another, but it is highly desirable to have those different coercions co-exist in the type system.

Between Books and Events (the example above, which is a favorite of Asher), there are more than one lexical coercion: at least two that refer to a reading and a writing Event, but also in suitable contexts printing, colouring. . . Distinguishing between the different lexical coercions is a necessary feature of any system for lexical semantics.

If such relations can be defined globally over types, it is not clear what the objects inhabiting those types are: when there are distinct ways to convert a Book into an Event, then the individual of type Book does not have a formal definition when used in a context requiring and Event! Moreover, it is easy to find a sequence of non-trivial coercions making a cycle. The ham sandwich left without paying is evidence of a coercion from Food to Human. There is an inclusion of Human in Animal, and the salmon we had yesterday was delicious is evidence of a grinding coercion from Animal to Food. If all coercions here were sub-typing coercions, we would have constructed a non-trivial morphism from Food to Food.

Most proposals that have been made, proposing that (general) linguistic coercions can be modelled with sub-typing relations, have ways to deal with these problems. [START_REF] Bekki | Logical polysemy and subtyping[END_REF] proposes a system for dealing with co-predications using injections that coerce the types of functors, rather of their argument. That strategy can of course work, but the authors of [START_REF] Bekki | Logical polysemy and subtyping[END_REF]] only hints at a formal system, that we think can be given within L TY n . Moreover, they propose a much more general system than is usual for sub-typing, using only functor modifiers and the polymorphic application rule, in which it would be hard to demonstrate the coherence and acyclicity of the relation.

In [START_REF] Asher | Formalization of coercions in lexical semantics[END_REF], on the other hand, that very solution seems to be rejected, and a very useful notion is proposed: local coercions. Here, the coercive judgements of the form A < B are not general axioms, but specified in the context of a specific derivation (as in A < B `f : A ! B a : A. . . ). Thus, at any given point, the sub-typing system can maintain its coherence and acyclicity. This means that the ontology does not specify a system of global relations between types, but that those are local to specific utterances and context-dependant when parsing utterances. [START_REF] Asher | Formalization of coercions in lexical semantics[END_REF] extends this proposal further by having coercions written as terms with dependent types. A full, and rather complex, formalisation is given in [START_REF] Luo | Coercive subtyping: Theory and implementation[END_REF] for general type theories, with no specific application to linguistics.

Our Montagovian System

The original goal of Montagovian Generative Lexical Theories such as L TY n and, to some extent, Asher's TCL, is to model lexical phenomena starting from restriction of selection added to the standard logical Montague analysis of syntax and semantics. Grammatical sentences such as the chair barked are infelicitous because the predicate bark apply to animals, and chairs are not animals. Restriction of selection needs a system of types that can distinguish between furniture and animals -the base types are selection classes, named sorts, and the types can also define relations of ontological inclusions. We believe the sorts involved to number a few hundred at most; this is not an absolute necessity and should be refined by linguistic studies. This intuition is based on several points of (admittedly anecdotical) linguistic evidence.

On the one hand, classifiers, in language that have them, form an interesting hierarchy of categories which, while not complete (as classifiers abound for physical objects and agents, but are sparse concerning abstract entities), is very broad, multi-scalar (with a few generic classifiers, a number of "intermediate" classifiers and many highly specifalised ones), and provide linguistic distinctions between classes of objects that are remarkably similar across languages (see [Mery and Retoré, 2013] and upcoming publications for details).

On the other hand, the readily available linguistic references (common dictionaries) provide some insight as for the possible selection classes for predicates.

In the early studies of lexical semantics, [Moravcsik, 1982], [Cruse, 1986] and [Pustejovsky, 1995], the tradictional dictionaries have been characterised as completely inadequate to represent language. To paraphrase Pustejovsky, as dictionaries are senseenumeration lexica, they can, at best, represent a fragment of language at a specific point of time.

Dictionaries are unable to account for the ability to give new meaning to words by combining them in novel, contextually-appropriate ways. We subscribe to this view, but argue that a perfect dictionary would give us the common restriction classes for predicates and thus a basic system of sorts, while our (generative) lexicon will provide the means to explore if a violation of selection is motivated by a lexical mechanism, or simply invalid. No dictionary would list all words that can apply to a specific predicate, and these restrictions of selection, when they are present, are by no means complete or easy to extract computationally. Nevertheless, Larousse in French commonly uses broad selection classes including person, man, woman, animal, (physical) body, etc. Littré refines categories more finely, with predicates applying to specific materials (glass, wool are base categories), places, instruments, etc.

Having different base types, we wish to be able to distinguish different cases of ill-formed utterances:

-A-grammatical utterances, when rules of syntax are not respected.

-A-semantic utterances, when the syntactic terms cannot be transformed into semantic terms. (This should not happen for sentences that are grammatical if the lexicon is complete.) -Infelicitous utterances, when there is an irresolvable type mismatch between terms (such as *The chair barked). -Utterances without interpretation, such as The king of France is bald.

In L TY n , lexical phenomena other than ontological inclusions, such as qualia adaptations or facets, are not defined by the type system, but rather licensed by the lexicon; thus the explicit addition of transformative terms in the definition of each lexeme, to be used at the stage of meaning composition by the means of simple application (of terms or types) and suitably polymorphic types. We will then be able to distinguish between semantic, but infelicitous sentences (when there is no possible way to resolve a type mismatch) and felicitous sentences (when there is a semantic term available to resolve every type mismatch). Note that infelicitous sentences might be felicitous in another context, where adaptative terms are provided.

We restrict the use of coercive sub-typing to ontological inclusions, as the two concepts can be modelled using similar constraints (most importantly unicity, which is guaranteed with identical objects); as illustrated above, coercive sub-typing, using only rules corresponding to the simply-typed l -calculus, is a special case of entailment, and the full calculus is thus well-defined and strongly normalising.

As we have seen, that system is designed to be practical, with an analysis completely defined from syntax to semantics. When the meaning assembly is finished, if the resulting l -term is of type t, it is a meaningful formula of the predicate calculus that can be used model-theoretically.

Disregarding the final typing, all lexical information, as well as quantification, selection, negation, and modalities, are presented in the form of predicates that can be evaluated. The process is also straightforward (if not trivial) to implement in functioning parsers for natural language.

Idiosyncratic Divergence

While we are confident that all the systems presented here can satisfactorily deal with similar issues of lexical semantics, the differences of formulations and interpretations suggest deeper divergences.

One obvious reason for the differences is with the background of their respective authors and the nature of the results. Systems inspired by Martin-Löf's Type Theory have successfully been implemented in proof-checkers and theorem-provers such as Coq and Plastic, or using pure functional languages such as Haskell, and are actively used to compute logical inferences adapted from natural language. However, these systems have no means to generate the semantic representation of utterances directly from syntax. On the other hand, L TY n is suitable to produce logical forms from syntax, but has not been used for tasks such as learning or inference. Several tests for complex phenomena (including co-predications, qualia adaptation, virtual traveler, plural and mass readings) have been made using syntax-semantics parsers such as Grail, a widecoverage, CG-based parser (using l -DRT rather than System-F) from Richard Moot, presented in [Moot, 2010].

Another reason is, perhaps, a difference of approach to the linguistic phenomena involved. We view general linguistic coercions as idiosyncratic by nature, with different coercions occurring in different languages (it has been proposed that similar words in different languages do sometimes not license the same lexical phenomena), communities (the same words in the same language acquire different meanings in professional contexts, such as operation for scientists, medical workers and military officiers) and literary genres (in Fable, Fantasy and Mythology, the type Person is usually much broader than Human). We reflected this by associating the transformations with the terms given in the lexicon and not their types: types serve to guide such systems, not specify every possible utterance. Our choice of having "a few" different sorts to act as selection classes imposes us to rely almost exclusively on lexical information, with terms having very different behaviours typed similarily. On the other hand, Martin-Löf-inspired systems rely heavily on typing to provide or block coercions; some systems provide information from the lexicon or different mechanisms, others encode everything in typing judgements or relations.

Those systems are of course able to give an account of idiosyncrasy as well, but we do not think that the implied typing mechanisms are similar: sorts are associated with words (with every common noun, or even more), which is subtly different from linguistic concepts -certainly at the granularity level, but also when defining selection classes and other lexical phenomena.

The most striking difference is thus the different options being pursued for the granularity of the basic sorts. This debate is not new, and certainly not settled; see [START_REF] Goddard | Semantic primes and cultural scripts in language learning and intercultural communication[END_REF] (among many other references). Common answers include:

-The single sort e, from [Montague, 1974]. This is not sufficient to provide mechanisms for the treatment of lexical phenomena; in fact, it is not sufficient to detect lexical phenomena at the phase of meaning assembly. -Some basic ontological categories (two rough dozens) distinguishing between physical/abstract, animated/inanimate, human/animal. . . that are most commonly distinguished in sentences. -Classes corresponding to different linguistic concepts (a few hundreds); we have suggested classifiers as a basis in [Mery and Retoré, 2013]. -Every common noun, as proposed in [Luo, 2012].

-Every constructible type as a "basic" sort.

In some of the proposals, the base types are further organised using a multi-scalar hierarchy, with type universes, kinds and sub-kinds allowing greater flexibility. The universal aspect of these type systems is also something that is seldom discussed.

We would argue that the types form an ontology of concepts, but can still be idiosyncratic: the "linguistic concepts". As a general principle, when it comes to computational systems, the less information is needed in order to proceed with basic tasks (such as listing the compatibilities between lexical items), the better. We feel that having a type for every word in a given language (about 200,000) would result in a final system that is very demanding in terms of spacial complexity, as well as of Kolmogorov complexity (if each lexical entry is described in terms of words that are compatible with it). Getting closer to the selection classes of language would mean that the type system will have less redundancy, be smaller, and factorise common behaviours; a system of sub-typing for ontological inclusions is, obviously, a strong asset. However, it can be objected that we are not, at this point, able to produce a definitive system of sorts that comply with these objectives, while words have the advantage of being easy to list and ready to use, and certainly form a set of linguistic concepts.

While we certainly do not wish to claim that our assumptions are the only possible or valid type-theoretic formulations of lexical semantics, we wanted to explore the differences between the systems presented in order to understand what might be done in order to render them compatible. It is our belief that, even though the basic hypotheses are different, it is possible to encode any phenomenon presented in one framework using the syntax of any other.

  Specialisation of the polymorphic e for the same type:) `e : P X.(X ! t) ! X `(e {RM}) : (RM ! t) ! RM C : (Derivation of began a book without qualia or sub-typing:) A B `b : RM ! t `(be {RM} (e {RM} b)) : (RM ! Evt) ! H ! t D : (The telic quale:) `fTelic : RM ! REvt E : (The sub-typing derivation:) REvt < Evt RM ! REvt < RM ! Evt (The full sentence:) C D E `(be {RM} (e {RM} b) f Telic ) : H ! t `j : H `(be {RM} (e {RM} b) f Telic j) : t

Many thanks to the attendants of the TYTLES workshop (ESSLLI Barcelona, August

2015), and to many people for long and helpful discussions on the topic of type-theoretic lexical semantics, including but not limited to Asher, Bekki, Chatzikyriakidis, Cooper, Icard, Kubota, Levine, Luo, Mineshima, Moot, Pustejovsky, Real, Satoh, Tanaka, . . . Special thanks to Frédéric Lang, who did a lot of preliminary work on the subject, and to Sergei Soloviev, for clarifications on the history of sub-typing formalisms.

Many other dictionaries note the differences between conventional meaning depending on whether the subject is a dog, an animal other than a dog, or an human being. More details on selection classes, dictionaries and lexical semantics are given in section 2.3.

The coercion can be described by a l -term, unique w.r.t. a suitable notion of equality, such as "modulo b h equivalence"; in practice, this term is often implicit and omitted.

Summary

We presented a suitable relation of sub-typing for L TY n that can be used with a system of types with many sorts and an organised lexicon to provide an analysis for different lexical phenomena from syntax to semantics. In order to provide a fully functional framework, acquiring the relevant sorts, lexical transformations, and ontological inclusions is the next step. Interesting perspectives for this have been provided by crowdsourced data in [START_REF] Chatzikyriakidis | Type Theories and Lexical Networks: Using Serious Games as the Basis for Multi-Sorted Typed Systems[END_REF] and enthymematic learning in [Breitholtz, 2015].

We also have discussed, using ontological inclusions and the common use of subtyping relations in different frameworks, some of the differences between several typetheoretic systems modelling lexical semantics.

Despite those differences, we think that all of these systems share a common understanding of the difficulties involved by the adaptability of human language and the context-dependence of the lexicon, and of the possible type-theoretic mechanisms that can be used to integrate such complex phenomena in formal, compositional frameworks.

As discussed at many occasions, such as the TYTLES workshop in ESSLLI Barcelona, August 2015, we believe that continued exchange of views between that growing community can result in one (or several) complete, functional, integrated frameworks that can successfully compete against big-data, statistical-based approaches to word sense disambiguation.


Livy Real

Christian Retoré

A Case Study of Copredication over a Deverbal that Reconciles Empirical Data with Computational Semantics

This paper is about the automated analysis of the lexical and compositional semantics of nominalisation, in particular of felicitous co-predicationsinfelicitous ones are rejected. We focus on Brazilian Portuguese nominalisation introduced by the suffix -ura but our discussion applies to other nominalisations and languages as well. Much of the theoretical work on deverbals, including ours, concluded that deverbal senses (process, result, location and so on) are rather unforeseeable from the verb and the suffix. The (in)felicity of co-predication, that is the possible conjunction of two predicates which applies to different senses, is also difficult to predict, and in the case of deverbals it is assumed to be (almost) impossible. We present here a study of the CHAVE corpus and show that CHAVE does actually contain some of the supposedly non-existent copredications. We explain our formalisation of sense variation and copredications in our logico-computational framework, Montagovean Generative Lexicon. We analyse the felicitous copredications and explain how the infelicitous ones are rejected. We give possible reasons for the felicity of the observed copredications, and finally we sketch out some guidelines for handling two notions that are necessary to a proper treatment of deverbals, namely local context and degree of semantic felicity.

Nominalisation and copredication

To account for the semantics of a compound expression, one needs to know which sense of its parts contributes, and how it contributes, to the meaning of the whole expression. For this aim, the predicates need to specify the nature of their arguments. For instance to bark applies to animals especially to dogs, while the following sentence is clearly infelicitous apart possibly in fictional contexts, as children stories.

(1) This chair barked.

Once a restricted and precise compositional framework is defined, meaning transfers and coercions require some flexibility in order to account of derived meaning, like:

(2) The head of department barked out orders to his secretary.

When a word has different meanings, an intriguing question is how do we deal with copredication, especially since the pioneering formal work of Pustejovsky [START_REF] Pustejovsky | The generative lexicon[END_REF].

The copredications on diverse senses of a given word is when we conjoin a predicate applied to one sense with another predicate applied to another sense, see next sentence:

(3) Your translation of Priscian's text, which has been revised many times, was placed on the desk.

Here one single token of the word translation refers both to the result of the act of translate and to its process.

Copredications are a challenge to the formal linguistic communities: their proper treatment, in a system that implements restrictions of selection, is difficult. Furthermore, as the senses themselves, they obey to language specific constraints that are possibly totally idiosyncratic and hard to learn from corpora -it is difficult to tell whether a copredication is rare or simply absent because it cannot be said. The computational semantics of co-predication, that is an automated semantic analysis which computes the semantics of meaningful sentences and rejects semantically ill-formed sentences, is quite difficult to achieve -let alone the difficulty to acquire the sophisticated lexicons that such an analyzer requires. Here are some examples of felicitous and infelicitous copredications: 3(4) Liverpool is a poor town.

(5) Liverpool is an important harbor. [START_REF] Grimshaw | Argument Structure[END_REF] Liverpool is an important club. [START_REF] Ježek | Nominals, Polysemy, and Co-predication[END_REF] Liverpool is a poor town and an important harbor.

(8) * Liverpool is an important harbor and an important club as well.

Another issue that has been intensively discussed, both in purely linguistic and formal studies [START_REF] Grimshaw | Argument Structure[END_REF][START_REF] Pustejovsky | The generative lexicon[END_REF], since at least [START_REF] Cruse | Lexical semantics. Cambridge textbooks in linguistics[END_REF], is the derivation of the sense of a nominalisation. Nominalisations are nouns derived from other part-of-speech words, as verbs (such nominalisations are called deverbals), adjectives or even nouns. How does one derive the possible senses of a nominalisation from the sense of the initial word? It seems there is not final answer to it, specially because many nominalisations have idiosyncratic meanings, as estacionamento ('parking') formed by estacionar ('to park') + -mento. The suffix -mento forms eventive nouns, but estacionamento has only the locative reading and not the eventive one and this cannot be compositionally predicted.

Nominalisations have been intensively studied in linguistics, but also in NLP and knowledge representation. Some rules managing their behavior have been proposed, but nothing fully satisfying appeared. Indeed, nominalisations cause a problem to every lexical theory that understands words (or parts of words, or lexicalised expressions) as static parts of discourse. Sometimes nominalisations behave as common nouns (without any special 'deep' structure), sometimes they seem to require specific arguments (and contexts) to achieve a given sense. According to some studies, as [START_REF] Grimshaw | Argument Structure[END_REF], those arguments correspond to the argument structure of the associated verb, but many lexical theories cannot account for this morpho-semantic relation (see [START_REF] Real | Nominalizac ¸ões[END_REF] for a deeper discussion).

Moreover, many examples indicate that the relationship between a verb and its nominalisation is not necessarily based on the argument structure of the verb, it may involve circumstantial complements as well, as noted by [START_REF] Real | Deverbal semantics and the Montagovian generative lexicon LTy n[END_REF]. This error comes from studies that only consider prototypical nominals (as construction, destruction and translation) with a set of possible senses restricted to process and result, which does not account for the many difference senses a nominalisation can have. Of course one may infer from corpora or lexical networks some information about the deverbal meaning but computing the lexical sense of the deverbal needed for compositional semantics from the one of the associated verb and the suffix is presently out of reach. Derivational relations, that are morphological relations that link a noun-verb pair can be extracted but extracting semantic information is much harder. Most of lexical resources do not have this kind of semantic information encoded as it has to be done manually and it is a very time consuming task that also needs to be done by experts. However, biggest resources have been encoding the semantic type of a nominalisation manually. In [START_REF] Osherson | Putting semantics into wordnet's morphosemantic links[END_REF], Princeton Wordnet team explains how they encoded semantic information about deverbals: starting from a previous list of morphological related items, they manually checked each pair to introduce in Princeton's Wordnet morpho-semantic links, that relate verb-noun pairs through typed relations, as agent, event and instrument.

To close this section, we present below two examples of copredications involving three senses of the Portuguese deverbal assinatura ('signature, signing') -i. discussion leading to an agreement, ii. writing act, iii. result all of them as a grapheme: (9) * A assinatura atrasou três dias e estava ilegível.

The signature took three days and was illegible.

(10) A assinatura furou a folha e estava ilegível.

The signature pierced the sheet of paper and was illegible.

Our computational-logic framework

We use a logical framework known as Montagovian Generative Lexicon (MGL), deeply presented in [START_REF] Retoré | The Montagovian Generative Lexicon L Ty n : a Type Theoretical Framework for Natural Language Semantics[END_REF] and firstly introduced in [START_REF] Mery | A Montagovian generative lexicon[END_REF]. Inspired by Pustejovsky's Generative Lexicon [START_REF] Pustejovsky | The generative lexicon[END_REF], MGL extends Montague semantics and can also be viewed as an extension of Muskens'Ty n , so it is also called as MGL L Ty n . As a formal definition of MGL would be too lengthy, we present here informally its principles, stressing the similarities and differences with related systems. As Montague's initial work [START_REF] Montague | English as a formal language[END_REF], this system is closely related to syntax, in particular to categorial grammar. If the semantic lexicon is known, i.e. if the association of typed terms to words is known, as in [START_REF] Montague | English as a formal language[END_REF], then the semantic representation can be automatically computed. First one substitutes in the parse tree the lexical lambda terms for the words, then one obtains a properly typed term, which possibly makes use of the optional terms. After reduction, this yields a logical formula of higher order predicate calculus (or nothing: if the type mismatch is not resolved, it is an infelicitous sentence). The reduction, which is a strongly normalizing process, guarantees that the semantic interpretation of a sentence always is a logical formula, i.e. something that makes sense.

There are no specific rules for combining those terms. Only applications and abstractions rules are needed -there can be some abstraction steps, because Lambek syntactic parse trees may contain some abstractions.

The semantic part of the lexicon associates each word with a main typed lambda term (the usual Montague lambda term) and several optional typed lambda terms (including at least the identity of the type of the main term). These typed terms are in system F: basically F is an easy generalization of simply typed lambda calculus which allows to factorise terms that act uniformly. 4 Here are some more details on types and terms:

-Base types includes t (type of propositions, that we would better call prop) but in addition to Montague there are many base types for entities e i that can be considered as sorts of a many sorted logic. -In addition to base types, we have type variables, and in addition to functional types, we have quantified types obtained by quantification over a type variable, i.e. we are using second order typed lambda calculus (Girard's system F, introduced in [START_REF] Girard | Une extension de l'interprétation de Gödel à l'analyse et son application: l'élimination des coupures dans l'analyse et la théorie des types[END_REF]). -Quantification over type variables is used to factorize operations like conjunction, or for determiners and quantifiers that acts uniformly over types -for instance the conjunction "and" used for copredication over a sense of a word w of a type B and another sense of the word w of another type A with two predicates of respective types a ! t and b ! t will apply to all types a and b, hence will contain a quantification over all types. But having many simply typed terms, e.g. for the conjunction used for copredication, would work just the same.5 -Reduction is defined as expected:

(l x U t Y )(u U ) ! t[x U := u U ] u is a term of type U and x a variable of type U (L Xt){T } ! t[X := T ] X is a type variable and U is a type
An important property is that reduction is a strongly normalising process, which preserves the types. Thus, the semantic lambda term associated with a sentence of type t reduces to a normal term of type t: it is easily seen (see e.g. [10, chap 3]) that this normal term, which only involves logical constants and predicates, corresponds to a logical formula.

Intuitively the optional lambda terms, which have a functional type, allow to view a word w of type A as a word of type B hence they give access to a sense of w different from the initial one that is described by the type of the main lambda term of w.

Semantic infelicity is described by type mismatch, i.e. when a functional term of type A ! X is applied to a term t of type B with B 6 = A.

The optional terms that the lexicon associate with the words of the analysed phrase are used to fix such type mismatches. To fix a type mismatch as above, the lexical entry of one of the words should have an optional term of type B ! A.

Incompatibilities are accounted for as folllows. When one sense associated with a word (that corresponds either to the main lambda term or to an optional lambda term) is incompatible with any other, the optional lambda term corresponding to this sense (or the identity type in the case of the main lambda term) is declared in the lexicon to be rigid -otherwise the term is said to be flexible.

Quantification over types allows a factored treatment of conjunction which is extremely useful for modeling copredications, as discussed in [15, Section 4.1] or [START_REF] Retoré | The Montagovian Generative Lexicon L Ty n : a Type Theoretical Framework for Natural Language Semantics[END_REF]: whenever an object x of type x can be viewed both as an object of type a to which a property P a!t applies and as an object of type b to which a property Q b !t applies (via two terms f 0 : x ! a and g 0 : x ! b ), the fact that x enjoys P ^Q can be expressed by the unique polymorphic term:

& p = L aL b l P a!t l Q b !t L x l x x l f x !a l g x !b . (& t!t!t (P ( f x))(Q (g x)))
Using F or any other Type Theory, as Martin Löf's theory, does not make a big difference: the impredicative quantification instead of predicative quantification does not make a big difference. The difference rather lies in the organization of the lexicon with optional lambda terms. The semantic representations are formulae of higher order many sorted logic and not type theoretical formulae -the later ones cannot be interpreted in usual set theoretic models.

A typical case: -ura in Brazilian Portuguese

In some previous papers [START_REF] Real | Deverbal semantics and the Montagovian generative lexicon LTy n[END_REF], we studied the nominalizations in Brazilian Portuguese (BP from now on) both formally and empirically and especially the ones that are constructed with the -ura suffix as abertura 'opening', assinatura 'signing/signature' and brancura 'whiteness'. The choice of the suffix -ura was made for many reasons: this work can certainly be used to understand other similar nominalisation suffixes at least in other neo-Latin languages as -ure in French (coupure 'cut'), -ura in Catalan (obertura 'opening') and -ura Spanish (blancura, 'whiteness') and possibly be extended to not so close languages, as German and its suffix -ung and -ní in Czech.

Our descriptive and theoretical previous work on these deverbals led us to the following conclusions:

-From a corpus based analysis [START_REF] Real | An overview on portuguese nominalisation[END_REF], we provided 13 types that -ura can mean: event, result, physical result, locative, collective, means, property, instrument, a given portion, rest, function, duration of a function, science/art. From theoretical words, it was known to have at least eight senses ( [START_REF] José | Formac ¸ão de palavras no português brasileiro contemporâneo[END_REF][START_REF] Luiz | A nominalizac ¸ão no português do brasil[END_REF][START_REF] Nonymous | This is one the papers involving at least one of the authors[END_REF]), including eventive, resultative, locative and collective readings, just as other well-known morphemes like -ung in German and -age in French. -It seems hopeless to predict the senses of -ura nominalisations, as well as the compatibility of these sense, at least for the time being no one knows how they could be acquired from corpora. Observe that senses are listed in a dictionary, while their pairwise compatibility is not.

-When the senses and their compatibility w.r.t. copredication are known, i.e. when someone can write the lexical entries, MGL faithfully models the deverbal, i.e. computes exactly the correct semantic representation(s), in particular it accepts exactly the possible copredications -and rejects the ones that are semantically illformed.

-Before the present corpus study, we concluded from examples in the literature that copredication was almost always impossible on the different senses of a deverbal, a possible exception being example [START_REF] Moot | The logic of categorial grammars: a deductive account of natural language syntax and semantics[END_REF] above.

Although the literature and our own work agreed that copredication on the different senses of a deverbal are almost impossible, we started to have doubts about this claim and wanted to confront it to corpus study. Indeed, while corpora cannot prove that a construction does not exist, they can prove that a construction exists, by providing meaningful examples of the construction.

As far as we know, [START_REF] Real | An overview on portuguese nominalisation[END_REF] was the first work that considers nominalisations in corpora, at least for Portuguese. It offers an enriched description of a relevant fragment of deverbal nominals in BP, considering all possible senses of each nominalisation, all possible co-predications over those senses and a large corpora analysis based on Corpus Brasileiro6 that did not find any co-predication with words formed by -ura. Therefore we generated a large list of possible co-predications and tested their acceptability with three native speakers of (Brazilian) Portuguese and most of the sentences were rejected by the three of them. The acceptable sentences were often using very frequent words of the common Brazilian lexicon. Thus [START_REF] Real | An overview on portuguese nominalisation[END_REF] claimed that the acceptability of co-predication structures is correlated to the frequency of the considered -ura word. Although this work provided some results, we decided to search for co-predication with words formed by -ura in another corpus keeping in mind the influence of the frequency on the felicity of the copredication. Indeed, the corpus Brasileiro is not a very reliable corpus 7 , it has much noise and does not seem to be correctly tagged.

A closer look at data

Thus, we searched for copredications involving -ura words in another corpus, we chose the CHAVE corpus 8 , which contains news from two relevant newspapers of Portugal (Publico) and of Brazil (Folha de São Paulo) from 1994 and 1995, with almost 1 million words.

Our search looked for the syntactic structures that may produce acceptable copredications, namely the ones of following the criterion given in [START_REF] Ježek | Nominals, Polysemy, and Co-predication[END_REF] ii. Temporal disjunction between the two predications; -The construction is taking ages and is of Victorian style. (no temporal disjunction) -The construction is taking ages and will be of Victorian style. (with temporal disjunction) iii. Omission of the internal argument.

-The construction took ages and is of Victorian style. (without internal argument) -The construction of the house took ages and is of Victorian style. (with internal argument) Here we bring an example from [START_REF] Ježek | Nominals, Polysemy, and Co-predication[END_REF] of a sentence that respects those constrains:

(11) La costruzione, che si protrasse fino al XVII secolo, rimane un'importante testimonianza della geniale tematica del Palladio.

'The building, which continued till the XVII century, represents an important evidence of Palladio's ingenious artwork.'

Above we have two split clauses that point to different periods of time and a possible internal argument of 'building' is missing.

Since the main structure of subordinate clauses in BP is formed by ', que' (, that), we captured sentences with main clause and subordinate clause (unfortunately, there was no way to search in the corpora temporal disjunction) by the formal expression:

[lema="-ura word"][func="N<.*"]*\,[pos="N"]*"," "que"

From our previous work [START_REF] Real | An overview on portuguese nominalisation[END_REF], we have found that the frequency of occurrence of a given word in the lexicon can have an important role on co-predication acceptability. So, from a list of all -ura nominalizations in Portuguese, obtained on OpenWordnet-PT 10 , we separated all the nominalizations into three blocks: very frequent, frequent and rare words, considering the total number of occurrences of each word in CHAVE. Then we arbitrarily picked 2 words from each of these groups and searched on CHAVE corpus for copredications with them 11 A legislature dur^f unc that [starts at the same time that a president probably gets elected in two rounds] dur!t [should wait for the key messages of the president to be sent] f unc!t , in order to substantiate the government program.

From 437 found sentences, 4 copredications represent 0,9% which is a high number if we consider that this kind of morpho-syntatic construction is expected to not normally happen.

Let us take 12 as an example. In 12, 'assinatura' ('signing/signature') in a assinatura que ele trac ¸a a partir do próprio nome 'the signature that he draws from his own name" has the physical result reading, as it means the mark on the paper that comes from the signing event. Therefore 'assinatura' in a assinatura coroa a chegada do patrimônio '[the signing] crowns the arrival of heritage' is related to the event of signing: it is not the mark on the paper that 'crowns the arrival of heritage', but the fact that the document was signed. It is interesting to note that the same lexical item in Portuguese that concomitantly belongs to the type event and to the type physical result is expressed in English by two different words: 'signature' (the physical result) and 'signing' (the event itself).

Here we consider the follow structure as the lexical entry for assinatura (that actually can hold much more types than these two types):

word main l -term optional l -terms assinatura l x v .sig(x) : v ! t Id v : v ! v ( f lexible) f j : v ! j ( f lexible)
Fig. 1. The lexical entry for assinatura

In Figure 1 the base types are defined as follows: v (events) and j (physical objects).

(16) the main term is l x v .(sig v!t x) [START_REF] Retoré | Typed Hilbert epsilon operators and the semantics of determiner phrases[END_REF] and the optional morphisms are a. Id v = l x v .x, the (always present) identity (referring to the agreement process) which is declared as rigid, b. f v!j j turning the event into a physical object and declared to be flexible.

For a semantic analysis of assinatura in those two clauses, the first step is the composition of assinatura with the definite article a ('the'), which we deeply explain in [START_REF] Real | Deverbal semantics and the Montagovian generative lexicon LTy n[END_REF][START_REF] Retoré | Typed Hilbert epsilon operators and the semantics of determiner phrases[END_REF]. Here we assume (sig) v as the formula of a assinatura, considering assinatura applied to a/the = il a.(a ! t) ! a .

Let us recall the polymorphic conjunction from section 2:

& P = L aL b l P a!t l Q b !t L x l x x l f x !a l g x !b .(t t!t!t (P ( f x))(Q (g x)))
Briefly, variable Q and P are the predicates to be copredicated, which apply to objects of types a and b respectively. Objects of type x can be mapped via f and g into a and b objects respectively ("viewed as"). The following term is able in any such situation to conjoin two different predicates that respectively apply to two different senses of the same lexical entry, if this lexical entry has a function ( f and g) that changes its previous type (a and b ) to the required type (x ). An easy example is a book which can be viewed as a physical object and as an informational content. Therefore a book can be both heavy and interesting, as the predicate heavy applies to the physical sense of book and the predicate interesting applies to the informational sense.

The instantiations for our example should be as follows:

-a = v, P = crown v!t , f = f v , -b = j, Q = draw j!t , g = f j , -x = v, x = sig v .
where: sig v = a assinatura crown v!t = [a assinatura] coroa a chegada do patrimônio '[the signing] crowns the arrival of heritage' draw j!t = trac ¸a [a assinatura] a partir do próprio nome 'he draws [the signature] from his own name'

The polymorphic "and" & P takes as arguments two properties P (here: crown v!t ) and Q (here: draw j!t ) of entities of respective type a (here: v) and b (here: f ), and returns a predicate that applies to a term x of type x . This predicate says that:

if x of type x (here sig v of type v):

• enjoys P (here crown( f v (x))) when viewed as an object of type a (here v) via f v • enjoys Q (here draw( f j (x))) when viewed as an object of type b (here j) via g (here f j ) then x endowed with the proper meanings has both properties that is crown( f v (x)) & P draw( f j (x)) Hence the copredication in example ( 12) can be derived.

Observe that if (as wrongly asserted in some papers) the senses of assinatura were incompatible, we could not derived this copredication. If they were incompatible then, in our semantic lexicon one of the two morphisms (identity and the function that maps the main sense to physical object sense) would be declared to be rigid. Here they are both flexible hence can be used both, as we did to derive the meaning of the copredication.

Consequences and ongoing work

These examples show that the copredications between different senses of a deverbal are not impossible as they are said to be in theoretical studies and in our previous corpus based analysis. Actually, our previous work considers only Brazilian Portuguese and CHAVE corpus is composed by European and Brazilian Portuguese, which leaves us the task of reproducing this experiment with two very reliable corpora of different kinds of Portuguese to check if there is a language specific constrain on copredications.

Our previous suggestion that the role of word frequency in the lexicon is also relevant to the felicity of copredications was not entirely proved, as from the six chosen words we work with, the most frequent word, abertura, does not appear in any copredication structure. However we do not consider this present work evidence enough to discard this suggestion and we still believe that the frequency of occurrence of a given word may play a role in copredication constructions.

Our MGL formalisation and computational analysis reject the infelicitous copredications and compute the semantics of felicitous copredications (just as we analysed the homemade and slightly far-fetched example [START_REF] Moot | The logic of categorial grammars: a deductive account of natural language syntax and semantics[END_REF] or rejected [START_REF] Montague | English as a formal language[END_REF] in [START_REF] Real | Deverbal semantics and the Montagovian generative lexicon LTy n[END_REF]).

The examples above of correct copredications suggest some hints on the semantic conditions that make copredications possible (in addition to the syntactic constraints reminded above). A likely semantic explanation, which is in the air at least for ordinary common nouns like institutions and the which we would like to confirm, is that the copredications are more felicitous when either:

1. The two predications belong to the same domain which gathers the two senses. Hence this possibility depends on the local context and this probably can be related to the local coercions discussed in [START_REF] Asher | Formalization of coercions in lexical semantics[END_REF]. Here we have local compatibilities, and once more for semantics the (local) context plays a central role. However since the relation between this model is up to now unrelated to syntax or discourse structure, it is presently hard to tell what is exactly the domain of locality of local coercions.

We can try to explain the felicity of the copredications over deverbals that we found in the CHAVE corpus by means of the two principles suggested above. This is just a try, which deserves to be discussed from data and psycholinguistic experiments. In the first CHAVE example, [START_REF] Pustejovsky | The generative lexicon[END_REF], the notion of achievement, supported by the word coroa ('crowns'), gathers in the same context both senses, i. resulting grapheme and ii. agreement process, that are used by the two predicates. In the second CHAVE example [START_REF] Real | Nominalizac ¸ões[END_REF], both senses of assinatura, as a grapheme and as the informational content expressed by the grapheme, share the notion of engagement (to defend the text). In the third CHAVE example [START_REF] Real | An overview on portuguese nominalisation[END_REF], the two aspects, result and process (brief, when compared to the time that elapsed since then) can be somehow identified when viewed from several thousand years later, the common feature being the notion of an instantaneous event, sense which is supported by the verb floresceu. In the fourth and last CHAVE example [START_REF] Real | Deverbal semantics and the Montagovian generative lexicon LTy n[END_REF], both predicates speak about a duration, since the people who should be waiting are precisely the ones elected for this duration. In the example [START_REF] Moot | The logic of categorial grammars: a deductive account of natural language syntax and semantics[END_REF], the opposition between the time to write the signature and its illegibility makes the copredication works, since world knowledge makes us to expect that things which take a long time to be made are good/useful.

As we may observe from these last examples or from example [START_REF] Moot | The logic of categorial grammars: a deductive account of natural language syntax and semantics[END_REF], the semantic felicity is not something that either holds or not. Hence a scalable notion of acceptance/felicity would be much welcome. For type theoretical framework like MGL and others, scalable type judgment (semantic representation of type t) as initiated by [START_REF] Cooper | A probabilistic rich type theory for semantic interpretation[END_REF] should be relevant. This scalability question is somehow orthogonal to the precise framework and therefore should be easy to implement for MGL once scales are integrated to type judgment.

We are presently exploring these possibilities.

9 :

 9 Syntactic constraints on co-predication on different meanings of a nominalisation: i. Split co-predication between main clause and subordinate clause; -The construction took ages and is of Victorian style. (non split) -The construction which took ages is of Victorian style. (split) -The construction which is of Victorian style took ages. (split)

( 18 )

 18 Barcelona organised the Olympiads and won four Champions Leagues. (institution/ football club) 2. There is an opposition between two predications.

( 19 )

 19 Huelva, a small southern town, defeated Barcelona. (institution/ football club)

  . Por outro lado e em um domínio totalmente diferente, a assinatura j^v que [ele trac ¸a a partir do próprio nome] j!t [coroa a chegada do patrimônio] v!t .On the other hand and in a totally different field, the signature j^v that [he draws from his own name] j!t [crowns the arrival of heritage] v!t .The pieces now in the Pushkin Museum in Moscow, and before the Prehistory and Ancient History Museum of Berlin, were found at the time of Troy 2, a town with an [imposing] r!e architecture v^r that [flourished 2500-2200 BC] v!t in the Bronze Age.(15) Function Duration + FunctionUma legislatura dur^f unc que [comec ¸a ao mesmo tempo em que comec ¸a o mandato de um presidente eleito provavelmente em dois turnos] dur!t [deverá aguardar o envio das principais mensagens do presidente] f unc!t , que consubstanciem o seu programa de governo.

			# occurrences
			in possible
		# occurrences co-predication
	W O R D	in CHAVE structures
	abertura	12900	221
	assinatura 4194	152
	arquitetura 1105	28
	legislatura 990	46
	moedura	1	0
	enxaguadura 0	0
	From the previous expression, we found 437 sentences we manually analysed
	searching for co-predications, and 4 of them can be considered co-predications:
	(12) Physical result (j)+ event(v)

[START_REF] Real | Nominalizac ¸ões[END_REF]

Physical Result + Informational Content Retomando o texto, mostra como [várias entidades se organizaram e recolheram] j!t em momentos diferentes milhares de assinaturas j^in f o que [reforc ¸aram e respaldaram as conquistas da populac ¸ão contra a empresa poluidora] in f o!t . Returning to the text, it shows how [various entities organized themselves and collected] j!t at different times thousands of signatures j^in f o which have [strengthened and endorsed the achievements of the population against the polluting company] in f o!t . (14) Result + Event As pec ¸as agora no Museu Pushkin, em Moscou, e antes no Museu de Pré-História e História Antiga de Berlim, foram achadas na altura de Tróia 2, uma cidade com arquitetura v^r [imponente] r!e que [floresceu de 2.500 a 2.200 a.C.] v!t , na Idade do Bronze.

A "*" indicates a bad sentence, either semantically of syntactically.

Let us take a programming example: a sorting function, like QuickSort, applies to a list of A object, and an order on A object and yields a sorted list of A objects: you can factor such a program/term over all types A, because it works just the same on every type.

We also have subtyping, in particular on the sorts e i , but we do not focus on it here.

Corpus developed by Tony Berber Sardinha & alii, trough AC/DC tool[START_REF] Santos | O projecto ac/dc: acesso a corpora/disponibilizac ¸ão de corpora[END_REF]. Freely available on http://www.linguateca.pt/ACDC.

We would like to thank Claudia Freitas for the discussions on corpora choice and the help with the search engines of Linguateca corpora.

Freely available on http://www.linguateca.pt/acesso/corpus.php?corpus=CHAVE.

As shown in[START_REF] Real | Deverbal semantics and the Montagovian generative lexicon LTy n[END_REF], this criterion is neither sufficient nor necessary for the felicity of copredications, but we thought that copredications following these patterns are more likely to occur.

Freely available on http://wnpt.brlcloud.com/wn/.

Abertura 'opening', assinatura 'signing/signature', arquitetura 'architecture', legislatura 'legislature', moedura 'milling', 'textitmoedura 'rinse'.