text:
Giorgio Gubbiotti
email: giorgio.gubbiotti@syndey.edu.au

DINH Nalini Joshi
email: nalini.joshi@syndey.edu.au

Thi Tran
email: dinhthi.tran@syndey.edu.au

Claude-Michel Viallet
email: claude.viallet@upmc.fr

COMPLEXITY AND INTEGRABILITY IN 4D BI-RATIONAL MAPS WITH TWO INVARIANTS

Keywords:

In this letter we give fourth-order autonomous recurrence relations with two invariants, whose degree growth is cubic or exponential. These examples contradict the common belief that maps with sufficiently many invariants can have at most quadratic growth. Cubic growth may reflect the existence of non-elliptic fibrations of invariants, whereas we conjecture that the exponentially growing cases lack the necessary conditions for the applicability of the discrete Liouville theorem.

Introduction

Bi-rational maps in two dimensions have played a crucial role in the study of integrable discrete dynamical systems since the seminal paper of [START_REF] Penrose | A quadratic mapping with invariant cubic curve[END_REF] and the introduction of the QRT mappings in [START_REF] Quispel | Integrable mappings and soliton equations[END_REF][START_REF] Quispel | Integrable mappings and soliton equations II[END_REF]. Elliptic curves and rational elliptic surfaces proved to be one of the main tools in understanding the geometry behind this kind of integrability, see [START_REF] Duistermaat | Discrete Integrable Systems: QRT Maps and Elliptic Surfaces[END_REF][START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé Equations[END_REF][START_REF] Tsuda | Integrable mappings via rational elliptic surfaces[END_REF]. In this letter we give examples of higherorder maps whose properties go beyond those of the two-dimensional maps, and show that the geometry of elliptic fibrations is no longer sufficient to explain their behaviour.

Up to now the QRT mappings appear to describe almost totality of the known integrable examples in dimension two. With some notable exceptions [START_REF] Duistermaat | Discrete Integrable Systems: QRT Maps and Elliptic Surfaces[END_REF][START_REF] Viallet | On the integrability of correspondences associated to integral curves[END_REF], no general framework exists for higher order maps. A generalization of the QRT scheme [START_REF] Quispel | Integrable mappings and soliton equations[END_REF][START_REF] Quispel | Integrable mappings and soliton equations II[END_REF] in dimension four was given in [START_REF] Capel | A new family of four-dimensional symplectic and integrable mappings[END_REF]. Certain maps obtained in [START_REF] Capel | A new family of four-dimensional symplectic and integrable mappings[END_REF] were shown in [START_REF] Hay | Hierarchies of nonlinear integrable q-difference equations from series of Lax pairs[END_REF] to be autonomous reductions of members of q-Painlevé hierarchies (multiplicative equations in Sakai's scheme [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé Equations[END_REF]). Since hierarchies are known also for the additive discrete Painlevé equations [START_REF] Cresswell | The discrete first, second and thirty-fourth Painlevé hierarchies[END_REF], it is clear that the cases considered in [START_REF] Capel | A new family of four-dimensional symplectic and integrable mappings[END_REF] cannot exhaust all the possible integrable autonomous maps in four dimensions, as already shown in [START_REF] Joshi | Rational maps with invariant surfaces[END_REF]. We mention that other examples of discrete mappings of higher orders were produced either by periodic or symmetry reduction of integrable partial difference equations [START_REF] Levi | Continuous symmetries of difference equations[END_REF][START_REF] Papageorgiou | Integrable mappings and nonlinear integrable lattice equations[END_REF][START_REF] Quispel | Integrable mappings derived from soliton equations[END_REF][START_REF] Van Der Kamp | The staircase method: integrals for periodic reductions of integrable lattice equations[END_REF] or as Kahan-Hirota-Kimura discretization [START_REF] Kahan | Unconventional numerical methods for trajectory calculations[END_REF][START_REF] Kimura | Discretization of the Lagrange top[END_REF] of continuous integrable systems [START_REF] Celledoni | Geometric properties of Kahan's method[END_REF][START_REF] Celledoni | Integrability properties of Kahan's method[END_REF][START_REF] Petrera | On integrability of Hirota-Kimura type discretizations: Experimental study of the discrete Clebsch system[END_REF][START_REF] Petrera | On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top[END_REF].

In this letter, we focus on the study of integrability properties of autonomous recurrence relations. Here a autonomous recurrence relation is given by bi-rational map of the complex projective space into itself: [START_REF] Arnol'd | Dynamics of complexity of intersections[END_REF] ϕ :

[x] ∈ CP n → [x ′ ] ∈ CP n ,
where n > 1 1 . We take [x] = [x 1 : x 2 : • • • : x n+1 ] and [x ′ ] = x ′ 1 : x ′ 2 : • • • : x ′ n+1 to be homogeneous coordinates on CP n . In this language non-autonomous recurrence relations of order n are represented by bi-rational maps Φ : CP n+1 → CP n+1 , where one of the variables x k , for k ∈ {1, . . . , n + 1} fixed, in affine coordinates has a linear evolution. Moreover we recall that a bi-rational map is a rational map ϕ : V → W of algebraic varieties V and W such that there exists a map ψ : W → V , which is the inverse of ϕ in the dense subset where both maps are defined [START_REF] Shafarevich | Basic Algebraic Geometry 1, volume 213 of Grundlehren der mathematischen Wissenschaften[END_REF].

Integrability for autonomous recurrence relations (discrete equations) can be characterized in different ways. In the continuous case, for finite dimensional systems, integrability is usually understood as the existence of a "sufficiently" high number of first integrals, i.e. of non-trivial functions constant along the solution of the differential system. In the Hamiltonian setting a characterization of integrability was given by Liouville [START_REF] Liouville | Note sur l'intégration des équations différentielles de la Dynamique, présentée au Bureau des Longitudes le 29 juin 1853[END_REF]. In the case of maps (1) the analogue of first integrals are the invariants. To be more precise we state the following: Definition 1. An invariant of a bi-rational map ϕ : CP n → CP n is a homogeneous function I : CP n → C such that it is left unaltered by action of the map, i.e. For n > 1, an invariant is said to be non-degenerate if:

(3)

∂I ∂x 1 ∂I ∂x n = 0.
Otherwise an invariant is said to be degenerate.

In what follows we will concentrate on a particular class of invariants:

Definition 2. An invariant I is said to be polynomial, if in the affine chart [x 1 : • • • : x n : 1] the function I is a polynomial function.
A polynomial invariant in the sense of definition 2 written in homogeneous variables is always a rational function homogeneous of degree 0. The form of the polynomial invariant in homogeneous coordinates is then given by: (4)

I ([x]) = I ′ ([x]) t d , d = deg I ′ ([x]
) , where deg is the total degree.

To better characterize the properties of these invariants we introduce the following: Definition 3. Given a polynomial function F : CP n → V , where V can be either CP n or C, we define the degree pattern of F to be: [START_REF] Byrnes | Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures[END_REF] dp

F = deg x1 F, deg x2 F, . . . , deg xn F . Example 1. Consider the following map in CP 2 : (6) ϕ : [x : y : t] → [-y(x 2 -t 2 ) + 2axt 2 : x(x 2 -t 2 ) : t(x 2 -t 2 )]
This map is known as the McMillan map [START_REF] Mcmillan | A problem in the stability of periodic systems[END_REF] and possesses the following invariant:

(7)

t 4 I McM = x 2 y 2 + (x 2 + y 2 -2axy)t 2
1 Bi-rational maps in CP 1 are just Möbius transformations so everything is trivial.

We have dp I McM = (2, 2), i.e. it is a bi-quadratic polynomial. We also note that the invariant of a QRT map [START_REF] Quispel | Integrable mappings and soliton equations[END_REF][START_REF] Quispel | Integrable mappings and soliton equations II[END_REF], I QRT , which is a generalization of the McMillan map [START_REF] Capel | A new family of four-dimensional symplectic and integrable mappings[END_REF], is the ratio of two bi-quadratic in the dynamical variables of CP 2 . Hence QRT mappings leave invariant a pencil of curves of degree pattern (2, 2).

Example 2. The invariants of the maps presented in [START_REF] Capel | A new family of four-dimensional symplectic and integrable mappings[END_REF], I CS , are are ratios of biquadratic in all the four dynamical variables of CP 4 , i.e. ratios of polynomial of degree pattern (2, 2, 2, 2). In this sense the classification of [START_REF] Capel | A new family of four-dimensional symplectic and integrable mappings[END_REF] is an extension of the one in [START_REF] Quispel | Integrable mappings and soliton equations[END_REF][START_REF] Quispel | Integrable mappings and soliton equations II[END_REF].

Finally we will consider invariants are not of the most general kind, but satisfy the following condition Definition 4. We say that a invariant I : CP n → C is symmetric if it is left unaltered by the following involution: [START_REF] Celledoni | Integrability properties of Kahan's method[END_REF] ι : [x 1 :

x 2 : • • • : x n : x n+1 ] → [x n : x n-1 : • • • : x 1 : x n+1 ] ,
i.e. ι * (I) = I.

We then have the following characterization of integrability for autonomous recurrence relations:

(i) Existence of invariants A n-dimensional
J li ∂J jk ∂w l-1 + J lj ∂J ki ∂w l-1 + J lk ∂J ij ∂w l-1 = 0, ∀i, j, k. and (10) d ϕJ(w) d ϕ T = J(w ′ ),
where d ϕ is the Jacobian matrix of the map ϕ, see [START_REF] Capel | A new family of four-dimensional symplectic and integrable mappings[END_REF][START_REF] Olver | Applications of Lie Groups to Differential Equations[END_REF]. The Poisson bracket of two smooth functions f and g is defined as

(11) {f, g} = ∇f J (∇g) T ,
where ∇f is the gradient of f . We can easily see that {w i-1 , w j-1 } = J ij . We note that in the case where the Poisson structure has full rank, i.e. n = 2r, we only need n/2 invariants which are in involution. In this case the Poisson matrix is invertible, and its inverse is called a symplectic matrix. A symplectic matrix give raise to a sympectic structure. (iii) Existence of a Lax pair [START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF] A n-dimensional map is integrable if it arises as compatibility condition of an overdetermined linear system. (iv) Low growth condition [START_REF] Bellon | Algebraic entropy[END_REF][START_REF] Falqui | Singularity, complexity, and quasi-integrability of rational mappings[END_REF][START_REF] Veselov | Growth and integrability in the dynamics of mappings[END_REF] A n-dimensional bi-rational map is integrable if the degree of growth of the iterated map ϕ k is polynomial with respect to the initial conditions [x 0 ]. Integrability is then equivalent to the vanishing of the algebraic entropy:

(12) ε = lim k→∞ 1 k log deg [x0] ϕ k .
Algebraic entropy is a measure of the complexity of a map, analogous to the one introduced by Arnol'd [START_REF] Arnol'd | Dynamics of complexity of intersections[END_REF] for diffeomorphisms. In this sense growth is given by computing the number of intersections of the successive images of a straight line with a generic hyperplane in complex projective space [START_REF] Veselov | Growth and integrability in the dynamics of mappings[END_REF]. We underline that the above list is not meant to be completely exhaustive of all the possible definitions of integrability. Since we are focused on autonomous recurrence relations we choosed to cover only the most used definition for these ones. Additional definitions of integrability have been proposed by other for nonautonomous systems.

Remark 1. In principle, the definition of algebraic entropy in equation ( 12) requires us to compute all the iterates of a bi-rational map ϕ to obtain the sequence

d k = deg [x0] ϕ k ∞ k=0
. Fortunately, for the majority of applications the form of the sequence can be inferred by using generating functions [START_REF] Lando | Lectures on Generating Functions[END_REF]:

(13) g (z) = ∞ n=0 d k z k .
A generating function is a predictive tool which can be used to test the successive members of a finite sequence. It follows that the algebraic entropy is given by the logarithm of the smallest pole of the generating function, see [START_REF] Grammaticos | How to detect the integrability of discrete systems[END_REF][START_REF] Gubbiotti | Integrability of difference equations through algebraic entropy and generalized symmetries[END_REF].

Remark 2. The condition of Liouville integrability [START_REF] Bruschi | Integrable symplectic maps[END_REF][START_REF] Maeda | Completely integrable symplectic mapping[END_REF][START_REF] Veselov | Integrable maps[END_REF] is stronger than the existence of integrals. Indeed, for a map, being measure preserving and preserving a Poisson/symplectic structure are very strong conditions. However, they lead to a great drop in the number of invariants needed for integrability. The same can be said for the existence of a Lax pair, since it is well known that a well posed Lax pair gives all the invariants of the system through the spectral relations. Finally, the low growth condition means that the complexity of the map is very low, and it is known that invariants help in reducing the complexity of a map. Indeed the growth of a map possessing invariants cannot be generic since the motion is constrained to take place on the intersection of hypersurfaces defined by the integrals. For maps in CP 2 , it was proved in [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF] that the growth can be only bounded, linear, quadratic or exponential. Linear cases are trivially integrable in the sense of invariants. We note that for polynomial maps, it was already known from [START_REF] Veselov | Growth and integrability in the dynamics of mappings[END_REF] that the growth can be only linear or exponential. It is known that QRT mappings and other maps with invariants in CP 2 possess quadratic growth [START_REF] Duistermaat | Discrete Integrable Systems: QRT Maps and Elliptic Surfaces[END_REF], so the two notions are actually equivalent for large class of integrable systems. Now we discuss briefly the concept of duality for rational maps, which was introduced in [START_REF] Quispel | Duality for discrete integrable systems[END_REF]. Let us assume that our map ϕ possesses L independent invariants, i.e.I j for j ∈ {1, . . . , L}. Then we can form the linear combination:

(14) H = α 1 I 1 + • • • + α L I L .
For an unspecified autonomous recurrence relation

(15) [x 1 : x 2 : • • • : x n+1 ] → [x ′ 1 : x 1 : • • • : x n ]
we can write down the invariant condition for H (14): ( 16)

H(x ′ 1 , [x]) = H ([x ′ ]) -H ([x]) = 0. Since we know that [x ′ ] = ϕ ([x]
) is a solution of ( 16) we have the following factorization:

(17) H(x ′ 1 , [x]) = A (x ′ 1 , [x]) B (x ′ 1 , [x]
) . We can assume without loss of generality that the map ϕ corresponds to the annihilation of A in [START_REF] Hay | Hierarchies of nonlinear integrable q-difference equations from series of Lax pairs[END_REF] 17) is non constant 2 . In general, since the map ϕ is bi-rational, we have the following equalities:

deg B x ′ 1 = deg x ′ 1 H -deg x ′ 1 A = deg x1 H -1, (18a) deg B xn = deg xn H -deg xn A = deg xn H -1. (18b)
H (x ′ 1 , [x]) = d i=1 A i (x ′ 1 , [x]
) , but we will not consider this case here. Now assume that the invariants (and hence the map ϕ) depends on some arbitrary constants I i = I i ([x]; a i ), for i = 1, . . . , M . Choosing some of the a i in such a way that there remains M arbitrary constants and such that for a subset a i k we can write equation [START_REF] Grammaticos | How to detect the integrability of discrete systems[END_REF] in the following way:

(20) H = a i1 J 1 + a i2 J 2 + • • • + a iK J ai K ,
where

J i = J i ([x]
), i = 1, 2, . . . , K are new functions. Then using the factorization [START_REF] Hay | Hierarchies of nonlinear integrable q-difference equations from series of Lax pairs[END_REF] we have that the J i functions are invariants for the dual maps.

Remark 4. It is clear from equation [START_REF] Kimura | Discretization of the Lagrange top[END_REF] that even though the dual map is naturally equipped with some integrals, it is not necessarily equipped with a sufficient number of integrals to claim integrability. In fact there exist examples of dual maps with any possible behaviour, integrable, superintegrable and non-integrable [START_REF] Gubbiotti | Integrability properties of a class of 4d bi-rational maps with two invariants[END_REF][START_REF] Joshi | Rational maps with invariant surfaces[END_REF].

In a recent paper [START_REF] Joshi | Rational maps with invariant surfaces[END_REF], the authors considered the autonomous limit of the second member of the dP I and dP II hierarchies [START_REF] Cresswell | The discrete first, second and thirty-fourth Painlevé hierarchies[END_REF]. We will denote these equations as dP II equations are given by autonomous recurrence relations of order four, and showed to be integrable according to the algebraic entropy approach. They showed that both maps possess two invariants, one of degree pattern (1, 3, 3, 1) and one of degree pattern [START_REF] Bellon | Algebraic entropy[END_REF][START_REF] Bruschi | Integrable symplectic maps[END_REF][START_REF] Bruschi | Integrable symplectic maps[END_REF][START_REF] Bellon | Algebraic entropy[END_REF]. Using these invariants, they showed that the dual maps of the dP II equations are integrable according to the algebraic entropy test and moreover, produced some 2 We remark that this assertion is possible because we are assuming that all the invariants are non-degenerate. It is easy to see that degenerate invariants can violate this property.

integrals, showing that these dual maps were actually superintegrable. Finally they gave a scheme to construct autonomous recurrence relations with the assigned degree pattern (1, 3, 3, 1) associated with I low and (2, 4, 4, 2) associated with I high and they provided some new examples out of this construction.

In a forthcoming paper [START_REF] Gubbiotti | Integrability properties of a class of 4d bi-rational maps with two invariants[END_REF] we consider the problem of finding all fourth order bi-rational maps ϕ : [x : y : z : u : t] → [x ′ : y ′ : z ′ : u ′ : t ′ ] possessing a polynomial a symmetric invariant I low such that dp I low = (1, 3, 3, 1) where the only non-zero coefficients are those appearing in the (1, 3, 3, 1) integral of both the dP In this letter we will present in detail four particular examples of this class. In Section 2, we will discuss two pairs of main-dual maps. We will discuss the integrability property of these maps in light of their invariants and of their growth. We will present maps possessing two invariants and integrable according to the algebraic entropy test with cubic growth. This implies that another rational invariant cannot exist. Indeed, the orbits of superintegrable maps with rational invariant are confined to elliptic curves and the growth is at most quadratic [START_REF] Bellon | Algebraic entropy of birational maps with invariant curves[END_REF][START_REF] Kh | Rational g-surfaces[END_REF]. From this general statement follows that a four-dimensional map with cubic growth can possess at most two rational first integrals. We note that some examples of cubic growth were already presented in [START_REF] Joshi | Rational maps with invariant surfaces[END_REF]. However, it was pointed out that these examples can be deflated to lower dimensional maps with quadratic growth. This also holds for our maps, i.e. we can deflate them to integrable maps in lower dimension. Furthermore, we will present a map with two invariants and exponential growth, that is non-integrable according to the algebraic entropy test. We discuss some possible reasons why this map is non-integrable even though it possesses two invariants. In the final Section, we will give some conclusions and an outlook on the future perspectives of this approach.

Notable examples

In this section we discuss two pairs of maps, which arise as part of a systematic classification to be presented in [START_REF] Gubbiotti | Integrability properties of a class of 4d bi-rational maps with two invariants[END_REF]. The interest in these particular maps arises since the relation between the invariants and the growth properties is non trivial. In both cases the main maps possess two functionally independent invariants, but one has cubic degree growth, and the other one has exponential degree growth. In both cases, the degree growth property of the dual maps reflect the growth of the main map. However, we note that not always the degree growth of the dual map reflects the main map one [START_REF] Gubbiotti | Integrability properties of a class of 4d bi-rational maps with two invariants[END_REF].

(P.i) and its dual map (Q.i). Consider the map

[x] → ϕ i ([x]) = [x ′ ]
given as follows:

(P.i)

x ′ = -{[νt 2 (x + z) + uz 2 ]y + t 2 µuz + (x + z) 2 y 2 }d -at 4 , y ′ = x 2 d(t 2 µ + xy), z ′ = yxd(t 2 µ + xy), u ′ = zxd(t 2 µ + xy), t ′ = txd(t 2 µ + xy).
This map depends on four parameters a, d and µ, ν.

From the construction in [START_REF] Gubbiotti | Integrability properties of a class of 4d bi-rational maps with two invariants[END_REF] we know that the map (P.i) possesses the following invariants: 22) is given by: (23) g P.i (s) = s 7 -3s 6 + s 5 -s 4 + 3s 3 + 3s 2 + s + 1 (s + 1)(s 2 + 1)(s -1) 4 .

t 6 I P.i low = at 4 yz + d νy 2 z 2 -yz(ux -uz -xy)µ t 2 -y 2 z 2 d(ux -xy -yz -uz), (21a)
Due to the presence of (s -1) 4 in the denominator we have that the growth of the map (P.i) is fitted by a cubic polynonomial. As discussed in the Introduction this means at once that the map is integrable according to the algebraic entropy test and that another rational invariant cannot exist. This simple observation on the degree of growth also shows that the geometry of the orbits of the map (P.i) is nontrivial, and goes beyond the existence of elliptic fibrations.

The dual map

[x] → ϕ ∨ i ([x]) = [x ′ ]
of (P.i) is given by: (Q.i)

x ′ = [β(2xy -2yz + uz)µ + (βν -α)y(x -z)] t 2
+ βy(z 2 y -x 2 y + uz 2 )

y ′ = x 2 β(t 2 µ + xy), z ′ = yxβ(t 2 µ + xy), u ′ = zxβ(t 2 µ + xy), t ′ = txβ(t 2 µ + xy).
This map depends on three parameters α, β, and µ, ν. The parameters µ and ν are shared with the main map (P.i).

The main map (P.i) possesses two integrals and depends on a and d whereas the dual map (Q.i) do not depend on them. Then according to [START_REF] Kimura | Discretization of the Lagrange top[END_REF] we can write down the invariants for the dual map (Q.i) as: [START_REF] Liouville | Note sur l'intégration des équations différentielles de la Dynamique, présentée au Bureau des Longitudes le 29 juin 1853[END_REF] αI P.i low + βI P.i high = aI Q.i low + dI Q.i high .

Therefore, we obtain the following expressions:

t 4 I Q.i low = [yzα + (µxy -yzµ -yνz + µuz)β]t 2 + βyz(xy + yz + uz), (25a)
t 8 I Q.i high = y 2 z 2 ν -yz(ux -uz -xy)µ α + (uz + xy -yz) 2 µ 2 + 2yz(ux -yz)νµ -ν 2 y 2 z 2 β
(27) g Q.i (s) = (s 3 -2s 2 -1)(s 3 -s 2 -s -1) (s 2 + s + 1)(s -1) 4 .
This means that the dual map is integrable according to the algebraic entropy test with cubic growth, just like the main map. Therefore, the pair of main-dual maps (P.i) and (Q.i) consists of two integrable equations with non-standard degree of growth. However, as remarked above the degree pattern of the invariants of the maps (P.i) and (Q.i) differ slightly.

We now consider the maps (P.i) and (Q.i) in affine coordinates which are given by [START_REF] Papageorgiou | Integrable mappings and nonlinear integrable lattice equations[END_REF] ϕ : (w 3 , w 2 , w 1 , w 0 ) → (w 4 , w 3 , w 2 , w 1 ),

where

w 4 = N 1 dw 3 (w 2 w 3 + µ) , (AP.i) w 4 = N 2 β w 3 (w 2 w 3 + µ) , (AD.i) with N 1 = -d w 0 w 2 1 w 2 + w 2 1 w 2 2 + 2w 1 w 2 2 w 3 + w 2 2 w 2 3 + µw 0 w 1 + νw 1 w 2 + νw 2 w 3 -a, (29)
N 2 = βw 0 w 2 1 w 2 + βµw 0 w 1 + βw 2 1 w 2 2 + (α -2βµ -βν) w 1 w 2 -βw 2 3 w 2 2 + (2βµ + βν -α) w 2 w 3 . ( 30
)
Invariants for these maps are obtained from I low and I high respectively by taking t = 1, u = w 0 , z = w 1 , y = w 2 , and x = w 3 . We note that when a Poisson structure has the full rank, using equation ( 10), one gets [START_REF] Petrera | On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top[END_REF] det(d ϕ)

2 = det J(w ′ ) det J(w) .
This implies that the map ϕ is either volume or anti-volume preserving.

We recall that a map ϕ is called (anti)volume preserving if there is a function Ω(w) such that the following volume form is preserved ( 32)

Ω(w) d w 0 ∧ d w 1 ∧ . . . ∧ d w n-1 = ±Ω(w ′ ) d w ′ 0 ∧ d w ′ 1 ∧ . . . ∧ d w ′ n-1 .
Thus, we can write

(33) ∂ w ′ 0 , w ′ 1 , . . . , w ′ n-1 ∂ w 0 , w 1 , . . . , w n-1 = ± Ω(w) Ω(w ′ ) ,
where the left hand side is the determinant of the Jacobian matrix of the map ϕ.

In [START_REF] Byrnes | Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures[END_REF] it was proved that if a map in n dimension is (anti) volume preserving and possesses n-2 invariant, then we can construct an (anti) Poisson structure of rank 2 from these invariants. However, these invariants turn out to be Casimirs (functions that Poisson commute with all other functions) of this Poisson bracket. Therefore, in order to have Liouville integrability we need an extra invariant apart from the known n -2 invariants if we want to use use Poisson structures constructed this way. In other words, the map is super integrable. Thus, to discuss about Liouville integrability of the maps (AP.i) and (AQ.i) we need to find either the third invariant or a Poisson bracket of rank 4. We do not have that information for these maps but we can show they reduce to three dimensional Liouville integrable maps via a process called inflation [START_REF] Joshi | Rational maps with invariant surfaces[END_REF]. This process will preserve the integrals, and in dimension three, two integrals are sufficient to claim integrability in the general sense as discussed in the Introduction.

It is easy to check that the maps (AP.i) and (AQ.i) are volume and anti-volume preserving, respectively, with respect to the same volume form:

(34) Ω = w 1 w 2 (w 1 w 2 + µ).
We now construct the (anti) Poisson structures for these two maps following [START_REF] Byrnes | Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures[END_REF].

We consider the dual multi-vector of the volume form

(35) τ = m ∂ w 0 ∧ ∂ w 1 ∧ ∂ w 2 ∧ ∂ w 3 ,
where m = 1/Ω. A degenerate Poisson structure for the map (AP.i) and a degenerate anti-Poisson structure for the map (AQ.i) are given by the following contraction

(36) J = τ ⌋ d I low ⌋ d I high ,
where I low and I high are invariants for these maps in affine coordinates. Since these (anti) Poisson structures are quite big, we do not present them here.

Remark 5. The Poisson structures which can be constructed using the method of [START_REF] Byrnes | Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures[END_REF] are degenerate and cannot be used to explain the integrability of the two maps (AP.i) and (AQ.i).

We also note that the maps (AP.i) and (AQ.i) can be reduced to three dimensional maps using a deflation v i = w i w i+1 . The recurrences for these maps are denoted by (DP.i) and (DQ.i) and are given as follows

dµ (v 0 + v 3 ) + dν (v 1 + v 2 ) + d v 0 v 1 + v 2 1 + 2v 1 v 2 + v 2 2 + v 2 v 3 + a = 0, (DP.i) βµ (-v 0 + 2 βv 1 -2 βv 2 + v 3 ) + (βν -α) (v 1 -v 2 ) (DQ.i) + β -v 0 v 1 -v 2 1 + v 2 2 + v 2 v 3 = 0.
Each of the maps (DP.i) and (DQ.i) has two functionally independent invariants which can be obtained directly from I low and I high even though they live in a different space. One can check that the map (DP.i) and (DQ.i) are anti-volume preserving and volume preserving with Ω = v 1 + µ. Therefore, we can construct their (anti) Poisson structure using the three dimensional version of [START_REF] Roberts | Birational maps that send biquadratic curves to biquadratic curves[END_REF]. Using the following invariant from I low for (DP.i) [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé Equations[END_REF]

I P.i 1 = dµv 0 v 1 -dµv 0 v 2 +dµv 1 v 2 +dνv 1 2 +dv 0 v 1 2 -dv 0 v 1 v 2 +dv 1 3 +dv 1 2 v 2 +av 1
we have found that the map (dP.i) has an anti-Poisson structure given by

J P.i 12 = d(v 1 -v 0 ), J P.i 2,3 = d(v 1 -v 2 ) J P.i 13 = -dµv 0 -dµv 2 -2dνv 1 -2dv 0 v 1 + dv 0 v 2 -3dv 1 2 -2dv 1 v 2 -a µ + v 1 .
Similarly, for the map (DQ.i) we obtain the invariant (38)

I DQ.i 1 = βµv 0 -βµv 1 + βµv 2 -νβv 1 + βv 0 v 1 + βv 1 2 + βv 1 v 2 + αv 1 ,
and the corresponding Poisson structure (39)

J Q.i =      0 β β (µ + ν -v 0 -2v 1 -v 2 ) -α µ + v 1 -β 0 β - β (µ + ν -v 0 -2v 1 -v 2 ) -α µ + v 1 -β 0      .
For these constructions, I P.i 1 and I Q.i 1 are Casimirs for their associated (anti) Poisson structures. Their second (anti) Poisson structures can be obtained from the invariant I high but we do not present here as they are quite big.

It is important to note that the (anti) Poisson structures of (AP.i)and (AQ.i) under inflation give us the trivial Poisson structures for (DP.i) and (DQ.i), i.e. J = 0, where 0 is the zero matrix. On the other hand, from the common factor that appears in the Poisson structure of (AP.i), we have found that there exists an anti-invariant K P.i for this map, i.e. K P.i (w) = -K P.i (w ′ )where (40) K P.i = 2d w 2 w 2 1 w 0 +w 2 2 w 2 1 +w 1 w 2 2 w 3 +µw 0 w 1 -µw 1 w 2 +µw 2 w 3 +νw 1 w 2 +a. However, K P.i is functionally dependent with I P.i low and I P.i high through the relation (41) K P.i 2 -4d I P.i high -8d νI P.i low = a 2 . Using this anti-invariant, we obtain the following anti-invariant for the map (DP.i)

(42) K DP.i = 2dµv 0 -2dµv 1 + 2dµv 2 + 2dνv 1 + 2dv 0 v 1 + 2dv 1 2 + 2dv 1 v 2 + a.
Therefore, using this anti-invariant, we get a Poisson structure for (DP.i) as follows (after factoring out a constant term) (43)

J P.i 2 =      0 1 µ -ν -v 0 -2v 1 -v 2 µ + v 1 -1 0 1 - µ -ν -v 0 -2v 1 -v 2 µ + v 1 -1 0      .
We can check directly that the invariants inherited from the affine map (AP.i) are in involution with respect to the Poisson structure [START_REF] Viallet | On the integrability of correspondences associated to integral curves[END_REF]. In the sense of the definition given in the Introduction, this means that the reduced maps (DP.i) and (DQ.i) are Liouville integrable.

Remark 6. We notice that we can always use the invariants ( 37) and [START_REF] Shafarevich | Basic Algebraic Geometry 1, volume 213 of Grundlehren der mathematischen Wissenschaften[END_REF] to reduce the three dimensional maps (DP.i) and (DQ.i) to two dimensional maps and relate them to QRT maps. To be more specific we have that the reduced map of (DQ.i) preserves a bi-quadratic curve so that it is of the QRT type. On the other hand, using the anti-invariant, the reduced map of (DP.i) sends a bi-quadratic to another bi-quadratic and fits in the framework of [START_REF] Roberts | Birational maps that send biquadratic curves to biquadratic curves[END_REF]. (P.ii)

x ′ = (x 2 + z 2 )y -uz 2 µ -t 2 (u -2y), y ′ = x(t 2 + µx 2 ), z ′ = y(t 2 + µx 2 ), u ′ = z(t 2 + µx 2 ), t ′ = t(t 2 + µx 2 ).
This map only depends on the parameter µ.

From the construction in [START_REF] Gubbiotti | Integrability properties of a class of 4d bi-rational maps with two invariants[END_REF] we know that the map (P.ii) has the following invariants:

t 5 I P.ii low = (x -z) (u -y) t 2 + z 2 µ µy 2 + t 2 , (44a) t 6 I P.ii high = (x -z) 2 y 4 + y 2 z 4 -2yz 4 u + u 2 z 4 µ 2 + 2t 2 x 2 -2xz + 2z 2 y 2 -2yz 2 u + u 2 z 2 µ + t 4 z 2 + u 2 + x 2 + y 2 -2uy -2xz . (44b)
Moreover, the map (P.ii) has the following degrees of iterates: .

This means that despite the existence of the two invariants (44) the map (P.ii) is non-integrable according to the algebraic entropy test: its entropy is positive and given by ε = log 2. Therefore we have that the map (P.ii) is an example of non-integrable admitting two invariants.

Again following [START_REF] Byrnes | Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures[END_REF] we can produce a Poisson structure of rank 2 for (P.ii) as the affine version of (P.ii) is volume preserving with Ω = (1 + µw 2 1 )(1 + µw 2 2 ), where we have taken t = 1, u = w 0 , z = w 1 , y = w 2 , and x = w 3 . By the construction, the two invariants (44) become Casimir functions for it, so again the existence of such Poisson structure do not imply any form of Liouville integrability. However, we notice that there are common factors appear at every non-zero entries of this structure. Thus, we have found the following anti-invariant for the map (P.ii) using these common factors (47)

K P.ii = µ w 0 w 2 1 -w 1 2 w 2 -w 1 w 2 2 + w 2 2 w 3 + w 0 -w 1 -w 2 + w 3 × µ w 0 w 2 1 -w 2 1 w 2 + w 1 w 2 2 -w 2 2 w 3 + w 0 + w 1 -w 2 -w 3 = F 1 F 2
This suggests that we should check each factor of K P.ii to see whether they are (anti) invariants of (Pii). By direct calculation we can see that the first factor F 1 is an anti-invariant and F 2 is an invariant for (P.ii), but they are not functionally independent with I low and I high . In fact, their relations are (48) I P.ii high -F 2 1 + 2I P.ii low = 0, and I P.ii high -F 2 2 -2I P.ii low = 0. Therefore, the map (P.ii) actually has two invariants of degrees (1, 2, 2, 1) and (1, 3, 3, 1). Nevertheless, despite the existence of such invariants the map (P.ii) is non-integrable in the sense of the algebraic entropy.

Remark 7. We can use F 1 and F 2 to construct an anti-Poisson structure for (P.ii) using the formula [START_REF] Roberts | Birational maps that send biquadratic curves to biquadratic curves[END_REF] (49) J 1,2 = -1, J 2,3 = 1, J 3,4 = -1

J 1,3 =

The dual map [x] → ϕ ∨

ii ([x]) = [x ′ ] of (P.ii) is given as follows:

(Q.ii)

x ′ = α x 2 -z 2 y + uz 2 µ + t 2 αu + βy 2 (x -z) µ + t 2 β (x -z) , y ′ = αx t 2 + µx 2 , z ′ = αy t 2 + µx 2 , u ′ = αz t 2 + µx 2 , t ′ = αt t 2 + µx 2 .

This map depends on three parameters α, β and B. The parameter µ is shared with the main map (P.ii). Since the main map (P.ii) possess two integrals depending only on one parameter µ then according to [START_REF] Kimura | Discretization of the Lagrange top[END_REF] we can write down only a single first integrals for the dual map (Q.ii):

(50) I Q.ii = αI P.ii high + βI P.ii low . The first integral (50) has degree pattern [START_REF] Bellon | Algebraic entropy[END_REF][START_REF] Bruschi | Integrable symplectic maps[END_REF][START_REF] Bruschi | Integrable symplectic maps[END_REF][START_REF] Bellon | Algebraic entropy[END_REF].

We have then that the dual map (Q.ii) has the following fast-growing degrees of iterates: This means that the dual map is non-integrable with same rate of growth as the main map. In this case we can show that the map is anti-volume preserving with the same measure as the main map (P.ii). Moreover, we proved that the map (Q.ii) do not possesses any addition integral up to order 14. Therefore at the present stage we cannot construct a Poisson structure using the method of [START_REF] Byrnes | Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures[END_REF].

( 2 )

 2 ϕ * (I) = I, where ϕ * (I) means the pullback of I through the map ϕ, i.e. ϕ * (I) = I (ϕ ([x])).

  II equations. These dP

  IIequation, and such that ϕ possesses a polynomial symmetric invariant I high such that dp I high = (2, 4, 4, 2). The two invariants I low and I high are assumed to be functionally independent and non-degenerate. Within this class we have found the known dP II equations as well as new examples of maps with these properties.

2. 2 .

 2 (P.ii) and its dual map (Q.ii). Consider the map [x] → ϕ ii ([x]) = [x ′ ] given as follows:

  {d n } P.ii = 1, 3, 9, 21, 45, 93, 189, 381, 765, 1533 . . . with generating function: (46) g P.ii (s) = 1 + 2s 2 (2s -1)(s -1)

  (51) {d n } Q.ii = 1, 3, 9, 21, 45, 93, 189, 381, 765, 1533, 3069 . . ..The growth (51) is clearly exponential and its generating function:(52) g Q.ii (s) = 1 + 2s 2 (2s -1)(s -1), confirms this showing that the algebraic entropy is positive and equal to ε = log 2.

  . Now since deg x ′ 1 H = deg x1 H and deg xn H = deg xn H we have that if if deg x1 H, deg xn H > 1 the factor B in (

  Therefore we have that if deg x1 H, deg xn H > 2, the annihilation of B does not define a bi-rational map in general, but an algebraic one. However when deg x1 H, deg xn H = 2 the annihilation of B defines a bi-rational projective map. We call this map the dual map and we denote it by ϕ ∨ . Remark 3. We note that in principle for deg x1 H = deg xn H = d > 2, more general factorizations can be considered:

	(19)

  t 8 I P.i high = [(uz + xy -yz)µ -νyz] at 6 + yz(xy + yz + uz)a + dµ 2 (uz + xy -yz) 2 + 2dµνyz(ux -yz) -dν 2 y 2 z 2 t 4 + 2dzy(uz + xy -yz)(xy + yz + uz)µ + 2dy 2 z 2 νux t 2 + dy 2 z 2 (xy + yz + uz) 2 .

	(21b)
	Moreover, the map (P.i) has the following degrees of iterates:
	(22)	{d n } P.i =1, 4, 12, 28, 52, 86, 130, 188, 260, 348, 452, 576, 720, 886, 1074, 1288, 1528, 1796, 2092 . . .

The generating function of the sequence (

  t 4 + z 2 y 2 (xy + yz -ux + uz)α + 2yz(uz + xy -yz)(xy + yz + uz)µ + 2y 2 z 2 νux β t 2 + z 2 y 2 (xy + yz + uz) 2 β.

	(25b)
	We remark that the invariant (25a) has degree pattern (1, 2, 2, 1) which differs from
	dp I P.i low .
	The map (Q.i) has the following degrees of iterates:
	(26)	{d n } Q.i = 1, 4, 12, 26, 48, 78, 118, 170, 234, 312, 406, 516, 644, 792 . . .
	with generating function:

  2µw 1 (w 2 -w 0 ) J 1,4 = -µ 2 w 1 w 2 [4 (w 0 w 1 -w 0 w 3 + w 2 w 3 ) -3w 1 w 2 ] + µ w 2We have checked that F 2 and I P.ii low are in involution with respect to this anti-Poisson structure. A Poisson structure can be obtained by multiplying this anti-Poisson structure with the anti-invariant F 1 .

	µw 2 1 + 1	, J 2,4 = -	2µw 2 (w 3 -w 1 ) µw 2 2 + 1
			1 + w 2 2 + 1
		(µw 2 1 + 1) (µw 2 2 + 1)

Conclusions and outlook

In this letter, we gave some examples of fourth order bi-rational maps with two invariants possessing interesting degree growth properties. These examples come from our forthcoming classification of all the fourth-order autonomous recurrence relations possessing two invariants in a given class of degree patterns [START_REF] Gubbiotti | Integrability properties of a class of 4d bi-rational maps with two invariants[END_REF] .

The first pair of bi-rational maps is given by the map (P.i) and its dual (Q.i) and consists of integrable maps with cubic growth. The interest in maps with cubic growth arises from geometrical considerations: maps with polynomial but higher than quadratic growth, can arise only in dimension greater than two [START_REF] Diller | Dynamics of bimeromorphic maps of surfaces[END_REF] and prove, in the case of superintegrable maps, the existence of non-elliptic fibrations of invariant varieties [START_REF] Bellon | Algebraic entropy[END_REF]. The interest in maps with this type of growth arose recently following the examples given in [START_REF] Joshi | Rational maps with invariant surfaces[END_REF] and we expect them to lead to many new and interesting geometric structures.

The second pair of fourth order bi-rational maps given by the map (P.ii) and its dual (Q.ii), consists of non-integrable maps with exponential growth. There are various possible reasons why the map (P.ii) is non-integrable despite possessing two invariants. To claim integrability with two invariants according to the discrete Liouville theorem [START_REF] Bruschi | Integrable symplectic maps[END_REF][START_REF] Maeda | Completely integrable symplectic mapping[END_REF][START_REF] Veselov | Integrable maps[END_REF] we need to prove that the map has a symplectic structure and that the two invariants commute with respect to this symplectic structure. Hence, either the map (P.ii) does not admit any symplectic structure, or the map (P.ii) admits only symplectic structures such that the two integrals (44) do not commute. Since, usually, from a set of non-commuting integrals it is possible to find a set of functionally independent commuting integrals we are more leaned to conjecture that equation (P.ii) is devoid of a non-degenerate Poisson structure.

Work is in progress to characterize the surfaces generated by the invariants in both integrable and non-integrable cases. We expect this to give new results in the geometric theory of integrable systems.


Irvin Dongo

Richard Chbeir
email: richard.chbeir@univ-pau.fr

S-RDF: A new RDF Serialization Format for better Storage without losing Human Readability

Keywords: Serialization Format, Semantic Web, Data Representation, RDF

Nowadays, RDF data becomes more and more popular on the Web due to the advances of the Semantic Web and the Linked Open Data initiatives. Several works are focused on transforming relational databases to RDF by storing related data in N-Triple serialization format. However, these approaches do not take into account the existing normalization of their databases since N-Triple format allows data redundancy and does not control any normalization by itself. Moreover, the mostly used and recommended serialization formats, such as RD-F/XML, Turtle, and HDT, have either high human-readability but waste storage capacity, or focus further on storage capacities while providing low human-readability. To overcome these limitations, we propose here a new serialization format, called S-RDF. By considering the structure (graph) and values of the RDF data separately, S-RDF reduces the duplicity of values by using unique identifiers. Results show an important improvement over the existing serialization formats in terms of storage (up to 71,66% w.r.t. N-Triples) and human readability.

Introduction

For the Semantic Web, RDF is the common format to describe resources, which are abstractions of entities (documents, abstract concepts, persons, companies, etc.) of the real world. It was developed by Ora Lassila and Ralph Swick in 1998 [START_REF] Lassila | Resource description framework (rdf) model and syntax specification[END_REF]. RDF uses triples in the form of subject, predicate, object expressions, also named statements, to provide relationships among resources.

Currently, RDF data available on the Web is increasing rapidly due to the promotion of the Semantic Web and the Linked Open Data (LOD) initiatives [START_REF] Sandro Hawke | W3C Semantic Web Activity[END_REF]. Governments, organizations and research communities are part of the LOD initiatives, providing their data to have a more flexible data integration, increasing the data quality and providing new services [START_REF] Hausenblas | Linked open government data[END_REF]. Since RDF does not restrict how data is converted, several RDF serializations are available in the literature [START_REF] Hernndez-Illera | Serializing rdf in compressed space[END_REF]. For instance, RDF/XML is historically the first W3C standard which serializes the RDF graph ( subject, predicate, object ) into XML. Other serializations, such as Turtle and N3, are also highly recommended [START_REF] Patrick | RDF 1.1 Semantics, W3C Recommendation 25 February[END_REF]. In the literature, several works have been proposed to convert different datasets to RDF/OWL. The works in [START_REF] Huang | Streaming transformation of xml to rdf using xpath-based mappings[END_REF][START_REF] Lacoste | An efficient xml to owl converter[END_REF][START_REF] O'connor | Acquiring owl ontologies from xml documents[END_REF][START_REF] Thuy | Dtd2owl: Automatic transforming xml documents into owl ontology[END_REF][START_REF] Vion-Dury | Using rdfs/owl to ease semantic integration of structured documents[END_REF] propose to convert XML data into RDF using XPath expressions, XSD Schemas, DTD 4 , etc. Other works provided in [START_REF] Bornea | Building an efficient rdf store over a relational database[END_REF][START_REF] Būmans | Rdb2owl: A practical approach for transforming rdb data into rdf/owl[END_REF][START_REF] Goasdoué | Getting more rdf support from relational databases[END_REF][START_REF] Konstantinou | Incremental export of relational database contents into rdf graphs[END_REF][START_REF] Salas | Rdb2rdf plugin: Relational databases to rdf plugin for eclipse[END_REF][START_REF] Sequeda | On directly mapping relational databases to rdf and owl[END_REF][START_REF] Stefanova | Scalable reconstruction of rdf-archived relational databases[END_REF][START_REF] Thuy | Rdb2rdf: Completed transformation from relational database into rdf ontology[END_REF] address RDF conversion of relational database models to publish huge quantity of information and linked to the Web. However, current adopted serialization formats are mainly focusing on document-centric view to increase human readability, while requiring important storage space and bandwidth resources [START_REF] Hernndez-Illera | Serializing rdf in compressed space[END_REF]. In essence, these formats do not control the redundancy of data by definition which also affects the conceptual model. The authors in [START_REF] Ticona-Herrera | Toward RDF Normalization[END_REF] address the syntactic redundancy of the data by applying a normalization methodology. Other authors as in [START_REF] Hernndez-Illera | Serializing rdf in compressed space[END_REF] propose a binary representation format called HDT, reducing the redundancy of data, but decreasing the human readability of the information.

To overcome these limitations, we propose here a new serialization format called S-RDF, which represents the RDF graph structure and the values separately for a better human readability. This serialization is available to manage medium-large datasets by reusing identifiers (keys) extracted from several ones. Moreover, the storage is reduced and some graph properties (e.g., degree centrality measure 5 ) can be easily analyzed. We validated our serialization format through several experiments. Results show an improvement over the existing serialization formats in terms of storage (up to 71.66% with respect to N-Triples) and human readability.

The rest of this paper is organized as follows. In Section 2, we present a motivating scenario to illustrate better the needs. Section 3 surveys the related literature. Terminologies and definitions are presented in Section 4. Section 5 describes our serialization format. In Section 6, we present the experiments conducted to evaluate the compression rate and the human-readability. Finally, we present conclusions in Section 7.

Motivating Scenario

As mentioned previously, RDF data can be represented in different ways (serializations), i.e., stored in a file system through several formats. In order to illustrate the limitations of existing serialization formats, we consider a scenario in which the information of Listing 1 is shared on the Web. This listing shows four Schools entities: S0991, S0992, S0993, and S0994, which have information such as rdf:type, ins:name, ins:postalCode, and ins:established. @ P r e f i x i n s : h t t p : / / i n s t i t u t i o n s . com / 0 . 2 / i n s : S0991 ; i n s : name " Lycee du Parc "ˆˆxsd : s t r i n g ; i n s : p o s t a l C o d e " 6 4 6 0 0 " ˆˆxsd : s t r i n g ; i n s : e s t a b l i s h e d 1985-05-19ˆˆxsd : d a t e ; r d f : t y p e h t t p : / /www. w3 . o r g / 2 0 0 2 / 0 7 / owl#Thing .

i n s : S0992 ; i n s : name " Napoleon B u s i n e s s "ˆˆxsd : s t r i n g ; i n s : p o s t a l C o d e " 6 4 1 0 0 " ˆˆxsd : s t r i n g ; i n s : e s t a b l i s h e d 1986-12-19ˆˆxsd : d a t e . r d f : t y p e h t t p : / /www. w3 . o r g / 2 0 0 2 / 0 7 / owl#Thing .

i n s : S0993 ; i n s : name " E c o l e n a t i o n a l de l ' e n e r g i "ˆˆxsd : s t r i n g ; i n s : p o s t a l C o d e " 6 4 5 0 0 " ˆˆxsd : s t r i n g ; i n s : e s t a b l i s h e d 1984-11-21ˆˆxsd : d a t e . r d f : t y p e h t t p : / /www. w3 . o r g / 2 0 0 2 / 0 7 / owl#Thing .

i n s : S0994 ; i n s : name " Grande V i l l e S c h o o l "ˆˆxsd : s t r i n g ; i n s : p o s t a l C o d e " 6 4 6 0 0 " ˆˆxsd : s t r i n g ; i n s : e s t a b l i s h e d 1977-08-22ˆˆxsd : d a t e . r d f : t y p e h t t p : / /www. w3 . o r g / 2 0 0 2 / 0 7 / owl#Thing .

Listing 1. School Information

Table 1 shows the serialization formats defined by the W3C (RDF/XML, Turtle, N-Triple, and N3). These formats are document-centric view since their data can be read and understood by humans; however, for a data that generates a graph with a considerable depth (more than three), the readability is reduced. For instance, according to our motivating scenario, one can easily observe the properties of the entity S0991 (ins:name, ins:postalCode, ins:established) and its respective values of the RDF/XML, Turtle , N-Triple and N3 serialization formats, since the depth of the generated graph is 2. If some blank nodes are added between the entity and the properties, the readability decreases by finding the properties in another part of the document, using the entity and blank nodes as references to search the values.

The RDFa, microdata and JSON-LD serialization formats are adopted as recommendation by the W3C. Table 2 shows and describes the three aforementioned formats. These formats are also document centric view as the previous ones; therefore, the same limitation is found. Moreover, since all serialization formats are document centric view, the storage is not taken into account by any of them. For small datasets, it is not a need, but for medium and large datasets, especially the ones obtained from relational databases, the storage represents a critical issue and has an impact on exchanging data.

In general, the first RDF serialization formats were proposed as documentcentric view (RDF/XML, Turtle), since RDF data describes mainly Web Pages as resources (e.g, DBpedia from Wikipedia) and the number of properties to described them is limited (About: Eiffel Tower is describe by 156 triples); however, as the resources can be linked on the Web, the number of triples increases exponentially by considering datasets that use several resources. Therefore, a format able to describe a resource or a set of resources is needed considering the storage as a main requirement for medium-large datasets.

N3 (Notation 3) [18]

It is an extension format of turtle language expressing a superset of RDF and has been designed with human readability in mind. Their current media type is text/rdf+n3. @prefix dc: <http://purl.org/dc/elements/1.1/> . @prefix ins: <http://institutions.com/0.2/> . @prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#> . @prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#> . @prefix xml: <http://www.w3.org/XML/1998/namespace> . @prefix xsd: <http://www.w3.org/2001/XMLSchema#> . ins:S0991 ins:established "1985-05-19"^^<xsd:date> ; ins:name "Lycee du Parc"^^<xsd:string> ; ins:postalCode "64600"^^<xsd:string> . ...

By regarding the limitations of existing serialization formats, we have identified three main requirements according to the challenges and objectives of this work:

-A high-human readability for easy understanding of data; -A high radio compression for minimizing the storage space and reducing exchanging delays; and -A format oriented to describe medium-large datasets.

The following section describes and compares the related work by using the identified requirements.

Related Work

To the best of our knowledge, several serialization formats have been also proposed in the literature other than the ones adopted or recommended by the W3C. The authors in [START_REF] Fernández | Binary rdf for scalable publishing, exchanging and consumption in the web of data[END_REF] present a binary RDF representation for large datasets. They

{

"@context": { "ins": "http://institutions.com/0.2/", "owl": "http://www.w3.org/2002/07/owl#", "rdf": "http://www.w3.org/1999/02/22-rdf-syntax-ns#", "rdfs": "http://www.w3.org/2000/01/rdf-schema#", "xsd": "http://www.w3.org/2001/XMLSchema#" }, "@id": "ins:S0991", "@type": "owl:Thing", "ins:established": { "@type": "xsd:date", "@value": "1985-05-19" }, "ins:name": { "@type": "xsd:string", "@value": "Lycee du Parc" }, ... } represent the RDF graph in three logical components: (i) Header, (ii) Dictionary, and (iii) Triples. The size of the datasets is reduced, improving the data sharing and the querying and indexing performance. In [START_REF] Hernndez-Illera | Serializing rdf in compressed space[END_REF], the authors improve their previous work up to 2 times for more structured datasets, and a significant improvement for semi-structured datasets as DBpedia. Other works, as in [START_REF] Chantrapornchai | Tripleid-c: Low cost compressed representation for rdf query processing in gpus[END_REF], have focused on compressed representation for RDF Querying. The authors highlight that the improvement is around 50% to 60% of the original HDT. This format is proposed for the use of GPU.

Table 3 shows our related work classification. RDF/XML, Turtle, N3 and JSON-LD focus on human readability since their formats can be easily read by humans. HDT, HDT++ and TripleID-C have been designed to improve the Before describing our serialization format, the following section introduces some common terminologies and definitions in the context of RDF.

RDF Terminologies and Definitions

RDF commonly uses triples in the form of subject, predicate, object expressions/statements, to provide relationships among resources. The RDF triples can be composed of the following elements: -An IRI, which is an extension of the Uniform Resource Identifier (URI) scheme to a much wider repertoire of characters from the Universal Character Set (Unicode/ISO 10646), including Chinese, Japanese, and Korean character sets [START_REF] Duerst | Internationalized Resource Identifiers (IRIs)[END_REF].

-A Blank Node, representing a local identifier used in some concrete RDF syntaxes or RDF store implementations. A blank node can be associated with an identifier (rdf:nodeID) to be referenced in the local document, which is generated manually or automatically -A Literal Node, representing values as strings, numbers, and dates. According to the definition in [START_REF] Cyganiak | RDF 1.1 Concepts and Abstract Syntax[END_REF], it consists of two or three parts:

• A lexical form, being a Unicode string, which should be in Normal Form C 6 to assure that equivalent strings have a unique binary representation • A datatype IRI, being an IRI identifying a datatype that determines how the lexical form maps to an object value • A non-empty language tag as defined by "Tags for Identifying Langua-ges" [START_REF] Phillips | Tags for Identifying Languages[END_REF], if and only if the datatype IRI is http://www.w3.org/1999/02/22rdf-syntax-ns#langString.

Table 4 shows the sets of RDF's elements that we use in our formal approach description. After the definition of sets of RDF'elements, we formally describe a triple in Def 1.

Definition 1. Triple (t):

A Triple, denoted as t, is defined as an atomic structure consisting of a 3-tuple with a Subject (s), a Predicate (p), and Object (o), denoted as t :< s, p, o >, where: -s ∈ I ∪ BN represents the subject to be described; -p is a predicate defined as an IRI in the form namespace prefix:predicate name, where namespace pref ix is a local identifier of the IRI, in which the predicate (predicate name) is defined. The predicate (p) is also known as the property of the triple; -o ∈ I ∪ BN ∪ L describes the object.

From Listing 1, one can observe the following triples with different RDF resources, properties, and literals:

-t3: <genid:S0991,rdf:type, http://www.w3.org/2002/07/owl#Thing> -t4: <genid:S0991,ins:name,"Lycée de la Plage"> -t5: <genid:S0991,ins:established,1985-05-19>

In this study, we also consider two types of properties (predicates): -Entity Property (ep): A predicate is an entity property when it is related to an IRI or a blank node. It is also known as Object property. For example, the property eni:locates is an entity property since it is related to a blank node.

-Value Property (vp): A predicate is a value property when it is related to a literal node. It is also known as Datatype property. For example, the property ins:established is a value property since it is related to a literal node.

An RDF document is defined as an encoding of a set of triples, using a predefined serialization format complying with an RDF W3C standards, such as RDF/XML, Turtle, N3, etc. Additionally, we use the term entity, formally described in Def.2, to identify an RDF resource (blank node and IRI). Definition 2. Entity (e): An entity in an RDF document, denoted as e, is represented as an IRI or a blank node (e.g., School, Power Plant).

For example, from Listing 1, the triple <S0991,ins:name,"Lycée de la Plage"> has the entity S0991.

In Def. According to Def. 6, the literal values from Listing 1 are: VA={"Lycée de la Plage", "64600", "1985-05-19", "Napoleon Business", "64100", "1986-12-19", " École National de l'energie", "64500", "1984-11-21", "Grande Ville School", "64200", "1977-08-22"}.

Table 5 summaries the sets of entities, entity and value properties, and literal values of an RDF document.

The following section presents and describes our new serialization format S-RDF.

S-RDF: Our Proposal

Our proposal mainly relies on a three step process: (i) Extraction of RDF elements, where the input, an RDF document in any format, is analyzed in order to extract the set of entities (E), entity properties (EP), value properties (VP), and literal values (LV); (ii) RDF Sequence-Value generation where entities, properties, and literal values are represented by unique identifiers (e.g., primary keys); and (iii) RDF Sequence-Structure generation where relations among entities, which define the RDF graph structure, are expressed using the Sequence-Value Representation. Thus, our serialization format (S-RDF ) consists in two parts: (i) Value Representation and (ii) Structure. Fig. 1 shows the framework of our proposal composed by three modules that materialize the three respective phases. In Def. 7, we formally describe the Value Representation part of our RDF sequence, called RDF Sequence-Value. This representation associates to each entity, entity and value property, and literal value a unique identifier to be used in the structure representation of the sequence. We propose four different identifiers to easily recognize the type of data in the second part of our sequence. The entities are represented by numbers of the decimal numeral system (base 10), starting from 1. In the case of entity and value properties, both identifiers correspond to the hexavigesimal numeral system (base 26), with a domain of lowercase and uppercase alphabet letters, respectively. For the literal values, the identifiers belong to the decimal numeral system as the ones of entities, but a symbol " " is added as a prefix. For instance, the 28 th element of the entities is represented as "28", "AB" for entity properties, 'ab" for value properties , while for literal values is " 28". Definition 7. RDF Sequence-Value (S-RDF-V ): Given a set of triples T = {t i | t i :< s, p, o >}, its RDF Sequence-Value is defined as a 4-tuple of:

S -RDF -V T = < Entities = { m i=1 < pk i , e i , type i >}, Entity properties = { n j=1 < pk j , ep j >}, V alue properties = { o k=1 < pk k , vp k , datatype k >}, Literal values = { p l=1 < pk l , lv l >} > where:
-Entities is a set of 3-tuples, where: Tables 6-9 represent the Entities, Entity properties, Value properties, and Literal values of Listing 1. The first element of the S-RDF-V is composed by the entities of Table 6. As only one relation among entities is shown in Listing 1, the second element (entity properties) of the 4-tuple is: {<A, http://www.w3-.org/1999/02/22-rdf-syntax-ns#type>}. The third and fourth elements are composed by the information in Table 8 and Table 9, respectively. The set of triples (T), obtained from Listing 1, has the following RDF Sequence-Value: For example, the set of triples (T) obtained from Listing 1, has the following RDF Sequence-Structure: S-RDF-S(T) ={ < 1, {(A, 5)}, {(a, 1), (b, 2), (c, 3)} >, < 2, {(A, 5)}, {(a, 4), (b, 5), (c, 6)} >, < 3, {(A, 5)}, {(a, 7), (b, 8), (c, 9)} >, < 4, {(A, 5)}, {(a, 10), (b, 11), (c, 12)} >, < 5, {}, {} >}, representing: entity "1" (http://institutions.com/0.2/S0991 according to Table 6), has an entity property " A" (http://www.w3.org/1999/02/22-rdf-syntax-ns#type according to Table 7), related to the entity "5" (http://www-.w3.org/2002/07/owl#Thing according to Table 6). It has a property value "a" (http://institutions.com/0.2/name according to Table 8) with a literal value " 1" ("Lycée de la Plage" according to Table 9), and so on.

S-RDF-V(T) = < Entities = { < 1, http : //institutions.com/0.2/S0991, 1 >, < 2
Once values and structure of the RDF data are defined, we formalize the whole RDF Sequence in Def. 9. Definition 9. RDF Sequence (S-RDF ): Given a set of triples T = {t i | t i :< s, p, o >}, its RDF Sequence is a 2-tuple consisting of two parts, defined as: S-RDF(T) = <S-RDF-V(T), S-RDF-S(T)> where:

-S-RDF-V(T) is the set of values of T defined in Def. 7.

-S-RDF-S(T) is the structure of T defined in Def. 8.

The S-RDF is built to represent triples considering the structure and values separately. Thus, an analysis over either the data or structure can be easily performed. Another benefit of this serialization format is the easy detection of some graph properties as the number of relationships (e.g., degree centrality measure) with respect to other serialization formats. Moreover, the storage space is reduced, since an IRI, which appears several times in an RDF document as a resource or property, is represented as a unique short key (e.g., key:1 represents value: http://institutions.com/0.2/S0991 or key:A represents value: http://www.w3.org/2002/07/owl#Thing, respectively). This new serialization format can be consider as part of RDF partition strategies where the models improve the storage and the querying; however, when the repository is exported/outsourced, the format is still the same (e.g., RDF/XML, Turtle). Our serialization is a new way to represent data to be shared on the Web, improving the storage without losing the readability.

In the following section, we evaluate our S-RDF with respect to the current serialization formats.

6 Experimental Evaluation

Experimental Environment and Datasets

In order to evaluate and validate our serialization format, we developed a desktop and online 7 prototype system based on Java and Jena 8 to manage the RDF data. Experiments were undertaken on a MacBook Pro, 2.2 GHz Intel Core(TM) i7 with 16.00GB, running a MacOS Mojave and using a Sun JDK 1.7 programming environment.

Our prototype was used to perform several experiments to evaluate the viability and the compression rate of our approach in comparison with the works proposed in the literature. To do so, we considered two datasets:

-Data 1: the DBpedia person data 9 with 16,842,176 triples; and -Data 2: the DBpedia geo coordinates 10 with 151,205 triples.

Note that some of the serialization formats (e.g., RDFa, HDT++) described in the related work section were not evaluated since there are no tools available that can manage huge quantity of triples. They are mainly document oriented converters (e.g., Easy-RDF 11 , RDF-Translator 12 ). For our readability test, HDT and HDT+ formats were analyzed since they have a binary representation and cannot be read by humans.

We describe as follows the tests performed to evaluate our proposal.

Evaluation

Test 1: We chose randomly 50,000 triples from Data 1 in order to measure the compression rate of the data with respect to the size of the input (6,102,029 For Data 2, we also chose 50,000 triples from this dataset, having a size of 7,356,637 bytes. Table 11 shows similar results as the ones of Data 1. HDT obtained the best result with 75.6508%, while for our serialization format was 70.7767%. The JSON-LD serialization format has a 59.6130% of compression rate with respect to the input size. In this test, we evaluated our human readability criterion by surveying 40 people that have under-and post-graduate degrees in computer science 13 . The participants evaluated the serialization formats through the three previous questions, choosing an option to answer them: Yes, No, and I do not know for the two first questions, and a value among 1 to 5 and "I do not know" option for the third one.

To evaluate the results, we calculated the F-measure, based on the Recall (R) and Precision (PR). These criteria are commonly adopted in information retrieval and are calculated as follows:

PR = A A + B ∈ [0, 1] R = A A + C ∈ [0, 1] F-measure = 2 × P R × R P R + R ∈ [0, 1]
where A is the number of correct answers; B is the number of wrong answers; and C is the number of "I do not know" options selected by the participants. Table 12 shows the results obtained for this evaluation. For Question 1, the N3 serialization format obtained the best Precision (84.62%), while the one for our serialization format was 84.00%. RDF/XML and JSON-LD obtained the lowest Precision (22.73% and 36.36%, respectively). By regarding the F-measure, we can observe that Turtle, N3, and our proposal (S-RDF) help user to identify some graph properties as the centrality measure, since they obtained a high result (over 68.00%). For Question 2, which is related to identify terminal nodes, most of the serialization formats obtained a similar F-measure (≈61.00%), but for the RDF/XML format, the F-measure was 43.14% due to the low Recall (35.48%). A low Recall can be interpreted as the serialization format is not easy-readable for the user. For Question 3, Turtle obtained the best F-measure (73.02%), while for S-RDF the value was 68.85%. By analyzing the answers, we noticed that some people confused the entity property and its respective value as a literal value since they only counted the number of elements associated to the entity. Table 13 shows the global results of this test. In this table, we can identify two groups: G1: RDF/XML, N-Triples, and JSON-LD with a F-measure around 43.00%, and G2: Turtle, N3, and S-RDF with a value around 68.00%. One of the reasons of the low F-measure obtained by G1, is that these formats were created to keep the interoperability among system, using XML and JSON formats for example. The results demonstrate that our serialization format (S-RDF) can improve the storage without losing the human-readability criterion.

Conclusion

In this paper, we propose a new serialization format, called S-RDF, which represents the RDF graph structure and values, separately. This format is focused on human readability, storage, and data redundancy to represent medium and large datasets. We evaluated our serialization format in terms of compression rate and human readability with respect to the state of the art. Results show a high compression without losing human readability, which is an advantage over the serialization formats created to minimize storage. According to the survey evaluation, our S-RDF allows identify easily the resources with more relations in the RDF graph (degree centrality measure) by identifying the entity with the bigger number of entity properties.

We are currently working on normalization methods over the S-RDF in order to provide a unique and deterministic output for similar inputs.

Fig. 1 .

 1 Fig. 1. Framework of our serialization format "S-RDF"

Table 1 .

 1 Serialization formats defined by the W3C

	S. Format Description	Example of Listing 1
	RDF/XML [18]	It is the first se-rialization format adopted by the W3C. This format serializes the RDF and XML files, where nodes and edges of the RDF document are represented using XML syntax. Their current media type is application/rdf+xml.	<?xml version="1.0"?> <rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:ins="http://institutions.com/0.2/"> <rdf:Description rdf:about="http://institutions.com/0.2/S0991"> <ins:name rdf:datatype="xsd:string">Lycee du Parc</ins:name> <ins:postalCode rdf:datatype="xsd:string">64600</ins:postalCode> <ins:established rdf:datatype="xsd:date">1985-05-19</ins:established> <rdf:type rdf:resource="http://www.w3.org/2002/07/owl#Thing" /> </rdf:Description> ... </rdf:RDF>
	Turtle (Terse RDF Triple Language) [18]	It is a textual serial-ization format to en-code RDF documents in a compact form and also readable for humans. Their cur-rent media type is application/x-turtle.	@prefix ns0: <http://institutions.com/0.2/> . @prefix xsd: <http://www.w3.org/2001/XMLSchema#> . <http://institutions.com/0.2/S0991> ns0:name "Lycee du Parc"^^xsd:string ; ns0:postalCode "64600"^^xsd:string ; ns0:established "1985-05-19"^^xsd:date . ...
	N-Triple (Notation of Triples) [18]	It is simple serializa-tion of RDF but not as compact as Turtle format. Their current media type is text/-plain.	<http://institutions.com/0.2/S0991> <http://institutions.com/0.2/-postalCode> "64600"^^<xsd:string> . <http://institutions.com/0.2/S0991> <http://institutions.com/0.2/-name> "Lycee du Parc"^^<xsd:string> . <http://institutions.com/0.2/S0991> <http://institutions.com/0.2/-established> "1985-05-19"^^<xsd:date> . ...

Table 2 .

 2 Serialization Formats recommended by the W3C

	S. Format Description		Example
				<div xmlns="http://www.w3.org/1999/xhtml"
				prefix="
				owl: http://www.w3.org/2002/07/owl#
				ns1: xsd:
	RDFa (Resource Description Framework in Attributes)	It is a serialization format that adds structured data to HTML or XHTML documents by extend-ing the attributes of elements.	rdf: http://www.w3.org/1999/02/22-rdf-syntax-ns# ins: http://institutions.com/0.2/ rdfs: http://www.w3.org/2000/01/rdf-schema#" <div typeof="owl:Thing" about="http://institutions.com/0.2/S0991"> > <div property="ins:name" datatype="ns1:string" content= "Lycee du Parc"></div> <div property="ins:postalCode" datatype="ns1:string" content=
				"64600"></div>
				<div property="ins:established" datatype="ns1:date" content=
				"1985-05-19"></div>
				</div>	...
				</div>
		It is a serialization
		format that describe
	Microdata	a simpler way of an-
	[1]	notating HTML ele-
		ments with machine-
		readable tags.
				</div>
		It is a concrete syntax
		format that extends
	JSON-LD [16]	the RDF data model to optionally allow JSON-LD to serialize
		Generalized	RDF
		Datasets.

<div> <div itemtype="http://www.w3.org/2002/07/owl#Thing" itemid="http://institutions.com/0.2/S0991" itemscope> <meta itemprop="http://institutions.com/0.2/postalCode" content="64600" /> <meta itemprop="http://institutions.com/0.2/established" content="1985-05-19" /> <meta itemprop="http://institutions.com/0.2/name" content="Lycee du Parc" /> </div> ...

Table 3 .

 3 Related Work Classification

	Serialization Format	Human Readability	Storage (Compression)	Non-Redundancy	Large-Medium Dataset	Media Type
	RDF/XML	Low+	Medium	Low	Low	RDF+XML
	Turtle	High	Medium	Medium	Low	-
	N-Triple	Low	Low	Low	Low	-
	N3	High	Medium	Medium	Low	-
	RDFa	Low	Low	Low	Low	HTML/XHTML
	Microdata	Low	Low	Low	Low	HTML
	JSON-LD	Medium	Medium	Medium	Low	JSON
	HDT [8]	Zero	High	High	High	Binary
	HDT++ [11]	Zero	High+	High	High	Binary
	TripleID-C [5]	Low	High	High	Medium	GPU
	RDF Sequence	High	Medium+	High	Medium+

storage, affecting the human readability. Note that none of the works satisfies all the defined requirements; thus, a new RDF serialization format is required.

Table 4 .

 4 Description of sets IRIs is defined as: I = {i 1 , i 2 , ..., in} | ∀i i ∈ I, i i is an IRI. L A set of literal nodes is defined as: L= {l 1 , l 2 , ..., ln} | ∀l i ∈ L, l i is a literal node. BN A set of blank nodes is defined as: BN = {bn 1 , bn 2 , ..., bnn} | ∀bn i ∈ BN , bn i is a Blank Node.

	Set	Description
	I	A set of

  3, Def. 4, Def. 5, and Def. 6, we formally describe the respective sets of entities, entity properties, value properties, and literal values of an RDF document. Definition 3. Entity Set (E ): Given a set of triples T = {t i | t i :< s, p, o >}, the entities of each t i define the set of all entities, denoted as E = n i=1 t i .s ∪ t i .o ⇐⇒ t i .o ∈ I ∪ BN , where n is the number of triples. The entity set according to Def. 3 of Listing 1 is: E={http://institutions.com-/0.2/S0991, http://institutions.com.com/0.2/S0992, http://institutions.com/0.2-/S0993, http://institutions.com/0.2/S0994, http://www.w3.org/2002/07/owl#-Thing}. Definition 4. Entity Properties (EP): Given a set of triples T = {t i | t i :< s, p, o >}, the predicates of all t i that are entity properties, define the set of entity properties, denoted as: EP = n i=1 t i .p ⇐⇒ t i .o ∈ I ∪ BN , where n is the number of triples. Given a set of triples T = {t i | t i :< s, p, o >}, the predicate of all t i that are value properties, define the set of value properties, denoted as: V P = n i=1 t i .p ⇐⇒ t i .o ∈ L, where n is the number of triples. Given a set of triples T = {t i | t i :< s, p, o >}, the literals of all t i define the set of literal values, denoted as: LV = n i=1 t i .o ⇐⇒ t i .o ∈ L, where n is the number of triples.

	The entity properties from Listing 1 are: EP={rdf:type} or EP={http://www-
	.w3.org/1999/02/22-rdf-syntax-ns#type}.
	Definition 5. Value Properties (VP): According to Def. 5, the value properties obtained from the triples of Listing 1
	are: VP={http://institutions.com/0.2/name, http://institutions.com/0.2/postal-
	Code, http://institutions.com/0.2/established}.
	Definition 6. Literal Values (LV ):

Table 5 .

 5 Description of sets of data of an RDF document E = n i=1 {t i .s} ∪ t i .o ⇐⇒ t i .o ∈ I ∪ BN . EP A set of entity properties is defined as: EP = n i=1 t i .p ⇐⇒ t i .o ∈ I ∪ BN . VP A set of value properties is defined as: V P = n i=1 t i .p ⇐⇒ t i .o ∈ L. LV A set of literal values is defined as: LV = n i=1 t i .o ⇐⇒ t i .o ∈ L.

	Set	Description
	E	A set of entities is defined as:

  .pk, entity property-entity = {ep j .pk, e k .pk}, value property-value = {vp l .pk, lv m .pk} >}. where: -entity * e i is an entity. -entity property-entity * ep j , e k represents t :< e i , ep j , e k >, such that ep j is an EntityP roperty where e i is the subject and e k the object. -value property-value * vp i , lv j represents t :< e i , vp l , lv m >, such that vp l is a P ropertyV alue where e i is the subject and lv m the object.

	S-RDF-S(T) = {	n i=1 <
	entity = e
	, http : //institutions.com/0.2/S0992, 1 >,
	< 3, http : //institutions.com/0.2/S0993, 1 >,
	< 4, http : //institutions.com/0.2/S0994, 1 >,
	< 5, http : //www.w3.org/2002/07/owl#T hing, 1 >},
	Entity properties = {
	< A, http : //www.w3.org/1999/02/22 -rdf -syntax -ns#type >},
	Value properties = {

i

Table 10 .

 10 Related Work Comparison for Data 1

	7 S-RDF: http://rdf-sequence.sigappfr.org
	8 Jena is a Java framework for building Semantic Web applications. It provides a
	extensive Java libraries for helping developers develop code that handles RDF,
	RDFS, RDFa, OWL and SPARQL in line with published W3C recommendations
	-https://jena.apache.org/about jena/about.html
	9 Information about persons extracted from the English and Germany Wikipedia, rep-
	resented by the FOAF vocabulary -http://wiki.dbpedia.org/Downloads2015-10.
	10 Geographic coordinates extracted from Wikipedia -https://wiki.dbpedia.org/
	downloads-2016-10.
	11 Easy-Converte: http://www.easyrdf.org/converter
	12 RDF-Translator: https://rdf-translator.appspot.com

Table 11 .

 11 Related Work Comparison for Data 2 Since there is no benchmark model for readability available in the literature to compare the existing serialization formats, we propose three questions which are related to several aspects of the RDF structure. (i) The first question is about relations, which can help to the end-user to recognize some important nodes according to the context, (ii) the second one is related to the terminal nodes, and (iii) the third one to literal values. The questions are presented as

	Serialization Format	Triples	Size	Compression Rate (%)
	RDF/XML	50,000	4,338,226	41.0298
	Turtle	50,000	5,908,228	19.6885
	N-Triple	50,000	7,356,638	-0.0001
	N3	50,000	5,908,228	19.6885
	JSON-LD	50,000	2,971,124	59.6130
	HDT	50,000	1,665,119 (HDT) 126,163 (Index)	75.6508
	S-RDF	50,000	2,149,852	70.7767
	Test 2: follows:
	1. Is the resource X the most related one of the data?
	2. Is the resource Y a terminal node in the data?
	3. How many literal values has the resource Z?
	where X, Y, and Z are resources that belong to the set of triples used to evaluate
	this test (see Listing 2).

Listing 2. List of triples used for Test 2, serialized in Turtle format

Table 12 .

 12 Number of correct, incorrect, and ambiguous values of each question per serialization format

	Serialization	Question 1		Question 2
	Format	C I A Precision Recall	F measure C I A Precision Recall	F measure
	RDF/XML	5 17 18 22.73% 21.74% 22.22% 11 9 20 55.00% 35.48% 43.14%
	Turtle	24 6 10 80.00% 70.59% 75.00% 17 5 18 77.27% 48.57% 59.65%
	N-Triple	10 6 24 62.50% 29.41% 40.00% 15 6 19 71.43% 44.11% 54.55%
	N3	22 4 14 84.62% 61.11% 70.97% 21 2 17 91.30% 55.26% 68.85%
	JSON-LD	8 14 18 36.36% 30.77% 33.33% 18 9 13 66.67% 58.06% 62.07%
	S-RDF	21 4 15 84.00% 58.33% 68.85% 19 8 13 70.37% 59.38% 64.41%
		Serialization		Question 3
		Format	C I A Precision Recall	F measure
		RDF/XML	15 6 19 71.43% 62.50% 66,67%
		Turtle	23 14 3 62.12% 88.46% 73.02%
		N-Triple	8 10 22 44.44% 26.67% 33.33%
		N3	19 10 11 65.52% 63.33% 64.41%
		JSON-LD	6 23 11 20.69% 35.29% 26.09%
		S-RDF	21 12 7 63.63% 75.00% 68.85%
		C = Correct
		I = Incorrect
		A = I do not know option (ambiguous).

Table 13 .

 13 Total number of correct, incorrect and ambiguous values per serialization format

	Serialization Format	C I A Precision Recall	F-measure
	RDF/XML	31 32 47 49.72% 39.91%	44.01%
	Turtle	64 25 31 73.14% 69.20%	69.22%
	N-Triple	33 22 65 59.46% 33.40%	42.63%
	N3	62 16 42 80.48% 59.90%	68.08%
	JSON-LD	32 46 42 41.24% 41.38%	40.50%
	S-RDF	61 24 35 72.67% 64.24%	67.37%
	C = Correct, I = Incorrect, A = I do not know option (ambiguous).

Document Type Definition (DTD) defines the structure and the legal elements and attributes of an XML document.

Centrality identifies the most related nodes within a graph, which have a high number of relations.

It is one of the four normalization forms, which consists on a Canonical Decomposition, followed by a Canonical Composition -http://www.unicode.org/reports/tr15/

The form is available here: https://forms.gle/DNMfsp5LL3nw1hW9A

Literal values = { < 1, "Lycée de la P lage" >, < 2, 64600 >, < 3, 1985 -05 -19 >, < 4, "N apoleon Business" >, < 5, 64100 >, < 6, 1986 -12 -19 >, < 7, " École N ational de l energie" >, < 8, 64500 >, < 9, 1984-11-21 >, < 10, "Grande V ille School" >, < 11, 64200 >, < 12, 1977 -08 -22 >} > The S-RDF-V represents the entities, properties, and values of an RDF document, but a document also has information about the relations among entities and literal values (node-edge-node); thus, the second part of our serialization is dedicated to represent the RDF graph structure, called RDF Sequence-Structure. It consists of a 3-tuple, where the first element is composed of an entity; the second element has all entities, which are related to the first element, preceded by its respective entity property; and the last element is used to represent value properties and its respective literal values. The RDF Sequence-Structure is defined in Def. 8. Definition 8. RDF Sequence-Structure (S-RDF-S ): Given a set of triples T = {t i | t i :< s, p, o >}, its RDF Sequence-Structure is defined as a set of 3tuples: