Gilles Trombettoni
email: gilles.trombettoni@sophia.inria.fr

Yves Papegay
email: yves.papegay@sophia.inria.fr

Gilles Chabert
email: gilles.chabert@emn.fr

Odile Pourtallier
email: odile.pourtallier@sophia.inria.fr

A Box-Consistency Contractor Based on Extremal Functions

Interval-based methods can approximate all the real solutions of a system of equations and inequalities. The Box interval constraint propagation algorithm enforces Box consistency. Its main procedure BoxNarrow handles one function f corresponding to the revised constraint, and one variable x, replacing the other variables of f by their current intervals. This paper proposes an improved BoxNarrow procedure for narrowing the domain of x when f respects certain conditions. In particular, these conditions are fulfilled when f is polynomial. f is first symbolically rewritten into a new form g. A narrowing step is then run on the non-interval extremal functions that enclose the interval function g. The corresponding algorithm is described and validated on several numerical constraint systems.

Motivation

Interval-based solvers can solve systems of numerical constraints (i.e., nonlinear equations or inequalities over the reals). Their reliability and increasing performance make them applicable to various domains such as robotics design and kinematics [START_REF] Merlet | Interval Analysis and Robotics[END_REF], proofs of conjectures [START_REF] Tucker | A Rigorous ODE Solver and Smale's 14th Problem[END_REF], robust global optimization [START_REF] Kearfott | Rigorous Global Search: Continuous Problems[END_REF][START_REF] Rueher | Capabilities of Constraint Programming in Rigorous Global Optimization[END_REF] and bounded-error parameter estimation [START_REF] Jaulin | Interval Constraint Propagation with Application to Bounded-error Estimation[END_REF].

Two main types of contraction algorithms allow solvers to filter variable domains, i.e., to reduce the intervals of each variable, without loss of solutions of the system: interval (numerical) analysis methods, like Interval Newton [START_REF] Moore | Interval Analysis[END_REF], and constraint propagation algorithms from constraint programming. The HC4 and Box algorithms [START_REF] Benhamou | Revising Hull and Box Consistency[END_REF][START_REF] Van Hentenryck | Numerica : A Modeling Language for Global Optimization[END_REF] are very often used in solving strategies. They perform a propagation loop and filter the variable domains with a specific revise procedure (called HC4-Revise and BoxNarrow) handling the constraints individually. For every pair (c, x) in the system, where c is the numerical constraint f (x, y 1 , . . . , y k-1 ) = 0, the BoxNarrow contraction procedure is applied to x by considering the uni-variate constraint:

f [Y ] (x) = f (x, [y 1 ], . . . , [y k-1 ]) = 0. That is, f [Y ]
is a function where each variable y i ∈ Y = {y 1 , . . . , y k-1 } of f has been replaced by its interval of variation. The important point is that f [Y ] is an interval function: to any x ∈ R, f [Y ] (x) is an interval. Thus, the iterative process run by BoxNarrow may be very slow in some cases. The main idea of PolyBox is to work with two non-interval functions instead of f [Y ] . These non-interval functions are obtained by a symbolic manipulation preprocessing that rewrites f into a new form g for which the two extremal functions enclosing g [Y ] can be easily extracted. Then, during constraint propagation, PolyBoxRevise calls BoxNarrow on the two extremal functions of g [Y ] . This implies a faster contraction. In addition, when g [Y ] is a low-degree polynomial, the computation of the new bounds of [x] follows simple evaluations using the real roots of the extremal functions identified analytically.

Background

Intervals allow reliable computations on computers by managing floating-point bounds and outward rounding.

v -v is the size of [v]. An interval vector, or box, [V ] = ([v 1 ], . . . , [v n ]) represents the Cartesian product [v 1 ] × . . . × [v n ]. Its size is the maximal size of its components [v i ], i = 1, . . . , n.
Interval arithmetic has been introduced to extend the real arithmetic to intervals [START_REF] Moore | Interval Analysis[END_REF]. For instance, we have straightforwardly

[v 1 ]+[v 2 ] = [v 1 +v 2 , v 1 +v 2 ].
This allows us to extend real valued functions to intervals. Such an extension must be defined so as to be conservative, i.e.,

∀V ∈ R k f (V ) = [f ](V ) and ∀[V ] ∈ IR k [f ]([V ]) ⊇ {f (V ), V ∈ [V ]}.
The natural extension [f ] N of a real function f replaces arithmetic over the reals by interval arithmetic. Consider for instance the real function [START_REF] Benhamou | Revising Hull and Box Consistency[END_REF]. Note that the natural extension of f depends upon its symbolic expression, and consequently is not unique. As a matter of fact, f may also be rewritten as (x 1 -x 2 ) 2 and yields the natural extension

f (x 1 , x 2 ) = x 2 1 -2x 1 x 2 + x 2 2 . The natural extension [f ] N from IR n to IR is defined by [f ] N ([x 1 ], [x 2 ]) = [x 1 ] 2 -2[x 1 ][x 2 ] + [x 2 ] 2 . Evaluated on the intervals [x 1 ] = [x 2 ] = [0, 1], we obtain [f ] N ([x 1 ], [x 2 ]) = [-2,
([x 1 ] -[x 2 ]) 2 . Note that ([x 1 ] -[x 2 ]) 2 = [0, 1] = {f (x 1 , x 2 ), x 1 ∈ [x 1 ], x 2 ∈ [x 2 ]} ⊂ [f ] N ([0, 1], [0, 1]
). This illustrates the dependency problem which is a major concern in interval arithmetic. f has multiple occurrences of variables that are handled as different variables by interval arithmetic. In general, the dependency problem implies an overestimation of the interval image. It renders NP-hard the problem of finding the optimal interval image of a polynomial [START_REF] Kreinovich | Computational Complexity and Feasibility of Data Processing and Interval Computations[END_REF]. This raises the need to symbolic manipulations of expression before calculations so as to reduce overestimation.

The PolyBox algorithm presented in this paper aims at solving nonlinear systems of constraints or Numerical CSPs. An NCSP P = (V, C, [V ]) contains a set of constraints C, a set V of n variables with domains [V ] ∈ IR n . A solution S ∈ [V ] to P satisfies all the constraints in C. To approximate all the solutions of an NCSP with interval-based techniques, the solving process starts from an initial box representing the search space and builds a search tree, following a Branch & Contract scheme. A Branching operation bisects the current box on one dimension (variable), generating two sub-boxes. At each node of the search tree, contraction/filtering algorithms improve the bounds of the current box with no loss of solutions. The process terminates with boxes of size smaller than a given positive ω.

The constraint programming community proposes constraint propagation algorithms that perform a propagation loop like AC3. Contracting optimally a box w.r.t. an individual constraint is referred to as hull-consistency problem. Similarly to the optimal interval image computation, due to the dependency problem, hull-consistency is not tractable. The main procedure of our algorithm is compared to two state-of-the-art revise algorithms that handle the constraints individually. HC4-Revise [START_REF] Benhamou | Revising Hull and Box Consistency[END_REF] is known to achieve the hull-consistency of constraints having no variable with multiple occurrences, provided that the function1 is continuous. It traverses twice the tree representing the mathematical expression of the constraint for narrowing all the involved variable intervals. BoxNarrow [START_REF] Benhamou | Revising Hull and Box Consistency[END_REF][START_REF] Van Hentenryck | Numerica : A Modeling Language for Global Optimization[END_REF] is stronger than HC4-Revise [START_REF] Collavizza | Extending Consistent Domains of Numeric CSP[END_REF] and can enforce hull-consistency of a constraint when it contains one variable with multiple occurrences. In the general case, it enforces the Box-consistency property [START_REF] Benhamou | Revising Hull and Box Consistency[END_REF].

Definition 2 An NCSP (X, C, [X]) is box-consistent if every pair (c, x), c ∈ C, x ∈ X is box-consistent. Consider the pair (c, x), where the constraint is described by c : f (x, y 1 , . . . , y k-1 ) = 0, f : R k → R, and the univariate interval function f [Y ] (x) = f (x, [y 1 ], . . . , [y k-1 ]). The pair (c, x) is box-consistent (with respect to the natural extension [f ] N ) on the domain [x] = [x, x], if: 0 ∈ [f [Y ] ] N ([x, +]) and 0 ∈ [f [Y ] ] N ([-, x]),
(resp. [x-η, x]) are discarded from [x] if 0 / ∈ [f [Y ] ] N ([x, x + η]) (resp. 0 / ∈ [f [Y ] ] N ([x -η, x]
)). This test sometimes uses a uni-variate interval Newton procedure. Observe that at the end of RightNarrow (step 16), [r] does not contain any zero of the function but an -zero. The slicing performed by BoxNarrow on a variable x limits the overestimation effect on x, but not on the other variables y i if they also occur several times.

[l]

[r] The aim of our new PolyBoxRevise procedure is to limit the overestimation due to multiple occurrences of variables y i and to speed up the iterative narrowing process introduced above. Before solving the system, in a preprocessing phase, for every pair (c, x) given by c : f (x, y 1 , . . . , y k-1 ) = 0, we use symbolic manipulation to rewrite f into a new form g(x, y 1 , . . . , y k-1 ). This preprocessing allows the PolyBoxRevise procedure to rapidly extract, during constraint propagation, non-interval extremal functions that enclose g [Y ] , before contracting [x].

Symbolic manipulation and extremal functions

For any pair (c, x) and any box ([x], [y 1 ], . . . , [y k-1 ]), the aim would be to extract the optimal extremal real (i.e., non-interval) functions h(x) and h(x) defined by h

(x) = min yi∈[yi] f (x, y 1 , . . . , y k-1 ) and h(x) = max yi∈[yi] f (x, y 1 , . . . , y k-1 ), ∀x ∈ [x]. Then a necessary and sufficient condition for x ∈ [x] to satisfy c is 0 ∈ [h(x), h(x)].
Unfortunately, it is generally not tractable to determine h and h due to the overestimation implied by the dependency problem (see Section 2 and [START_REF] Kreinovich | Computational Complexity and Feasibility of Data Processing and Interval Computations[END_REF]), and we have to be satisfied with functions g and g such that for any

x ∈ [x] we have g(x) ≤ h(x) ≤ f (x, y 1 , . . . , y k-1 ) ≤ h(x) ≤ g(x), ∀x ∈ [x], ∀y i ∈ [y i ], i ∈ {1, . . . , k-1}. Now the test 0 ∈ [g(x), g(x)
] is only a necessary condition. We will refer to g and g as minimal and maximal extremal functions.

The aim is to automatically and rapidly identify extremal functions. This can clearly not be done for any function f and we have restricted our attention to the class of functions that can be described by f (x, y 1 , . . . , y k-1 ) = d i=0 f i (y 1 , . . . , y k-1 ).h i (x) where d is a positive integer and h i has a finite number of zeros in [x] that can be computed exactly. In addition, the sign of h i (x) is known for any x ∈ [x]. We have in mind elementary functions such as x i , log(x) or e x .

We have used the symbolic manipulation tool Mathematica [START_REF] Wolfram | Mathematica[END_REF] for automatically identifying functions f i and h i , and to rewrite them in the most appropriate manner. The procedure FullSimplify of Mathematica computes automatically several possible forms for every f i (heuristically) and selects the form minimizing a given criterion. The criterion we have specified is the number of occurrences of each variable y i . During the solving, like BoxNarrow, the PolyBoxRevise procedure first replaces, in the new analytic form g, the variables y i by their domains. We thus obtain g

[Y ] (x) = d i=0 [f i ] N ([Y ])h i (x), [Y ] = ([y 1 ], . . . , [y k-1 ]). Given the box [Y ], the coefficients [f i ] N ([Y ]), denoted [c i ]
, are now numerical intervals. Due to the assumptions on h i , the following two functions g [Y ] and g [Y ] respectively minimal and maximal extremal functions, computed at the bounds of the interval coefficients [c i ]: [START_REF] Araya | Making Adaptive an Interval Constraint Propagation Algorithm Exploiting Monotonicity[END_REF][START_REF] Collavizza | Extending Consistent Domains of Numeric CSP[END_REF]x + [0.84147, 1] and then are not comparable. That is why our contractor starts by calling systematically the cheap HC4-Revise procedure on the initial form f before performing the process described below. For a given [Y ], once the extremal functions have been determined, we proceed with the contraction part of PolyBoxRevise (during constraint propagation). Starting with an initial interval [x], let us detail how the new and improved left bound l ([l] is 1 u.l.p. large) of [x] is determined. (A symetric process is performed for the right bound.) PolyBoxRevise first determines with which extremal function to work with. Three cases occur:

g [Y ] (x) = d i=1 c - i (x)h i (x) and g [Y ] (x) = d i=1 c + i (x)h i (x) with c - i (x) = c i , c + i (x) = c i , if h i (x) ≥ 0 c - i (x) = c i , c + i (x) = c i , if h i (x) ≤ 0 Example. Consider the function: f (x, y 1 , y 2 ) = (y 1 + y 2 )x 2 + (2y 1 y 2 )x + sin(y 2 ). For the domains [y 1 ] = [0.5, 1], [y 2 ] = [1, 2], we have g [Y ] (x) = [1.5, 3]x 2 +
g [0.5,1],[1,2] (x) = 1.5x 2 + x + 0.84147, if x ≥ 0 1.5x 2 + 4x + 0.84147, if x ≤ 0 g [0.5,1],[1,2] (x) = 3x 2 + 4x + 1, if x ≥ 0 3x 2 + x + 1, if x ≤ 0 Remark. g [Y
1. If g [Y ] (x) ≤ 0 and 0 ≤ g [Y ] (x) : l = x (no contraction) 2. If g [Y ] (x) > 0 : g [Y ] is selected 3. If g [Y ] (x) < 0 : g [Y ]
is selected (situation depicted in the left side of Fig. 1)

The smallest root [l] of g [Y ] (x) = 0 in [x] can now be computed using the standard BoxNarrow (i.e., LeftNarrow) procedure applied to the extremal function selected. The advantage is a faster convergence since BoxNarrow is run with a non-interval function.

We have implemented the polynomial case, where h i (x) = x i , i = 0, . . . , d. In particular, when the degree d is smaller than 4, instead of using BoxNarrow to determine the real roots of g [Y ] (x) = 0, we have used explicit analytical expressions of the roots. For d = 3, we have used the Cardano's expressions 4 of the real roots. We have adapted these symbolic methods to manage rounding errors due to floating point calculation by first replacing all the coefficients by a degenerate interval (of null size).

Finally, we have implemented a new procedure PolyBoxRevise based on the Box algorithm variant called BC4 [START_REF] Benhamou | Revising Hull and Box Consistency[END_REF]. If f (x, y 1 , . . . , y k-1 ) has a single occurrence of x, PolyBoxRevise calls HC4-Revise (like BC4 does). Otherwise, it uses the rewritten form g (with appropriate symbolic expressions for the f i (y 1 , . . . , y k-1 )) of f produced automatically by the FullSimplify procedure of Mathematica [START_REF] Wolfram | Mathematica[END_REF] in the preprocessing. Four cases occur: Remarks. The second case above can be illustrated by an equation of the system Caprasse (tested below): -2x + 2txy -z + y 2 z = 0 that our symbolic tool rewrites into : (-2 + 2ty)x + (-1 + y2)z = 0 (for the contraction of [x] or [z]).

Observe that the new form makes disappear the multiple occurrences of x and z. The decrease in occurrences of x and z illustrates a successful transformation leading to a gain in CPU time. An equation of the instance 6body shows a counterproductive transformation of 5(b Comparison with the Box algorithm of Numerica Van Hentenryck, Michel et Deville have also used extremal functions (without using this vocabulary) in their interval-based solver Numerica [START_REF] Van Hentenryck | Numerica : A Modeling Language for Global Optimization[END_REF]. The principle is introduced in one page in a technical article [START_REF] Van Hentenryck | Solving Polynomial Systems Using a Branch and Prune Approach[END_REF]. Numerica manages different forms of the handled system, and a separate constraint propagation is run on the system in an entirely expanded form for using extremal functions.

1 -d 1 ) + 3(b 2 -d 2 )(b 1 + d 1 -2f 1 ) = 0 into b 1 (5+3(b 2 -d 2 ))+(-5+3b 2 -3d 2 )d 1 +6(-b 2 +d 2 )f 1 = 0. Indeed,
PolyBox follows on the contrary a scheme close to BC4. It manages a unique system with revise procedures adapted to every pair (f, x), which causes an overestimation smaller than the entirely expanded form used by Numerica. In addition, like BC4, PolyBox also uses HC4-Revise when x occurs only once in f . Finally, the analytic solving of low degree polynomials is added. We have compared our PolyBox algorithm to BC4 and HC4. The symbolic manipulation of all pairs (f, x) is achieved in a fraction of a second in a preprocessing by Mathematica [START_REF] Wolfram | Mathematica[END_REF]. All the contractors have been implemented in the free Ibex interval-based C++ library [START_REF] Chabert | Ibex -An Interval Based EXplorer[END_REF]. To find all the solutions to the tested NCSPs, the solving strategy bisects the variables in a round-robin way. Between two branching points in the search, constraint propagation (i.e., PolyBox, HC4 or BC4) is performed before an interval Newton.

Among the 44 polynomial systems with isolated solutions found in COPRIN's Web page 5 , we have selected the 12 instances that are solved by at least one of the 3 strategies in a time comprised between 1 second and 1 hour (on a Pentium 3 GHz) and have equations with multiple occurrences of the variables.

Table 1 reports interesting speedups brought by PolyBox on these instances. The column PolyBox--in the table corresponds to a variant of PolyBox in which the low degree polynomials are not handled analytically but by BoxNarrow. The additional gain brought by the analytic process is significant in only two NCSPs.

Our first results are promising, so that it should be worthwhile hybridizing PolyBox with other algorithms, especially those achieving the Box-consistency or a weaker form of it [START_REF] Araya | Making Adaptive an Interval Constraint Propagation Algorithm Exploiting Monotonicity[END_REF][START_REF] Goldsztejn | Box Consistency through Adaptive Shaving[END_REF]. An idea would be to keep the rewritten forms only if they are of degrees 2 and 3, and add them as global and redundant constraints in the system for improving the constraint propagation.

Definition 1 (

 1 Basic definitions, notations) IR denotes the set of intervals [v] = [a, b] ⊂ R where a, also denoted v, and b, also denoted v, are floating-point numbers.

  where [x, +] and [-, x] denote intervals of size one u.l.p. 2 at the bounds of [x]. In practice, for every pair (f, x), starting with an interval [x], the BoxNarrow procedure returns a reduced interval [x ] ⊆ [x] such that [x , +] (resp. [-, x ]) is the smallest (resp. largest) -solution 3 of the equation f [Y ] (x) = 0. Existing procedures use a shaving principle to narrow [x]: "Slices" [x, x+η]

Figure 1

 1 illustrates that f [Y ] is an interval function. It also shows the steps followed by BoxNarrow. The top (resp. the bottom) side of the figure details the "dichotomic" work performed by LeftNarrow (resp. RightNarrow) on slices/intervals of decreasing size, starting from [x]. For LeftNarrow, if the size of the current interval [l] is less than or equal to 1 u.l.p. and 0 ∈ [f [Y ] ] N ([l]), then the procedure returns [l]. The last step 17 replaces the interval [x] by the new interval [l, r].

Fig. 1 .

 1 Fig. 1. The BoxNarrow procedure. The algorithm returns the interval computed in the step 17. (The additional contraction performed by Newton is not considered.)

1 .

 1 f is not polynomial w.r.t. x: the procedure calls BoxNarrow (or HC4-Revise in a hybrid version because it is less time consuming). 2. g [Y ] (x) contains only one occurrence of x: HC4-Revise is applied to g [Y ] (x). 3. g [Y ] (x) has multiple occurrences of x and d < 4: analytic determination of the smallest root of g [Y ] (x) in [x]. 4. g [Y ] (x) has multiple occurrences of x and d ≥ 4: numerical determination of the smallest root of g [Y ] (x) in [x], using BoxNarrow.

for obtaining an expanded form on b 1 or d 1 ,

 1 the transformation increases the overestimation because of the additional occurrences of variables a 2 , b 2 and d 2 .

Table 1 .

 1 Results. The entries in the last four columns are the CPU time in second (first row) and the number of nodes in the search tree (second row).

	4 Experiments
	Name	#var #sol	HC4	BC4	PolyBox--PolyBox
	Caprasse	4	18	5.53	37.2	2.34	2.16
				9539	6509	2939	2939
	Yamamura1	8	7	34.3	13.4	5.79	2.72
				42383	4041	2231	2231
	Extended Wood	4	3	0.76	1.94	1.34	1.12
				4555	1947	3479	3479
	Broyden Banded	20	1	> 3600	0.62	0.16	0.09
				?	1	1	1
	Extended Freudenstein	20	1	> 3600	0.19	0.22	0.11
				?	121	121	121
	6body	6	5	0.58	2.93	0.73	0.73
				4899	4797	4887	4887
	Rose	3	18	> 3600	> 3600	4.00	4.10
				?	?	12521	12521
	Discrete Boundary	39	1	179	29.5	41.8	16.1
				185,617	3279	3281	3281
	Katsura	12	7	102	404	103	104
				14007	11371	13719	13719
	Eco9	8	16	66	191	71	71
				132,873 125,675	131,911	131,911
	Broyden Tridiagonal	20	2	470	495	403	350
				269,773 163,787	164,445	164,445
	Geneig	6	10	3657	> 7200	3508	3363
				79,472,328	?	4,907,705 4,907,705

along with projection functions used during the second top-down tree traversal...

One Unit in the Last Place is the gap between two successive floating-point numbers.

x ∈ R n is an -solution of f (x) = 0, if [-, ] ∩ f (x) = ∅.

G. Cardano. Ars magna, sive de regulis algebraicis liber unus, Nuremberg, 1545.

See www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/benches.html. These benchmarks have been proposed by the interval community and some of them correspond to real problems.


Bara Florence

Exploratory procedures employed by visually impaired and blind children during joint book reading

Keywords: visually impaired, blind, children, tactile book, exploratory procedures, joint reading

Illustrations in storybooks not only provide a source of interest in children but support language development and literacy skills. A better understanding of how to communicate stories through tactile illustrations is a central issue for practitioners and researchers who work with visually impaired children. Recognizing tactile illustrations implies precisely perceiving the shapes and being able to associate meaning to theses representations. This study focused on the way tactile pictures were haptically explored by visually impaired and blind children during joint book reading. The aim was to determine how the task and the type of illustrations promote the haptic exploration. Children seemed to be more active and used more often the contours following procedure when they have to find the meaning of the pictures than when the meaning was given to them through reading. The 3D illustrations led to the use of more different exploratory procedures than the 2D illustrations.

Increasing language level, acquiring vocabulary and understanding stories orally told are important components of child development in the literacy domain. Picture books reading is a common way to develop these abilities in young children both at school and at home, and has been shown to be an important activity to promote children's language and literacy skills [START_REF] Abbott | Structural equation modeling of relationships among developmental skills and writing skills in primary and intermediate grade writers[END_REF][START_REF] Bus | Joint book reading makes for success in learning to read: A meta-analysis on intergenerational transmission of literacy[END_REF][START_REF] Collins | ELL preschoolers' English vocabulary acquisition from storybook reading[END_REF][START_REF] Mol | Interactive book reading in early education: A tool to stimulate print knowledge as well as oral language[END_REF][START_REF] Sénéchal | Testing the home literacy model: mother involvment in kindergarten is differentially related to grade 4 reading comprehension, fluency, spelling, and reading for pleasure[END_REF]. Picture books reading at home has been linked to children's development of language [START_REF] Debaryshe | Joint picture book reading correlates of early oral language skill[END_REF][START_REF] Payne | The role of home literacy environment in the development of language ability in preschool children from lowincome families[END_REF][START_REF] Snow | Turn the page please: Situation-specific language acquisition[END_REF], print concepts, and emergent reading [START_REF] Snow | The contracts of literacy: What children learn from learning to read books[END_REF][START_REF] Sulzby | Emergent literacy[END_REF]. Most books published for young children are illustrated. It is widely assumed that such illustrations increase children's motivation and attention, promote their creativity, serve as mental scaffolds (making the story more concentrated, more concrete, more coherent and comprehensible), clarify the text content and serve to organize verbal information [START_REF] Levin | Understanding illustrations in text[END_REF][START_REF] Peeck | Increasing picture effects in learning from illustrated text[END_REF]. According to [START_REF] Fang | Illustrations, text, and the child reader. What are pictures in children's story books for?[END_REF] pictures in storybooks serve to help to establish the setting, develop the characters and the plot, provide a different viewpoint, contribute to the text coherence and reinforce the text. In complementing the text, pictures can help the children to perceive, understand and remember the stories [START_REF] David | News concreteness and visual-verbal association: Do news pictures narrow the recall gap between concrete and abstract news[END_REF][START_REF] Rubman | A, B seeing: The role of reconstructive processes in children 's comprehension monitoring[END_REF]. Exposure to information both verbally and visually results in the construction of separate verbal and pictorial representations that are connected in memory (dual coding hypothesis proposed by [START_REF] Paivio | Mental representations: A dual-coding approach[END_REF], which leads to the establishment of stronger, more elaborated and more organized memory traces [START_REF] Digdon | Preschoolers' learning when pictures do not tell the whole story[END_REF][START_REF] Levin | Understanding illustrations in text[END_REF][START_REF] Peeck | Increasing picture effects in learning from illustrated text[END_REF][START_REF] Pressley | Picture content and preschoolers' learning from sentences[END_REF]. [START_REF] Levin | On functions of pictures in prose[END_REF] described five functions of pictures in books: decorational (little relationship with the text), representational (illustrate the text content), organizational (provide a framework for the text content), interpretational (clarify the text content), and transformational (include mnemonic components). A majority of the illustrations are representational in that they literally depict or overlap part or all of the text content, with the basic purpose of making the story concrete [START_REF] Carney | Pictorial illustrations still improve students' learning from text[END_REF].

Within the picture book reading interaction, there are three components: a book, a child and an adult [START_REF] Martinez | Read it again: The value of repeated readings during storytime[END_REF][START_REF] Van Kleeck | Research on book sharing: Another critical look[END_REF]. The benefits of picture books reading depend on the type of illustrations, the children characteristics (age, abilities, learning potential), and the quality of the interaction between the adult and the child during reading [START_REF] Alder | Directed picture processing: The effects for learners on recall of related text[END_REF]Aram, Fine, & Ziv, in press;[START_REF] Deloache | Joint picturebook interactions of mothers and 1-year-old children[END_REF][START_REF] Ninio | The achievement and antecedents of labelling[END_REF][START_REF] Senechal | Age-related differences in the organization of parent-infant interactions during picture book reading[END_REF] Illustrations enhance preschoolers' memories of auditory prose only under some conditions. Even though a particular illustration has been designed to be cognitively useful, its effectiveness depends on how the learner perceives and processes the picture. Some specific features in book support more learning than other.

Considering the type of picture, the benefits on the stories' recalling showed effect size from moderate (representational pictures) to high (transformational pictures) [START_REF] Levin | On empirically validating functions of picture in prose[END_REF]. The ability of children to transfer knowledge from pictures in books to reality depends on the physical similarity between the picture and their real world referent, with greater iconicity leading to better learning [START_REF] Ganea | Transfer between picture books and the real world by very young children[END_REF][START_REF] Tare | Less is more: How manipulative features affect children's learning from picture books[END_REF]. [START_REF] Carney | Pictorial illustrations still improve students' learning from text[END_REF] wrote guidelines to create text illustrations: they recommended to select pictures that provide congruent or supporting information to the text, to add pictures only when it is necessary (the more complex is the text, the more helpful the pictures are), to guide children to do something with the pictures (labeling the features of the illustration), and to choose pictures as text supplements rather than as text substitutes. The learner characteristics, as age, working memory resources, reading ability and ability to extract visual information from pictures may constrain the benefits of picture books reading. Picture facilitation effects require that individuals encode and connect in memory both verbal and pictorial information. Storybook pictures may interfere with words reading and extracting meaning from text if not enough working memory resources are available [START_REF] Mayer | A split-attention effect in multimedia learning: Evidence for dual processing systems in working memory[END_REF]. [START_REF] Tare | Less is more: How manipulative features affect children's learning from picture books[END_REF] showed that children's learning from pictures was impeded by manipulative features, which are intended to increase young children's engagement with books. They suggested that children were distracted by manipulative illustrations that focused their attention toward the sensory stimuli and away from the text content. The children's manipulation might have increased their cognitive load such that they could not process what the adult was telling. Picture books tell stories with a combination of texts and illustrations, providing children with a narrative language and a visual experience (Yu, 2012). Thus the child's visual information processing has also an impact on the extraction and interpretation of meaning. If perceptual processing and memory resources are crucial in picture books reading in sighted children, it is the same in visually impaired and blind children. The tactile skills and manipulative experiences that are essential for blind children development and learning are also the foundation of their emergent literacy [START_REF] Stratton | Emergent literacy: A new perspective[END_REF]. The haptic system that involves active manual exploration is necessary for blind and visually impaired individuals to collect information about the objects in their environment.

Even though some aspects of literacy development naturally fit with the ways blind and visually impaired children learn, others need some adaptations. This need of adaptations was highlighted by [START_REF] Crespo | Storybooks for blind infants and children[END_REF] who showed that some parents did not read stories to their blind children because they thought that their children might be confused by visual references because they could not see the pictures. After participating in making tactile story books, these parents were more positive about reading stories to their children. The purpose of tactile books is to communicate information through touch. Tactile illustrations are scanned with the fingertips and the reader had to interpret and associate meaning to the picture. Most of books are based on an existing printed book and usually a tactile picture is transferred from a visual one, and the printed text is transferred into Braille. The illustrations in tactile books come in a wide variety of materials and can be produced by several different techniques, such as swell paper (lines, dots and surfaces are used to create raised line drawings), thermoform (technique of molding plastic in a matrix), textiles, textures, 3D objects... Tactile pictures that are literally transferred from visual pictures might not fit with the cognitive and haptic abilities of young blind children and bidimensional representations can be very hard to understand [START_REF] Lewi-Dumont | L'apprentissage de la lecture chez les enfants aveugles: Difficultés et évolution des compétences [Reading acquisition in blind children[END_REF]. To facilitate tactile exploration and interpretation, pictures need to be simplified and some details of the visual picture need to be deleted, leaving only the most important features [START_REF] Eriksson | Livres d'images pour enfants déficients visuels [Tactile picture books for visually impaired children[END_REF]. As it is the case with sighted children, tactile pictures (if correctly identified) improve the understanding of the text [START_REF] Stratton | On the way to literacy: Early experiences for young visually impaired children[END_REF], develop creativity and imagination [START_REF] Meuwes | Livres tactiles [Tactile books[END_REF] and improve the retention and recalling of the story [START_REF] Pring | Pictures for the blind: an investigation of the influence of pictures on the recall of texts by blind children[END_REF]. Even if tactile pictures are often hardly recognizable by blind children, they could be helpful for their cognitive, language and literacy development [START_REF] Hatwell | La lecture tactile des cartes et des dessins en relief par les aveugles [Haptic reading of raised cards and drawings by blind individuals[END_REF][START_REF] Lewi-Dumont | L'apprentissage de la lecture chez les enfants aveugles: Difficultés et évolution des compétences [Reading acquisition in blind children[END_REF][START_REF] Miller | Reading comes naturally: A mother and her blind child's experiences[END_REF]. Selecting stories that relate to a child's interest and experiences [START_REF] Stratton | On the way to literacy: Early experiences for young visually impaired children[END_REF] and adapting books and reading strategies [START_REF] Crespo | Storybooks for blind infants and children[END_REF][START_REF] Miller | Reading comes naturally: A mother and her blind child's experiences[END_REF] will increase children's understanding of stories.

The type of illustrations, the child characteristics (age, memory resources, knowledge level) and how the book is read and haptically explored, should constrain the benefits of tactile picture books reading. A main difficulty leads in the cognitive load generated by tactile pictures identification and interpretation. Haptic processing allows to collect sequential and fragmented information that need to be integrated, which places an important load on attention and working memory [START_REF] Hochberg | Representation of motion and space in video and cinematic displays[END_REF][START_REF] Loomis | Similarity of tactual and visual picture recognition with limited field of view[END_REF]. As the haptic system is more sequential than vision and more adapted for processing material properties of objects, such as texture and temperature [START_REF] Lederman | Hand movements: A window into haptic object recognition[END_REF], 1993), it could be difficult for blind children to access shape information contained in tactile pictures.

Even if haptic can process geometric properties of objects (such as shape and size), it cannot be done with the same accuracy than with vision. Taking into account the small haptic perceptual field, it is difficult for blind people to keep in memory raised drawing [START_REF] Hatwell | La lecture tactile des cartes et des dessins en relief par les aveugles [Haptic reading of raised cards and drawings by blind individuals[END_REF]. Movement control during exploration interferes with identification, and sequential processing in haptic is slower than parallel processing in vision. The dual task of exploration and identification constrain the identification latencies. If the exploration is guided, it relieves the observer (adults and children) partly from the exploration task and lead to better identification scores (D'Angiulli, [START_REF] D'angiulli | Blind children recognizing tactile pictures respond like sighted children given guidance in exploration[END_REF]. This guidance can be made by the experimenter who move the observer hand [START_REF] Magee | Exploring pictures tactually[END_REF], or can be facilitated by the material used, like an embossed surface that guides the finger more easily along the picture than only a line [START_REF] Thompson | The role of pictorial convention in haptic picture perception[END_REF]. The verbal guidance could also induce some exploratory procedures in blind children and help them to understand tactile pictures [START_REF] Theurel | The effect of joint reading on the tactile comprehension of a tactile illustratde book by early blind children[END_REF] The type of manual procedures children employed could determine, at least in part, the accuracy of tactile pictures identification. The haptic strategies used for object exploration might be influenced by task demands, prior knowledge [START_REF] Alexander | Knowledge is not everything: Analysis of children's performance on a haptic comparison task[END_REF][START_REF] Millar | Network models for haptic perception[END_REF], age and level of vision. How the hands move and how long determine the sensory information that is obtained through haptic exploration and what is perceived [START_REF] Lederman | Hand movements: A window into haptic object recognition[END_REF]. In adult and children, it has been shown that bimanual exploration leads to better performance than unimanual exploration [START_REF] Ballasteros | Haptic discrimination of bilateral symetry in 2-dimensional and 3-dimensional unfamilair displays[END_REF][START_REF] Russier | Haptic discrimination of two-dimensional raised-line shapes by blind and sighted adults[END_REF][START_REF] Wijntjes | Look what i have felt: Unidentified haptic line drawings are identified after sketching[END_REF]. Exploration using several fingers rather than just one seems to improve recognition of 2D raised line drawings [START_REF] Klatzky | Haptic identification of objects and their depictions[END_REF]. [START_REF] Lederman | Hand movements: A window into haptic object recognition[END_REF], 1993, 2009) categorized six different exploratory procedures (EPs) and showed how each of these is associated with a specific property and a specific goal. When grouping object by texture, individuals generally performed "lateral motion" (sideways back and forth movements of the fingers), whereas when grouping objects by shape, they often produced "contour following" (tracing the object contours with their fingertips). Even if [START_REF] Bushnell | The development of haptic perception during infancy[END_REF], 1993[START_REF] David | News concreteness and visual-verbal association: Do news pictures narrow the recall gap between concrete and abstract news[END_REF] suggested that preschool children might have the capacity to display mature haptic abilities, the EP they used are not always well adapted to the task. [START_REF] Schwarzer | learning categories by touch: On the development of holistic and analytic processing[END_REF] showed that 4 ½ years old children were able to produce EPs as adults when apprehending surface texture information, however, despite being asked to focused on texture when exploring and classifying 3D objects, they also employed procedures such as contours following and enclosure that are specific to shape extraction. Furthermore, children tend to use global strategies (lateral motion) in a shape discrimination task and for perceiving 2D tactile patterns, since this procedure seems to be inappropriate for the encoding of shape information but optimal for the processing of texture information [START_REF] Hatwell | Manual exploratory procedures in children and adults[END_REF][START_REF] Vinter | Exploratory procedures of tactile images in visually impaired and blindfolded sighted children: How they relate to their consequent performance in drawing[END_REF]. Up to around 5 years, children's manual exploration remains partial and incomplete and is unsuitable for the extraction of precise shape information [START_REF] Piaget | Child conception of space[END_REF]. Exploratory procedures developed gradually with age and when children grow older, they shift to using adult like patterns of exploration such as contours following [START_REF] Hatwell | Manual exploratory procedures in children and adults[END_REF]. The results of [START_REF] Morrongiello | Tactual object exploration and recognition in blind and sighted children[END_REF] revealed the emergence of a developmental pattern between the ages of 3 and 8 years in object exploration and recognition for sighted children. Older sighted children were able to recognize more objects and more quickly than younger children, and also were more thorough in their exploration patterns.

With increasing age, children appear to change their representation of objects from one based predominantly on global shape to one that incorporates both global shape and specific local parts. The nature of the exploratory procedures used by children influence the quality of their perception. The thoroughness of the object exploration by blind children was positively correlated with their recognition scores [START_REF] Morrongiello | Tactual object exploration and recognition in blind and sighted children[END_REF]. [START_REF] Vinter | Exploratory procedures of tactile images in visually impaired and blindfolded sighted children: How they relate to their consequent performance in drawing[END_REF] showed strong relationships between the exploratory procedures employed by sighted, blind and visually impaired children, when exploring bidimensional tactile patterns and their consequent performance in drawing. The greater the number of strategies was, the more the drawings resembled the model and/or were characterized by salient features and/or were produced using contour lines. The lateral motion procedure, that is not suitable for shape extraction led to poorly recognizable drawings. These authors emphasized the fact that blind children need to be trained in exploratory procedures at an early age and suggested that specific educational program of exploratory activities could be designed for young blind children when they are introduced to illustrated tactile books.

To be able to create and propose pictures that are suited for tactile scanning, there is a need to analyze what is to be shown and how the pictures need to be explored. Considering the importance of exploratory procedures in picture identification, this study focused on the way pictures in tactile books are haptically explored by visually impaired and blind children during joint reading. The aim was to determine how the type of task and the nature of the illustrations promote haptic exploration of tactile illustrated books. In experiment 1, the exploratory procedures used by children when exploring the book illustrations, following two different goals, were analyzed: (1) exploring the picture to infer its meaning and (2) exploring the picture while listening to the text and attempting to link the text content with the illustration. In experiment 2 the exploratory procedures used by children with visual handicap when exploring two kinds of illustrations (2D and 3D illustrations) were compared.

Study 1

The aim of this study was to assess the effect of the goal pursued by children when exploring tactile pictures. The effect of the task on the exploratory procedures employed by children was studied. The exploratory procedures used by children when they explored the pictures for the first time and tried to find their meaning were compared to the exploratory procedures employed when the adult read the text and the child has to link the illustration with the text content.

Method

Participants

The participants of this study were six children, two girls and four boys, aged between 5 years 7 months and 8 years 5 months. Four of them were right-handed and two left-handed.

Five were visually impaired and one was totally blind. The main causes of visual impairment were bilateral retinal atrophy, glaucoma and nystagmus. The cause of blindness was prematurity. The children did not present any other known cognitive or psychiatric disabilities. Four of them were attended specialized classroom, and two were in regular classes. None of them was academically retarded. Written consent was obtained from the parents of each child participating in the study. The adults who read the book were the teachers and the educators in charge of the children.

Material

The book "Ali and Leo", produced by "Les Doigts Qui Rêvent", a publishing company specialized in books adapted for visually impaired children, was used. The pictures were produced in swell paper. The pictures were not representational in that they do not exactly depict the reality. They consisted in raised line drawings and each picture can be associated with different meanings. This book is assumed to develop imagination and creativity in blind children. One page consisted in the Braille and printed text and the other page consisted in the raised line drawing. No color was added in the pictures.

Procedure

Children worked in pairs with their teacher/educator. Each of them had a book in front of him. Each page of the book was explored in two steps. First of all, the child had to explore the illustration in the manner that he chose and had to guess its meanings. He knew that one picture can represent different meanings. He had to propose some ideas and the adult and the other child could react to the proposals. Then the text was read in a loud voice by the teacher, and the child had to explore the picture at the same time. In this second step he had to find in the picture what can represent the text content. Thus two different goals were pursued by the children, finding the meaning of a raised drawing and associating the prose contained in the book to the illustration. Each reading session was videotaped. In order to preserve children's attentional capacity, only five illustrations were shown. Each session lasted around 20 minutes.

The use of different exploratory procedures, the total exploration time for each procedure as well as whether the exploration was bimanual or unimanual were coded. The following criteria were used to code the exploratory procedures [START_REF] Davidson | Haptic judgment of curvature by blind and sighted humans[END_REF][START_REF] Lederman | Hand movements: A window into haptic object recognition[END_REF]:

Contour following: dynamic edge following using finger movements Lateral motion: dynamic and repetitive movement of one or more fingers Static contact: stationary contact with the surface without molding Enclosure: dynamic molding of the fingers and/or the palm to the shape contours Pinch: holding edges in a pincer grip between thumb and one or more fingers No exploration: no hand in contact with the picture "Surface sweeping with the palm": It is similar to lateral motion but children lay their palm flat on the page and do not use their fingers. This procedure (movement of the palm over the whole page surface) was added because it was often used when the children discovered a new tactile picture.

Results

As the type and the proportion of exploratory procedures used by the six children were similar, we chose to calculate the mean scores in function of the goal. The mean percentage of time employed for each exploratory procedure is reported in figures1 and 2.

When the children have to find the meaning of the pictures, they used most of the time the lateral motion (54%) and the contour following (28%) procedures. They were very active in exploring the picture and they spent only 10% of the time with no hand in contact with the surface of the page. The surface sweeping with the palm is a global procedure only used at the beginning, when the children discovered the illustration. It was systematically used by three children and concerned 6% of the haptic exploration duration. On overall children used more often unimanual exploration (59%) than bimanual exploration (41%).

When they have to link the text read by the adult to the illustration, they are less active and spent almost half of the time (49%) without exploring the picture. This result has to be nuanced because one child tried to follow the text in Braille while the adult was reading the text. Instead of exploring the picture (as it was recommended by the teacher), he was scanning the Braille characters with his finger trying to read by himself. Children used more often unimanual exploration (62%) than bimanual exploration (38%).

Whatever was the task, three EPs were used (lateral motion, contour following and static contact). The lateral motion was the preferred procedure. In both tasks the children used bimanual and unimanual exploration, and used alternatively right hand and left hand when producing unimanual exploration.

On average, children made 0,6 proposals for the meaning of each drawing (between 0 to 6 for each page). The correlations between the duration of each exploratory procedures and the number of proposals were not significant.

Discussion

The type of exploratory procedures and the amount of time employed for each procedure depend on the goal pursued by children. Children seem to be more active when they have to find the meaning of the picture than when the meaning was given to them through the reading text. However this finding needs to be nuanced because the exploration to find meaning was always the first step of the book exploration. When the text was read, they already have some representation of the illustration because they explored and interpreted it before. A procedure of surface sweeping was often use by three children when they discovered the pages. This procedure allows rapid scanning of the page that brings global information about the size, the texture and the global shape of the drawing. This global procedure might guide the specific exploratory procedures (contours following) employed after. Whatever was the task, lateral motion was the preferred procedure, even if it brings global information about shape and is more appropriate for texture features than for geometric features. Contours following, that allows precise extraction of shape, was more used when children have to find meaning of pictures. Static contact was more often used when children listen to the text. It is difficult to know if this procedure is an active procedure used by children to link the content of the text and the drawing or if they laid their hand on the page while only listening and not inferring information by touch. "No exploration" corresponds to no hand in contact with the illustration; however, it is sometimes difficult to separate the absence of exploration and the static contact procedure. Cognitive and attentional resources might have been on overall focused on the story listening and understanding, it is possible that no tactile information was interpreted from the static contact. It could explain why children spend more time using static contact when listening to the story than when exploring the pictures in the aim to find their meaning.

Study 2

This study aimed at assessing the effects of the type of illustration on the exploratory procedures used by children. Two types of illustration, 2D illustrations and 3D illustrations, were compared.

Method

Participants

The participants of this study were four children, three boys and one girl, aged from 5 years 7 months to 8 years 5 months. One was visually impaired and three were blind, with reduced residual visual capacity, and no perception of shapes. Three were right-handed and one left-handed. The main causes of blindness were congenital cataract, congenital retinal detachment, albinism, nystagmus, and optic atrophy. None of the children presented any other known cognitive or psychiatric disabilities. They were attended regular classrooms, but were supported by specialized teacher and educators. The adults who read the story were the educators in charge of the children.

Material

Two books that tell almost the same story "Goldilocks and the tree bears" but with different illustrations were used. In each book, every double page contain the text on the left page (in Braille and printed) and the illustration on the right page. In one book the illustrations were bidimensional tactile pictures (this book was produced by "Les Doigts Qui Rêvent").

The illustrations were simplified pictures made with different textures. The characters were the three bears (brown bear shape with soft fur) and Goldilocks (little girl, with textile skirt and long blond synthetic hair). Different objects were illustrated, bowls, table, house, threes, chairs and beds. This book contained 13 pages. The other book proposed 3D illustrations (realized by M. Carrier-Panellati, who illustrates books for blind children). The illustrations consisted in 3D objects glued to the page (plates, seats, beds, windows) or free from the pages (the three bears, the little girl, three spoons) that the child can held and manipulate all along the story. This book contained 6 pages. It tells almost the same story than the first book but with some added details about the material features like: "a plastic and a metal spoon", "a thin and a warm blanket", "embossed bed and soft bed", "a smooth and rough chair" in order to focus sometimes the children's attention on material properties of objects that can be more easily extracted by haptic exploration than geometric ones.

Procedure

The story was read by the educator in charge of the child. The child was seated beside the educator at a table, the book in front of him. The educator read aloud the text, while the child explored the illustration. For two children the story with 2D illustrations was read before the story with 3D illustrations, whereas the stories were read in the reverse order for the two other children. There was at least an interval of two weeks between the two reading sessions.

The reading sessions were videotaped and analyzed. Each EP employed was coded for each child. Comparing to experiment 1, one exploratory procedure well adapted for 3D objects exploration was added: "grasping and holding". The total duration of each exploratory procedures employed by the children during joint reading was assessed for each book.

When analyzing the videos the contents of the text were connected to the exploratory movements initiated at the same time. In the book with 3D illustrations, 53 words or sentences in the text could be linked with specific exploratory movement. In the book with 2D illustrations the text and the picture could be linked 35 times. For example: the story tells "once upon a time there were three bears" and the child is expected to touch the three bears represented by 3D bears or by 2D bear shapes. If the story tells "a small bed", the child is expected to touch the small bed. For each story, the number of exploratory movements directly linked with the story content (e.g.: the story told about a big bear and at the same time, the child haptically explored the big bear…), the number of exploratory movements not directly linked with the text (e.g: exploration of different elements of the page before the beginning of the reading, or exploration of the trees whereas the story told about the house) and the number of exploratory movements forgotten (the child did not explore the small bowl whereas the story told about a small bowl) were determined.

Results

The mean durations of each exploratory procedure in function of the type of illustration are reported in figures 3 and 4.

On overall, more EPs were employed when children explored 3D illustrations than 2D illustrations. When they explored the book with 3D illustrations, children employed six different EPs (grasp and hold, lateral motion, contours following, static contact, enclosure, pressure and pinch), whereas when they explored the book with 2D illustrations, they only used three different EPs. Among these EPs, the grasp and hold procedure is specific to 3D patterns and can't be used for 2D patterns. However, pinch, enclosure, static contact, lateral motion and pressure could have been used for 2D patterns. The grasp and hold procedure was the preferred one when exploring 3D illustrations (55%) whereas the lateral motion procedure was the most often employed when children explored 2D patterns (71%). When they explored the book with 3D illustrations, children spent less time without exploring (3%) than when they explored the book with bidimensional illustrations (13%). Whatever was the type of illustrations children used more often bimanual exploration, 70% of the time for 3D illustrations and 66% of the time for 2D illustrations, than unimanual exploration.

For each book, the percentage of exploratory movements directly linked with the text content, the percentage of exploratory movement not directly linked with the text and the percentage of exploratory movement forgotten was calculated. These percentages are presented in table 1.

Discussion

The type of illustration determines the exploratory procedures used by children. When three-dimensional illustrations were used, children employed more EPs than when bidimensional illustrations were used. Whereas for 3D illustrations the grasp and hold procedure was mainly used, for 2D illustrations, the lateral motion was the preferred procedure. The grasp and hold procedure is a very complete procedure in that it allows a lot of redundant and different information about objects very quickly (Klatsky, 1990). It can be noticed that sometimes this procedure was associated with another procedure, performed at the same time with the free hand. For example the child held the little girl in his right hand and performed lateral motion movement on the bed with his left hand. Whatever was the type of illustration, half or more of the EPs were directly linked to the text content. However, when 3D illustrations were proposed, children made on average more exploratory movements directly linked to the text content and less EP with no link with the story. Sometimes, the text content guided the EP that was used and sometimes, the type of illustration guided the EP used. For example, the pressure procedure was employed when the text told about a soft mattress. When the text told about a rough seat, it provoked the lateral motion procedure in most cases. The use of 3D objects provoked in most cases the grasp and hold procedure. The pinch procedure was often used with Goldilocks' hair, the bed edges, the blankets and the window.

General discussion

Illustrations in storybooks not only provide a source of interest in children but support literacy development. Picture books are learning and informational resources that can promote children's knowledge about the surrounding world. Picture books reading support language and cognitive development and is a first entry in literacy development [START_REF] Adams | Beginning to read: Thinking and learning about print[END_REF][START_REF] Bus | Joint book reading makes for success in learning to read: A meta-analysis on intergenerational transmission of literacy[END_REF][START_REF] Collins | ELL preschoolers' English vocabulary acquisition from storybook reading[END_REF][START_REF] Mol | Interactive book reading in early education: A tool to stimulate print knowledge as well as oral language[END_REF][START_REF] Sénéchal | Testing the home literacy model: mother involvment in kindergarten is differentially related to grade 4 reading comprehension, fluency, spelling, and reading for pleasure[END_REF]. Considering the impact of picture books on literacy development, it is crucial that children with a visual handicap can access to this way of conveying information. Literacy environment is accessible for visually impaired children through touch. Thus how to communicate ideas and stories through touch is a central issue for practitioners and researchers interested in this field. Tactile illustrations are the link between the child's concrete, tactile experiences of the world (what he can hold and explore with his hands) and a symbolic representation of that experience. Recognizing tactile illustrations implies precisely perceiving the shapes and being able to associate meaning to theses representations. Whereas vision and audition are recognized for providing highly precise spatial and temporal information respectively, the haptic system is especially effective at processing the material characteristics of surfaces and objects. Thus this is challenging to find ways to help blind individuals to perceive precise shape of object representing in illustrated tactile books. Haptic perception of object depends on the nature of contact (which exploratory procedures) and the duration of exploration, that means how and how long the picture is explored [START_REF] Klatzky | Haptic identification of objects and their depictions[END_REF]. The exploratory procedures are strongly linked with the extraction of certain properties of objects. For example, static contact (placing a large skin surface against an object without motion) is associated with queries about warmth or coolness, pressure is associated with hardness, unsupported holding with weight… For shape, the enclosure procedure makes it possible to extract global shape information while the contours following procedure is suitable for precise extraction of shapes. The various EPs have different costs in terms of their execution time and their interference with other exploration patterns occurring at the same time [START_REF] Klatzky | Haptic identification of objects and their depictions[END_REF][START_REF] Lederman | Hand movements: A window into haptic object recognition[END_REF]. They have also benefits in that some EP can provide incidental information about objects properties for which they are not optimal. An EP is optimal when it provides the most precise discrimination along a given dimension. Static contact is quick to execute, provides incidental information about texture, volume, shape and temperature, and can co-occurs with unsupported holding and enclosure [START_REF] Klatzky | Haptic identification of objects and their depictions[END_REF][START_REF] Lederman | Haptic perception: A tutorial[END_REF]. The type of exploratory procedures determines performance in recognizing, matching objects on shape, and drawing [START_REF] Kalagher | Developmental change in young children's use of haptic information in a visual task: The role oh hand movements[END_REF][START_REF] Vinter | Exploratory procedures of tactile images in visually impaired and blindfolded sighted children: How they relate to their consequent performance in drawing[END_REF]. As shown by [START_REF] Vinter | Exploratory procedures of tactile images in visually impaired and blindfolded sighted children: How they relate to their consequent performance in drawing[END_REF] the diversity of procedures used and the number of appropriate procedures for global and local shape encoding determines the quality of drawings. Observing and describing which EPs are used by children when they explored a tactile illustrated book might bring information about how to construct an efficient illustration, recognizable and easily explored by visually impaired and blind children, and how to teach them to explore tactile pictures.

In the present studies, the EPs were influenced by the goal pursued by children, the type of illustrations used and the content of the text. The EPs employed when the child has to imagine the meaning of what he was touching and the EPs when he has to associate the drawing with the text content were compared. On overall the children were more active, spending more time exploring the illustrations when they have to find meaning than when they have to associate the text content with the drawing. However, the task of associating the drawing with the text content always happened after the task of searching meaning. It was expected that the children would have quickly scanned the page to find some elements memorized before that they could have linked to the text content. Instead of that, they spent half of the time only listening at the story without initiating any exploratory movement.

Children who already knew the drawing might have been less involved in this task and might have used all their attentional resources to listen to the proposals and to link these proposals to their own proposals formulated the step before. When they discovered a new illustration half of them tend to use a procedure that was called "surface sweeping with the palm", which consists of quickly and globally scan the whole surface of the page with the palm lay flat on the illustration, moving in lateral or circular motion. This global procedure initiated at the beginning of the exploration might bring global information about the size, the texture and the global shape of the illustration. [START_REF] Klatzky | Identifying objects from a haptic glance[END_REF]showed that a brief "haptic glance" lasting about 200 ms was sometimes sufficient for haptic identification of familiar objects with highly diagnostic geometric or material features. The duration of manual exploration has been shown to influence haptic processing in terms of the distinction between featural and global processing of object structure. After this global procedure of surface sweeping with the palm, they alternated between global procedures like lateral motion and specific ones like contours following. In both tasks lateral motion was largely employed since this procedure is not the most appropriate for encoding the shape but is optimal for the processing of texture information [START_REF] Lederman | Hand movements: A window into haptic object recognition[END_REF]. The common use of this procedure was highlighted in previous studies [START_REF] Theurel | The effect of joint reading on the tactile comprehension of a tactile illustratde book by early blind children[END_REF][START_REF] Vinter | Exploratory procedures of tactile images in visually impaired and blindfolded sighted children: How they relate to their consequent performance in drawing[END_REF].

Young children tend to often use the lateral motion procedure when they have to explore bidimensional drawings in the aim to memorize it or to infer their meaning even if this is not optimal for encoding precise shape. However, lateral motion used in combination with the contour following procedure makes it possible to avoid the confusion between foreground and background because it enables children to explore the filled internal region [START_REF] Kennedy | Pictorial foreground-background reversal reduces tactual recognition by blind subjects[END_REF]. Concerning the exploration of these illustrations in swell paper children chose more often to use unimanual exploration than bimanual exploration, even though bimanual activity is assumed to enlarge the tactile field and to help reducing the costs in terms of memory and information integration processes [START_REF] Ballasteros | Haptic discrimination of bilateral symetry in 2-dimensional and 3-dimensional unfamilair displays[END_REF][START_REF] Craig | Attending the two fingers: Two hands are better than one[END_REF][START_REF] Wijntjes | Look what i have felt: Unidentified haptic line drawings are identified after sketching[END_REF].

The use of the different EPs was also influenced by the text content and the type of illustrations. Two ways of illustrating books, one using bidimensional representations and the other using tridimensional objects, glued or freed from the page, were compared. The use of 3D illustrations led to a high diversity of EPs. Children used different procedures like grasp and hold, lateral motion, pinch, enclosure, pressure and static contact. The grasp and hold procedure, which is specific to 3D objects and which was largely employed, allows to collect different and redundant information about objects. Klatzky (1990) predicted that the most efficient way to process an object's properties was to grasp and lift it, because it provokes at the same time the exploratory procedures of static contact, unsupported holding and enclosure. This action permits to provide at least global information about material and structural properties. Children need redundancy in their information processing to interpret haptic information [START_REF] Millar | Understanding and representing space[END_REF] and this redundancy can be provoked by the use of different procedures for exploring the same object. When exploring tridimensional illustrations, children alternated between coarse and specific information extraction steps. Exploratory movements were oriented toward the extraction of the whole picture characteristics (surface sweeping with the palm, lateral motion, grasp and hold, enclosure) and toward the extraction of local features (pinch, contour following). The diversity of procedures has been linked with the quality of exploration and interpretation of pictures. Vinter 2012 showed that the large number of procedures used was linked with the quality of drawings produced by children after the exploration step. For bidimensional pictures lateral motion was often used. This procedure is optimal for perceiving the texture of pictures. In some case, it might be suitable to the understanding of the story because each character was produced in different texture. For example the bears were in soft fur whereas the little girl wore a skirt in fabric. Thus the lateral motion procedure permitted to easily distinguish the bears from the girl. For the objects that could only been distinguished by the size (big, medium and small bowls) a global procedure might be sufficient. However, to understand, that it is a house or a tree or a table that is represented, a specific contours following procedure is needed. Thus, children were expected to alternate between global and local procedures for more informational exploration of the picture. Some EPs were specific to particular illustrations. The pinch procedure was observed for the long hair of Goldilocks, and for the exploration of the house windows, the blankets and the door. The type of illustrations and the materials used to make the illustrations may constraint the exploratory procedures employed during joint storybook reading.

The EPs employed by children were also largely constrained by the content of the text.

In one of the story the author added material information in the text. The text by emphasis some characteristic of objects features induced the use of exploratory strategies. The description of embossed mattress led to lateral motion or pressure procedures in all children.

The descriptions of soft bed led to pressure whereas the description of rough seat led to lateral motion. The blanket described as thin led to pinch. The description of the warm chocolate generally produced static contact even if the bowls were not hot in reality. A child played with this concept and put his hand in contact with the drawing of the bowl and quickly removed it simulating the hot sensation. The warmth of the bowls was represented by a drawing of smoke above the bowls. This representation was difficult to understand for children and needed explanations from the educator. As it was shown by [START_REF] Theurel | The effect of joint reading on the tactile comprehension of a tactile illustratde book by early blind children[END_REF]

  , the verbal descriptions delivered by the adult along the reading guide the EPs employed by children. Descriptions of pictures with tactile vocabulary led to the use of specific exploratory procedures directly linked with the qualifier. The verbal interactions during joint book reading improved the quality of the exploration initiated by the child. Adult by reading the text and providing additional information guides the exploratory procedures and could consequently enhance the understanding of the story. should not be considered as separate entity and are strongly linked to the text content. Information containing in the picture has to be congruent with and to correspond to the information brought by the text. Both picture and text are interrelated and guide the exploratory movements. Despite the fact that picture books reading is an important part of literacy development, tactile books are less present in the environment of children with visual handicap than in the environment of sighted children. When picture books are available, it is not easy for blind children to understand and interpret the tactile pictures that are often hardly recognizable. The understanding of the pictures depends in part of the quality of the haptic exploratory procedures used. Determining which of the EPs are beneficial or detrimental to the extraction of shape information contained in tactile pictures might help to define how illustrated books can be elaborate, which type of illustrations can be chosen and how the adult can read the book and guide the children's exploration. Children need to be taught how to explore the pictures and which EPs are optimal in which context, to support their understanding of illustrations and stories.

 Yu, X. (2012). Exploring visual perception and children's interpretations of picture books. Research,34,[292][293][294][295][296][297][298][299]

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