Justin Salez

The Mézard-Parisi equation for matchings in pseudo-dimension d > 1

Keywords: recursive distributional equation, random assignment problem, mean-field combinatorial optimization, cavity method. 2010 MSC: 60C05, 82B44, 90C35

We establish existence and uniqueness of the solution to the cavity equation for the random assignment problem in pseudo-dimension d > 1, as conjectured by Aldous and Bandyopadhyay (Annals of Applied Probability, 2005) and Wästlund (Annals of Mathematics, 2012). This fills the last remaining gap in the proof of the original Mézard-Parisi prediction for this problem (Journal de Physique Lettres, 1985).

Introduction

The random assignment problem is a now classical problem in probabilistic combinatorial optimization. Given an n × n array {X i,j } 1≤i,j≤n of iid nonnegative random variables, it asks about the statistics of

M n := min σ n i=1 X i,σ(i) ,
where the minimum runs over all permutations σ of {1, . . . , n}. This corresponds to finding a minimum-length perfect matching on the complete bipartite graph K n,n with edge-lengths {X i,j } 1≤i,j≤n . Using the celebrated replica symmetry ansatz from statistical physics, Mézard and Parisi [START_REF] Mézard | Replicas and optimization[END_REF][START_REF] Mézard | Mean-field equations for the matching and the travelling salesman problems[END_REF][START_REF] Mézard | On the solution of the random link matching problems[END_REF] made a remarkably precise prediction concerning the regime where n tends to infinity while the distribution of X i,j is kept fixed and satisfies P (X i,j ≤ x) ∼ x d as x → 0 + , 1 for some exponent 0 < d < ∞. Specifically, they conjectured that

M n n 1-1/d P ---→ n→∞ -d R f (x) ln f (x)dx, (1)
where the function f : R → [0, 1] solves the so-called cavity equation:

f (x) = exp - +∞ -x d(x + y) d-1 f (y)dy . (2)
Aldous [START_REF] Aldous | Asymptotics in the random assignment problem[END_REF][START_REF] Aldous | The ζ(2) limit in the random assignment problem[END_REF] proved this conjecture in the special case d = 1, where the term (x + y) d-1 simplifies and makes the cavity equation exactly solvable, yielding

f (x) = 1 1 + e x and -d R f (x) ln f (x)dx = π 2 6 .
Since then, several alternative proofs have been found [START_REF] Linusson | A proof of Parisi's conjecture on the random assignment problem[END_REF][START_REF] Nair | Proofs of the Parisi and Coppersmith-Sorkin random assignment conjectures[END_REF][START_REF] Wästlund | An easy proof of the ζ(2) limit in the random assignment problem[END_REF]. This stands in sharp contrast with the case d = 1, where showing that the Mézard-Parisi equation ( 2) admits a unique solution has until now remained an open problem [START_REF] Aldous | A survey of max-type recursive distributional equations[END_REF]Open Problem 63]. Wästlund [START_REF] Wästlund | Replica symmetry of the minimum matching[END_REF] circumvented this issue by considering instead the truncated equation

f λ (x) = exp - λ -x d(x + y) d-1 f λ (y)dy , 0 < λ < ∞. (3)
Using an ingenious game-theoretical interpretation of this equation, he showed the existence of a unique, global attractive solution f λ : [-λ, λ] → [0, 1] for every 0 < λ < ∞, provided d ≥ 1. He then used this fact to establish that

M n n 1-1/d P ---→ n→∞ lim λ→+∞ ↑ -d λ -λ f λ (x) ln f λ (x)dx. (4)
Wästlund [START_REF] Wästlund | Replica symmetry of the minimum matching[END_REF] explicitly left open the problem of completing the proof of the original Mézard-Parisi prediction by showing (i) that the untruncated cavity equation admits a unique solution f and (ii) that f λ → f as λ → ∞. The purpose of this short paper is to establish this conjecture.

f λ (x) ln f λ (x)dx ----→ λ→+∞ R f (x) ln f (x)dx.
Consequently, the two limits in (1) and (4) coincide.

In addition, we provide a short alternative proof of the crucial result of [START_REF] Wästlund | Replica symmetry of the minimum matching[END_REF] that the truncated equation (3) admits a unique, attractive solution.

Remark 1. Very recently, a proof of uniqueness for the truncated equation (3) has been announced [START_REF] Larsson | The Minimum Perfect Matching in Pseudo-dimension 0[END_REF] for the case 0 < d < 1. It would be interesting to see if the result of the present paper can be extended to this regime.

Remark 2. For a random variable Z with P (Z > x) = f (x), the cavity equation ( 2) simply expresses the fact that Z solves the distributional identity

Z d = min i≥1 {ξ i -Z i } , (5)
where {ξ i } i≥1 is a Poisson point process with intensity dx d-1 ∂x on [0, ∞), and {Z i } i≥1 are iid with the same distribution as Z, independent of {ξ i } i≥1 . Such recursive distributional equations arise naturally in a variety of models from statistical physics, and the question of existence and uniqueness of solutions plays a crucial role for the rigorous understanding of those models.

We refer the interested reader to the comprehensive surveys [START_REF] Aldous | The objective method: probabilistic combinatorial optimization and local weak convergence[END_REF][START_REF] Aldous | A survey of max-type recursive distributional equations[END_REF] for more details. In particular, [4, Section 7.4] contains a detailed discussion on equation ( 5), and [4, Open Problem 63] raises explicitly the uniqueness issue. We note that the refined question of endogeny remains a challenging open problem. Recursive distributional equations for other mean-field combinatorial optimization problems have been analysed in e.g. [START_REF] Gamarnik | Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method[END_REF][START_REF] Parisi | Mean field matching and traveling salesman problems in pseudo-dimension 1[END_REF][START_REF] Khandwawala | Solutions to recursive distributional equations for the mean-field TSP and related problems[END_REF].

The remainder of the paper is organized as follows. Section 2 deals with the truncated equation (3) for fixed 0 < λ < ∞ and is devoted to the alternative analytical proof that there is a unique, globally attractive solution f λ . Section 3 prepares the λ → ∞ limit by providing uniform controls on the family {f λ : 0 < λ < ∞} and by characterizing the possible limit points. This reduces the proof of Theorem 1 to establishing uniqueness in the untruncated Mézard-Parisi equation (λ = ∞), which is done in Section 4.

The truncated cavity equation

(λ < ∞) Fix a parameter 0 < λ < ∞. On the set F of non-increasing functions f : [-λ, λ] → [0, 1], define an operator T by (T f )(x) = exp -d λ -x (x + y) d-1 f (y)dy . ( 6
)
The purpose of this section is to give a short and purely analytical proof of the following result, which was the main technical ingredient in [START_REF] Wästlund | Replica symmetry of the minimum matching[END_REF] and was therein established using an ingenious game-theoretical framework.

Proposition 1. T admits a unique fixed point f λ and it is attractive in the sense that

|T n f (x) -f λ (x)| ---→ n→∞ 0, uniformly in both x ∈ [-λ, λ] and f ∈ F. Proof. Write f ≤ g to mean f (x) ≤ g(x) for all x ∈ [-λ, λ]. In particular, 0 ≤ f ≤ T 0
for every f ∈ F, where 0 denotes the constant-zero function. Note also that the operator T is non-increasing, in the sense that

f ≤ g =⇒ T f ≥ T g.
Those two observations imply that the sequences {T 2n 0} n≥0 and {T 2n+1 0} n≥0 are respectively non-decreasing and non-increasing, and that their respective pointwise limits f -and f + satisfy

f -≤ lim inf n→∞ T n f ≤ lim sup n→∞ T n f ≤ f + ,
for any f ∈ F. Moreover, the dominated convergence Theorem ensures that T is continuous with respect to pointwise convergence, allowing to pass to the limit in the identity T n+1 0 = T (T n 0) to deduce that

T f -= f + and T f + = f -. (7)
Therefore, the proof boils down to the identity f -= f + , which we now establish. By definition, we have for any f ∈ F,

(T f )(x) = exp -d λ -λ (x + y) d-1 1 (x+y≥0) f (y)dy .
Since d > 1, we may differentiate under the integral sign to obtain

(T f ) ′ (x) = -d(d -1)(T f )(x) λ -λ (x + y) d-2 1 (x+y≥0) f (y)dy.
Integrating over [-λ, λ] and noting that (T f ) (-λ) = 1, we conclude that

1 -(T f ) (λ) = d(d -1) [-λ,λ] 2 (x + y) d-2 1 (x+y≥0) (T f )(x)f (y)dxdy.
Let us now specialize to f = f ± . In both cases, the right-hand side is

d(d -1) [-λ,λ] 2 (x + y) d-2 1 (x+y≥0) f + (x)f -(y)dxdy,
by [START_REF] Kolmogorov | Introductory real analysis[END_REF]. Therefore, we have

(T f + ) (λ) = (T f -)(λ), i.e. λ -λ d(λ + y) d-1 f + (y)dy = λ -λ d(λ + y) d-1 f -(y)dy.
Since we already know that f -≤ f + , this forces f -= f + almost-everwhere on [-λ, λ], and hence everywhere by continuity. Finally, the convergence

T n 0 → f λ = f ± is automatically uniform on [-λ, λ]
, by Dini's Theorem.

3 Relative compactness of solutions (λ → ∞)

In order to study properties of the family {f λ : 0 < λ < ∞}, we extend the domain of f λ to R by setting f λ (x) = 1 for x ≤ -λ and f λ (x) = 0 for x > λ.

Proposition 2 (Uniform bounds). For all 0 < λ < ∞ and x ≥ 0,

f λ (x) ≤ exp - x d e 1 -f λ (-x) ≤ exp - x d e f λ (-x) ln 1 f λ (-x) ≤ exp - x d e f λ (x) ln 1 f λ (x) ≤ 1 + x d e exp - x d e .
Proof. Let 0 < λ < ∞. We may assume that x ∈ [0, λ], otherwise the above bounds are trivial. By definition, we have

f λ (x) = exp - λ -x d(x + y) d-1 f λ (y)dy . (8)
Now, since x ≥ 0 and f λ is non-increasing, we have

λ -x (x + y) d-1 f λ (y)dy = 0 -x (x + y) d-1 f λ (y)dy + λ 0 (x + y) d-1 f λ (y)dy ≥ f λ (0) x d d + λ 0 y d-1 f λ (y)dy.
Applying u → exp(-du) to both sides and using (8), we obtain

f λ (x) ≤ f λ (0) exp(-f λ (0)x d ). (9)
In turn, this inequality implies that for all x ≥ 0,

λ x d(y -x) d-1 f λ (y)dy ≤ f λ (0) +∞ x dy d-1 e -f λ (0)y d dy = exp(-f λ (0)x d ).
Applying u → exp(-u) to both sides, we conclude that

f λ (-x) ≥ exp -e -f λ (0)x d . ( 10
)
In particular, taking x = 0 yields f λ (0) ≥ e -1 , and reinjecting this into ( 9) and ( 10) easily yields the first three claims. For the last one, observe that u → u ln 1 u increases on [0, e -1 ] and decreases on [e -1 , 1], with the value at u = e -1 being precisely e -1 . Therefore, if exp(-x d /e) ≤ e -1 , we may use the bound f λ (x) ≤ exp(-x d /e) to deduce that

f λ (x) ln 1 f λ (x) ≤ x d e exp - x d e .
On the other hand, if exp(-x d /e) ≥ e -1 , then

f λ (x) ln 1 f λ (x) ≤ e -1 ≤ exp - x d e .
In both cases, the last inequality holds, and the proof is complete.

Proposition 3. The family {f λ : 0 < λ < ∞} is relatively compact with respect to the topology of uniform convergence on R, and any sub-sequential limit as λ → ∞ must solve the cavity equation (2).

Proof. Let {λ n } n≥0 be any sequence of positive numbers such that λ n → ∞ as n → ∞. By Helly's compactness principle for uniformly bounded monotone functions (see e.g. [START_REF] Kolmogorov | Introductory real analysis[END_REF]Theorem 36.5]), there exists an increasing sequence {n k } k≥0 in N and a non-increasing function f : R → [0, 1] such that

f λn k (x) ---→ k→∞ f (x), (11)
for all x ∈ R. Thanks to the first inequality in Proposition 2, we may invoke dominated convergence to deduce that for each x ∈ R,

λn k -x f λn k (y)(x + y) d-1 dy ---→ k→∞ +∞ -x f (y)(x + y) d-1 dy.
Applying u → exp(-du) and recalling (8), we see that

f (x) = exp -d +∞ -x f (y)(x + y) d-1 dy ,
which shows that f must solve the cavity equation ( 2). This identity easily implies that f is continuous. Consequently, the convergence (11) is uniform in x ∈ R, by Dini's Theorem.

The un-truncated cavity equation (λ = ∞)

To conclude the proof of Theorem 1, it now remains to show that the untruncated equation

f (x) = exp -d +∞ -x (x + y) d-1 f (y)dy . (12)
admits at most one fixed point f : R → [0, 1]. Proposition 3 will then guarantee the convergence f λ ---→ λ→∞ f , which will in turn imply

λ -λ f λ (x) ln f λ (x)dx ----→ λ→+∞ R f (x) ln f (x)dx,
by dominated convergence, thanks to the last inequalities in Proposition 2.

A quick inspection of the proof of Proposition 2 reveals that it remains valid when λ = ∞. In particular, any solution f to [START_REF] Mézard | On the solution of the random link matching problems[END_REF]

must satisfy max(f (x), 1 -f (-x)) ≤ exp - x d e , (13)
for all x ≥ 0. It also clear from [START_REF] Mézard | On the solution of the random link matching problems[END_REF] that f must be (0, 1)-valued and continuous. We will use those properties in the proofs below.

Lemma 1. If f, g solve ( 12), then there exists t ≥ 0 such that for all x ∈ R,

f (x + t) ≤ g(x) ≤ f (x -t).
Proof. [START_REF] Nair | Proofs of the Parisi and Coppersmith-Sorkin random assignment conjectures[END_REF] ensures that for any t ∈ R, y → (1 + |y|)(f (y -t) -g(y)) is integrable on R, so that by dominated convergence,

1 x d-1 +∞ -x (y + x) d-1 (f (y -t) -g(y)) dy ----→ x→+∞ ∆(t), (14)
where

∆(t) := R (f (y -t) -g(y)) dy. ( 15
)
Observe that t → ∆(t) increases continuously from -∞ to +∞, as can be seen from the decomposition

∆(t) = +∞ 0 (1 -g(-y) -g(y))dy + +∞ -t f (y)dy - +∞ t (1 -f (-y))dy.
In particular, we can find t 0 ≥ 0 such that ∆(-t 0 ) < 0 < ∆(t 0 ). In view of ( 14), we deduce the existence of a ≥ 0 such that for all x ≥ a,

+∞ -x (y + x) d-1 g(y)dy ≥ +∞ -x (y + x) d-1 f (y + t 0 )dy (16) +∞ -x (y + x) d-1 g(y)dy ≤ +∞ -x (y + x) d-1 f (y -t 0 )dy. ( 17
)
Applying u → exp(-du), we conclude that for all x ≥ a,

f (x + t 0 ) ≤ g(x) ≤ f (x -t 0 ). (18)
In turn, this implies that ( 16)-( 17) also hold when x ≤ -a, so that (18) actually holds for all x outside (-a, a). On the other hand, since g is (0, 1)-valued and f has limits 0, 1 at ±∞, we can choose t 1 ≥ 0 large enough so that

f (-a + t 1 ) ≤ g(a) ≤ g(-a) ≤ f (a -t 1 ).
Since f, g are non-increasing, this inequality implies that for all x ∈ [-a, a],

f (x + t 1 ) ≤ g(x) ≤ f (x -t 1 ). (19)
In view of ( 18)-( 19), taking t := max(t 0 , t 1 ) concludes the proof.

Proof of Proposition 3. Let f, g solve equation ( 12) and let t be the smallest non-negative number satisfying for all x ∈ R,

f (x + t) ≤ g(x) ≤ f (x -t). (20)
Note that t exists by Lemma 1 and the continuity of f . Now assume for a contradiction that t > 0. Clearly, each of the two inequalities in (20) must be strict at some point x ∈ R (and hence on some open interval by continuity), otherwise we would have g ≥ f or g ≤ f and (12) would then force g = f , contradicting the assumption that t > 0. Consequently, the function ∆ defined in [START_REF] Wästlund | An easy proof of the ζ(2) limit in the random assignment problem[END_REF] must satisfy ∆(-t) < 0 < ∆(t). By continuity of ∆, there must exists t 0 < t such that ∆(-t 0 ) < 0 < ∆(t 0 ). As we have already seen, this inequality implies

f (x + t 0 ) ≤ g(x) ≤ f (x -t 0 ), (21)
for all x outside some compact [-a, a]. In particular, we now see that the inequalities in (20) must be strict for all large enough x. Thus, for all x ∈ R, Applying u → exp(-du) now shows that the inequalities in (20) must actually be strict everywhere on R, hence in particular on the compact [-a, a]. By uniform continuity, there must exists t 1 < t such that f (x + t 1 ) ≤ g(x) ≤ f (x -t 1 ),

for all x ∈ [-a, a]. In view of ( 21)-( 22), the number t ′ := max(t 0 , t 1 ) now contradicts the minimality of t.

Theorem 1 .

 1 For d > 1, the Mézard-Parisi equation (2) admits a unique solution f : R → [0, 1]. Moreover, f λ → f pointwise as λ → +∞, and λ -λ

  x) d-1 f (y + t)dy +∞ -x (y + x) d-1 g(y)dy < +∞ -x (y + x) d-1 f (y -t)dy.


Christophe Danjou
email: christophe.danjou@utc.fr

OntoSTEP-NC for integration from CAD to CNC

Keywords: Interoperability, OntoSTEP-NC, Integrated Design and Manufacturing, CAD-CAM-CNC, PDM-MPM-ERP

OntoSTEP-NC for integration from CAD to CNC

Introduction

Face to the globalization and the growing competition concerning product costs, companies must constantly increase their productivity. In a "Rapid-development" context, companies have to improve their industrialization ability. In fact, they must industrialize their products and processes with an increasing speed, lower costs and maintaining a high level of quality.

French FUI project called ANGEL (Atelier Numérique coGnitif intEropérable et ag-iLe) will focus on the capitalization of cuts know-how in order to improve the competitiveness of enterprises for developing tools and methods to retrieve information from the CNC machine. To achieve the information flow bi-directionality (machines control and capitalization), the systems must be able to exchange information and to use these information.

Defining interoperability as the exchange of information between systems and the use of their functionality, interoperability can be a solution to treat the continuity of digital flow. It also ensures capitalization of technical know-how linking the different stages from design to manufacturing.

This paper presents one way to achieve interoperability between systems. The first section gives an outline of the literature review with a common interoperability definition and then exposes interoperability solutions for industrialization. Section 3 deals with the research question and tell more about the chosen OntoSTEP-NC solution. Section 4 concludes this paper and gives an overview on the future work to be done.

2

Literature review

Interoperability definition

According to Kosanke [START_REF] Kosanke | ISO Standards for Interoperability : a Comparison[END_REF] a careful chosen web search produces 22 definitions of interoperability. Following the interpretation and the people who use it, the term interoperability can make different senses. This explains why a large number of definitions can be found in the literature. IEEE (IEEE, 1990) defines interoperability as "the ability of two or more systems or components to exchange information and to use the information that has been exchanged". ATHENA [START_REF] Athena | Enterprise Modelling in the Context of Collaborative Enterprises ATHENA[END_REF] and INTEROP NoE [START_REF] Interop | Knowledge Map of Research in Interoperability in the INTEROP NoE[END_REF] projects define interoperability for enterprises. In this case, interoperability is the ability of interaction between companies or at least between parts of them. Vernadat [START_REF] Vernadat | Enterprise Modelling and Integration: Principles and Applications[END_REF] defines interoperability as the "ability to communicate with peer systems and access their functionality". This definition highlights the need to exchange functionalities. But here we will use Wegner's [START_REF] Wegner | Interoperability[END_REF] definition which describe interoperability as "The ability of two systems (or more) to communicate, cooperate and exchange services and data, thus despite the differences in languages, implementations, executive environments and abstraction models". ANGEL project will focus on the industrialization phase interoperability between the different CAX systems and Information Systems.

According to EIF [START_REF] Eif | European Interoperability Framework[END_REF] there exist three levels of interoperability: Technical level, Semantic level and Organizational level. A system is interoperable if and only if it satisfies the three levels of interoperability at every moment:

The technical level ensures the continuity of the information flow through tools and technological solutions.

The semantical level ensures the information sharing and services to keep the semantics flow. According to the standard ISO 14258 there are three different approaches to achieve semantic interoperability: Integration, Unification and Federation.

The organizational level deals with the processes, users and those involved in the operation of the system [START_REF] Paviot | Méthodologie de résolution des problèmes d'interopérabilité dans le domaine du Product Lifecycle Management[END_REF].

Interoperability for industrialization

This section exposes a study of the link between the different industrialization phases [START_REF] Danjou | Integrated Platform from CAD to CNC : State of the Art[END_REF]. Focus on interoperability point of view, these links are stud-ied to determine the interoperability levels and the methods used. The integrated design/industrialization platforms contain various expert software (CAD, Simulation, CAM ...) and different systems to support these expertises (PDM (Product Data Management), MPM (Manufacturing Process Management), ERP (Enterprise Resource Planning Information), and MES (Manufacturing Execution System)). In the chain of industrialization three types of links are identified:

-The Design / Simulation link covering links between CAD software, Simulation and PDM -The Design / Manufacturing link covering links between CAD, CAM, PDM, MPM, ERP and MES -The Design / Assembly link covering the same software links and information systems as above, but taking into account a different expertise (assembly and nonmanufacturing) The survey lead shows that literature extensively addresses the barriers related to technical and semantic levels. However barriers related to organizational level as associated human nature problems are less treated in the literature [START_REF] Chen | Enterprise Interoperability Framework[END_REF]. On the technical level, SOA systems and mediators are mainly used. On the semantic level, federative approach and ontologies are the most exploited.

3

Integrated platform from CAD to CNC

Standard for interoperability

Following the study carried out in the previous section, federative approach and standard appears as a solution to enhance interoperability between the industrialization stages. STEP standard is the most use standard for interoperability from design to simulation. Going in the same way STEP-NC (STEP standard with CNC features) is an interoperability solution for the link design to manufacturing.

As explained in Xu [START_REF] Xu | STEP-Compliant NC Research: The Search for Intelligent CAD/CAPP/CAM/CNC Integration[END_REF], Newman [START_REF] Newman | Strategic Advantages of Interoperability for Global Manufacturing Using CNC Technology[END_REF], and Nassehi [START_REF] Nassehi | Toward Interoperable CNC Manufacturing[END_REF], STEP-NC can wear through its rich data format, much information that can be integrated in the digital flow CAD-CAM-CNC. In opposition to the multi-interfaces needs [START_REF] Van Houten | Manufacturing Interfaces[END_REF] this standard let compile all the information from the design, simulation and manufacturing. Indeed STEP-NC allows having in the same file the CAD requirements, the manufacturing features, the machining process and the cutting tools description. Moreover, STEP-NC format provides bidirectional digital flow because it allows feedback from the manufacturing and simulation stages to the design.

The STEP-NC standard is led by two norms. Both norms define two different level AIM for the ISO 10303 -AP238 and ARM level for the ISO 14649. Covering the whole manufacturing technologies chain (CAD, CAM, CNC) STEP-NC ensures data exchanges as shown in figure 1.

Fig. 1. STEP-NC for data exchange between manufacturing technologies

Research question

Although the STEP-NC format wears all the information for CAD, CAM and CNC, the following question arises: How to integrate manufacturing expertise in the design / industrialization process for mechanical parts? Some problems have been identified to ensure the continuity of the semantical flow. In figure 1, it clearly appears that the information systems chain, the bi-directionality of the semantic flow is not achieved. The identified problems are listed below:

-There is no formalization of business expertise form the user's knowledge and from information in the Post-Processor) -There is a lack in the completeness of standards (Ex: STEP-NC) -Interoperability between information systems is not achieved Given the state of the art already completed for interoperability between design and simulation phases, it appears that interoperability between PDM / MPM / ERP / MES has been little addressed. Indeed, MPM appears as an essential element for lifecycle management of production lines. According to [START_REF] Elheni-Daldoul | Enterprise Information Systems. Interoperability : Focus on PLM Challenges[END_REF], the MPM is the only software able to provide a link between the early phases such as CAD and PDM, and downstream applications, such as ERP and MES.

Developing OntoSTEP-NC

The ontology model allows a large range of combination thanks to its modular structure. This is why in order to face these problems, it has been chosen to use an ontology (OntoSTEP-NC) based on the STEP-NC standard as support PDM-MPM-ERP-MES trade. As shown in figure 2, OntoSTEP-NC seems a possible way to improve the design / industrialization chain interoperability. Considering this two solutions, the interoperability is achieved in both information flow levels (CAX tools and IS). For OntoSTEP-NC model the ISO 14649 standard has been chosen as a basis. Indeed, this standard format is a meta-level model for CNC information so on On-toSTEP-NC describes a meta-level data.

Fig. 2. OntoSTEP-NC for data exchange between information systems

To develop OntoSTEP-NC, we have chosen to use Protégé a free OWL ontology editor (developed by Standford University) [START_REF] Noy | Ontology Evolution: Not the Same as Schema Evolution[END_REF]. OntoSTEP-NC will be first developed on ARM meta-level based on ISO 14649, then the model will be enriched to support required information. Moreover the ontology lets a large range of possibilities to modify the model structure and to add new entities and features. This ontology presents the duals aspects to be standard as STEP-NC ISO 14649 and to be flexible as an ontology structure. Hence the interoperability will be realized at the information system level between PDM-MPM-ERP-MES systems.

Once the integrated platform from CAD to CNC implemented, we aim to enable the feedback from the CNC to CAD. This becomes possible with the bi-directionality of the information flow and the semantical flow continuity. We propose a knowledge database based on OntoSTEP-NC. Thanks to the recognition of the manufacturing features and the business rules extraction through data mining we could be able to complete this knowledge databases. Coupled to the OntoSTEP-NC we would be able to have all the feedbacks at the early design concepts aiming at drawing up a quote and to have a well-born product. Indeed the product has to be validated since the first time it is manufactured in order to have the best manufacturing efficiency. This is why the OntoSTEP-NC must cover all the need in the different stages of the industrialization chain.

Conclusion and future work

This paper focuses on interoperability from CAD to CNC. First a literature review on interoperability is led. These literatures review first precise that Wegner interoperability definition has been chosen in our case study. Then the paper proposes to analyze the major works led onto interoperability between the main stages of industrialization phases: Considering the link CAD/Simulation, CAD/Manufacturing, and CAD/Assembly. This study points out that SOA and mediators are the most common used solution to ensure technical interoperability whereas for semantical interoperability the most used solutions are federative approach and ontologies.

Based on the previous conclusion, to achieve interoperability and data exchange, the chosen solution is OntoSTEP-NC. This solution presents dual aspect as the neutral format and the ability to modify the model according the needs. OntoSTEP-NC will help us to have feedbacks at the early design concepts in order to draw up a quote and to improve the rate of well-born products. In fact coupled to OntoSTEP-NC, the manufacturing features recognition and business rules extraction will allow completing the cutting knowledge databases.

The work plans for the next two years is firstly to study the industrial partners' information systems in order to fit their data models. Then the major work will focus on OntoSTEP-NC to support all the expertise and the business rules extracted from the manufacturing data. Finally, the connection from OntoSTEP-NC and information systems has to be developed to definitively ensure the semantical flow.

Table 1

 1 highlights the most significant papers analyzed following the three levels of interoperability seen previously are used: Semantical level, Technical level and Organizational level. Based on the 32 papers analyzed, the literature review is exposed in a double-entry table to classify the different jobs in the category that corresponds to it.

		Technical	Semantical	Organizational
		Level	Level	Level
	Design -Simulation	SOA Conception Loops	OntoSTEP PPO RsM PDO	PSS
	Design -Manufacturing	Mediators UbiDM	INFELT STEP-NC	PSS
	Design -Assembly	MUVOA AOD	CPM	PSS

Table 1 .

 1 Synthesis of the literature survey[START_REF] Danjou | Integrated Platform from CAD to CNC : State of the Art[END_REF]

Acknowledgements: This work is done in the French FUI project ANGEL. We also thank all consortium partners for their contribution during the development of ideas and concepts proposed here. I also want to thank Julien Le Duigou and Benoït Eynard for their scientific contribution and their discussion challenges.