Bernard P Zeigler
email: zeigler@rtsync.com

Alexandre Muzy
email: alexandre.muzy@cnrs.fr

Gianfranco Minati

Eliano Pessa

Ignazio Licata

Emergence at the Fundamental Systems Level: Existence Conditions for Iterative Specifications

Keywords: emergence, uniqueness, existence of solutions, input/output system, system specifications, Discrete Event System Specification

Conditions under which compositions of component systems form a well-defined system-of-systems are here formulated at a fundamental level. Statement of what defines a well-defined composition and sufficient conditions guaranteeing such a result offers insight into exemplars that can be found in special cases such as differential equation and discrete event systems. For any given global state of a composition, two requirements can be stated informally as:

(1) the system can leave this state, i.e., there is at least one trajectory defined that starts from the state; and (2) the trajectory evolves over time without getting stuck at a point in time. Considered for every global state, these conditions determine whether the resultant is a well-defined system and, if so, whether it is non-deterministic or deterministic. We formulate these questions within the framework of iterative specifications for mathematical system models that are shown to be behaviorally equivalent to the Discrete Event System Specification (DEVS) formalism. This formalization supports definitions and proofs of the afore-mentioned conditions. Implications are drawn at the fundamental level of existence where the emergence of a system from an assemblage of components can be characterized. We focus on systems with feedback coupling where existence and uniqueness of solutions is problematic.

Introduction

Emergence has been characterized as taking place in strong and week forms. Mittal [START_REF] Mittal | Emergence in stigmergic and complex adaptive systems: A formal discrete event systems perspective[END_REF] pointed out that strong emergent behavior results in generation of new knowledge about the system representing previously unperceived complex interactions. This can occur in the form of one or more of new abstraction levels and linguistic descriptions, new hierarchical structures and couplings, new component behaviors, and new feedback loops. Once understood and curated, the behavior returns to the weak form, as it is no longer intriguing, and then can begin to be treated in regularized fashion. Emergent behavior is likely an inherent feature of any complex system model because abstracting a continuous real-world system (e.g., any complex natural system) to a constructed system-model must leave gaps of representation that may diverge in unanticipated directions. In [START_REF] Mittal | Harnessing emergence: The control and design of emergent behavior in system of systems engineering[END_REF] philosophically, following Ashby [START_REF] Ashby | An Introduction to Cybernetics[END_REF] and Foo and Zeigler [START_REF] Foo | Emergence and computation[END_REF], the perceived global behavior (holism) of a model might be characterized as: Components (reductionism) + interactions (computation) + higher-order effects where the latter can be considered as the source of emergent behaviors [START_REF] Kubik | Toward a formalization of emergence[END_REF][START_REF] Szabo | Formalization of weakemergence in multiagent systems[END_REF].

The Discrete Event Systems Specification (DEVS) formalism has been advocated as an advantageous vehicle for researching such structure-behavior relationships because it provides the components and couplings for models of complex systems and supports dynamic structure for genuine adaption and evolution. Furthermore, DEVS enables fundamental emergence modeling because it operationalizes the closure-under-coupling conditions that form the basis of well-defined resultants of system composition especially where feedback coupling prevails [START_REF] Ören | System Theoretic Foundations of Modeling and Simulation: A Historic Perspective and the Legacy of A[END_REF][START_REF] Zeigler | Some Modeling & Simulation Perspectives on Emergence in System-of-Systems[END_REF].

In this paper, we formulate conditions under which a composition of component systems form a well-defined system-of-systems at a fundamental level. Formal statement of what defines a well-defined composition and sufficient conditions guaranteeing such a result offer insight into exemplars that can be found in special cases such as differential equation and discrete event systems. Informally stated, we show that for any given global state of a composition, two requirements can be stated as: [START_REF] Mittal | Emergence in stigmergic and complex adaptive systems: A formal discrete event systems perspective[END_REF] the system can leave this state, i.e., there is at least one trajectory defined that starts from the state; and (2) the trajectory evolves over time without getting stuck at a point in time. Considered for every global state, these conditions determine whether the resultant is a well-defined system and if so, whether it is non-deterministic or deterministic. We formulate these questions within the framework of iterative specifications for mathematical system models that are shown to be behaviorally equivalent to the Discrete Event System Specification (DEVS) formalism. This formalization supports definitions and proofs of the afore-mentioned conditions and allows us to exhibit examples and counter-examples of condition satisfaction. Drawing on Turing machine halting decidability, we investigate the probability of legitimacy for randomly constructed DEVS models. We close with implications for further research on the emergence of new classes of well-defined systems.

Let us start with a well-known concept, the Turing Machine (TM) (cf.

[9]). Usually, it is presented in a holistic, unitary manner but as shown in Figure 1b, we can decompose it into two stand-alone independent systems: the TM Control (S1) and the Tape System, (S2). Foo and Zeigler [START_REF] Foo | Emergence and computation[END_REF] argued that the re-composition of the two parts was an easily understood example of emergence wherein each standalone system has very limited power but their composition has universal computation capabilities. Examining this in more depth, the Tape system shown in Figure 1a is the dumber of the two, serving a memory with a slave mentality, it gets a symbol (sym) and a move (mv) instruction as input, writes the symbol to the tape square under the head, moves the head according to the instruction, and outputs the symbol found at the new head location. The power of the Tape system derives from its physicality-its ability to store and retrieve a potentially infinite amount of data-but this can only be exploited by a device that can properly interface with it. The TM Control by contrast has only finite memory but its capacity to make decisions (i.e., use its transition table to jump to a new state and produce state-dependent output) makes it the smarter executive. The composition of the two exhibits "weak emergence" in that the resultant system behavior is of a higher order of complexity than those of the components (logically undecidable versus finitely decidable), the behavior that results can be shown explicitly to be a direct consequence of the component behaviors and their essential feedback coupling-cross-connecting their outputs to inputs as shown by the arrows of Figure 1b. We are going to use this example to discuss the general issues in dealing with such compositions. The interdependent form of the interaction between the TM control and the tape system illustrates a pattern found in numerous information technology and process control systems and recognized early on by simulation language developers [START_REF] Uhrmacher | Dynamic structures in modeling and simulation: A reactive approach[END_REF] (though not necessarily in the modular form described here). In this interaction, each component system alternates between two phases, active and passive.

When one system is active the other is passive-only one can be active at any time. The active system does two actions: (1) it sends an input to the passive system that activates it (puts it into the active phase); and (2) it transits to the passive phase to await subsequent re-activation. For example, in the re-composed Turing Machine, the TM control starts a cycle of interaction by sending a symbol and move instruction to the tape system then waiting passively for a new scanned symbol to arrive. The tape system waits passively for the sym, mv pair. When it arrives, it executes the instruction and sends the symbol now under the head to the waiting control.

Such active-passive compositions provide a class of systems from which we can draw intuition and examples for generalizations about system emergence at the fundamental level. We will employ the modeling and simulation framework based on system theory formulated in [START_REF] Uhrmacher | Dynamic structures in modeling and simulation: A reactive approach[END_REF] especially focusing on its concepts of iterative specification and the Discrete Event Systems Specification (DEVS) formalism. Special cases of memory-less systems and the pattern of active-passive compositions are discussed to exemplify the conditions resulting ill-definition, deterministic, and non-deterministic as well as probabilistic systems. We provide sufficient conditions, meaningful especially for feedback coupled assemblages, under which iterative system specifications can be composed to create a well-defined resultant and that moreover can be simulated in the DEVS formalism.

However, to address more fundamental issues, we need to start with a more primitive and perhaps more intuitive notion of a system. As in Figure 2, consider a concept of system with states, transitions, and times associated with transitions. For example, there are transitions from state S1 to state S3 and from S3 to S4 which each takes 1 time unit and there is a cycle of transitions involving S4, . . . , S7 each of which take zero time. There is a self-transition involving S2 which consumes an infinite amount of time (signifying that it is passive, remaining in that state forever.) This is distinguished from the absence of any transitions out of S8. A state trajectory is a sequence of states following along existing transitions, e.g., S1, S3, S4 is such a trajectory. This example gives us a quick understanding of the conditions for system existence at the fundamental level.

We say that the system is:

• not defined at S8 because there is no trajectory emerging from it;

• non-deterministic at S1 because there are two distinct outbound transitions defined for it; and

• deterministic at S2 and S4 because there is only one outbound transition for each.

We say that the system is well-defined if it is defined at all its states. These conditions are relative to static properties, i.e., they relate to states not how the states follow one another over time. In contrast, state trajectories relate to dynamic and temporal properties. When moving along a trajectory, we keep adding the time advances to get the total traversal time, e.g., the time taken to go from S2 to S4 is 2. Here, a trajectory is said to be progressive in time if time always advances as we extend the trajectory. For example, the cycle of states S4 . . . S7 is not progressive because as we keep adding the time advances the sum never increases. Conceptually, let us start a clock at 0 and, starting from a given state, we let the system evolve following existing transitions and advancing the clock according to the time advances on the transitions. If we then ask what the state of the system will be at some time later, we will always be able to answer if the system is well-defined and progressive. A well-defined system that is not progressive signifies that the system gets stuck in time and after some time, it becomes impossible to ask what the state of the system is in after that time. Zeno's paradox offers a well-known metaphor where the time advances diminish so that time accumulates to a point rather than continue to progress and offers an example showing that the pathology does not necessarily involve a finite cycle. Our concept of progressiveness generalizes the concept of legitimacy for DEVS [START_REF] Uhrmacher | Dynamic structures in modeling and simulation: A reactive approach[END_REF] and deals with the "zenoness" property which has been much studied in the literature [START_REF] Nutaro | Building Software for Simulation: Theory and Algorithms with Applications in C++[END_REF]. We return to it in more detail later.

Thus, we have laid the conceptual groundwork in which a system has to be well-defined (static condition) and progressive (temporal dynamic condition) if it is to have achieved independent existence when emerging from a composition of components.

Formal Background

Hereafter are presented basic and coupled specifications.

Basic Discrete Event System Specification

As presented in [START_REF] Zeigler | Theory of Modeling and Simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems[END_REF], a Discrete Event System Specification (DEVS) is a structure

M =< X, S, Y, δ int , δ ext , λ, ta > where X is the set of input values, S is a set of states, Y is the set of output values, δint : S → S is the internal transition function, δ ext : Q × X → S is the external transition function, where Q = {(s, e)|s ∈ S, 0 ≤ e ≤ ta(
s)} is the total state where e is the time elapsed since last transition, λ : S → Y is the output function, ta : S → R + ∞ is the time advance function. In the Turing machine example of Figure 1, the tape system and control engine are each atomic DEVS. For the tape system, a state is a triple (tape, pos, mv) where the tape is an infinite sequence of zeros and ones (symbols), pos represents the position of the head, and mv is a specification for moving left or right. An external transition accepts a symbol, move pair, writes the symbol in the square of the current head position and stores the move for the subsequent internal transition that executed the specified move. For the control engine, a state is a pair (st, sym) where st is a control state and sym is a stored symbol. An external transition stores the received symbol for subsequent use. An internal transition applies the TM transition table to the (st, sym) pair and transitions to the specified control state.

The core of this concept simplifies when the system is input-free (or autonomous, i.e., not responding to inputs) and outputs are not considered. Then, we have a transition system M =< S, δ, ta > where δ ⊆ S × S and ta : S × S → R + ∞ . Here, we are allowing the transition system to be non-deterministic so that rather than δ and ta being functions they are presented as relations.

For example, in Figure 2, we have S = {S1, S2, S3, S4, S5, S6, S7} , δ = {(S1, S3), (S3, S4), ...}, and ta(S1, S3) = 1, ta(S3, S4) = 1,...

In this formal version of the informal statements above, the transition system, M, is

• not defined at S8 because there is no transition pair with S8 as the left member in δ;

• non-deterministic at S1 because it is a left member of two transition pairs (S1, S2) and (S1, S3); and • deterministic at S2 and S4 because there is only one transition pair involving each one as a left member.

This gives concrete and formal expression to the earlier definition of these concepts.

Coupled DEVS Models

DEVS models can be coupled to form coupled models that themselves are DEVS models manifesting closure under coupling. To keep the presentation as straightforward as possible in this paper we will limit the discussion to coupling of two components without external inputs or outputs. This is illustrated in Figure 1b where the coupling recipe maps the output of the control engine (symbol, move pair) to the input of the tape system and likewise, the output of the tape system (symbol) to the input of the control engine. This mapping is assumed to take zero time (any delay in a real manifestation can be modeled by inserting delay components). To sketch the way the resultant of coupling is computed and show that it is expressed as a DEVS, let M =< X, S, Y, δ int , δ ext , λ, ta > and M =< X , S , Y , δ int , δ ext , λ , ta >.

Then, the resultant is a DEVS:

M coup =< Q coup , δ coup , ta coup > where Q coup = Q × Q , ta coup : Q coup → R + ∞ , δ coup : Q coup → Q coup and ta((s, e), (s , e )) = min{ta(s) -e, ta (s ) -e }.
To explain, the next event will occur according the time advance and will be driven by the imminent component (whose next event time is the minimum of the two-neglecting ties for simplicity here). For illustration, let t * be the time advance and let the second component be imminent. This component generates its output and sends it to the first via the coupling which reacts to it using its external transition function establishing its next state δ ext (s, t * , λ (s )); at the same time, the imminent component applies its internal transition function to establish its next state δ int (s ). Thus, we have δ coup ((s, e), (s , e )) = (δ ext (s, t * , λ (s )), δ int (s )).

Having briefly reviewed the basics of DEVS, we are ready to consider the general case of iterative specification of which DEVS is an example.

Iterative System Specifications

We briefly review an approach to Iterative Specification of Systems that was introduced in [START_REF] Zeigler | Theory of Modeling and Simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems[END_REF] and provide more detail in Appendix B. I/O systems describe system behavior with a global transition function that determines the final state given the initial state and the applied input segment. Since input segments are left segmentable and closed under composition, we are able to generate the state and output values along the entire input interval. However, such an approach is not very practical. What we need is a way to generate state and output trajectories in an iterative way going from one state along the trajectory to the next.

Iterative specification of systems is a general scheme for defining systems by iterative applications of generator segments. These are elementary segments from which all input segments of a system can be generated. Having such a concept we define a generator state transition function and iteratively apply generator segments to the generator state transition function. The results produced by the generator state transitions constitute the state trajectory for the input segment resulting from the composition of the generator segments. The general scheme of iterative specification forms a basis for more specialized types of specifications of systems. System specification formalisms are special forms of iterative specifications with their special type of generator segments and generator state transition functions.

Consider (Z, T) the set of all segments {ω :< 0, t 1 >→ Z | t 1 ∈ T}. Here, the notation, ω :< 0, t 1 >→ Z} means that ω is a mapping from an interval of time base, T to a set of values Z. For a subset Γ of (Z, T), the concatenation closure of Γ is denoted Γ + . Example generators are bounded continuous segments, and constant segments of variable length generating bounded piecewise continuous segments piecewise constant segments, respectively. Unfortunately, if Γ generates Ω, we cannot expect each ω ∈ Ω to have a unique decomposition by Γ. A single representative, or canonical decomposition can be computed using a Maximal Length Segmentation (MLS). First we find ω 1 , the longest generator in Γ that is also a left segment of ω. This process is repeated with what remains of ω after ω 1 is removed, generating ω 2 , and so on. If the process stops after n repetitions, then ω = ω 1 ω 2 . . . ω n . We say that Γ is an admissible set of generators for Ω if Γ generates Ω and for each ω ∈ Ω, a unique MLS decomposition of ω by Γ exists.

The following is the basis for further analysis in this paper:

Theorem 1. Sufficient Conditions for Admissibility. If Γ satisfies the following conditions, it admissibly generates Γ + :

1. Existence of longest initial segments:

ω ∈ Γ + ⇒ max{t | ω t> ∈ Γ} exists 2. Closure under right segmentation: ω ∈ Γ ⇒ ω <τ ∈ Γ for all τ ∈ dom(ω)
An iterative specification of a system is a structure

G = < T, X, Ω G , Y, Q, δ G , λ >
where T, X, Y, and Q have the same interpretation as for I/O systems, Ω G is an admissible set of input segment generators, δ G : Q × Ω G →Q is the single segment state transition function, and λ :

Q × X→Y is the output function. An iterative specification G = < T, X, Ω G , Y, Q, δ G , λ > specifies a time invariant system, S G = < T, X, Ω + G , Y, Q, δ + G , λ >. The system is well-defined if δ + G , the extension of δ G , has the composition property, i.e., δ + G (q, ω 1 • ω 2 ) = δ + G (δ + G (q, ω 1 ), ω 2 ), for all ω 1 , ω 2 ∈ Ω + G .
For more details, see Appendix B.

DEVS Simulation of Iterative Specification

As reviewed in Appendix B, the notion of iterative specification was introduced to characterize diverse classes of systems such as differential equation systems and discrete time systems. With the motivation of including discrete event systems under the same umbrella as more familiar systems, DEVS was defined using iterative specification. Here we show that the converse is also true, namely, that DEVS can directly represent iterative specifications. Given an iterative specification G = < T, X, Ω G , Y, Q, δ G , λ > , we construct a DEVS model M = < X, S, Y, δ int , δ ext , λ, ta > that can simulate it in a step-by-step manner moving from one input segment to the next. The basic idea is that we build the construction around an encoding of the input segments of G into the event segments of M as illustrated in Figure 3. This is based on the MLS which gives a unique decomposition of input segments to G into generator subsegments. The encoding maps the generator subsegments. ω i in the order they occur into corresponding events which contain all the information of the segment itself. To do the simulation, the model M stores the state of G. It also uses its external transition function to store its version of G's input generator subsegment when it receives it. M then simulates G by using its internal transition function to apply its version of G's transition function to update its state maintaining correspondence with the state of G. Details of the construction are provided in Appendix C. We note that the key to the proof is that iterative specifications can be expressed within the explicit event-like constraints of the DEVS formalism, itself defined through an iterative specification. Thus DEVS can be viewed as the computational basis for system classes that can be specified in iterative form satisfying all the requirements for admissibility.

Coupled Iterative Specification

Although closure of coupling holds for DEVS, the generalization to iterative specification does not immediately follow. The problem is illustrated in Figure 4 where two iterative system specifications, ISP1 and ISP2, are cross-coupled such as exemplified by the Turing machine example in Figure 1. The cross-coupling introduces two constraints shown by the equalities in the figure, namely, the input of ISP2, ω2 must equal the output of ISP1, ρ1, and the input of ISP1, ω1 must equal the output of ISP2, ρ2. Here we are referring to input and output trajectories over time as suggested graphically in Figure 4. Since each system imposes its own constraints on its input/output relation, the conjunction of the four constraints (two coupling-imposed, two system-imposed) may have zero, one, or multiple solutions. As indicated, in the sequel, we deal with the simplified case of two coupled components. However, the results can be readily generalized with use of more complex notation. We begin with a definition of the relation of input and output segments that a system imposes on its interface. We need this definition to describe the interaction of systems brought on through the coupling of outputs to inputs. Definition 2. The I/O Relation of an iterative specification is inherited from the I/O Relation of the system that it specifies. Likewise, the set of I/O Functions of an iterative specification is inherited from the system it specifies. Formally, let S G = < T, X,

Ω + G , Y, Q, δ + G , λ > be the system specified by G = < T, X, Ω G , Y, Q, δ G , λ >. Then the I/O Functions associated with G is β : Q × Ω + G → Ω + G where for q ∈ Q, ω ∈ Ω + G , β(q, ω) = λ(δ + G (q, ω)).
Let β1 and β2 represent the I/O functions of the iterative specifications, ISP1 and ISP2, respectively. Applying the coupling constraints expressed in the equalities above, we make the definition: Definition 3. A pair of output trajectories (ρ1,ρ2) is a consistent output trajectory for the state pair (q1,q2) if ρ1 = β1(q1,ρ2) and ρ2 = β2(q2,ρ1). Definition 4. A Coupled Iterative Specification has unique solutions if there is a function, F:Q1 × Q2 → Ω; with F(q1,q2) = (ρ1,ρ2), where there is exactly one consistent pair (ρ1,ρ2) with infinite domain for every initial state (q1,q2). Infinite domain is needed for convenience in applying segmentation. The coordinator receives the current states of the components and applies the F function to compute unique consistent output segments. After segmentation using the MLS as in the proof of Theorem D in Appendix D, it packages each as a single event in a DEVS segment as shown. Each DEVS component computes the state of its Iterative Specification at the end of the segment as in Theorem D. Then it sends this state to the coordinator and the cycle repeats. This completes an informal version of the proof which would formally proceed by induction.

The solution function F represents an idealization of the fixed point solutions required for Differential Equation System Specification (DESS) and the local solution approaches of Discrete Time System Specification (DTSS) and Quantized DEVS [START_REF] Zeigler | Theory of Modeling and Simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems[END_REF]. Generalized Discrete Event System Specification (GDEVS) [START_REF] Giambiasi | A generalized discrete event specification for accurate modeling of dynamic systems[END_REF] polynomial representation of trajectories is the closest realization but the approach opens the door to realization by other trajectory prediction methods.

Special Case: Memoryless Systems

Consider the case where each component's output does not depend on its state but only on its input.

Let gr represent the ground state in which the device is always found (all states are represented by this state since output does not depend on state.) In the following R1 and R2 are the I/O relations of systems 1 and 2, respectively. In this case, they take special forms:

ρ1 = β1(gr, ω1) ⇔ (ω1, ρ1) ∈ R1 ρ2 = β2(gr, ω2) ⇔ (ω2, ρ2) ∈ R2 ω2 = ρ1 ω1 = ρ2 ρ2 = β2(q2, ω2) ⇔ (ρ1, ω1) ∈ R2 ⇔ (ω1, ρ1) ∈ R2 -1 i.e., (ρ2, ρ1) is consistent ⇔ (ρ2, ρ1) ∈ R1 ∩ R2 -1
Let f and g be defined in the following way:

f (ρ2) = β1(gr, ρ2) g(ρ1) = β2(gr, ρ1)
So for any ρ1, ρ1 = f (ρ2) = f (g(ρ1)) and g = f -1 (considered as a relation). Finally, F(q1, q2) = (ρ1, f -1 (ρ1)) has:

• no solutions if f -1 does not exist, yielding no resultant; • a unique solution for every input if f -1 exists, yielding a deterministic resultant; and • multiple solutions for a given segment

ρ if f -1 (ρ) is multivalued, yielding a non-deterministic resultant.
For examples, consider an adder that always adds 1 to its input, i.e.,

β(gr, ρ) = f (ρ) = ρ + 1 i.e., ∀t ∈ T, β(gr, ρ)(t) = ρ(t) + 1
Cross-coupling a pair of adders does not yield a well-defined resultant because f -1 (ρ) = ρ -1 = f . However, coupling an adder to a subtracter, its inverse, β(gr, ρ) = ρ -1, yields a well-defined deterministic system.

For other examples, consider combinatorial elements whose output is a logic function of the input and consider a pair of gates of the same type connected in a feedback loop:

• A NOT gate has an inverse 0 → 1, 1 → 0, so the composition has two solutions one for each of two complementary assignments, i.e., F(gr, gr) = {(1, 0), (0, 1)} yielding a non-deterministic system. • An AND gate with one of its input held to 0 always maps the other input into 0 so has a solution only for inputs of 0 to each component, i.e., F(gr, gr) = (0, 0), yielding a deterministic system. • An AND gate with a stuck-at input of 1 is the identity mapping and has multiple solutions i.e., F(gr, gr) = {(0, 0), (1, 1)}, yielding a non-deterministic system.

Active-Passive Systems

We now show that active-passive systems as described earlier offer a class of systems for which the iterative specifications of components satisfy the admissibility conditions specified in Theorem 4. As in Figure 6a, consider a pair of cross-coupled systems each having input generators that represent null and non-null segments. A null generator represents the output of a passive system whereas a non-null generator represents the output of an active system. For example in the TM case, a non-null generator is a segment that represents transmission of a symbol by the tape unit and of a symbol, move pair for the TM control. As in Figure 6b let S1 and S2 start as active and passive, respectively. Then, they output non-null and null generators, respectively. Since the null generator has infinite extent, the end-time of the non-null generator determines the time of next event, t* (as defined earlier) and we apply the transition functions in Figure 6a to find that at t* the systems have reversed phases with S1 in passive and S2 in active. Definition 6. Define a triple < Q, δ G , λ > by Q = {active, passive} × {active, passive} with F(active, passive) = (non-null, null) δ G (active, passive) = (δ(active, null t * > ) = δ(passive, non-null t * > )) = (passive, active) with t * (active, passive)= endtime of the non-null generator segment λ(active, passive) = (non-null, null)

Let the non-null and null generators stand for sets of concrete segments that are equivalent with respect to state transitions and outputs. Then, such a scheme can define a deterministic or non-deterministic resultant depending on the number of consistent pairs of (non-null, null) generator segments are possible from the (active, passive) state. Furthermore, mutatis-mutandis for the state (passive, active).

Temporal Progress: Legitimacy, Zenoness

Although we have shown how a DEVS coupled model can simulate a coupling of iterative specifications, it does not guarantee that such a model is legitimate [START_REF] Zeigler | Theory of Modeling and Simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems[END_REF]. Indeed, backing up, legitimacy of the simulating DEVS is reflective of the temporal progress character of the original coupling of specifications. We see that the time advance has a value provided that the function F is defined for current state. However, as in the case of zenoness, a sequence of such values may accumulate at a finite point thus not allowing the system to progress beyond this point. Consequently, we need to extend the requirements for coupled iterative specifications to include temporal progress in order to fully characterize the resultant composition of systems. Thus, we can extend the definitions: Zeigler et al. [START_REF] Zeigler | Theory of Modeling and Simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems[END_REF] showed that the question of whether a DEVS is legitimate is undecidable. The approach was to express a Turing Machine as a DEVS and showing that solving the halting problem is equivalent to establishing legitimacy for this DEVS. We can transfer this approach to the formulation of the TM in Figure 1 by setting the time advances of both components to zero except for a halt state which is a passive state (and has an infinite time advance). Then a TM DEVS is legitimate just in case the TM it implements ever halts. Since there is no algorithm to decide whether an arbitrary TM will halt or not the same is true for legitimacy. Since DEVS is an iterative specification, the problem of determining whether an Iterative Specification is progressive is also undecidable. Likewise, the question of whether a coupled iterative specification is progressive is undecidable.

Fundamental Systems Existence: Probabilistic Characterization of Halting

The existence of an iteratively specified system requires temporal progress and there is no algorithm to guarantee in a finite time that such progress is true. The underlying point is that, given an arbitrary TM, we have to simulate it step-by-step to determine if it will halt. Importantly, if a TM has not halted after some time, however long, this gives us no further information on its potential halting in the future. However, while this is true for individual TMs, the situation may be different for the statistics of subclasses of TMs. In other words, for a given class of TMs, the probability of an instance halting may be found to increase or decrease for the future given that it has not yet halted.

We performed an empirical study of randomly generated TMs described in Appendix E. The results seem to show that TMs break into two classes-those that halt within a small number of steps (e.g., [START_REF] Steiniger | Intensional couplings in variable-structure models: An exploration based on multilevel-DEVS[END_REF]) and those that do not halt. Indeed, the data suggest that approximately one-third of the sampled TMs halt, while two-thirds do not. For those that halt, the probability of halting at the first step is greatest and then decreases exponentially and so is well-described by a geometric distribution with probability of success approximately 1/6. In other words the probability of halting is 1/6 at each step given the TM has not halted earlier for the set that will eventually halt.

Summarizing, the existence of an iteratively specified system is not algorithmically decidable but there may be useful probabilistic formulations that can be applied to sub-classes of such specifications. For example, it seems to be that with high probability the halting of a two-symbol, three-state TM can be decided within 16 simulation steps. Therefore, it might be tractable for an assemblage of components to find a way to solve its composition problems in a finite time for certain classes of components couplings.

Discussion: Future Research

The discussion presented here was limited to compositions having two components with cross-coupled connections. However, this scope was sufficient to expose the essential effect of feedback in creating the need for consistent assignments of input/output pairs and the additional-distinct-requirement for time progression. Further development can extend the scope to arbitrary compositions generalizing the statement of consistent assignments to finite and perhaps, infinite sets of components.

Further, when extending to arbitrary composition sizes, the analysis revealing fundamental conditions for the emergent existence of systems from component systems suggests new classes of systems that can be defined using iterative specifications. For example, we might consider the active-passive systems in a more general setting. First allowing any finite number of them in a coupling can satisfy the requirement for at most one active component at any time by restricting the coupling to a single influencer for each component. In fixed coupling exhibiting feedback, this amounts to a cyclic formation in which activity will travel around the cycle from one component to the next. Another common pattern is a single component (root) influencing a set of components (branches) each of which influences only the root. Several possibilities for interaction of the root with the branches can result in admissible compositions. For example, the root might select a single branch to activate to satisfy the single active component requirements. Alternatively, it might activate all branches and count the number of branches as they successively go passive, becoming active when all the branches have become passive. These alternative configuration require knowledge of the protocol underlying the interaction but this only means that proofs of emergent well-defined resultants will be conditioned to narrower sub-classes of systems rather than to broad classes such as all differential equation systems.

After much progress in understanding how dynamic structure can be managed in DEVS-based simulations [START_REF] Barros | Modeling and Simulation of Dynamic Structure Heterogeneous Flow Systems[END_REF][START_REF] Barros | Dynamic Structure Multi-Paradigm Modeling and Simulation[END_REF][START_REF] Steiniger | Intensional couplings in variable-structure models: An exploration based on multilevel-DEVS[END_REF], a framework that incorporates several insights has recently been developed [START_REF] Muzy | Specification of dynamic structure discrete event systems using single point encapsulated control functions[END_REF]. This framework, formulated in the DEVS formalism, can be extended to iterative specifications and investigated for conditions of existence as done here. The extension will probably increase the complexity required to state and establish the conditions but would, if successful, bring us closer to understanding the emergence of systems from assemblages of components in the real world. and mv is a specification for moving left or right. An internal transition moves the head as specified; an external transition accepts a symbol, move pair, stores the symbol in the current position of the head and stores the move for subsequent execution. The slot for storing the move can also be null, which indicates that a move has taken place.

Each function can be described as follows: The results are shown in the first two columns of Table E2. They seem to show that TMs break into two classes-those that halt within a small number of steps (e.g., [START_REF] Steiniger | Intensional couplings in variable-structure models: An exploration based on multilevel-DEVS[END_REF]) and those that do not halt. Indeed, the data in the table suggest that approximately one-third of the sampled TMs halt, while two-thirds do not. For those that halt, the probability of halting at the first step is greatest and then decreases exponentially (cf. Figure E1). The third row shows the probabilities computed from a geometric success model using probability of success as 0.161. The agreement of the observed and predicted are in close agreement. This suggests that for the set that will eventually halt, the probability of a TM halting at each step is approximately 1/6, given that it has not halted earlier. Summarizing, the existence of an iteratively specified system is not algorithmically decidable but there may be useful probabilistic formulations that can be applied to sub-classes of such specifications. For example, it seems to be that with high probability the halting of a two-symbol, three-state TM

Figure 1 .

 1 Figure 1. Turing Machine decomposition (a) and modular composition (b).

Figure 2 .

 2 Figure 2. System with tuned transitions.

Figure 3 .

 3 Figure 3. DEVS Simulation of an Iterative Specification.

Figure 4 .

 4 Figure 4. Input and output trajectories of coupled systems.

Definition 5 .Theorem 2 .Theorem 3 .Theorem 4 .

 5234 A Coupled Iterative Specification is admissible if it has unique solutions. The following theorems are proved in Appendix D. An admissible Coupled Iterative Specification specifies a well-defined Iterative Specification at the I/O System level. The set of Iterative Specifications is closed under admissible coupling. DEVS coupled model can component-wise simulate a coupled Iterative Specification. Proof. The coupled model has components, which are DEVS representations of the individual Iterative Specifications (according to Theorem D in Appendix D) and also a coordinator as shown in Figure 5.

Figure 5 .

 5 Figure 5. DEVS Component-wise simulation of Coupled Iterative Specifications.

Figure 6 .

 6 Figure 6. Active-Passive Example of Admissible Progressive Coupled Iterative Specifications.

Definition 7 .

 7 An Iterative Specification is progressive if every sequence of time advances diverges in sum. Definition 8. A Coupled Iterative Specification is progressively admissible if its components are each progressive and the resultant Iterative Specification at the I/O System level is progressive. Definition 9. A progressive Iterative Specification can be simulated by a legitimate DEVS. Definition 10. A progressively admissible Coupled Iterative Specification can be component-wise simulated by a legitimate DEVS.

δ

  int (tape, pos, mv) = (tape, move(pos, mv), null) δ ext ((tape, pos, null), e, (sym, mv)) = (store(tape, pos, sym), pos, mv) ta(tape, pos, mv) = 1 ta(tape, pos, null) = ∞ λ(tape, pos, mv) = getSymbol(tape, pos) where move(pos.mv) = pos + mv store(tape, pos, sym) = tape where tape (pos) = sym, tape (i) = tape(i) getSymbol(tape, pos) = tape(pos)

Table E1 .

 E1 Template for two-symbol, three-state TMs. State Symbol Next State {A, B, C} Move {1, -1} Print Symbol {0, 1}

	A	0	x	x	x
	A	1	x	x	x
	B	0	x	x	x
	B	1	x	x	x
	C	0	x	x	x
	C	1	x	x	x

Table E2 .

 E2 Empirical Frequency Distribution for halting step vs. Geometric Model.

	Halts at N Frequency Geometric Model
	1	0.161	0.161
	2	0.1	0.09
	3	0.042	0.039
	4	0.023	0.022
	5	0.016	0.015
	6	0.005	0.00488
	7	0.004	0.00390
	11	0.001	0.000990
	16	0.001	0.000985
	10000	0.647

Figure E1. Empirical frequency vs. geometric model.

Appendix A.1. Tape System

A state is a triple (tape, pos, mv) where tape : I → {0, 1}, pos ∈ I, mv ∈ {-1, 1} and I is the integers; in other words, the tape is an infinite sequence of bits, where pos represents the position of the head, Appendix A. [START_REF] Mittal | Harnessing emergence: The control and design of emergent behavior in system of systems engineering[END_REF]

. TM Control

A state is a pair (st, sym) where st is a control state and sym is a stored symbol. An internal transition applies the TM transition table to the (st, sym) pair and transitions to the specified control state. An external transition stores the received symbol for subsequent use.

Each function can be described as follows:

δ int (st, sym) = (TMState(st, sym), null) δ ext ((st, null), e, sym) = (st, sym) ta(st, sym) = 1 ta(st, null) = ∞ λ(st, sym) = TMOutput(st, sym)

where TMState(st, sym) = st TMOutput(st, sym) = sym

Appendix B. Iterative Specification of Systems

The following is extracted from [START_REF] Zeigler | Theory of Modeling and Simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems[END_REF] Chapter 5.

Appendix B.1. Generator Segments

Consider (Z,T) the set of all segments {ω :< 0, t 1 > → Z | t 1 ∈ T}, which is a semigroup under concatenation. For a subset Γ of (Z,T), we designate by Γ + the concatenation closure of Γ (also called the semigroup generated by Γ). Example generators are bounded continuous segments, and constant segments of variable length generating bounded piecewise continuous segments piecewise constant segments, respectively. Unfortunately, if Γ generates Ω (a subset of segments), we cannot expect each ω ∈ Ω to have a unique decomposition by Γ; that is, there may be distinct decompositions ω 1 , ω 2 , . . . , ω n and ω 1 , ω 2 , . . . , ω n such that ω1

A single representative, or canonical decomposition can be computed using a maximal length segmentation. First, we find ω 1 , the longest generator in Γ that is also a left segment of ω. This process is repeated with what remains of ω after ω 1 is removed, generating ω 2 , and so on. If the process stops after n repetitions, then ω = ω 1 ω 2 . . . ω n .

It is not necessarily the case that MLS decompositions exist (i.e., that the just-mentioned processes will stop after a finite number of repetitions). Thus, we are interested in checkable conditions on a set of generators that will guarantee that each segment generated has an MLS decomposition. Fortunately, a segment can have at most one MLS decomposition. We say that Γ is an admissible set of generators for Ω if Γ generates Ω and for each ω∈Ω, a unique MLS decomposition of ω by Γ exists. (We also say Γ admissibly generates Ω.) The following is proved in [START_REF] Zeigler | Theory of Modeling and Simulation: Integrating Discrete Event and Continuous Complex Dynamic Systems[END_REF]: Theorem B1. Sufficient Conditions for Admissibility. If Γ satisfies the following conditions, it admissibly generates Γ + :

. Generator State Transition Systems

Having established a terminating process for obtaining MLS decompositions, we wish to use these decompositions to help us generate a transition function given only that function's action on the generators. In other words, let Ω G be an admissible generating set for Ω and suppose we have defined a function δ G : Q × Ω G → Q, which we call a single segment transition function.

Let ω 1 , ω 2 , . . . , ω n be the MLS decomposition of ω. Having δ defined for each segment ω i , we wish to piece together these parts to obtain compound transition associated with ω itself.

An iterative specification of a system is a structure

where T, X, Y, and Q have the same interpretation as for I/O systems; Ω G is the set of input segment generators;

G is the extension of δ G which is well-defined according to the following: Theorem B2. Sufficient Conditions for Iterative Specification. Let G = < T, X, Ω G , Y, Q, δ G , λ > be a structure as just defined. Then, if the following conditions hold, G is an iterative specification and

DEVS Atomic Model Simulation of an Iterative Specification

Theorem C1. A DEVS atomic model can simulate an Iterative Specification.

Proof. As shown in Figure 3 and in Figure 5, given an iterative specification G

According to the definition of the simulating DEVS:

In the Iterative Specification:

In the DEVS, let δ DEVS be the transition function of the system specified by the DEVS. We want to show that h(δ + G (q, ωω )) = δ DEVS (h(q), g(ωω )), i.e.,

Thus, Proof. Given two iterative specifications, G i and their state sets,

eventually the state set of the iterative specification to be constructed. For any pair (q1,q2) in Q, let (ρ1, ρ2) be a consistent output trajectory for (q1, q2), i.e., ρ1 = β(q1, ρ2) and ρ2 = β(q2, ρ1) and F(q1, q2) = (ρ1, ρ2).

Define an autonomous Iterative Specification < Q, δ G , λ > by δ G (q1, q2) = (δ(q1, ρ2 t * (q1,q2)> ), δ(q2, ρ1 t * (q1,q2)> ))

where t * (q1, q2) = min{t * 1 , t * 2 } and t * 1 , t * 2 are the times of the MLS for ρ1 and ρ2. In other words, since each of the component iterative specifications have maximum length segmentations we take the time of next update of the constructed specification to be determined by earliest of the times of these segmentations for the consistent pair of output trajectories. This allows us to define a step-wise transition for the constructed transition function. Closure under composition for this transition function can be established using induction on the number of generators in the segments under consideration. Similarly, the output function is defined by: λ(q1, q2) = (λ(q1), λ(q2)).

Theorem D2. The set of Iterative Specifications is closed under admissible coupling.

Proof. This theorem is a corollary of the previous theorem.

Appendix E. A Statistical Experiment Sampling from 2-Symbol, 3-State TMs

We performed a statistical experiment sampling from two-symbol, three-state TMs. Each sampled TM fills in the xs in Table E1 with values from the sets shown, where one of the rows is chosen at random and its next state change to the halt state (e.g., the first row can be (A,-1,0) meaning jump to state A, move back, and print 0). The number of TMs in this class is (2 × 2 × 3) 2×3 = 12 6 . We generated 1000 samples randomly and simulated them until they halted or for a maximum of 1000 time steps. can be decided within 16 simulation steps. Therefore, it might be tractable for an assemblage of components to find a way to solve its composition problems in a finite time for certain classes of components couplings.


Sébastien Ménigot
email: sebastien.menigot@univ-tours.fr

Jean-Marc Girault
email: jean-marc.girault@univ-tours.fr

Harmonic Magnification by Time Reversal based on a Hammerstein Decomposition

In tissue harmonic imaging, the contrast has been increased by using nonlinearities. However, enhancing the contrast requires a good harmonic level. Time reversal process is well-known to enhance the backscattered signal by optimizing the transmitted wave. However, it is not well-adjusted for second harmonics, since the nonlinearities of the time reversed harmonic signal will be double the frequency and will be removed by the transducer. To take into account the harmonics, one way consists in modelling the ultrasound system by parallel subsystems with a Hammerstein model. Therefore finding the optimal wave for harmonic generation means to create a matched filter for the subsystem describing the nonlinearity. As described by the time reversal, the first wave propagates to the medium. The backscattered second harmonic components are extracted from a Hammerstein model. The second harmonic signal is time reversed, frequency shifted by a demodulation and sent in the medium. In this study, the nonlinear components could be increased by 15 dB in comparison to the time reversal process and conventional imaging. Moreover, the SNR can be increased by 7 dB. The second harmonic time reversed waves focus better on the inclusion and wave coherence is preserved. From this point of view, the optimization process can be viewed as an extension of the matched filtering feature of the time reversal principle to second harmonics. 1

I. INTRODUCTION

The medical ultrasound imaging systems have been improved by taking into account the nonlinear wave propagation. When ultrasound sinus waves at frequency f 0 propagate nonlinearly, the echoes received are composed of harmonic components (2f 0 , 3f 0 , ...). By extracting each harmonic component, it is possible to obtain ultrasound images with high contrast [START_REF] Burns | Instrumentation for contrast echocardiography[END_REF]. Given this success, tissue harmonic imaging has become the native imaging modality in conventional ultrasound scanners.

Since the backscattered nonlinearities are a function of the transmitted signal, enhancing the image quality in harmonic imaging means to find the best transmitted wave. Waveform design can be decomposed in two ways: optimizing the wave in space or optimizing the wave in time. In the first way, beamforming has improved the contrast and the resolution [START_REF] Szabo | Diagnostic Ultrasound Imaging: Inside Out[END_REF], even if many studies focused on a simple transmitter and a synthetic beamforming on reception [START_REF] Jensen | Synthetic aperture ultrasound imaging[END_REF]. The second way consists in finding the temporal waveform. Several studies have proposed a solution for nonlinear imaging. An iterative algorithm searches the features of the wave [START_REF] Ménigot | Optimization of contrast to tissue ratio by frequency adaptation in pulse inversion imaging[END_REF] or directly the values of the samples [START_REF] Ménigot | Optimization of contrast resolution by genetic algorithm in ultrasound tissue harmonic imaging[END_REF].

Moreover, time reversal optimizes the signal-to-noise ratio (SNR) by combining a waveform design in space and in time thanks to a physical matched filter. It makes possible to focus the wave on small inhomogeneities, without information on the medium and even if the medium is aberrating [START_REF] Fink | Time-reversal of ultrasonic fields .1. basic principles[END_REF]. Its principle consists in sending a first wave at the frequency f 0 and in retro-propagating the time reversed echoes. Nevertheless, as the wave relives its past propagation, time reversal reduces the harmonic components [START_REF] Tanter | Breaking of time reversal invariance in nonlinear acoustics[END_REF]. This property has been used to reduced the tissue harmonic components in ultrasound contrast imaging [START_REF] Couture | Suppression of tissue harmonics for pulse-inversion contrast imaging using time reversal[END_REF]. Therefore, this method is not adjusted to maximize the harmonics components from propagation in tissue.

However, other approaches based on the time reversal principle have been proposed, as for instance the method combining time reversal and harmonics [START_REF] Goursolle | Non-linear based time reversal acoustic applied to crack detection: Simulations and experiments[END_REF]. The second harmonics at 2f 0 is extracted by filtering, even if this filtering do not guarantee that the received harmonics are equal to the extraction, because of the phase. However, this phase difference makes possible that the time reversed second harmonics focus on the small inhomogeneity by a liner propagation at 2f 0 . Therefore, a direct time reversal of second harmonics is not the matched filter which optimizes the SNR for harmonic components, since the propagation at 2f 0 is linear. To understand it, one way consists in modelling the ultrasound system by simple parallel subsystems. Such model as a Hammerstein system is sufficient to describe the nonlinear behaviour [START_REF] Narendra | An iterative method for the identification of nonlinear systems using a hammerstein model[END_REF], [START_REF] Sbeity | Contrast improvement in sub-and ultraharmonic ultrasound contrast imaging by combining several hammerstein models[END_REF]. Thus the system usually can be decomposed by parallel subsystems where the nonlineary is separated of the linearity. For example, a second order model (shown in Fig. 1) can describe the output from the input. The first subsystem is dedicated to linear output by a linear filter h 1 . In the second subsystem, in order to create the second harmonics, a polynomial function is added to a linear filter h 2 . In time reversal, only the first channel is considered. Therefore, after the transmission of the time reversed fundamental component at f 0 , the SNR of the linear output is optimal thanks to a matched filter of the first channel. However, finding the optimal wave for harmonic generation means to create a matched filter for the second channel and its solution is not trivial. Indeed, after the transmission of the time reversed second harmonics at 2f 0 , on the one hand, the 2f 0 retropropagated component is due to a linear process (first channel) and so not optimal (since it is different to the time reversal solution). On the other hand, the nonlinearities double the frequency at 4f 0 because of the square function in the second channel. The matched filter for the second harmonics is thus disrupted by the nonlinear effect (modelled here by a square function). Our objectives is thus to find the optimal wave to optimize the harmonic generation by taking into account this frequency doubling and without increasing the transmit power. We proposed to adjust time reversal by adding a new step. By assuming that the ultrasound system can be modelled as a Hammerstein decomposition, the matched filter for harmonics could be obtained by reversing time and by annihilating the harmonic effects. This annihilation could be simply obtained by inversing the function responsible to the nonlinear effect in the Hammerstein model. In this paper, this method was applied in simulation. Moreover, its feasability was just evaluated in order to magnify the second harmonics only.

II. HARMONIC MAGNIFICATION

As previously explained, the harmonic magnification is based on the time reversal principle according to two steps (Fig. 2). The first step (switches in position 1) consists in sending a first standard excitation x standard (n) and in extracting the second harmonic backscattered components y NL (n) by a Hammerstein model. However, by retro-propagating this second harmonics directly, the components at 2f 0 are due to a linear effect, since the subsystem describing the harmonic behaviour double the frequency. Thus, in a second step (switches in position 2), it is required to annihilate the nonlinear effects by inverting the nonlinear contribution and then by reversing time. This optimal signal x opt (n) is finally retro-propagated into the medium.

A. Harmonic extraction

The harmonic extraction is based on a Hammerstein model. Each model is usually decomposed by parallel subsystems as a polynomial function and a linear filter. However, the polynomial functions are inverse functions only on R + . Thus, to garantee the inversion of subsystems, the polynomial functions are replaced by frequency shifting (Fig. 2) thanks to a modulation. Note that this change is also possible, since the continuous component usually modelled thanks to the square function is here filtered by the transducer. The signal modelling ŷ(n) of the basckattered signal y(n) can be thus written as:

ŷ(n) = P p=1 M m=1 h p (m)x standard (n -m) • C p (n), ( 1
)
where n is the discrete time, M the memory of the Hammerstein model, P its order, x standard (n) the input model and C p (n) = cos 2π (p-1)f0n

Fs

with F s the sample frequency. Note that here the order P has to be set 2 in order to extract the second harmonics. Therefore, the Hammerstein modelling is solved to find the linear filter coefficients h p .

This problem can be written with an algebraic formula, such as :

ẑ = X T h, (2)
where ŷ = [y(M + 1), . . . , y(N )] T with N the sample quantity and T the symbol of the vector transposition, h the vector of coefficients and the matrix of input signals

h = [h 1 (1), . . . , h 1 (M ), h 2 (1), . . . , h 2 (M )] , (3)
X = [x 1 x 2 ] with x p =        v p (M ) v p (M + 1) . . . v p (N ) v p (M + 1) v p (M + 2) 0 . . . . . . v p (N ) 0 0        , (4)
and

v p (n) = x(n)C p (n).
Finding a solution minimum the error between the signal modelling ŷ(n) and the backscattered signal y(n) can be solved with a pseudo-inversion, such as:

h = X T X -1 X T y. (5)
Finally, the second harmonic extraction is based on the coefficients h 2 = [h 2 (1), . . . , h 2 (M )] T . The second harmonic signal y NL can be modelled as:

y NL = x T 2 h 2 . (6)
Note that a Matlab program of a Hammerstein decomposition is available on RunMyCode.org (http://www.runmycode.org/coder/view/2395

B. Time Reversal for Second Harmonics

After the second harmonic extraction, the second harmonic signal y NL is time reversed. Moreover, since we assume that the second harmonic can be modelled by a frequency shifting and a linear filter, annihilating the effects of the second harmonics consists in an "opposite" frequency shifting by a demodulation. Moreover, this inverse frequency shifting has to take into account the phase of the backscattered second harmonic signal y NL . Thus the best transmitted signal to magnify the second harmonics can be written as follow:

x opt (n) = A • y NL (n) • cos 2πf 0 n F s + φ , (7)
where A is a coefficient in order to preserve the transmit power equal to the initial transmitted signal x standard (n). The phase difference φ is obtained by comparing the phase of the second harmonic signal y NL and the phase of the cosinus:

φ = arctan ϕ(y NL ) ϕ(cos) , (8)
with ϕ(y NL ) is the phase of the the second harmonic signal y NL and the ϕ(cos) the phase of a cosinus at the frequency f 0. Note these phases are numerically obtained by a fast Fourier transform.

III. SIMULATION MODEL

In order to validate the feasibility, we used realistic simulations which have already been proven in medical ultrasound imaging [START_REF] Ménigot | Optimization of contrast to tissue ratio by frequency adaptation in pulse inversion imaging[END_REF]. The simulation model is composed of different phases: transmission, 2D nonlinear propagation and reception.

At the transmitter, a Gaussian pulse x standard (n) at a centre frequency f 0 of 2 MHz and with a bandwidth of 50% is generated digitally with Matlab (The Mathworks, Natick, MA, USA). It is transmitted to an 8-element probe with a pitch of 400 µm and filtered by its transfer function from a realistic transducer, centred at f c = 4 MHz with a fractional bandwidth of 80% at -3 dB. Note that the signal x standard (n) is set in order to guarantee the reception of second harmonics. The pulse wave generated is propagated nonlinearly into an attenuating medium (Fig. 3), i.e. a 2 mm-broad inclusion of a second tissue sample is inserted into the first tissue at 8 mm below the surface). Note that these small sizes were chosen to reduce the computation time of the propagation simulation. Moreover a small target with a high impedance is inserted at 12 mm. The 2D nonlinear wave propagation is solved using the Anderson model based on a pseudo-spectral derivative and a time-domain integration algorithm [START_REF] Anderson | A 2d nonlinear wave propagation solver written in open-source matlab code[END_REF]. Fig. 3. Grid of medium properties. The ultrasound transducer was at a depth of 0 mm, here at the top. Note that this scheme is not to scale.

Finally, the signals y(n) basckattered by tissue are recorded and filtered by the transfer function of the transducer. The harmonic magnification process is separately applied for basckattered signal y(n) of each element of the probe. Finally the best signals x opt (n) to magnify the second harmonics are transmitted to the medium and the backscattered signal y opt (n) are recorded.

IV. RESULTS

Fig. 4 shows the input and second harmonic output signals in the case of standard time reversal and in the case of time reversal for second harmonics. Note that the second harmonic extraction was made by a second-order Hammerstein decomposition. Their respective spectra are depicted in Fig. 4a. As an illustration, the spectra of the initial Gaussian pulse at 2 MHz and the the harmonic basckettered signal after focusing are shown. Note that this focusing is obtained by a standard beamforming using Euclidean geometry [START_REF] Szabo | Diagnostic Ultrasound Imaging: Inside Out[END_REF].

By using time reversal, the second harmonic waves focused on the target position. However, if the time reversal was dedicated to the second harmonics, the amplitude of the second harmonic signal was higher. This results was confirmed on the spectra with a second harmonic components higher than 15 dB by using time reversal for second harmonics than standard time reversal or than standard beamforming. Moreover, on the harmonic backattered signals, the SNR was measured in case of time reversal for second harmonics using frequency shifting. This SNR is compared to the method using time reversed second harmonics without frequency shifting as in [START_REF] Goursolle | Non-linear based time reversal acoustic applied to crack detection: Simulations and experiments[END_REF], standard time reversal and a focusing by beamforming. They are shown in Fig. 5. The SNR could increase sighlty by using time reversal rather a standard beamforming. However, the SNR increase could reach 7 dB in comparison the time reversal for second harmonics. Note that no harmonic extraction were used to evaluated the SNR on time reversed second harmonics without frequency shifting, since the retropropagation was a linear process.

Finally, as an illustration, pressure propagating in the medium by using time reversal for second harmonics are shown at the beginning of the propagation and at the focusing. As soon as the beginning of the propagation, the wavefront was deformed to take into account the inclusion. Thus the wave focused even with the presence of this inclusion. Fig. 6. Image of pressure propagating in the medium by using time reversal for second harmonics at the beginning of the propagation (a) and at the focusing (b). As an illustration, the target is depicted with a red cross and the inclusion with a white line. Note that the ultrasound probe was located at 0 mm (on the bottom in the figures).

V. DISCUSSIONS AND CONCLUSION

In contrast with the standard time reversal, the method presented here is adjusted to magnify the second harmonics. By finding a wave which could take into account the medium complexity, the wave focused on the target and wave coherence is preserved. Therefore it generated a higher quantity of second harmonics and it made possible to highly increase the SNR.

Fig. 1 .

 1 Fig. 1. Block diagram of a second order polynomial Hammerstein model.

Fig. 2 .

 2 Fig. 2. Block diagram of harmonic magnification by time reversal based on a Hammerstein decomposition. The yellow functions are the standard imaging. By setting the switches in position 1, the second harmonic components can be extracted by the Hammerstein model in green. Finally the second harmonic components are used to optimize the the basckettered harmonics in blue.

Fig. 4 .

 4 Fig. 4. (a) Driving pressures transmitted to the medium thanks to the center element of the probe in case of time reversal and of time reversal for second harmonics. (b) Harmonic backscattered signals received by the center element of the probe in case of time reversal and and of time reversal for second harmonics. (c) Spectra of Harmonic backscattered signals in case of time reversal and and of time reversal for second harmonics. As an illustration, the spectra of the initial Gaussian pulse and the harmonic basckettered signal after focusing by beamforming are depicted.

Fig. 5 .

 5 Fig. 5. SNR measured on harmonic backscattered signals obtained with time reversal (TR) for second harmonics using frequency shifting, without frequency shifting, standard time reversal and a focusing by beamforming.

DOI : 10.1109/ULTSYM.2016.7728742 c

IEEE. Reprinted, with permission, from Sébastien Ménigot and Jean-Marc Girault, Harmonic magnification by time reversal based on a Hammerstein decomposition, 2016 IEEE International Ultrasonics Symposium (IUS), 2016. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the Université François Rabelais de Tours' products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org.

Moreover, this magnification lies in the annihilation of the harmonic effects before sending the time-reversed signal. This annihilation was possible because we assumed that the ultrasound system could be modelled by a Hammerstein model using frequency shifting. However, this assumptions could become wrong for high nonlinearity. Moreover, taking into account the phase was crucial for the demodulation. However, the phase estimation was not precise and yet was sufficient to improve the SNR.

From this point of view, the optimization process can be viewed as an extension of the matched filtering feature of the time reversal principle to second harmonics. This method could help to improve the harmonic imaging in the case where the aberration phase can be high. We will carry on validating this concept experimentally. Moreover, the method will be generalized at high order of nonlinearity by adjusting the nonlinear annihilation.


Laëtitia Diez
email: laetitia.diez@univ-lorraine.fr

Pascale Marangé
email: pascale.marange@univ-lorraine.fr

Éric Levrat
email: eric.levrat@univ-lorraine.fr

Maintenance best way for meeting the challenge of regeneration

Keywords: Maintenance, maintenance services, maintenance management, sustainability, regeneration paradigm

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

To resolve the problem of the climate change, the increase in the greenhouse gases and the consumption of goods, the exhaustion of natural resources, the accumulation of wastes, etc., the political powers orientate their strategy towards sustainable development. This development is defined as a "development that meets the needs of the present without compromising the ability of future generations to meet their own needs" (WCED, 1987). In this sense, several conferences and projects expand such as United Nations Conference on Sustainable Development (UNCSD), Rio+20 and COP21, and the Europe 2020 project. Thus, the current economic model, the linear economy (takemake-jump), does not meet new requirements of sustainable development. It becomes obsolete. Therefore, we must change paradigm and turn towards, for example, the circular economy (make-use-return indefinitely). The circular economy considers that natural resources are limited and that waste does not exist [START_REF] Foundation | Report Version 1, 2 & 3 : Towards the Circular Economy[END_REF]. The idea of this economy is to create closed loops by reusing the manufacturing goods so that waste becomes resources. It is also called the "paradigm of regeneration". The principle of the circular economy in based on several schools of thought:

-Biomimicry [START_REF] Benyus | Biomimicry[END_REF] is the study of nature to design goods and to develop new technologies. For example, the form of eagle has inspired the highspeed train, and the skin of sharks has inspired the swimsuit; -Industrial Ecology [START_REF] Despeisse | Industrial ecology at factory level: a prototype methodology[END_REF] is the study of material and energy flows through the industrial system to add value to wastes produced by a firm. For this, a firm uses wastes produced by another firm as resources. Kalundborg area [START_REF] Zhu | The development of regional collaboration for resource efficiency: A network perspective on industrial symbiosis[END_REF] is a good example; -Regenerative Design [START_REF] Du Plessis | Towards a regenerative paradigm for the built environment[END_REF] is the study of environment and communities of a place to design in harmony with them. The permaculture is an application of this concept; -Cradle-to-Cradle [START_REF] Mcdonough | Cradle to Cradle: Remaking the Way We Make Things[END_REF] is the popular term for regenerative design, which uses a biomimicry approach to design a good. These authors use the term of nutrients to talk about resources. [START_REF] Diez | Maintenance as a Cornerstone for the Application of Regeneration Paradigm in Systems Lifecycle[END_REF] have approached this notion. These schools of thought are oriented design and consider a perfect exchange of wastes between different firms. The model regrouping all these concepts does not exist, but we can approach it. Indeed, research areas such as maintenance turn towards the sustainable problematic with new concepts including the three sustainable pillars. They include:

-Green maintenance focuses on the elimination of all wastes produced and associated with maintenance, by practicing the pollution prevention, toxic use reduction and design for environment. [START_REF] Ajukumar | Evaluation of green maintenance initiatives in design and development of mechanical systems using an integrated approach[END_REF] propose an approach to evaluate it by basing on a multi-criteria decision-making and on green maintenance requirements (environmental compatibility, energy efficiency, and human health and safety risks);

-Maintenance-centered Circular Manufacturing [START_REF] Takata | Maintenance-centered Circular Manufacturing[END_REF] is based on the circular manufacturing but sees maintenance as a core activity in a system. In this vision, manufacturing, reuse and maintenance merge to create a cohesive flow forward/feedback loop and to increase the product longevity.

Our works are in the same logic as [START_REF] Takata | Maintenance-centered Circular Manufacturing[END_REF], but our point of view is directly inspired by nature. In this sense, the purpose of this paper is to highlight the major challenges brought by the paradigm of regeneration to maintenance. For this, section 2 develops the concept of regeneration in nature and the industrial world. Then, section 3 describes the industrial processes emerging from this concept and their impact on maintenance. Section 4 focuses on the regeneration influence on maintenance, and vice versa. Finally, section 5 concludes this paper.

PARADIGM OF REGENERATION

The biology literature defines "regenerate" as a process "to grow again". The term of regeneration has used in two different contexts. The first is the regeneration of organic tissue. Indeed, we talk about this regeneration when, for example, the lizard's tail grows back, or when your skin heals. Scientists have studied this phenomenon to create a selfhealing rubber, metal and concrete. The second context corresponds to the trophic organization, namely, the food web. This organization is circular and reuse all that is produced by each entity. We are interested on this regeneration context and we apply it to the organization of the industrial world. The concept of waste does not exist in nature. For it, all is food [START_REF] Kormondy | Concepts of Ecology[END_REF]; each entity (living being) feeds another entity (Fig. 1). Indeed, vegetables eat solar energy and mineral salts. Herbivores eat vegetables and carnivores eat herbivores and other carnivores. Each entity produces organic matters that are eaten by decomposers (fungi, worms, bacteria). These decomposers produce mineral salts to feed vegetables, and the cycle continues. To summarize, all resources produced by and in nature are regenerated, except solar energy. The essential entities are decomposers, without them, nature would be a big rubbish dump and would not be circular. Therefore, to become circular the industrial world must add the notion of decomposer.

Trophic organization

Towards a regeneration of industrial world

Contrary to the trophic organization, the industrial organization stays linear (Fig. 2). Indeed, primary industries extract natural resources and supply secondary industries. These industries transform natural resources in goods and supply human activities. These industries and activities produce wastes. However, no or very few industries (recycling process) exist to transform these wastes into resources. No recycled waste is very numerous and is landfilled. Therefore, if we want to meet sustainable requirements in circular economy, we must be inspired by the trophic organization [START_REF] Geng | Scavengers and decomposers in an eco-industrial park[END_REF]. We must add more processes, called "technical decomposers", to transform unusable items to add value to them. They create new companies and new business.

NEW FUNCTIONS OF RETIREMENT PROCESS

In the case of the regeneration paradigm, the functions of retirement process change and it becomes polyvalent (emergence of several sub-processes). From the Takata's works (Fig. 4), we propose to develop the emerging subprocesses that we call technical decomposers. Fig. 4. "Concept of maintenance-centered circular manufacturing" [START_REF] Takata | Maintenance-centered Circular Manufacturing[END_REF] In this part, we consider that all equipment, subassemblies and components must be regenerated. [START_REF] Foundation | Report Version 1, 2 & 3 : Towards the Circular Economy[END_REF][START_REF] Govindan | Reverse logistics and closed-loop supply chain: A comprehensive review to explore the future[END_REF] have identified four technical decomposers:

Identification of technical decomposers

-Repair process: the repair is a physical action taken to restore the required function of a faulty item. This process makes a transformation of time, and preservation of item functionalities; -Reuse process: the reuse is defined as an exhaustion of the items lifetime through multiple users or multiple products, to reduce the number of items needed to satisfy the requirements of a certain users groups for a certain period [START_REF] Takata | Maintenance-centered Circular Manufacturing[END_REF]. This process makes a transformation of time and space, and preserves the items functionalities -Remanufacturing process: the remanufacturing term is not clearly defined and includes many processes as disassembly, cleaning, control and reassembly [START_REF] Barquet | An integrated approach to remanufacturing: model of a remanufacturing system[END_REF]. In our works, we limit this concept as disassembly process. It makes a transformation of time, space and shape, and does not preserve the item functionalities. By contrast, the functionalities of its subassemblies are preserved. -Recycling process: The recycling term includes the material and energy recovery. Making energy recovery does not allow regenerating an item indefinitely; there is loss of materials. We choice the term of destructuring to represent the material recovery by grinding. This process makes a transformation of time, space and shape, and does not preserve any item functionalities. Finally, in this part, we consider the repair/adjustment, reuse, disassembly and destructuring processes as technical decomposers (Fig. 5).

Evolution in the technical decomposers

Depending on the incoming item type in a decomposer, possible transformations and outgoing items are not the same. Indeed, the point of view is very important. An equipment, a subassembly and a component will not have the same decomposers and transformations (Fig. 5 and Tab. 1).

Tab. 1. Legend of Fig. 5 Fig. 5. Maintenance management and decomposition process Firstly, we focus on the equipment. The equipment can be repaired, reused, disassembled or destructured:

-If it is repaired, then the repair process transforms a faulty equipment (E) in an equipment meeting operational requirements (Er). For this, the process needs subassemblies (SA*) and components (C*) to replace items to change. Therefore, it produces also subassemblies (SA') and components (C') to regenerate; -If it is reused, then the reuse process can sell a faulty or repaired equipment; -If it is disassembled, then the disassembly process transforms an equipment in subassemblies and/or components; -If it is destructured, then the destructuring process transforms an equipment in a secondary raw material (SRM). Next, we focus on the subassembly. Although technical decomposers are same as the equipment, the subassembly is considered as a spare part after repair and reuse processes. Finally, we focus on the component. The component can be repaired, reused and destructured. A component cannot be disassembled because it is the lowest item in the parts list. So:

-If it is repaired, then the repair process transforms a faulty component (C) in a component meeting operational conditions (C'). For this, the process needs raw material (RW); -If it is reused, then the reuse process can sell a faulty or repaired component; -If it is destructured, then the destructuring process transforms a component in a secondary raw material.

Currently, during the operation phase, maintenance monitors an equipment, decides when and how to maintain it, and when to stop its use, and landfill it. In the regeneration paradigm, landfill is not a solution, so maintenance must choice among the technical decomposers.

Importance of maintenance in the decomposition

In Fig. 5, maintenance has two roles: decomposer and manager. Indeed, by definition, maintenance is able to repair an item and, in the same time, to manage it. We have seen above the repair decomposer, so in this section, we focus on the management process.

Here, maintenance is the first step of decomposition. If it estimates that an item must be regenerated, it stop its use, and orientates it towards a decomposer. As a result, maintenance must determine which decomposer is able to regenerate an item. This involves knowing each decomposer, their objectives, their actions, their productions, and their requirements. Moreover, by knowing items outgoing of a decomposer, maintenance can optimize the regeneration and anticipate actions to take. For example, if a system is fixed and that subassemblies and components are changed, then maintenance knows that should also regenerate these items. On the other hand, that allows determining which future purchaser can buy the regenerated item, and by extension, identifying purchaser requirements. These requirements affect directly decomposers processes and theirs actions, and by consequently, affect maintenance. Maintenance must consider these requirements, in addition to decomposers requirements. Thus, it can stop the use of an item when it respects again all requirements and increasing probabilities of sale.

Finally, through the regeneration paradigm, this process becomes one of the most important processes in the industry world. However, even if, maintenance is the cornerstone of the decomposition, it must adapt to meet the regeneration requirements.

MAINTENANCE AND REGENERATION

Maintenance and regeneration are closely related.

Maintenance allows optimizing decomposition possibilities, and so, serves interests of the regeneration. Whereas, the regeneration allows creating new business skills and new jobs for maintenance.

Regeneration influence on maintenance

In (AFNOR, 2010), maintenance is "the combination of all technical, administrative and management actions during the lifecycle of an item intended it in, or restore it to, a state in which it can perform the required function." With the paradigm of regeneration, this definition becomes "the combination of … the required function and a state in which it can be regenerated by at least a technical decomposer." Thus, this new role raises the following research question: "How to preserve the potential of regeneration of an item?"

Regeneration Failures

In this new approach, new failures emerge, namely, regeneration failures. We define theses new failures as the fact that an item cannot meet decomposer requirements, and consequently, remove regeneration possibilities. These failures are difficult to manage because they are easily affected by technical failures. Indeed, if a technical failure occurs in an equipment, maintenance must be careful that its actions do not affect the potential of regeneration of the equipment. For example, a technician removes a faulty engine and replaces it by a new engine. This engine has the same properties as the original engine. However, the system capacities and properties are not the same as their origins, because the system is already deteriorated. Then, if the system operates based on the engine capacities, it is possible that it is unable meet the engine requirements. It could wear out faster or break. Consequently, the regeneration possibilities of the system and their components could be reduced, and the regeneration potential of the system would be negatively modified. Thus, maintenance must manage two types of failures (technical and regeneration), and monitor that one type will not affect another (more particularly in the case of a technical failure). Consequently, it must put into question the current corrective action plan.

Monitoring and decision-making

To detect regeneration failures, maintenance must monitor new elements and new performances, or monitor elements by other means. For example, to regenerate an equipment with the "disassembly" decomposer, maintenance must monitor interfaces, i.e., its subcomponents must always be separable. Indeed, if a component melts and merges with another component, then they cannot separate and regenerate by the disassembly decomposer. Consequently, merged components are considered as a single component.

Consider another example; a future purchaser accepts to buy a second-hand system if and only if the system is able to produce 40 pieces/hour. Therefore, if the current user wants to sell his system, he should not exceed this threshold. For this, the user must monitor production performances and stop the system use when the threshold is attained. Others requirements could have been added, as the energy consumption, the energy efficiency [START_REF] Godichaud | Sustainable management of end-of-life systems[END_REF], the compatibility with another system, etc. Moreover, current maintenance monitors items and decides when and how to maintain them, and when to stop their use and landfill them. In the regeneration paradigm, maintenance skills can be used to choice among decomposers. Indeed, if possibilities of retirement increase, then possibilities of "endof-life" increase too. Maintenance must not only decide to repair or not an item but also decide to reuse, to disassemble or to destructure it. Consequently, required skills to maintain the potential of regeneration are very various; maintenance does not neglect any decomposers. Finally, the role of monitoring and decision-making processes are especially complex. They must combine regeneration failures with technical failures; as a result, there is an increase of variables and retirement possibilities to manage. Furthermore, many requirements appears with the regeneration. These processes must be monitored in real time to meet requirements of sustainable development, decomposers and futures purchasers. Moreover, monitoring should also monitor different requirements depending on each possible decomposer. Thus, the regeneration paradigm adds new business skills and jobs. Regeneration maintenance needs experts in regeneration monitoring and decision-making.

Anticipation

A solution to answer the preservation of regeneration problem is not waiting for the end-of-life of the equipment. In that, for example, [START_REF] Song | Proactive remanufacturing timing determination method based on residual strength[END_REF] study the most critical component of a system to determine when to stop the exploitation of this system and remanufacturing it. In other words, the authors propose to make proactive remanufacturing. Whereas [START_REF] Hu | Remanufacturing Decision Based on RUL Assessment[END_REF] have proposed a methodology to choice a remanufacturing process according to the system's RUL. Another approach has used the capacity of maintenance process to predict a failure. [START_REF] Iung | Advanced Maintenance Services for Promoting Sustainability[END_REF] propose to use the prognostic to study flows properties, and thus securing the respect of sustainability requirements. By anticipating a regeneration failure, we can regenerate an equipment by another decomposer than the destructuring decomposer. The prediction can be made based on the decomposer selected by the user or based on available decomposers. For this, we can base on processes used by the PHM (Prognostics and Health Management) community, and adapt them to regeneration failures. Indeed, the prognostic is defined as a process that "analyses the symptoms of faults to predict future condition and residual life within design parameters" [START_REF] Afnor | NF ISO 13381-1 : Condition monitoring and diagnostics of machines -Prognostics[END_REF]. With the regeneration view, the prognostic process must also analyze the regeneration requirements to predict when the use of an item must be stopped to meet decomposer requirements. Thus, making prognostic will not be only limited to the calculation of a conventional RUL, but will also include the estimation of time before which a studied item cannot be regenerated by a decomposer. However, it is a complex estimation because this prognostic calls in different parameters that evolve with the decomposer. Requirements are not the same if the decomposer is the reuse or destructuring process.

Finally, the regeneration paradigm sets new objectives for maintenance. It becomes the regeneration coordinator and is even more omniscient than today. It also becomes a sector that creates jobs and business skills.

Maintenance influence on regeneration

To manage all this, we need a manager, and we propose maintenance as a regeneration manager, who meets the requirements of many stakeholders. Thus, the regeneration exists only if a market exists, i.e., a regenerated item must have the same properties than a new item. To maximize the resale possibilities, we must meet the purchaser requirements. Other regeneration actors are the technical decomposers. As already pointed out, they regenerate an unusable item to adding value to it. Consequently, an unusable item must meet the decomposer requirements. For example, by definition, the disassembly decomposer cannot regenerate only an item, which can be dismantled. Next, we consider biosphere in the stakeholders. This stakeholder is the basis of the regeneration. Indeed, the regeneration paradigm exists only to preserve the human life, which depends on the biosphere. Therefore, the biosphere must be also preserved. We do not neglect its requirements.

All these requirements can be functional, operational, interactional and constraining. One of the goals of maintenance is to maintain an item in operational conditions. In other words, maintenance is able to meet and to preserve functional and operational requirements. It monitors also interactions between some critical items. The regeneration requirements add and/or modify requirement repository, but as they are the same types of current requirements. Therefore, maintenance is able to manage these requirements.

On the other hand, maintenance uses methods to preserve an item in operational condition, as for example, Reliability Centered Maintenance, PHM and Total Productive Maintenance. Obviously, these methods can be used and adapted to meet the regeneration requirements. The research of PHM community can be also used to predict the time when an item will not be able to regenerate by a decomposer, and when to stop its use to decompose it. Likewise, Computerized Maintenance Management System could be also adapted to help maintenance to manage at best regeneration.

Besides methods and tools used in maintenance, the Product Lifecycle Management (PLM) can help to improve the regeneration. This management allows managing all data relating to the design, production, support and retirement of products. In the case of the regeneration, PLM could be used to follow the different lives of a product, and the evolution of its regeneration requirements. For example, after each repairs, the data about the actions taken are saved. Then, when the user wants to sell his system, he sells in the same time the data about its use, its repairs, its consumption, etc. The purchaser will know all about his new acquisition. The same logic can be apply to the subassemblies and components removed during the maintenance operations.

Actually, maintenance wants to all about an equipment and their components: their RUL, their state, repairs made and their composition, etc. With these skills, it is able to monitor an item to regenerate it, to select the best decomposer for an item depending on this state. In other words, maintenance is the best candidate to maintain the potential of regeneration.

CONCLUSIONS

Nature is a good source of inspiration to move from linear economy to circular economy. Indeed, it allows highlighting missing elements to apply the principle of natural regeneration in the industrial world. In adding technical decomposers, the industrial regeneration becomes possible. This new concept creates new processes and new works, and more particularly, for decomposers. Decomposition processes transform wastes in resources and maintenance process becomes the manager of the regeneration paradigm. This process is the only process able to monitor, to diagnose, to predict and to provide a decision-making. Finally, the paradigm of regeneration adds a new job to maintenance. This process does not only maintain an equipment in operational condition but must also maintain its potential for regeneration. Maintenance allows optimizing the regeneration of an item, and consequently, meeting sustainable requirements.

Our future work will focus on the regeneration requirements.

In first, we must identify these requirements and determine their effects on requirements repository have to. Thus, we can determine and monitor the critical elements. We can regenerate them when they satisfy a maximum of stakeholder requirements depending on each decomposer. Next, we can determine an optimum point in terms of regeneration costs, between resale cost, decomposition cost, etc., and regeneration failures.

Fig. 1 .

 1 Fig. 1. Trophic organization

Fig. 2 .

 2 Fig. 2. Linear industrial organization