Sergio Chibbaro
email: chibbaro@ida.upmc.fr

Angelo Vulpiani

Compressibility, Laws of Nature, Initial Conditions and Complexity

Keywords: Chaos, Information theory, Complexity B

We critically analyse the point of view for which laws of nature are just a mean to compress data. Discussing some basic notions of dynamical systems and information theory, we show that the idea that the analysis of large amount of data by means of an algorithm of compression is equivalent to the knowledge one can have from scientific laws, is rather naive. In particular we discuss the subtle conceptual topic of the initial conditions of phenomena which are generally incompressible. Starting from this point, we argue that laws of nature represent more than a pure compression of data, and that the availability of large amount of data, in general, is not particularly useful to understand the behaviour of complex phenomena.

Introduction

It is not necessary to stress too much the fact that the external world is not just a jungle of irregular events. There is a quite clear evidence of our ability to understand (at least partially) the many regularities of our physical world. Then, it is quite natural to ponder about the origin of such a success. A very general question is: why is the physical world comprehensible? In particular one can wonder about the existence and the status of mathematical laws which allow us quantitative or qualitative predictions in agreement with experiments [START_REF] Sève | Penser avec Marx aujourd'hui: philosophie? La Dispute[END_REF][START_REF] Bailly | Mathématiques et sciences de la nature[END_REF].

In the past some scientists (and philosophers) stated that the aim of science is to organise in the most economical fashion the data collected from experiments. In this view, laws are just a very effective way to compress disparate data. Likely the most important champion of such a view of the science has been Mach [START_REF] Mach | On the Economical Nature of Physical Inquiry[END_REF][START_REF] Mach | The Science of Mechanics: A Critical and Historical Account of Its Development[END_REF]: The so-called descriptive sciences must chiefly remain content with reconstructing individual facts … But in sciences that are more highly developed, rules for the reconstruction of great numbers of facts may be embodied in a single expression.

Thus, instead of noting individual cases of light-refraction, we can mentally reconstruct all present and future cases, if we know that the incident ray, the refracted ray, and the perpendicular lie in the same plane and that sin α/ sin β = n. Here, instead of the numberless cases of refraction in different combinations of matter and under all different angles of incidence, we have simply to note the rule above stated and the values of n, which is much easier. The economical purpose is here unmistakable.

This point of view has been shared by many scientists of the positivism or neopositivism currents. Interestingly, such an approach has been recently reconsidered in the framework of algorithmic complexity [START_REF] Li | An Introduction to Kolmogorov Complexity and Its Applications[END_REF] by researchers without specific philosophical interests. For instance, Solomonoff, one of the fathers of the theory, considers (without any reference to Mach) a scientific law, and more generally a theory, as an algorithm for compressing the results of experiments, providing a mathematical formalisation of the idea of science as an economy of thought [START_REF] Solomonoff | A formal theory of inductive inference. Part I-II[END_REF]: The laws of science that have been discovered can be viewed as summaries of large amounts of empirical data about the universe. In the present context, each such law can be transformed into a method of compactly coding the empirical data that gave rise to the law. We can cite other similar opinions, e.g. The existence of regularities may be expressed by saying that the world is algorithmically compressible. Given some data set, the job of the scientist is to find a suitable compression, which expresses the causal linkages involved. For example, the positions of the planets in the solar system over some interval constitute a compressible data set, because Newton's laws may be used to link these positions at all times to the positions (and velocities) at some initial time. In this case Newtons laws supply the necessary algorithm to achieve the compression [START_REF] Davies | Why is the physical world so comprehensible[END_REF]. The intelligibility of the world amounts to the fact that we find it to be algorithmically compressible. We can replace sequences of facts and observational data by abbreviated statements which contain the same information content. These abbreviations we often call laws of Nature. If the world were not algorithmically compressible, then there would exist no simple laws of Nature [START_REF] Barrow | New Theories of Everything[END_REF].

As an interesting exception to the idea of science as economy of thought, we may recall Born [START_REF] Born | Natural Philosophy of Cause and Chance[END_REF] who ironically noted:

if we want to economise thinking, the best way would be to stop thinking at all, and then the expression economy of thinking may have an appeal to engineers or others interested in practical applications, but hardly to those who enjoy thinking for no other purpose than clarify a problem.

In our opinion the idea of economy of thought removes all objectivity to the scientific laws and mathematical constructions. For reasons which we do not discuss here, this approach has gained much interest and broad success in the last decades under the vaguely-defined concept of complexity. The word "complexity" has become rather a "logo" for mainstream analysis; like in the 30's to be "modern" was mandatory, to be "complex" seems required to be fashionable today [START_REF] Sokal | Postmodern Intellectuals[END_REF]. While we can agree that the arising of nonlinear physics and mathematics since the pioneering works of Poincaré has represented a major change in science, perhaps even a change of paradigm, the recent insistence on the "complex" often appears preposterous [START_REF] Cencini | Chaos. World Scientific[END_REF]. Loosely speaking, complexity studies share the idea to apply the same tools, mainly from dynamical systems and statistical mechanics, to a very large spectrum of phenomena, from social and human sciences to astrophysics, regardless of the specific content of each problem. Therefore it is implicitly assumed that the laws underlying these phenomena are not important. More recently, this point of view has become extremely radical with the "big data" philosophy, which presents many conceptual and technical problems [START_REF] Cencini | Chaos. World Scientific[END_REF][START_REF] Coveney | Big data need big theory too[END_REF][START_REF] Crutchfield | The dreams of theory[END_REF]. In that framework, laws should be supplanted by the statistical analysis of a large amount of data, again carelessly of any specificity. In this sense, science becomes a technical compression of data.

The point of view of science as economy of thought seems to be in agreement with the idea that the central goal of science has been thought to be "prediction and control". As a relevant example of this opinion we can mention the von Neumann's belief that powerful computers and a clever use of numerical analysis would eventually lead to accurate forecasts, and even to the control of weather and climate:

The computer will enable us to divide the atmosphere at any moment into stable regions and unstable regions. Stable regions we can predict. Unstable regions we can control. 1We know now that the great von Neumann was wrong: he did not take into account the role of chaos. About half a century ago, thanks to the contribution of M.Hénon, E. Lorenz and B.V. Chirikov (to cite just some of the most eminent scientists in this field), we had the (re)discovery of deterministic chaos. This event was scientifically important, e.g. to clarify topics as the different possible origin of the statistical laws and the intrinsic practical limits of predictions. Yet one has to admit that the topic of "chaos" induced a certain confusion about concepts as determinism, predictability, stochastic laws and the understanding of a phenomenon in terms of compression; for instance, Davies [START_REF] Dyson | Birds and frogs[END_REF] writes there is a wide class of physical systems, the so-called chaotic ones, which are not algorithmically compressible. We will see that what is not compressible is the time sequence generated by chaotic systems, and this is due to the non-compressibility of a generic initial condition. The aim of the present paper is to clarify the real relevance of the ideas as "compression" and algorithmic complexity in science, in particular in physics. For such an aim, a detailed analysis of of laws, initial conditions and data is necessary. Section 2 is devoted to a general discussion on evolution laws, initial conditions and data. In Sect. 3 we treat in details the role of the initial conditions and the compressibility. The question of the relation between compression and laws of natures is discussed in Sect. [START_REF] Mach | The Science of Mechanics: A Critical and Historical Account of Its Development[END_REF]. In order to clarify the role of the algorithmic complexity in the research, two case studies are presented in Sect. 5. Some final remarks and considerations in Appendix section.

About the Laws and Data

The aim of this Section is to clarify the distinction between phenomena, evolution laws, initial conditions, and series of data; the level of the discussion will not be technical, the precise notion of complexity will be discussed in Sect. 3 and in the Appendix.

Natural phenomena, roughly speaking, can be divided in two large classes:

(a 1 ) simple, e.g. stable and predictable, an example is the pendulum; (a 2 ) complex, i.e. irregular and unpredictable, a paradigmatic example is turbulence in fluids.

The evolution laws can be:

(b 1 ) known; (b 2 ) unknown.
The initial conditions can be:

(c 1 ) simple; (c 2 ) complex.

Finally, the series of data generated by a certain phenomenon appear:

(d 1 ) regular; (d 2 ) irregular.

The Simplest Case: The Law Is Known

First, let us consider the case b 1 , which is, from a methodological point of view, the simplest one. The possibilities are the following:

a 1 + c 1 (S.1)
a 1 + c 2 (S.
2)

a 2 + c 1 (C.1)
a 2 + c 2 (C.2)
The cases S.1 and S.2 are quite clear: independently from initial conditions, the system will display a regular behaviour, thus

a 1 + c 1 → d 1 , and a 1 + c 2 → d 1 .
The case C.1 is not typical (i.e. rather rare); on the contrary the case C.2 must be considered generic, i.e. considering a large ensemble of natural phenomena, almost all of them will join this class. In turn, "almost all" indicates that the probability to find one behaving differently will be basically zero if we consider a large enough ensemble. Indeed the situation C.1 can be observed only under particular circumstances, most of natural phenomena are part of the category C.2. Why initial conditions are almost always complex will be explained in Sect. [START_REF] Mach | On the Economical Nature of Physical Inquiry[END_REF].

Considering now the case C.2, since the evolution law of the system is known, the irregularity in the outcome of the dynamics has to be necessarily hidden in the initial conditions. To better clarify this point, one can consider that any initial condition x 0 can be written, in a unique way, in terms of a binary sequence {i 1 , i 2 , . . .}. Such a sequence can be compressible (for instance periodic, or periodic after a certain initial part), corresponding to rational x 0 ; in such a case the dynamics generates a regular sequence, as in the C.1 case:

a 2 + c 1 → d 1 .
incompressible (e.g. aperiodic), in such a case the dynamics generates an irregular sequence, as in the C.2 case:

a 2 + c 2 → d 2 .

The Evolution Law Is Not Known

Of course only in few lucky situations (mainly in physics) we know the laws ruling a certain phenomena with a good precision. In ecology, biology and many other applied sciences, it is not possible to write down the equations describing a certain phenomenon on the basis of well established theoretical frameworks and it is unavoidable to use a combination of intuition and experimental data [START_REF] Chibbaro | Reductionism, Emergence and Levels of Reality[END_REF][START_REF]Time Series Prediction: Forecasting the Future and Understanding the Past[END_REF].

Often the importance of the concept of state of the system, i.e. in mathematical terms, the variables which describe the phenomenon under investigation is not enough stressed. The relevance of this aspect is usually underestimated; only in few cases, e.g. in mechanical systems, it is easy to identify the variables which describe a given phenomenon. On the contrary, in a generic case, there are serious difficulties; we can say that the main effort in building a theory on nontrivial phenomena concerns the identification of the appropriate variables. Such a difficulty is well known in physics, for instance in the context of statistical mechanics Onsager and Machlup, in their seminal work on fluctuations and irreversible processes [START_REF] Onsager | Fluctuations and irreversible processes[END_REF], stressed the problem with the caveat: how do you know you have taken enough variables, for it to be Markovian? 2 2 As paradigmatic example let us consider the Langevin equation

d 2 x dt 2 + γ dx dt = -ω 2 x + cη
In a similar way, Ma notes that [START_REF] Ma | Statistical Mechanics[END_REF]: the hidden worry of thermodynamics is: we do not know how many coordinates or forces are necessary to completely specify an equilibrium state.

Usually we have no definitive method for selecting the proper variables and only a deep theoretical understanding can suggest the "good ones".

Takens [START_REF] Takens | Detecting strange attractors in turbulence[END_REF] showed that from the study of a time series {u 1 , u 2 , . . . , u M }, where u is an observable, it is possible (if we know that the system is deterministic and is described by a finite dimensional vector) to determine a proper set of variables fully describing the system . Unfortunately, in practice, the method has rather severe limitations:

(a) it works only if we know a priori that the system is deterministic; (b) because of the finite size M of the time series, in practice the protocol fails if the dimension of the attractor is large enough (say more than 5 or 6).

Therefore this method cannot be used, apart from special cases (with a small dimension), to build a model for the evolution law from data [START_REF] Eckmann | Fundamental limitations for estimating dimensions and lyapunov exponents in dynamical systems[END_REF]. We will discuss in some details the difficulties of the method in Sect. 5.2.

Evolution Laws and Initial Conditions

Let us now consider a topic which should be carefully analyzed: the role of initial conditions, which are usually independent of the laws of nature. Such an important point had been already realised by Newton [START_REF] Newton | Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World[END_REF] who noted that all the planets move in the same direction on concentric orbits, while the comets move in eccentric orbits, concluding that such a property of the solar system cannot be a mere coincidence, but it is due to the initial condition. Wigner considers the understanding of the distinction between laws and initial conditions as the most important contribution that Newton made to science, even more important than the laws of gravitation and dynamics [START_REF] Wigner | Events, laws of nature and invariant principles[END_REF].

Two Examples of Deterministic Systems

Let us analyse the very different behaviour of two deterministic systems; we will see how the initial conditions can play a basic role.

Example A The pendulum of length L:

d 2 θ dt 2 = - g L sin θ . ( 1
)
From well known mathematical theorems on differential equations we know that:

Footnote 2 continued where η is a white noise, i.e. a Gaussian stochastic process with η = 0 and η(t)η(t ) = δ(tt ), and γ > 0. It is worth emphasising that the vector y = (x, dx/dt) is a Markov process, i.e. its stochastic evolution at t > 0 is determined only by y(0), on the contrary the scalar variable x is not a Markovian process, and thus its dynamics depends on its past history.

(a) the initial condition (θ (0), dθ(0)/dt) determines in a unique way the state of the system (θ (t), dθ(t)/dt) at any time t; (b) the motion is periodic, i.e. there exists a time T (depending on the physical parameters) such that

θ(t + T ), dθ(t + T ) dt = θ(t), dθ(t) dt ;
(c) the time evolution can be expressed in terms of a function of t and the initial conditions:

θ(t) = F t, θ(0), dθ(0) dt .
The function F can be written in an explicitly way if θ(0) and dθ(0)/dt are small, in such case T = 2π √ L/g:

θ(t) = θ(0) cos(ωt) + 1 ω dθ(0) dt sin(ωt) dθ(t) dt = -θ(0)ω sin(ωt) + dθ(0) dt cos(ωt)
where ω = 2π/T . In the generic case, F can be easily determined with the wished precision by numerical resolution. Example B The Bernoulli's shift:

x t+1 = 2x t mod 1 .
(

) 2
Where the operation mod 1 corresponds to take the fractional part of a number, e.g. 1.473 mod 1 = 0.473. It is possible to show that the Bernoulli's shift is chaotic: a small error in the initial conditions doubles at every step. Consider an initial condition x 0 in the interval [0, 1], it can be expressed by an infinite sequence of 0 and 1:

x 0 = a 1 2 + a 2 4 + • • • + a n 2 n + • • • (3)
where every a n takes either the value 0 or the value 1. The above binary notation allows us to determine the time evolution by means of a very simple rule: at every step, one has just to move the "binary point" of the binary expansion of x 0 by one position to the right and eliminate the integer part. For example, from

x 0 = 0.11010000101110101010101100 . . .

one has

x 1 = 0.1010000101110101010101100 . . . x 2 = 0.010000101110101010101100 . . .

x 3 = 0.10000101110101010101100 . . . and so on. In terms of the sequence{a 1 , a 2 , . . .} it becomes quite clear how crucially the temporal evolution depends on the initial condition. If the binary sequence associated to x 0 is not complex, e.g. x 0 is a rational number, then the sequence {x 0 , x 1 , . . .} will be regular; on the contrary if the binary sequence associated to x 0 is complex, the sequence {x 0 , x 1 , . . .} will remain complex.

Let us consider two initial conditions x (1) 0 and x (2) 0 such that |x (1) 0x (2) 0 | < 2 -M for some arbitrary (large) integer number M, this means that x (1) 0 and x [START_REF] Bailly | Mathématiques et sciences de la nature[END_REF] 0 have the first M binary digits identical, and they may differ only afterwards. The above discussion shows that the distance between the points increases rapidly: for t < M one has an exponential growth of the distance between the two trajectories

|x (1) t -x (2) t | ∼ |x (1) 0 -x (2) 0 | 2 t , (4)
As soon as t > M one can only conclude that |x

(1)

t -x (2) t | < 1.
We can say that our system is chaotic: even an arbitrarily small error in the initial conditions eventually dominates the solution of the system, making long-term prediction impossible.

From the above brief discussion, we see how in deterministic systems one can have the following possible cases (in decreasing order of predictability):

1. Explicit possibility to determine the future (pendulum in the limit of small oscillations); 2. Good control of the prediction, without an explicit solution (the pendulum with large oscillations); 3. Chaos and practical impossibility of predictability (Bernoulli's shift).

About Initial Conditions and Compression

Let us consider again the dynamical system (2), which is chaotic, i.e. the distance between two trajectories initially very close, increases exponentially in time:

δ t = |x (1) t -x (2) t | ∼ δ 0 e λt , ( 5
)
where λ, called Lyapunov exponent, is positive. In the system (2), λ = ln 2, this means that a small error in the initial conditions doubles at every step. Suppose that x 0 is a real number in the interval [0, 1], it can be expressed by an infinite sequence of 0 and 1 as in [START_REF] Mach | On the Economical Nature of Physical Inquiry[END_REF]. We already saw that looking at the sequence {a 1 , a 2 , . . . , a n , . . .}, it becomes quite clear how crucially the temporal evolution depends on the initial condition.

Let us now make a brief digression on the notion of "complexity" of a binary sequence. Generally speaking, different types of sequences are possible, for example consider the following ones: 11111111111111 . . . [START_REF] Solomonoff | A formal theory of inductive inference. Part I-II[END_REF] 10101010101010 . . . [START_REF] Davies | Why is the physical world so comprehensible[END_REF] 00101000110100 . . . [START_REF] Barrow | New Theories of Everything[END_REF] It is quite natural to say that sequences ( 6) and ( 7) appear to be "ordered", whereas sequence (8) seems "complex". Why should one classify the sequences in this way?

In the case of ( 6) and ( 7) the knowledge of the first n values {a 1 , . . . , a n } appears to be sufficient to predict the following values {a n+1 , a n+2 , . . .}. This is not true for sequence [START_REF] Barrow | New Theories of Everything[END_REF], which seems to be generated by a stochastic, rather than deterministic rule. In this case, one could think that the sequence of 0 and 1 is generated tossing a coin, and writing 1 for heads and 0 for tails. One way to formalise this intuitive concept of complex behaviour is to associate it with the lack of a constructive rule; then the cases ( 6) and ( 7) are not complex because they can be generated by means of very simple rules. On a computer, for instance, ( 6) can be generated through a single statement:

WRITE 1 N TIMES
and similarly for [START_REF] Davies | Why is the physical world so comprehensible[END_REF]:

WRITE 10 N/2 TIMES
By contrast, (8) seems to require a program of the kind:

WRITE 0 WRITE 0 WRITE 1 WRITE 0 . . .
We can conclude that the sequences ( 6) and ( 7) can be considered "simple" because they can be obtained with a short computer code; on the contrary the length of computer code which generates ( 8) is proportional to the size of the sequence. The rationalization of the above remarks needs the introduction of a precise mathematical formalisation of the algorithmic complexity of a sequence [START_REF] Li | Inductive reasoning and Kolmogorov complexity[END_REF][START_REF] Li | Kolmogorov complexity arguments in combinatorics[END_REF][START_REF] Li | Kolmogorov complexity and its applications[END_REF], a brief introduction is given in the Appendix.

Do Scientific Laws Compress Empirical Data?

Surely it is fair to say that once a scientific law has been established one has a sort of compression, however such a conclusion deserves a careful analysis. In order to clarify such a topic we briefly discuss two examples of scientific laws, namely the Newton's equations for the classical mechanics and the Schrödinger equation.

Classical Mechanics and Astronomy

It is well know that from Newton's equations and the gravitation law, one can derive many important astronomical facts, for instance Kepler's laws.

On the other hand it is not completely correct to conclude that Newton's equations and the gravitation law are able to compress all the astronomical behaviors. After the seminal contribution of Poincaré, we know that a system of three bodies interacting with the gravitational force is usually chaotic [START_REF] Cencini | Chaos. World Scientific[END_REF]. Such a celebrated system is an example of the case (C.2).

In the following we will show how the presence of chaos implies the failure of the possibility to compress astronomical evolution. For sake of simplicity, instead of discussing such a difficult problem, we reconsider the system (2) which shares many features with the three body problem.

Let us analyse the problem of transmission to a friend, with accuracy , of a sequence x t 0 < t < T , generated by the rule [START_REF] Bailly | Mathématiques et sciences de la nature[END_REF]. At first glance, the problem seems quite simple: we could opt for transmitting x 0 and the rule (2), which costs a number of bits independent of T . Our friend would then be left with the task of generating the sequence x 1 , x 2 , . . . , x T . However, we must also choose the number of bits to which x 0 should be specified. From [START_REF] Mach | On the Economical Nature of Physical Inquiry[END_REF], the accuracy at time T requires accuracy δ 0 ∼ 2 -T for x 0 , hence the number of bits specifying x 0 grows with T . Again, we have to tackle the problem of the complexity of a sequence of symbols, {a 0 , a 1 , . . .}. The fact is that there are "simple" initial conditions, of the type ( 6) or [START_REF] Davies | Why is the physical world so comprehensible[END_REF], which can be specified by a number of instructions independent of the length of the sequence, but there are complex sequences as well.

We saw that the evolution law of ( 2) is nothing but a shift of the binary point of the sequence {a 1 , a 2 , . . . , }. Therefore we have that the evolution of x 0 is regular (e.g. periodic) if its sequence {a 1 , a 2 , . . . , } is not complex while it is irregular if {a 1 , a 2 , . . . , } cannot be compressed.

So we have that both in systems with regular behavior (the pendulum) and chaos (the Bernoulli's shift), it is straightforward to compress the evolution law. The difference between the two systems is in the output which is always regular in the pendulum, whereas in the Bernoulli's shift it can be regular or irregular depending on the initial condition.

The conclusion, somehow rather intuitive, is that in deterministic systems the details of the time evolution are well hidden in the initial condition which turns out to be typically complex. The complexity of initial conditions follows from an important mathematical result of Martin-Löf [START_REF] Martin-Löf | The definition of random sequences[END_REF] who showed that almost all infinite binary sequences, which express the real numbers in [0, 1], are complex. We do not enter into details of such a topic which involves rather subtle points related to the infinity and the Gödel theorem [START_REF] Calude | Classical, quantum and biological randomness as relative unpredictability[END_REF].

Coming back to astronomy, from the previous result we can conclude that, in presence of chaos, the knowledge of the basic laws ruling the time evolution of the astronomical bodies (i.e. Newton's laws and the gravitational force) is not enough to compress the complex time behaviour which is hidden in the (almost surely) complex initial condition.

Quantum Mechanics and Chemistry

Consider now the Schrödinger equation and its relation with chemistry; P.A.M. Dirac wrote the following celebrated sentence [START_REF] Dirac | Quantum mechanics of many-electron systems[END_REF]:

The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.

We are able to find the explicit solution of the Schrödinger equation for the hydrogen atom, and such a result has been the starting point to explain with high accuracy the phenomena observed in experimental spectroscopy. So at first glance it seems fair to say that the Schrödinger equation is able to compress the spectroscopic data.

On the other hand the relation between quantum mechanics and chemistry is rather controversial, and surely much weaker than the link between Newtonian mechanics and astronomy. For instance, in the latter case the theory was able to predict the existence of a previously unknown planet (Neptune). On the contrary, as far as we know, there is nothing similar in chemistry regarding the prediction of a new element solely on the basis of quantum mechanics.

Let us briefly discuss an issue which allows us to understand the severe limitation of the predictive power of quantum mechanics [START_REF] Chibbaro | Reductionism, Emergence and Levels of Reality[END_REF][START_REF] Primas | Chemistry, Quantum Mechanics and Reductionism[END_REF][START_REF] Scerri | Collected Papers on Philosophy of Chemistry[END_REF]. Consider the pyramidal molecules, e.g. ammonia (N H 3 ), phosphine (P N 3 ) or arsine (As H 3 ). The three isolated molecules are described by the same Hamiltonian with the unique numerical difference of a parameter (namely the masses of the different chemical species N , P and As). From an analysis of the quantum problem of the isolated molecule one obtains that the pyramidal molecules are delocalised, in clear disagreement with experiments which show that arsine is localised [START_REF] Primas | Chemistry, Quantum Mechanics and Reductionism[END_REF][START_REF] Jona-Lasinio | Spontaneous symmetry breaking-variations on a theme[END_REF]. The localization does not follow in a straightforward way from quantum mechanics but is a consequence of the interaction of single molecules with an external environment consisting of a very large number of components. The emergence of molecular structures can be understood only considering the interaction with an environment containing a large number of microscopic constituents [START_REF] Primas | Chemistry, Quantum Mechanics and Reductionism[END_REF][START_REF] Jona-Lasinio | Spontaneous symmetry breaking-variations on a theme[END_REF]. In this case, we could say that it is not fair to speak of compression because of the (very) complex boundary conditions to be supplied to Schrödinger equations.

About the Role of Algorithmic Complexity: Two Case Study

In order to clarify the role of chaos, initial conditions and algorithmic complexity in real scientific activity we discuss in some details two important topics. Namely we consider the features of fully developed turbulence (FDT) [START_REF] Frisch | Turbulence: The Legacy of AN Kolmogorov[END_REF], and how to reconstruct the evolution law from time series in the cases it is not possible to use some well established theory.

Turbulence

Turbulent flows, a paradigmatic case of complex system, are governed by the Navier-Stokes equations (NSE) which can be written in one line. In the incompressible case one has:

∂ t u + (u • ∇)u = - 1 ρ ∇ p + ν u + f, ∇ • u = 0,
where u is the velocity field, ρ the (constant) density, p the pressure, ν the kinematic viscosity and f an external force [START_REF] Frisch | Turbulence: The Legacy of AN Kolmogorov[END_REF][START_REF] Bohr | Dynamical Systems Approach to Turbulence[END_REF].

So, naively, one could conclude that, since we know the equation for the time evolution of the velocity field, somehow, the phenomenon of turbulence has been compressed, as well as most of fluid mechanics. The study of some specific aspects allows for the understanding of the precise meaning and limitation of such a conclusion. First, let us consider the problem of the initial conditions: of course in any experiment they are necessarily known with a limited precision. A rather severe limitation is due to the fact that in the limit of very large Reynolds numbers R e 3 , for a proper description of the turbulent velocity field it is necessary to consider a huge number of degrees of freedom: a rough estimate is N ∼ R 9/4 e [START_REF] Frisch | Turbulence: The Legacy of AN Kolmogorov[END_REF][START_REF] Bohr | Dynamical Systems Approach to Turbulence[END_REF]. Therefore for the typical values of R e in FDT (∼ 10 6 -10 9 ), because of the gigantic amount of data necessary to describe the involved degrees of freedom, we have an obvious impossibility to access to the initial conditions with the proper accuracy.

In addition at large R e the NSE are chaotic: the distance between two initially close initial conditions increases very fast. Therefore, as a consequence of the practical impossibility to access the initial conditions with high accuracy, and the presence of deterministic chaos, even with a very powerful computer and accurate numerical algorithms, it is not possible to perform a simulation of the NSE for a long term and compare the single-trajectory prediction with experimental results.

Because of the practical impossibility to compare the experimental results with the numerical computation of the field u(x, t), we cannot say that the NSE are able to compress the turbulent behaviors. Nevertheless there is a general consensus on the validity of the NSE for the FDT.

We can mention at least four items supporting the opinion that NSE are able to describe FDT, the agreement of the results observed in FDT and those obtained by the NSE for: (a) short time prediction of the velocity field; (b) long time prediction of averaged (e.g. spatially caorse grained) quantities; (c) the scaling laws, and more generally, the statistical features; (d) the qualitative and quantitative spatio-temporal features (e.g. large scale coherent structures).

A general discussion can be found in the literature [START_REF] Frisch | Turbulence: The Legacy of AN Kolmogorov[END_REF][START_REF] Bohr | Dynamical Systems Approach to Turbulence[END_REF].

When the Evolution Law Is Not Known

Let us note that the NSE have been derived on a theoretical basis using the Newton equations, assuming the hypothesis of the continuity of matter, and some thermody- 3 The Reynolds number

R e = U L ν ,
being U and L the typical velocity and length of the flow respectively, indicates the relevance of the non linear terms. At small R e we have a laminar flow, while the regime R e 1 is called fully developed turbulence. namic considerations. One can wonder about the possibility to obtain the NSE just looking directly at experimental data.

Since in the NSE one deals with fields (i.e. infinite dimensional quantities), it is natural to expect formidable difficulties. A less ambitious (but conceptual similar) task is to build models in finite dimension on the basis of experimental data [START_REF] Kantz | Nonlinear Time Series Analysis[END_REF]. Only for the sake of simplicity we assume the most favorable case, i.e. the time is discrete, the system is deterministic and we know that the state of the system at time k is a finite dimensional vector x k .

Consider the problem of the prediction from the available data, i.e. a long time sequence. A quite natural approach is to search for a past state similar to the present state of a given phenomenon of interest, then, looking at the sequence of events that followed the past state, one may infer by analogy the evolution that will follow the present state. In more precise terms, given a known sequence of "analogues", i.e. of past states x 1 , . . . , x M which resemble each other closely in pairs, so that |x k -x M | < with reasonably small, one makes the approximate prediction: [START_REF] Cencini | Chaos. World Scientific[END_REF].

x M+1 = x k+1 if x k is an analogue of x M
In the case the above protocol can be used, one may then proceed to build a model of the phenomenon, i.e. to determine a function f(x) such that the sequence of states is well approximated by the dynamical system

x k+1 = f(x k ). ( 9
)
The application of this method requires knowledge of at least one analogue. It is possible to state4 that such knowledge requires sufficiently long sequences, at least of duration of order T R ∼ (L/ ) D , where L is the typical length scale of the system, and D is the dimension of the attractor 5 . The exponential growth of T R as a function of D has a severe impact on our ability to make predictions, and the building of a model for the evolution law [START_REF] Born | Natural Philosophy of Cause and Chance[END_REF], solely relying on previously acquired data. One can say that D larger than 6 renders the approach described here useless, because it makes it practically impossible to observe the "same" state twice, i.e. within an acceptable accuracy .

As already stressed in Sect. 2.2, the state of the system, i.e. the variables which describe the phenomenon under investigation, is typically not known. Therefore an unavoidable technical aspect is the determination of the proper state of the system from the study of a time series {u 1 , u 2 , . . . , u M }, where u is an observable. The most relevant result for such a problem is due to Takens who has been able to show that, at least from a mathematical point of view, it is possible (if we know that the system is deterministic, described by a finite dimensional vector, and M is arbitrarily large) to determine a proper state-variable X. In a nutshell: there is a finite integer m such that the delay coordinate vector (of dimension m)

y (m) k = (u k , u k-1 , .., u k-m+1 ) (10)
can faithfully reconstruct the properties of the underlying dynamics. 6Of course the practical limitation due to the exponential increasing of T R as a function of D, is present also in the Takens's method; therefore we have rather severe practical limitations [START_REF] Cencini | Chaos. World Scientific[END_REF]. Indeed, the conceptual idea behind all these inductive approaches is always to try a reconstruction of the relevant phase-space, which, at a resolution level , has roughly a volume of (L/ ) D . To be explicit, and stress the limit of the method, consider a system ruled by a deterministic law, for which the dimension of the attractor is D, and we know a time series {u 1 , u 2 , . . . , u M } of an observable; the method of Takens allows to find (an approximation of ) the evolution law only if M is larger compared with A D . The value of A depends on the wished accuracy; just to give an idea let us assume A = 100, corresponding to just a fair accuracy, for D = 6, 7, and 8 we have A D = 10 12 , 10 14 and 10 16 respectively. Therefore also in the case we know that the system is ruled by a deterministic law, such a knowledge does not imply the actual possibility to perform an explicit compression.

Discussion

McAllister [START_REF] Mcallister | Algorithmic randomness in empirical data[END_REF] has observed that empirical data sets are algorithmically incompressible, concluding that the task of scientific laws and theories does not consist in compressing empirical data. We share such an opinion on the incompressibility of generic empirical data, even though his argument is maybe too sharp. As previously discussed in the context of chaotic deterministic systems (e.g. the Bernoulli's shift), the typical output is incompressible and, from a mathematical point of view, such a result is a consequence of the important result obtained by Martin-Löf [START_REF] Martin-Löf | The definition of random sequences[END_REF]: almost all the initial conditions correspond to incompressible sequences.

Regarding the opinion that scientific laws constitute a compression of empirical data, McAllister claims that no scientist has ever made such a statement. We do not enter into the historical aspects. However we want to discuss the following example: consider a series of light-refraction experiments, in which {α 1 , . . . , α N } are the angles of the incident rays, and {β 1 , . . . , β N } the angles of the refracted rays. The sequences {α 1 , . . . , α N } and {β 1 , . . . , β N } may or may not be compressed. This is a frozen accident which depends on the protocol followed by the scientist while preparing the experiment, for instance, in the case of the protocol α n+1 = α n + δ, the sequences can be compressed, on the contrary if each α n is selected according to a random rule, the sequences are not compressible. However, once the values {α 1 , . . . , α N } are known, the sequence {β 1 , . . . , β N } is simply determined by the Snell's law: sin α/ sin β = n, and this is a genuine form of compression. Therefore the example about the Snell's laws, which is often cited (likely because mentioned by Mach), is not particularly deep.

A less trivial instance concerns the Navier-Stokes equation for fluids. We saw how in such a chaotic systems, although the time sequences are not compressible, the NSE have a predictive power, in the sense that they are able to generate results in good qualitative and quantitative agreement with the experiments.

The claim that the world is comprehensible because it is algorithmically compressible is, in our opinion, a truism, which is equivalent to saying that laws of nature exist. We note that the actual possibility to understand the world arises mainly from a series of lucky facts, in particular:

• Typically physical laws obey spatially and temporally local rules, i.e. a given phenomenon is not affected too much by events which are distant in time and/or in space. Practically the main laws of physics, like the equations of Maxwell, Schrödinger, Newton etc., obey the locality assumption and are described by differential equations. • Despite the enormous complexity and the intricate interconnections of different phenomena, often there is a scale separation which allows us for a description in terms of effective theories of the different levels on which reality may be considered.

A celebrated example of an effective theory coming from the use of the separation of scales which characterises the microscopic, and the macroscopic realms, is the Langevin equation describing the Brownian motion. On the other hand, if the laws are not known and we have just the possibility to study time series, the scenario is quite pessimistic. If the effective dimensionality is (relatively) large, even in the most simple case of deterministic system, it is not possible to find the evolution laws and therefore to perform an explicit compression [START_REF] Ruelle | Deterministic chaos: the science and the fiction[END_REF]. Therefore the idea law = possibility of compression of data, must be (re)considered with many caveats. Perhaps we live in a "big data" era, but actually not big enough to model complex phenomena, without the help of some theory.

We stress again that disregarding the distinction between initial conditions and laws of nature can lead to great confusion. In the Introduction we cited Davies [START_REF] Davies | Why is the physical world so comprehensible[END_REF] who claims that chaotic systems are not algorithmically compressible. The discussion in Sect. 3 shows how chaotic systems can be trivially compressible (in the sense that it is easy to write down the evolution laws, as, e.g., for the Bernoulli's shift or the NSE). What can be not compressible is the output and this is related to the complexity of the sequence associated to the initial condition.

McAllister after an analysis of the relevance of compressibility in science concludes [START_REF] Mcallister | Algorithmic randomness in empirical data[END_REF]:

In sum, a scientific law or theory provides an algorithmic compression not of a data set in its entirety, as Mach, Solomonoff and others believed, but only of a regularity that constitutes a component of the data set and that the scientist picks out in the data. The remaining component of the data set, which is algorithmically incompressible, is regarded as noise in the sense of classical information theory.

We may agree with the previous sentence, if we keep our eyes open. To be able to distinguish a regularity (the law) from underlying noise, a proper resolution and, capitally, the separation of scale which permits to build coarse-graining description are necessary. Moreover, in our opinion, for the understanding of any nontrivial topic it is too naive to hope in an approach based on data and algorithms, and the study of the phenomenal framework is unavoidable [START_REF]Large Deviations in Physics: The Legacy of the Law of Large Numbers[END_REF]. Some remarks are in order. First, we would like to stress an issue which, although rather important, is often not discussed. The relevance of the scale resolution is closely linked with the proper effective variables which are able to describe the phenomenon under investigation. Let us consider a fluid which can be described, at microscopic level, in terms of its molecules; in such an approach the correct variables are the positions and momenta of the molecules. So we have a very accurate description containing a lot of informations. However, the microscopic level sometimes is not interesting. For instance in engineering (or geophysical) problems it is much more relevant to adopt an hydrodynamical description in terms of few fields (velocity, temperature and so on). Of course using such a macroscopic description one has a huge decreasing of the amount of information and an increasing of the possibility to compress data.

Then, few words on the qualitative aspects of science. Often qualitative results are considered less important than the quantitative ones. That is an unfair view, since although some results cannot be expressed in terms of numerical sequences, they can be interesting and rigorous. For instance it can be important to know that a phenomenon is periodic or some variables are bounded in a certain domain. We can mention the Lotka-Volterra like equations, for which sometimes one can show that the time behaviour is periodic, even though it is not possible to find the explicit solution. In a similar way, in some celestial mechanics problems, it is enough to be sure that the motion (e.g. of an asteroid) remains in a bounded region [START_REF] Cencini | Chaos. World Scientific[END_REF]. The previous qualitative results, although they cannot be formalized in terms of algorithmic compression (which involves sequences) are genuine forms of compression of information.

Finally, we would like to indicate an important example of a complex system which has been understood rather successfully thanks to a traditional theory/data approach: weather forecasting. This problem is related to the dynamics of atmosphere which is characterized by (i) huge degrees of freedom;(ii) dynamical interaction with a complex environment; (iii) chaos and many non-linear feedback mechanisms. Nevertheless, the modern developments of weather forecasting are based on the basic theory of such a system, fluid mechanics as pioneered by Richardson [START_REF] Chibbaro | Reductionism, Emergence and Levels of Reality[END_REF][START_REF] Vulpiani | Lewis Fry Richardson: scientist, visionary and pacifist[END_REF]. In particular, the use of the theory and the analysis of many phenomenological data have led to the recognition of different separated scales, which has been key for the development of a hierarchy of models adequate at different scales today solved by numerical integration. Those models are statistical and mainly qualitative, but rigorous to some extent thanks to the separation of scale. This example should point out that the path to the understanding of complex phenomena is a brilliant interplay of deep empirical analysis, creative theoretical developments and technical developments.

Cited by Dyson[START_REF] Dyson | Birds and frogs[END_REF].

This is the essence of Kac's lemma, a well know result of ergodic theory[START_REF] Cencini | Chaos. World Scientific[END_REF].

In conservative cases, e.g. Hamiltonian systems, D is the number of variables involved in the dynamics; if the system is dissipative, D can be a fractional number and is smaller than the dimension of the phase-space.

A rigorous result states: m ≥ 2[D] + 1; from heuristic arguments on can expect that m = [D] + 1 is enough[START_REF] Cencini | Chaos. World Scientific[END_REF].

Acknowledgements We thank M. Falcioni for his remarks and suggestions and we thank in a special way A. Decoene for her careful reading of the manuscript.

Appendix: The Algorithmic Complexity in a Nutshell

The rationalization of the idea of "randomness" needs the introduction of a precise mathematical formalisation of the complexity of a sequence.

This has been proposed independently in 1965 by Kolmogorov, Chaitin and Solomonoff, and refined by Martin-Löf [START_REF] Li | An Introduction to Kolmogorov Complexity and Its Applications[END_REF][START_REF] Martin-Löf | The definition of random sequences[END_REF].

Given the sequence a 1 , a 2 , . . . , a N , among all possible programs which generate this sequence one considers with the smallest number of instructions. Denoting by K (N ) the number of these instructions, the algorithmic complexity of the sequence is defined by

Therefore, if there is a simple rule that can be expressed by a few instructions, the complexity vanishes. If there is no explicit rule, which is not just the complete list of 0 and 1, the complexity is maximal, that is 1. Intermediate values of K between 0 and 1 correspond to situations with no obvious rules, but such that part of the information necessary to do a given step is contained in the previous steps.

To give an intuitive idea of the concept of complexity, let us consider a situation related to the transmission of messages [START_REF] Chaitin | Information, Randomness & Incompleteness: Papers on Algorithmic Information Theory[END_REF]: A friend on Mars needs the tables of logarithms. It is easy to send him the tables in binary language; this method is safe but would naturally be very expensive. It is cheaper to send the instructions necessary to implement the algorithm which computes logarithms: it is enough to specify few simple properties, e.g.

However, if the friend is not interested in mathematics, but rather in football or the lottery, and wants to be informed of the results of football matches or lottery draw, there is no way of compressing the information in terms of an algorithm whose repeated use produces the relevant information for the different events; the only option is the transmission of the entire information. To sum up: the cost of the transmission of the information contained in the algorithm of logarithms is independent of the number of logarithms one wishes to compute. On the contrary, the cost of the transmission of football or lottery results increases linearly with the number of events. One might think that the difference is that there are precise mathematical rules for logarithms, but not for football matches and lottery drawings, which are then classified as random events.


Mireille Chiroleu-Assouline
email: mireille.chiroleu-assouline@univ-paris1.fr.

Mouez Fodha

Double Dividend and Distribution of Welfare : Advanced Results and Empirical Considerations

Keywords:

The objective of this paper is to analyze the efficiency and equity consequences of the implementation of an environmental tax. Do the two purposes collide or can they be simultaneously achieved? Using results of the related literature, we show that precise conditions have to be met in order to achieve three goals: increase of the environmental quality, increase of the economic efficiency and improvement of the intergenerational equity. Are such theoretical conditions likely to occur? What are the rooms of manoeuvre for an environmental tax reform in the European countries? In each of them, the revenues of the existing environmental taxes are low in comparison to the weight of the labor taxes (which are highly distortionary). Using European data, we show that, among all the European countries, Belgium, France and, surprisingly, Sweden, exhibit the less green tax system and offer the easiest opportunity to use the environmental tax as a mean to alleviate the tax burden on labor.

1-Introduction

"A rising price on carbon emissions is the essential underlying support needed to make all other climate policies work. [...] A rising carbon price is essential to "decarbonize" the economy, i.e., to move the nation toward the era beyond fossil fuels. The most effective way to achieve this is a carbon tax (on oil, gas, and coal) at the well-head or port of entry. [...] The public will support the tax if it is returned to them [...]" This warm apology of the carbon tax is not due to an European economist, nor a European politician, but to an American scientist, James Hansen 1 who became in the United States one of the main advocates of the tax against the cap and trade approach. This quite surprising attitude corresponds to a larger movement in favor of the price instrument because of the relative failure observed until now of the emissions trading system as an instrument for the GHG regulation. In fact, this movement takes place in Europe too, where the Swedish presidency of the European Union (second semester of 2009) encouraged the other member countries to implement carbon taxes bearing on all sectors of activity which are not regulated by the emission quotas system.

As well as the cap and trade mechanism, the tax allows to achieve the environmental objectives while minimizing the global cost. One of the advantages of an environmental tax is that it provides public revenues which can be recycled. This is a reason why it can be preferred to subsidies or emission quotas. It has been argued that, as governments use these revenues to decrease other distortionary taxes, an environmental tax may lead to a double dividend, according to Goulder's definition, by improving the environmental quality and achieving a less distortionary tax system [START_REF] Goulder | Environmental Taxation and the "Double Dividend": A Reader's Guide[END_REF]). This can be a strong argument in favor of an increasingly green tax system. But one of its disadvantages is that, like any consumption tax, it often appears to be regressive, i.e. more harmful for the welfare of the poorest households than for the richest ones.

What would be the inequality consequences of the European Carbon Tax Project or of the Climate-Energy Contribution planned by the French government as from 2010? Hence, as Hansen suggests it, an environmental tax can hardly be considered without adequate revenue recycling in order to enhance the acceptance of the environmental policy. But the aim of such a recycling can therefore be twofold to reduce, or even annihilate the gross cost of the policy, as 1 Eminent climatologist, he is director of the NASA Goddard Institute of Space Studies. After the election of U.S. President Barak Obama in November 2008, J. Hansen sent him a letter to urge him to support a carbon tax. measured by the global welfare loss, or to compensate the generated vertical inequity.

The objectives of this paper are to analyze the efficiency and distributional consequences of the implementation of a carbon tax and to enlighten the links between the different purposes of the recycling of its revenue. Do the two purposes collide or can they be simultaneously achieved?

What are the rooms of manoeuvre in the European countries? Does the present situation not constitute an opportunity to undertake a tax reform of major scale?

This article suggests some answers to these questions by giving insights of the results in the double dividend theory. We compare the main contributions of the existing literature, focusing on our previous works, and we show that the conditions to obtain a successful environmental tax policy rely on the objectives and constraints of the policy: welfare equity, environmental quality, economic efficiency, and employment... Nevertheless, the Tinbergen rule cannot be set aside and achieving all these goals simultaneously seems very hard and require strong structural conditions to be met. These conditions concern, as usual, specific characteristics of the production functions' elasticities, agents' preferences and initial level of the tax rates. Secondly, in the lights of these theoretical results, this paper gives some empirical considerations in order to determine the opportunities for the European countries to adopt a global environmental tax policy. Using European data, we show that, among all the European countries, Belgium, France and, surprisingly, Sweden, exhibit the less green tax system and offer the easiest opportunity to use the environmental tax as a mean to alleviate the tax burden on labor.

The paper is organized as follows. Section 2 presents the concept of double dividend and studies the conditions of its achievement by the mean of a balanced "green" tax reform. Section 3 emphasizes the distributional effects of an environmental tax between generations, especially when there is involuntary unemployment. Section 4 enlightens the distributional effects between different categories of households and puts forward an appropriate policy mix to compensate them. Section 5 gives some empirical considerations using European data and the last section concludes.

2-In quest of the double dividend: theoretical results insights

Can an environmental tax reform be designed without negatively affecting the economic welfare?

This still open question has given rise to a huge literature on the double dividend. [START_REF] Terkla | The Efficiency Value of Effluent Tax Revenues[END_REF] first had the intuition that the recycling of the revenue of an environmental tax could reduce or even eliminate the gross cost of its implementation. [START_REF] Baumol | The Theory of Environmental Policy[END_REF], [START_REF] Pearce | The Role of Carbon Taxes in Adjusting To Global Warming[END_REF] and [START_REF] Oates | Pollution Charges As a Source of Public Revenues, Resources of the Future Discussion Paper[END_REF] suggest that these efficiency gains could be a powerful argument in favor of environmental taxation. After some prior refutations of the double dividend hypothesis (Bovenberg and de Mooij [1994] and their followers), [START_REF] Goulder | Environmental Taxation and the "Double Dividend": A Reader's Guide[END_REF] and [START_REF] Ligthart | The Macroeconomic Effects of Environmental Taxes : A Closer Look at the Feasibility of Win-Win Outcomes[END_REF] show that the existence of the double dividend essentially depends on the possibility of transferring the global tax burden from the wage earners to some fixed production factors or to other consumers, thus emphasizing the role of heterogeneity. Following this line, in Chiroleu-Assouline and Fodha [2005] and [2006], we study the existence conditions of a long term double dividend, taking into account the distinction between wage earners and retired consumers, by means of overlapping generations models.

A necessary condition to obtain a double dividend lies in the Laffer-efficiency2 of the initial tax system: the increase in the environmental tax rate has to allow a decrease of the labor tax rate, which is not always and everywhere guaranteed. It depends on the elasticity of each fiscal base to its tax rate and more precisely on the relative sizes of the fiscal base effects and of the tax rate effects. If one of the taxes (labor or environment) would not be Laffer efficient, the double dividend could not be obtained because, instead of reducing the existing tax distortions, the tax reform would enhance them.

Although these theoretical results are obtained in a static framework, conditions to obtain a double dividend are already hard to verify. Nonetheless, as global environmental policies concern the very long term, one cannot neglect the dynamic consequences of these policies. We will see that the take into account of intergenerational considerations, like advocated in the Bruntland's report when defining the sustainable development concept, increases the number of constraints.

Seeking for a third dividend: the long-term dynamic efficiency

Results concerning the dynamic properties of the double dividend objective can be found in Chiroleu-Assouline-Fodha [2006]. Our analyze of the double-dividend and welfare distribution issues takes place within a framework of overlapping generations models which seems more convenient for analyzing such environmental problems. In the absence of altruism, the behaviour of private agents is responsible for an intergenerational environmental externality. The solution which is usually advocated in order to internalize this kind of externality is the implementation of environmental taxation. But such taxes imply non-environmental welfare losses for the generations bearing them. It is the reason why generally the generation which has to implement the reform is the one which mainly bears its costs. The main contribution of Chiroleu-Assouline-Fodha [2006] is then to examine, with an appropriate demographic framework, whether a revenue-neutral increase in the pollution tax compensated by a change of the labor tax can yield a double dividend, by (i) achieving the long term environmental objective and (ii) improving the long term non environmental welfare.

More generally, the intertemporal double dividend could be defined as an improvement of the discounted intertemporal social welfare of all generations. It takes into account the welfare of the generations alive during the transition between initial and final steady-states. This welfare criterion leads to the modified golden rule which is preferred to the golden rule criterion because of both ethical and practical problems: the first generations would be sacrificed to the future generations while these ones will fully benefit from the environmental policy; moreover, decisions about future environmental improvements are to be made by present generations which support their economic costs. The arguments given by [START_REF] Goulder | An eye on the future[END_REF], for instance, in order to justify discounting apply here to justify the use of the modified golden rule.

Our results about the intertemporal double dividend are then linked to the general discounting debate. Even if some generations are made worse off along the transition path, any increase in the social discount factor rises the weight of the welfare of the infinite number of generations alive in the final steady-state equilibrium and (in case of a long term double dividend) increases the occurrence probability of an intertemporal double dividend. We focus then on the comparison of long term steady state equilibria because a necessary condition to obtain an intertemporal double dividend is that the generations on the new steady state equilibrium are made better off with respect of both environmental and non-environmental welfare. We show that, according to the discounted social welfare criterion, a double dividend can occur.

Nevertheless, the issue of acceptability by the present generations induces us to study the necessary conditions not only for environmental policy to be beneficial in the long term but also for not damaging the welfare of the generation which has to take the decision to implement the policy.

In this paper [START_REF] Chiroleu-Assouline | Double Dividend Hypothesis, Golden Rule and Welfare Distribution[END_REF]), the economy consists of two periods lived individuals being affected by the quality of the environment during their all life. We assume that overall consumption is polluting. The government is financing its spending with a labor tax and the pollution tax. We study the necessary conditions for environmental policy to be beneficial in the long term and we examine the compatibility conditions of both dividends: these necessary conditions depend on the capital intensity of the initial steady state equilibrium relative to the golden rule. We show that the fiscal change does not always harm the welfare of the present generations and that, under some peculiar assumptions; it is possible to obtain both a long term double dividend and the respect of short term intergenerational equity.

More precisely, the existence conditions of a double dividend rely on the initial per capita capital stock, on the initial tax rates and on the intertemporal elasticity of substitution. Our first conclusion is that, whatever the initial capital intensity of the economy, the double dividend cannot be obtained if the intertemporal elasticity of substitution is equal or greater than unity. Furthermore, if this elasticity is less than unity, the existence of the double dividend depends on the position of the economy relative to the golden rule and to the modified golden rule.

In case of over-accumulation of capital, the double dividend cannot occur. Indeed the consumption of the young generation should increase while the one of the old one should decrease and both taxes should be Laffer-efficient. In this last case, the environmental fiscal reform may be justified from an ecological point of view, when the initial pollution level is greater than the optimal one, but not from an economic point of view as the research of a double dividend gets the economy farther from the social optimum.

In case of under-accumulation of capital, the existence of a double dividend needs a reduction of the consumption of the old generation and an increase of the young one and it can occur only if one of the taxes is Laffer-inefficient and if the initial capital stock is greater than the one defined by the modified golden rule.

The analysis of the distributional effects between younger and older coexisting at a given date shows that the environmental fiscal reform always induces a non-environmental welfare loss for one of the existing generations: the young one when the reform bears ex post on wages, the old one when it bears on capital. For nobody to be harmed, the first dividend needs to be high enough.

3-Distributional intergenerational effects in case of involuntary unemployment

The existing literature before the clarification paper by [START_REF] Goulder | Environmental Taxation and the "Double Dividend": A Reader's Guide[END_REF] usually considered many kinds of second dividend: rise in employment, improved efficiency, improved distributional equity (the three kinds of dividend distinguished by [START_REF] Ekins | On the Dividends from Environmental Taxation[END_REF]) or rise in private profits, cut in the cost of public funds, and so on. But all these dividends, except the distributional one, are part of the one and only convenient criterion that refers to economic efficiency. Nevertheless, an employment dividend can also be defined, which plays in favour of the efficiency dividend without coinciding with it.

Moreover, beside potential efficiency properties, environmental decisions have an impact on the welfare of both current and future generations, as explained before. These intergenerational issues on environmental externalities or on taxation have been quite widely studied in the economic literature. In an overlapping generations framework, John et al. [1995] examine the effect of an environmental tax whose revenue is financing a public pollution abatement activity. [START_REF] Fisher | Pollution and Economic Growth[END_REF], using an endogenous growth model with pollution, derive the conditions for a pollution tax not to slow economic growth. More generally, the literature on the double dividend issue leads to some clear results [START_REF] Bosello | The Double Dividend Issue: Modelling Strategies and Empirical Findings[END_REF]): when the economy is made of one productive sector, using only one productive factor (labor), and one representative consumer, the strong version of the double dividend is rejected. Instead, when there are several productive factors and/or several consumer groups, the double dividend can be obtained but at the expense of equity (Bovenberg and van der Ploeg [1995], [START_REF] Proost | The Double Dividend and the Role of Inequality Aversion and Macroeconomic Regimes[END_REF], Bovenberg and Heijdra [1998]). For instance, Bovenberg and Heijdra [1998] examine the effects of a green tax on polluting capital when the tax revenue is redistributed by lump-sum intergenerational transfers and find that this tax benefits the younger generation but harms the older ones. This property might question the relevance of the fiscal reform if it harms some generations. Nevertheless, any environmental tax is based on the equity and intergenerational solidarity principle: it aims to give to the future generations the same environmental amenities as to the present generations. But such a fiscal reform will only be acceptable if it improves the global welfare of all generations, the present like the future ones.

All these papers conclude that environmental taxation implies such a welfare loss for older generations that its implementation cannot be wished: one of the generations which would decide it would also bear the heaviest burden.

We study the existence conditions of a double dividend (according to Goulder's definition), but also the role of the employment dividend and the distributional equity issue in Chiroleu-Assouline and Fodha [2005]. The distributional concern can be viewed between different classes of households (employed and unemployed) or between different generations. This paper integrates both the efficiency and intergenerational distributional aspects of environmental taxes. Under the assumption of involuntary unemployment, we examine whether a revenue-neutral increase in the pollution tax compensated by a change of the labor tax can yield a double dividend. This general framework can be related to those of Bovenberg and Heijdra [1998] and [START_REF] Bovenberg | Optimal Taxation, Public Goods and Environmental Policy with Involuntary Unemployment[END_REF] but differs from them in several ways. Bovenberg and Heijdra [1998] do not explicitly address the double dividend issue but investigate whether a higher pollution tax can be Pareto welfare improving by benefiting all generations. In their paper, pollution is due to capital utilization while we assume that the fiscal base of the environmental tax is consumption (which causes pollution) rather than capital; their paper focus mainly on optimal capital taxation which allows them to pay little attention to the employment issue. But we show that the existence of unemployment adds another way for obtaining a double dividend. Lastly, we assume that involuntary unemployment is caused by an exogenous minimum wage rate, like [START_REF] Bovenberg | Optimal Taxation, Public Goods and Environmental Policy with Involuntary Unemployment[END_REF] (who do not consider more than one generation).

In Chiroleu-Assouline and Fodha [2005], the economy consists of two periods lived individuals working and consuming when young and consuming and being affected by the quality of the environment only when being old. The government is financing its spending (public investment and unemployment benefits) with a labor tax and a pollution tax. We characterize the specification of the fiscal reform when the revenue of the pollution tax is recycled by a variation of the rate of social contributions instead of by lump-sum transfer.

We show that the fiscal change does not always harm the welfare of the younger generation and, under certain assumptions about agents' preferences, it is possible to obtain both a double dividend and the respect of intergenerational equity. We show finally that, under some conditions about the variations of the unemployment rate and of the interest rate, an inter-generations distributional dividend can also be obtained. More precisely, the employment dividend is warranted and, under some conditions, a double dividend can be obtained. It depends on the relative magnitude of the effects on the income, on the interest rate and on the fiscal base of the pollution tax. Indeed, if the environmental enforcement of the fiscal system yields a decrease of the labor tax rate, the environmental dividend is more likely to occur but the second and third dividends are less likely to occur that the fall in unemployment and the rise in interest rate are lower.

4-In quest of fairness and acceptability by heterogeneous households

One of the most publicly debated issues about environmental regulation concerns its effects on the poorest agents and the fairness of such a policy. How could these detrimental consequences be smoothed or even eliminated?

Environmental policies are indeed an issue in which income class plays a significant role. First, the poor and the rich seem to assign different degrees of priority to environmental protection [START_REF] Baumol | The Theory of Environmental Policy[END_REF]). This is quite intuitive since assuming environmental quality as a normal good, one would expect that wealthier individuals would want to buy more of it. Second, distributive elements also matter when we consider how the costs of a policy of environmental protection are likely to be distributed among individuals with differing incomes. Since indirect taxes have been proved to be regressive, any environmental tax policy is likely also to be regressive. In particular, in the French case, a tax on energy or transport consumption harms the lowest wage households three times more than the highest wage households [START_REF] Ruiz | Le caractère régressif des taxes indirectes : les enseignements d'un modèle de microsimulation[END_REF]). Moreover, the usual recycling of the environmental tax revenues through a decrease in the labor tax rate could also be regressive [START_REF] Metcalf | A Distributional Analysis of Green Tax Reforms[END_REF]). This second point is of interest in a world in which inequality and poverty have assumed high priority among social issues. In addition, without adequate consideration of this aspect of the matter, it would be hardly possible to design policies that can obtain the support they require for adoption. Thus, by ignoring their distributive effects, environmental policies may unintentionally harm certain groups in society.

While positive net benefits from environmental policy imply that the gains have exceeded the losses for society as a whole, this may not be true for all members of society. Some segments of society may bear a disproportionate share of costs. Paying attention to environmental justice makes sense for two reasons, one ethical and the other pragmatic. The ethical dimension concerns whether the distribution of benefits and costs is in accordance with the norms of social justice.

Desirable policies are both efficient and fair. The pragmatic dimension emphasizes the relationship between the distributional burden and both the likelihood that environmental legislation will pass and its ultimate form. Policies and programs that are perceived as unfair will stand little prospect of passage even if they enhance the prospects for efficiency and sustainability. Identifying the sources of unfairness and restructuring programs to eliminate them increases the likelihood that otherwise desirable programs can proceed.

It is well known that the norms of social justice are not sufficiently well defined such that no norm is beyond reproach. Nonetheless, some conventional approaches have arisen that can serve to guide our inquiry [START_REF] Atkinson | Social Justice and Public Policy[END_REF]). These involve two concepts known as horizontal and vertical equity. Horizontal equity occurs when people with equal income are treated equally.

With respect to pollution control, the principle of horizontal equity is satisfied if all persons with the same income receive the same net benefit. Vertical equity deals with the treatment of unequal or, using income as a basis, with the treatment of those with different income levels. The first step in assessing whether a particular policy satisfies vertical equity is to calculate how the net benefits are distributed among income groups: progressively, regressively or proportionally. A regressive allocation occurs only if the ratio of net benefit to income is larger for the rich than the poor. Hence, regressive policies violate the vertical equity principle.

We noticed that the large body of literature which has deeply analyzed the double dividend issue has surprisingly neglected the distribution issue of the welfare gain, although it is usually obtained at the expense of some groups of agents. The potential contradiction between efficiency and distributional concern has yet already been emphasized by some works in other frameworks (Bovenberg and van der Ploeg [1995], [START_REF] Proost | The Double Dividend and the Role of Inequality Aversion and Macroeconomic Regimes[END_REF], Bovenberg and Heijdra [1998]).

Our most recent paper [START_REF] Chiroleu-Assouline | Environmental Tax and the Distribution of Income among Heterogeneous Workers[END_REF]) intends to fill this peculiar gap and to analyze whether an environmental tax policy can respect the two equity principles simultaneously, the vertical as well as the horizontal one. We consider the possibility of designing an environmental policy in order to ensure a non-decreasing welfare for each class of workers. Compared to the standard double dividend literature which pursues two objectivesimprove the environment by increasing an environmental tax (first dividend) and the economic welfare by decreasing another distortionary tax (second dividend) -we add the third objective of Pareto improvement. It corresponds to an acceptability/unanimity criterion for the policy when agents are heterogeneous. The aim of this paper is therefore to design a balanced environmental tax reform able to correct these regressive properties of taxes and to leave all classes of worker better off.

This paper relies on an overlapping generations model, with polluting capital, but sharing the mean features of Chiroleu-Assouline and Fodha [2005] and [2006].

As in [START_REF] Chao | Greenhouse Gas Abatement: How Much? and Who Pays?[END_REF] or [START_REF] Williams | Environmental Tax Interactions when Pollution Affects Health or Productivity[END_REF] or [2003], we assume that the degradation of environmental quality has a negative impact on the total productivity of factors. This assumption is justified by the results of an increasing number of empirical studies measuring the health effects of pollution (OECD [2008]) and the impact of the health of workers on labor productivity [START_REF] Bloom | The Effects of Health on Economic Growth: A Production Function Approach[END_REF], in a sample consisting of both developing and industrial countries, found that good health, proxied by life expectancy, has a sizable, positive effect on economic growth).

Since [START_REF] Ostro | The Effects of Air Pollution on Work Loss and Morbidity[END_REF], many papers have emphasized the loss of productivity caused by the health effects of pollution, e.g. [START_REF] Samakovlis | Valuing Health Effects of Air Pollution -Focus on Concentration-response Functions[END_REF], or [START_REF] Pervin | Societal Costs of Air Pollution-Related Health Hazards: A Review of Methods and Results[END_REF] for air-pollution, and also [START_REF] Bosello | Economy-wide Estimates of the Implications of Climate Change: Human Health[END_REF] or [START_REF] Hübler | Costs of Climate Change -The Effects of Rising Temperatures on Health and Productivity[END_REF] for the health effects of climate change.

We assume that the production technology is a function of capital and heterogeneous labor.

Heterogeneous workers live two periods (young and old) and earn wages corresponding to their skill and consequently to their productivity. The labor tax is a very general one that could be either a progressive or proportional tax. Our demographic assumptions allow us to take into account several income classes; indeed, we consider (i) the heterogeneity characteristics of the labor market (high wages -skilled workers, middle wages, low wages -non skilled workers...), (ii) the heterogeneity of the individual income sources (wages for workers, savings for retirees).

The environmental policy consists of increasing the environmental tax on savings, in a secondrank framework. We then characterize the necessary conditions for the obtaining of a double dividend, i.e. an improvement of the environmental quality and an improvement of the welfare when the revenue of the pollution tax is recycled by a change in the labor tax rates. Previous studies show that the existence of a double dividend requires economic conditions such that the double dividend hypothesis seems unrealistic. Conversely, we show that the conditions for the double dividend existence lie in the distributive properties of the labor taxes. Even when the double dividend is not possible, the cost of the pollution regulation can be minimized by a new designing of the progressivity of the labor tax instead of an homogenous cut in the labor tax rates.

The results are dependent on the initial tax system. We also emphasize that (i) an increase of the environmental tax deteriorates the welfare of all and is regressive, (ii) the low paid workers prefer an environmental tax reform balanced by a decrease in the flat rate component but the high paid workers prefer a decrease of the progressivity. We conclude that the distributive properties of the tax policy could be one of the instruments of internalization of the intergenerational externalities.

To a certain extent, our paper highlights the gap between economic efficiency and vertical equity and illustrates the problem of the aggregation of positive and negative compensatory variations: the usual method of aggregation gives a higher weight to the wealthiest classes and introduces a bias when assessing the desirability or the acceptability of any environmental tax reform.

-In quest of the double dividend: some empirical considerations

What are the rooms of manoeuvre for the European countries? In each of them, the revenues of the existing environmental taxes are low in comparison to the weight of the social contributions (employees' and employers') and of the income tax (see Figure 1): from 4,6% for France or Belgium up to 10% for the Netherlands. Labor taxes are widely thought to be highly distortionary in European countries (EC, 1998 or Fiorito and[START_REF] Fiorito | Distortionary Taxation and Labour Market Performance[END_REF], at the opposite of United States where the tax bearing on capital exerts the highest distortion and was targeted by the first studies addressing the double dividend issue [START_REF] Goulder | Environmental Taxation and the "Double Dividend": A Reader's Guide[END_REF][START_REF] Bovenberg | Green Tax Reforms and the Double Dividend: an Updated Reader's Guide[END_REF]. For example, [START_REF] Scott | Optimal Taxation and OECD Labor Taxes[END_REF] finds that the excess burden of labor taxes in European countries (France, Germany, Italy) is about twice than in USA, Canada or Japan (Fig 5,p 939). In line with these findings, [START_REF] Hansson | Taxpayers' Responsiveness to Tax Rate Changes and Implications for the Cost of Taxation in Sweden[END_REF] shows that in Sweden the deadweight loss associated with using the existing labor tax rather than a lump-sum tax is in the order of 30% of its revenue. In order to enlighten the possibilities of greening the tax structure of the European countries, we define the potential transformation ratio of a tax system as the ratio of the revenues of the environmental taxes to the revenues of the other taxes bearing essentially on labor (income tax and social contributions). For EU15, the revenue of the environmental taxes account approximately to 13,5% of the revenues of labor and personal income taxes. Compared to this mean, among all the European countries, the Netherlands, Ireland and Denmark are characterized by the "greeenest" tax structure (see Figure 2) because they already have transferred a significant part of their fiscal revenues on environmental taxes. At the other side, Belgium, France and, surprisingly, Sweden exhibit the less green tax system and offer the easiest opportunity to use the environmental tax as a mean to alleviate the tax burden on labor and to reduce the welfare loss due to the inefficiency of the present tax system. This indicator reflects only a part of the tax choices of a country since it can small also because the share of the other taxes (like capital tax or consumption tax) is itself relatively small.

To correct from this bias, we normalize the potential transformation ratio by the sum of the shares of all the taxes considered here: income tax, social contributions and environmental taxes. This normalized indicator is graphed in Figure 3, which shows that the ranking between the European countries is not drastically modified by the normalization: the last four countries remain the same, (even the evidence is now less for Sweden) and the four "greenest" countries also. As we have pointed out in the last section concerning the theoretical acceptability of the tax policy by heterogeneous agents, the results depend on the initial progressivity characteristics of the labor tax rate; this kind of policy mix will be more effective when the initial tax system is not so much progressive. On the basis of OECD publications (OECD, 2009), we use the mean wage tax rate (Income tax plus employee contributions less cash benefits as a % of gross wage earnings) to compute a progressivity index of the European tax systems: this is the ratio between the wage tax rate observed at 167% and at 67% of median wages. Because we wish to set the family policy considerations aside, we focus on the tax rates bearing on single persons without children. As it is showed by Figure 4, Germany, France and Sweden typify the less progressive tax structure: their progressivity index are respectively 1.2, 1.3 and 1.6 which means that, for Germany for example, the mean tax rate at 167% of median wages is 20% greater than at 67% of median wages, since the difference of wages equals 250%. These countries could therefore benefit in a similar manner from a policy mix bearing on a large increase of progressivity index allowing a significant decrease in the flat component of the income tax rate.

-Conclusion

In September 2009, France, following Sweden and other Scandinavian countries, has officially proposed 3 a carbon tax at a rate equal to 17€/ton CO 2 . This could be a more general European movement towards a green tax reform. But, this goal is not easy to achieve. Indeed, the European countries are still heterogeneous, characterized by huge differences regarding long term energy solutions (like nuclear in France vs Germany), tax policy harmonization, coordination of public policies… The French debate is also still open. The intensive debates concern the recycling of the environmental tax revenues, showing at first a preference for compensating the short-term equity detrimental consequences instead of seeking to enhance the global efficiency, by paying lumpsum compensations rather than by reducing the labor taxes nor by re-designing the progressivity characteristics of the tax system. The need to also tax the firms which are already participating in the EU ETS will certainly make the compensation through labor tax cuts more appealing. All these debates also reflect the uncertainties around the short term consequences of the tax reform but, most of all, they express the reluctance of the societies regarding any green taxation reform.

The latter should invite us to analyze more deeply the determinants of the individual environmental sensitivity, in order to better define the marketing of the future green policies. In this paper, we give the main conditions such that an environmental tax reform may achieve three goals: increase of the environmental quality, increase of the economic efficiency and improvement of the equity. The last point is an important one. The short-term costs of the reform are most important than the long-term consequences for the policy to be adopted. We show that the long term benefits could be compatible with the short term ones. These precise conditions have to be precisely evaluated for the European countries.

Figure 1 Tax

 1 Figure 1

  Figure 2

Figure 3 Normalized

 3 Figure 3

  Figure 4

A tax is Laffer-efficient if its revenue grows when the tax rate increases.

The decision of the Conseil Constitutionnel of the 29th December 2009 requires some modifications of the perimeter of the taxed firms but it should not constitute a durable obstacle to the implementation of such a tax in France.