Fabrizio Bianchi
email: fabrizio.bianchi@math.univ-toulouse.fr

Johan Taflin
email: johan.taflin@u-bourgogne.fr

BIFURCATIONS IN THE ELEMENTARY DESBOVES FAMILY

Introduction and result

It is a classical result due to Lyubich [START_REF] Lyubich | Some typical properties of the dynamics of rational maps[END_REF] and Mané-Sad-Sullivan [START_REF] Mané | On the dynamics of rational maps[END_REF] that, for a family of rational maps, the continuity of the Julia set (in the Hausdorff topology) with the parameter is equivalent to the holomorphic motion of the repelling periodic points and of the Julia set itself. This allows one to decompose the parameter space of the family into the stability locus (where such holomorphic motions exist) and its complement, the bifurcation locus. De Marco [START_REF] Demarco | Dynamics of rational maps: a current on the bifurcation locus[END_REF] considered the canonical current dd c L on the parameter space (where L is the Lyapounov function, given by the integration of the logarithm of the Jacobian with respect to the equilibrium measure, i.e. the unique measure of maximal entropy) and proved that it is exactly supported on the bifurcation locus. Thus, stability is equivalent to the harmonicity of the Lyapounov function. An important feature of this picture is that the stability locus is dense in the parameter space.

This description was recently extended to higher dimension by Berteloot, Dupont, and the first author in [START_REF] Berteloot | Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of P k[END_REF]. It is proved there that the complex laplacian of the Lyapounov function still detects the bifurcation of the repelling cycles and that, up to a zero measure subset with respect to the equilibrium measure, the Julia set moves holomorphically precisely out of the support of dd c L. This lead to a coherent definition of stability and bifurcation in general dimension. Two main questions were left open in that work: the equivalence between the above stability conditions and the Hausdorff continuity of the Julia set, and the possible density of stability. The goal of this paper is to present an example answering by the negative to both of the above questions. More precisely, we prove the following.

Main Theorem. The family of endomorphisms of P 2 given by

f λ = -x(x 3 + 2z 3 ) : y(z 3 -x 3 + λ(x 3 + y 3 + z 3 )) : z(2x 3 + z 3 )
with λ ∈ C * satisfies the following properties:

(1) the Julia set of f λ depends continuously on λ, for the Hausdorff topology;

(2) the bifurcation locus coincides with C * .

The family above (referred to as the elementary Desboves family) was studied by Bonifant-Dabija [START_REF] Bonifant | Self-maps of P 2 with invariant elliptic curves[END_REF] and Bonifant-Dabija-Milnor [START_REF] Bonifant | Elliptic curves as attractors in P 2 . Part 1: Dynamics[END_REF], who revealed the dynamical richness of these maps. One of their main properties is that the small Julia set J 2 (λ) of f λ (the support of the equilibrium measure) and its large Julia set J 1 (λ) (the complement of the Fatou set, which coincides with the support of the The first author was partially supported by the ANR project LAMBDA, ANR-13-BS01-0002 and by the FIRB2012 grant "Differential Geometry and Geometric Function Theory", RBFR12W1AQ 002.

Green current) coincide (see Theorem 2.2). We can thus refer to any of them as the Julia set. It would be interesting to know whether the bifurcations in our example are robust. Let us mention that Romain Dujardin has announced the existence of several mechanisms leading to such phenomena.

The paper is organized as follows. In Section 2 we describe in details the Desboves maps and collect (in Theorem 2.2) the properties that we shall need in the construction, deducing in particular the continuity of the Julia set and the presence of bifurcations. This already proves that continuous and holomorphic dependence of the Julia set on the parameter are not equivalent in general dimension. Then, in Section 3 we prove that the bifurcation locus coincides with C * , by proving that Misiurewicz parameters (see Definition 3.1) are dense in the parameter space. The appendix is devoted to the proof of Theorem 2.2. Since in our particular setting the proof can be simplified with respect to the work by Bonifant, Dabija and Milnor, we include it for sake of completeness.

Elementary Desboves maps

In this section we define and give the main properties of the elementary Desboves maps. See [START_REF] Bonifant | Self-maps of P 2 with invariant elliptic curves[END_REF] and [START_REF] Bonifant | Elliptic curves as attractors in P 2 . Part 1: Dynamics[END_REF] for a more detailed description. We start considering the classical Desboves map given by

f D := x(y 3 -z 3 ) : y(z 3 -x 3 ) : z(x 3 -y 3 ) .
This is a rational map on P 2 , with non-empty indeterminacy locus. The important feature we are interested in is the following: the map f D leaves invariant a pencil of elliptic curves and in particular the Fermat curve C := {x 3 + y 3 + z 3 = 0}. We will now consider a well-chosen perturbation of f D , still leaving the curve C invariant. A natural way to do this is to consider the following Desboves family (studied in [START_REF] Bonifant | Elliptic curves as attractors in P 2 . Part 1: Dynamics[END_REF]), parametrized by (a, b, c) ∈ C 3 . We set Φ(x, y, z) := x 3 + y 3 + z 3 for convenience. The Desboves family is thus given by f a,b,c := x(y 3 -z 3 + aΦ) : y(z 3 -x 3 + bΦ) : z(x 3 -y 3 + cΦ) .

By construction, every such map leaves invariant the Fermat curve C. Moreover, f a,b,c is actually a well defined endomorphisms of P 2 (i.e., its indeterminacy locus is empty) with (a, b, c) avoiding the union of the seven hyperplanes abc(a

+ b + c)(a + 1 -b)(b + 1 -c)(c + 1 -a) = 0.
We will now specialize the parameters in order to recover a last property that we shall need in the sequel: we want every element of our perturbation to preserve a pencil of lines (i.e., to be elementary, see [START_REF] Dabija | Algebraic and geometric dynamics in several complex variables[END_REF]). This can be done by specializing the coefficients a = -1 and c = 1. We thus get the following definition (we rename the parameter b to λ). Definition 2.1. Let λ ∈ C. An elementary Desboves map is a map of the form

f λ = -x(x 3 + 2z 3 ) : y(z 3 -x 3 + λ(x 3 + y 3 + z 3 )) : z(2x 3 + z 3 ) .
The elementary Desboves family is the holomorphic family f :

C * × P 2 → C * × P 2 given by f (λ, z) := (λ, f λ (z)).
When λ = 0, the map f λ is an endomorphism of P 2 . With λ = 0, the map f 0 has an indeterminacy locus equal to the point ρ 0 := [0 : 1 : 0]. Moreover, we have f -1 λ ({ρ 0 }) = ({ρ 0 }) for every λ ∈ C * . So, f can also be seen as a family of endomorphisms of P 2 \ {ρ 0 }, parametrized by λ ∈ C.

We denote by P the pencil of lines passing through the point ρ 0 . For every λ ∈ C, f λ preserves P and leaves invariant the coordinate lines X := {x = 0}, Y := {y = 0}, Z := {z = 0}, and the Fermat curve C. The line Y is transversal to P. Hence, the action on P can be identified to the restriction g of f λ to Y , which is independent from λ. To be more precise, by construction the restriction h of f λ to C is independent from λ. It lifts to a map of the form z → αz + β on C and a simple computation of the multiplier at a fixed point gives that α = -2. Hence, h has degree 4 and is uniformly expanding. If π : P 2 \ {ρ 0 } → Y denotes the projection along P on Y then π semiconjugates g to h, i.e. π • h = g • π. Therefore, the map g is a degree 4 Lattès map. An explicit computation gives that, in the coordinate w := x z , the map g takes the form

(1) w → -w w 3 + 2 2w 3 + 1 .

It has 3 critical points, all with double multiplicity, which are given by [1 : 0 :

ω j ],
where ω is a primitive third-root of 1 and j ∈ {0, 1, 2}. It is worth pointing out that these three points are mapped by g to the three (fixed) points of intersections between Y and C, [1 : 0 : -ω j ]. Since the intersection of C and Y are transverse, and the dynamics is expansive on both curves, each of these points is repelling for

f λ , λ ∈ C.
In the next Theorem we collect the properties of the elementary Desboves maps that we shall need in the sequel. The first statement in the Main Theorem is then a consequence of Theorem 2.2.

Corollary 2.3. Let (f λ ) λ∈C * be the elementary Desboves family. Then the Julia set depends continuously on λ for the Hausdorff topology.

Proof. As the equilibrium measure depends continuously on λ, its support J 2 (λ) varies lower semicontinuously with the parameter for every family of endomorphisms of P 2 (see [START_REF] Dinh | Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings[END_REF]). It is thus enough to prove that, in the elementary Desboves family, this dependence is also upper semicontinuous. This is equivalent to prove that the set ∪ λ {λ} × J 2 (λ) is closed in the product space C * × P 2 . This immediately follows since its complement Ω is the basin of attraction of C * × {ρ 0 } in the product space, which is open. More precisely, if (λ 0 , z 0 ) ∈ Ω then by Theorem 2.2, z 0 is in basin of ρ 0 for f λ0 . Therefore, there exists an open subset U of P 2 included in this basin such that f λ0 (U ) ⊂ U and z 0 ∈ U . Hence, if V ⊂ C * is a small neighborhood of λ 0 then f λ (U ) ⊂ U for all λ ∈ V, i.e. V × U ⊂ Ω.

Remark 2.4. In our setting, the point ρ 0 has to be thought of as the point at infinity, and the Julia set shares a lot of similarities with the filled Julia set of a polynomial map. In particular, being the Julia set J 2 (λ) the complement of the "basin at infinity" of f λ , the above proof is essentially the same of the classical fact that the filled Julia set depends upper semicontinuously on the parameter.

We study now the critical set C λ of the maps f λ . This is a curve in P 2 of degree 9. More precisely, in homogeneous coordinates the Jacobian of f λ is equal to Jac(f λ ) = 8(x 3 -z 3 ) 2 (-x 3 + z 3 + λ(4y 3 + x 3 + z 3 )).

The term (x 3 -z 3 ) gives the three lines L j passing thought ρ 0 and [1 : 0 : ω j ], with j ∈ {0, 1, 2}, respectively, each with multiplicity 2. This corresponds to the critical set of the Lattès map g. The other terms gives a (possibly non irreducible) curve of degree 3, given in homogeneous coordinates by

C λ := {-x 3 + z 3 + λ(4y 3 + x 3 + z 3 ) = 0}. We thus have C λ = C λ ∪ (L 0 ∪ L 1 ∪ L 2 ).
Since C λ has degree 3, it intersects Y in three points. These are given by the points [x : 0 : z] with (x, z) solution of (1 + λ)z 3 + (λ -1)x 3 = 0. For λ = 1, we get the point [1 : 0 : 0] = Z ∩ Y (as a triple solution). Analogously, for λ = -1 we get the point [0 : 0 : 1] = X ∩ Y . Otherwise, we have the three points [w : 0 : 1], where w 3 = 1+λ 1-λ . So, every point w ∈ Y except the six points [1 : 0 : ±ω j ] (i.e., the critical points for the Lattès map and their images, which are always repelling for f λ , see above) there exists a λ = 0 such that w ∈ C λ ∩ Y . Moreover, the description above implies the following observation.

:= ∪ λ∈V C λ ∩ Y is open in Y .
For λ = 0 the critical set is very special: it is reduced to the union of the three lines L i , each with multiplicity 3. Thus, the intersection of C 0 with Y is again given by the three points [1 : 0 :

ω j ].
Finally, we study the two intersection points x 0 := [0 : 0 : 1] = X ∩ Y and z 0 := [1 : 0 : 0] = Z ∩ Y . We focus on x 0 , since the arguments are the same for z 0 . Since both the lines X and Y are fixed, the point x 0 is a fixed point. Since the restriction of f to Y is a Lattès map, the differential of f at x 0 has one repelling eigenvalue. To compute the other one, notice that the restriction of f λ to X has the form (in the coordinate t := y z )

(2) t → (1 + λ)t + λt 4 .

In particular, the point ρ 0 corresponds to the superattracting fixed point at infinity. The differential at 0 (corresponding to x 0 ) being (1 + λ), we get that x 0 is a repelling fixed point for |1 + λ| > 1. The analogous computation on z 0 gives that z 0 is repelling if and only if |1 -λ| > 1. In particular,

(3) for every λ = 0 one among x 0 and z 0 is repelling.

Remark 2.6. Observe that the transverse multiplier 1 + λ at x 0 can almost be chosen arbitrarily. If 1 + λ satisfies the Brjuno condition, the system f λ admits a Siegel disc (see [START_REF] Berteloot | Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of P k[END_REF]) passing through x 0 . By Theorem 2.2, this disc is contained in the support of the equilibrium measure. This answers to a question stated in [START_REF] Berteloot | Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of P k[END_REF]. A different example of this phenomenon was given by the first author in [START_REF] Bianchi | Motions of Julia sets and dynamical stability in several complex variables[END_REF]. For other values of λ, it can give a parabolic petal or an attracting disc in J 2 (λ). It is even possible to have a set of positive Lebesgue measure covered by such attracting discs, see [BDM07, Theorem 6.3] and [START_REF] Taflin | Invariant elliptic curves as attractors in the projective plane[END_REF]. In particular, the small Julia set has positive measure for these parameters.

Misiurewicz parameters and bifurcations

In this section we prove that the bifurcation locus of the elementary Desboves family coincides with the parameter space C * , completing the proof of the Main Theorem. By [START_REF] Berteloot | Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of P k[END_REF], Misiurewicz parameters (as defined below) are dense in the bifurcation locus. We will thus prove that these parameter are dense in C * . Recall that C f and C λ denote the critical set of the family f and of f λ , respectively. We shall set C n f := f n (C f ) and C n λ := f n λ (C λ ). Definition 3.1. A parameter λ 0 ∈ C * is called a Misiurewicz parameter if there exist a neighbourhood N λ0 ⊂ C * of λ 0 and a holomorphic map σ : N λ0 → P 2 such that:

(1) for every λ ∈ N λ0 , σ(λ) is a repelling periodic point;

(2) σ(λ 0 ) is in the Julia set of f λ0 ;

(3) there exists an n 0 such that (λ 0 , σ(λ (1) c 1 is a critical point for f λ1 ;

0 )) ∈ f n0 (C f ); (4) σ(N λ0 ) f n0 (C f ).
(2) f n1 λ1 (c 1 ) = N . Since N is repelling (and thus contained in the Julia set, by Theorem 2.2), to prove that λ 1 is a Misiurewicz parameter it is enough to check that N is not persistently contained in f n1 (C f ). This follows from Lemma 3.3 below, and the proof is completed as V can be chosen arbitrarily small. Lemma 3.3. The points x 0 = [0 : 0 : 1] and z 0 = [1 : 0 : 0] are not persistently contained in any set C n λ .

Proof. Since for every λ ∈ C we have f -1 λ (ρ 0 ) = {ρ 0 }, we can see the family as a family of endomorphisms of P 2 \ {ρ 0 }, with parameter space C. We still denote by C n λ and C n f the iterates of the critical sets. Assume, for sake of contradition, that there exists a n 0 such that x 0 ∈ C n0 λ for every λ ∈ C * . Since C n0 f is closed, this would imply that x 0 ∈ C n0 0 . But, for every n ≥ 1, we have C n 0 ∩ Y = {[1 : 0 : -ω j ] : j = 0, 1, 2}. This gives the desired contradiction.

Remark 3.4. Theorem 4.6 in [START_REF] Berteloot | Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of P k[END_REF] states that, in any holomorphic family f of endomorphisms of P k , the set of parameters λ for which P k coincides with the closure of the postcritical set of f λ is dense in any open subset of the bifurcation locus. It follows that, in the elementary Desboves family, this property holds for a dense subsets of parameters in C * . Remark 3.5. It follows from [START_REF] Berteloot | Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of P k[END_REF] that, for a dense subset of parameters λ ∈ C * , the map f λ admits a Siegel disc contained in the support of the equilibrium measure.

  Theorem 2.2. [BD02, Proposition 6.16] Let λ ∈ C * . If R(λ) denotes the closure of the repelling periodic points of f λ then (1) J 1 (λ) = J 2 (λ) = R(λ); (2) the Fatou set coincides with the basin of the superattracting fixed point ρ 0 = [0 : 1 : 0].

  Lemma 2.5. Let V be an open subset of C. Then the set W

  Theorem 3.2. Let f be the elementary Desboves family. Then Misiurewicz parameters are dense in C * .Proof. Let λ 0 ∈ C * . If V ⊂ C * issmall enough open neighbourhood of λ 0 , by (3) one of the two points [0 : 0 : 1] and [1 : 0 : 0] is repelling. Call N this point. By Lemma 2.5, the setW := ∪ λ∈V C λ ∩ Y is open in Y .Since the preimages of N by g are dense in Y , there exists a λ 1 ∈ V, a point c 1 ∈ Y and a number n 1 such that:

Appendix A. Proof of Theorem 2.2

This section consists of a simplified proof, in our setting, of Theorem 2.2. The original proof can be found in [START_REF] Dabija | Algebraic and geometric dynamics in several complex variables[END_REF][START_REF] Bonifant | Self-maps of P 2 with invariant elliptic curves[END_REF][START_REF] Bonifant | Elliptic curves as attractors in P 2 . Part 1: Dynamics[END_REF].

Lemma A.1. For every λ = 0, the exceptional E set of f λ is reduced to the point ρ 0 .

Proof. Since the point ρ 0 is backward invariant, it is contained in the exceptional set. Let us prove the other inclusion. The intersection of E with the invariant line X = {x = 0} must be reduced to ρ 0 (the exceptional set for the restriction, by (2)). So, if E were of dimension 1, it had to be a line passing thought ρ 0 . This is excluded since the exceptional set of the Lattès restriction g to Y is empty. Once we know that E has dimension 0, the fact that is reduced to ρ 0 again follows since the exceptional set of g is empty.

Lemma A.2. There exists a repelling point r 0 on the curve C contained in the small Julia set J 2 (λ).

Proof. Since the intersection of C and Y are transverse, and the dynamics is expansive of both curves, every intersection point of C and Y is repelling. Take as r 0 any such point (e.g., the fixed point [1 : 0 : -1]). In order to prove that r 0 ∈ J 2 (λ) we prove that, given any sufficiently small open neighbourhood U of r 0 , we have (4)

Indeed, take any small ball V disjoint from ρ 0 and intersecting the small Julia set J 2 (λ). By (4), preimages of V must accumulate r 0 . Since J 2 (λ) ∩ V = ∅, this proves that r 0 ∈ J 2 (λ), as desired.

Let us thus prove the claim (4). Suppose that E := P 2 \ ({ρ 0 } ∪ ∪ n f n λ (U )) = ∅ and take a line l 0 through ρ 0 intersecting E in a point e 0 . Since the exceptional set of the Lattès restriction of f λ to Y is empty, there exists a sequence of preimages of the line l 0 converging to the line l r through ρ 0 and r 0 . This gives a sequence of preimages e n of e 0 that, up to a subsequence, converge to a point ẽ ∈ l r . Since f -1 λ (E) ⊂ E and E is closed in P 2 \ {ρ 0 }, we have ẽ ∈ l r ∩ E. Moreover, ẽ = ρ 0 since ρ 0 is attracting. On the other hand, the point ẽ must be contained in the exceptional set of the restriction of f λ to l r . An explicit computation yields that this exceptional set is reduced to ρ 0 and gives the desired contradiction.

We can now prove Theorem 2.2.

Proof of Theorem 2.2. The first assertion is a consequence of Lemma A.2. Indeed, since backward preimages (in C) of any point in C are dense in this curve, this implies that all the curve C is included in the Julia set J 2 (λ). Since the exceptional set ρ 0 (see Lemma A.1) is not contained in C, preimages of this curve equidistribute (see [START_REF] Dinh | Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings[END_REF]) the Green current (supported on the large Julia set J 1 (λ)). This gives that the preimages of r 0 ∈ J 2 (λ) are dense in J 1 (λ), yielding J 1 (λ) ⊆ J 2 (λ). The assertion follows since the inclusions J 2 (λ) ⊆ R(λ) ⊆ J 1 (λ) always hold (the first by Briend-Duval equidistribution result [START_REF] Briend | Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de CP k[END_REF]).

The equality between the Fatou set and the basin Ω of ρ 0 simply comes from the fact that f λ is uniformly expanding in the transverse direction to P in P 2 \ Ω.


Mahadi Ddamulira
email: mddamulira@tugraz.at

On a problem of Pillai with Fibonacci numbers and powers of 3

Keywords: Fibonacci numbers, Linear forms in logarithms, Baker's method Mathematics Subject Classification 11B39, 11J86

Consider the sequence {F n } n≥0 of Fibonacci numbers defined by F 0 = 0, F 1 = 1, and F n+2 = F n+1 + F n for all n ≥ 0. In this paper, we find all integers c having at least two representations as a difference between a Fibonacci number and a power of 3.

Introduction

We consider the sequence {F n } n≥0 of Fibonacci numbers defined by F 0 = 0, F 1 = 1, and F n+2 = F n+1 + F n for all n ≥ 0.

The first few terms of the Fibonacci sequence are {F n } n≥0 = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . . . In this paper, we are interested in studying the Diophantine equation

F n -3 m = c (1)
for a fixed integer c and variable integers n and m. In particular, we are interested in finding those integers c admitting at least two representations as a difference between a Fibonacci number and a power of 3. This equation is a variant of the Pillai equation

a x -b y = c (2)
where x, y are nonnegative integers and a, b, c are fixed positive integers.

In 1936 and again in 1937, Pillai (see [START_REF] Pillai | On a xb y = c[END_REF][START_REF] Pillai | A correction to the paper on a xb y = c[END_REF]) conjectured that for any given integer c ≥ 1, the number of positive integer solutions (a, b, x, y), with x ≥ 2 and y ≥ 2 to the Eq. ( 2) is finite. This conjecture is still open for all c = 1. The case c = 1 is Catalan's conjecture which was proved by Mihȃilescu (see [START_REF] Mihȃilescu | Primary cyclotomic units and a proof of Catalan's conjecture[END_REF]). Pillai's work was an extension of the work of Herschfeld (see [START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF][START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF]), who had already studied a particular case of the problem with (a, b) = (2, 3). Since then, different variants of the Pillai equation have been studied. Some recent results for the different variants of the Pillai problem involving Fibonacci numbers, Tribonacci numbers, Pell numbers and the k-generalized Fibonacci numbers with powers of 2 have been intensively studied in [START_REF] Bravo | On a problem of Pillai with Tribonacci numbers and powers of 2[END_REF][START_REF] Chim | On a variant of Pillai's problem[END_REF][START_REF] Chim | On a variant of Pillai's problem II[END_REF][START_REF] Ddamulira | On a problem of Pillai with Fibonacci numbers and powers of 2[END_REF][START_REF] Ddamulira | On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2[END_REF][START_REF] Hernane | On Pillai's problem with Pell numbers and powers of 2[END_REF][START_REF] Hernández | On Pillai's problem with the Fibonacci and Pell sequences[END_REF].

Main result

The main aim of this paper is to prove the following result. Since F 1 = F 2 = 1, we discard the situation when n = 1 and just count the solutions for n = 2.

Theorem 1

The only integers c having at least two representations of the form F n -3 m are c ∈ {-26, -6, -1, 0, 2, 4, 7, 12}. Furthermore, all the representations of the above integers as F n -3 m with integers n ≥ 2 and m ≥ 0 are given by

-26 = F 10 -3 4 = F 2 -3 3 ; -6 = F 8 -3 3 = F 4 -3 2 ; -1 = F 6 -3 2 = F 3 -3 1 0 = F 4 -3 1 = F 2 -3 0 ; 2 = F 5 -3 1 = F 4 -3 0 ; 4 = F 7 -3 2 = F 5 -3 0 ; 7 = F 9 -3 3 = F 6 -3 0 ; 12 = F 8 -3 2 = F 7 -3 0 . ( 3
)

Auxiliary results

To prove our main result Theorem 1, we need to use several times a Baker-type lower bound for a nonzero linear form in logarithms of algebraic numbers. There are many such in the literature like that of Baker and Wüstholz from [2]. We use the one of Matveev from [START_REF] Matveev | An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II[END_REF]. Matveev [START_REF] Matveev | An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers, II[END_REF] proved the following theorem, which is one of our main tools in this paper.

Let γ be an algebraic number of degree d with minimal primitive polynomial over the integers

a 0 x d + a 1 x d-1 + • • • + a d = a 0 d i=1 (x -γ (i) ),
where the leading coefficient a 0 is positive and the η (i) 's are the conjugates of γ . Then, the logarithmic height of γ is given by

h(γ ) := 1 d log a 0 + d i=1 log max{|γ (i) |, 1} .
In particular, if γ = p/q is a rational number with gcd( p, q) = 1 and q > 0, then h(γ ) = log max{| p|, q}. The following are some of the properties of the logarithmic height function h(•), which will be used in the next sections of this paper without reference:

h(η ± γ ) ≤ h(η) + h(γ ) + log 2, h(ηγ ±1 ) ≤ h(η) + h(γ ), h(η s ) = |s|h(η) (s ∈ Z). (4)
Theorem 2 (Matveev) Let γ 1 , . . . , γ t be positive real algebraic numbers in a real algebraic number field K of degree D, b 1 , . . . , b t be nonzero integers, and assume that

:= γ b 1 1 • • • γ b t t -1, ( 5
)
is nonzero. Then log | | > -1.4 × 30 t+3 × t 4.5 × D 2 (1 + log D)(1 + log B)A 1 • • • A t , where B ≥ max{|b 1 |, . . . , |b t |}, and
A i ≥ max{Dh(γ i ), | log γ i |, 0.16}, for all i = 1, . . . , t.
During the calculations, we get upper bounds on our variables which are too large; thus, we need to reduce them. To do so, we use some results from the theory of continued fractions.

For the treatment of linear forms homogeneous in two integer variables, we use the well-known classical result in the theory of Diophantine approximation.

Lemma 1 Let τ be an irrational number, p 0 q 0 , p 1 q 1 , p 2 q 2 , . . . be all the convergents of the continued fraction of τ and M be a positive integer. Let N be a nonnegative integer such that q N > M. Then, putting a(M) := max{a i : i = 0, 1, 2, . . . , N }, the inequality

τ - r s > 1 (a(M) + 2)s 2 ,
holds for all pairs (r , s) of positive integers with 0 < s < M.

For a nonhomogeneous linear form in two integer variables, we use a slight variation of a result due to Dujella and Pethő (see [START_REF] Dujella | A generalization of a theorem of Baker and Davenport[END_REF], Lemma 5a), which is itself a generalization of the result of Baker and Davenport [START_REF] Baker | The equations 3x 2 -2 = y 2 and 8x 2 -7 = z[END_REF]. For a real number X , we write ||X || := min{|X -n| : n ∈ Z} for the distance from X to the nearest integer.

Lemma 2 Let M be a positive integer, p q be a convergent of the continued fraction of the irrational number τ such that q > 6M, and A, B, μ be some real numbers with A > 0 and B > 1. Let further ε := ||μq|| -M||τ q||. If ε > 0, then there is no solution to the inequality

0 < |uτ -v + μ| < AB -w , in positive integers u, v, and w with u ≤ M and w ≥ log(Aq/ε) log B .
Finally, the following lemma is also useful. It is Lemma 7 in [START_REF] Sánchez | Linear combinations of factorials and s-units in a binary recurrence sequence[END_REF].

Lemma 3 (Gúzman, Luca) If m ≥ 1, T > (4m 2 ) m , and T > x/(log x) m , then x < 2 m T (log T ) m .

Proof of Theorem 1

Assume that there exist nonnegative integers n, m, n 1 , m 1 with min{n,

n 1 } ≥ 2 and min{m, m 1 } ≥ 0 such that (n, m) = (n 1 , m 1 )
, and

F n -3 m = F n 1 -3 m 1 .

Without loss of generality, we can assume that

m ≥ m 1 . If m = m 1 , then F n = F n 1 , so (n, m) = (n 1 , m 1 )
, which gives a contradiction to our assumption. Thus, m > m 1 . Since

F n -F n 1 = 3 m -3 m 1 , (6)
and the right-hand side is positive, we get that the left-hand side is also positive and so n > n 1 .

Using the Binet formula

F k = α k -β k √ 5 for all k ≥ 0, (7)
where

(α, β) := 1+ √ 5 2 , 1- √ 5 2
are the roots of the equation x 2x -1 = 0, which is the characteristic equation of the Fibonacci sequence. One can easily prove by induction that

α k-2 ≤ F n ≤ α k-1 for all k ≥ 1. ( 8
)
Using the Eq. ( 6), we get

α n-4 ≤ F n-2 ≤ F n -F n 1 = 3 m -3 m 1 < 3 m , (9) α n-1 ≥ F n ≥ F n -F n 1 = 3 m -3 m 1 ≥ 3 m-1 , ( 10
)
from which we get that

1 + log 3 log α (m -1) < n < log 3 log α m + 4. (11)
If n ≤ 300, then m ≤ 127. We ran a Mathematica program for 2 ≤ n 1 < n ≤ 300 and 0 ≤ m 1 < m ≤ 127 and found only the solutions from the list [START_REF] Bravo | On a problem of Pillai with Tribonacci numbers and powers of 2[END_REF]. From now, we assume that n > 300 and from [START_REF] Hernández | On Pillai's problem with the Fibonacci and Pell sequences[END_REF] we have that m > 127. Therefore, to solve the Diophatine Eq. ( 1), it suffices to find an upper bound for n.

Bounding n

By substituting the Binet formula [START_REF] Ddamulira | On a problem of Pillai with k-generalized Fibonacci numbers and powers of 2[END_REF] in the Diophantine Eq. ( 1), we get

α n √ 5 -3 m = β n √ 5 + α n 1 -β n 1 √ 5 -3 m 1 ≤ α n 1 + 2 √ 5 + 3 m 1 ≤ 2α n 1 √ 5 + 3 m 1 < 3 max{α n 1 , 3 m 1 }.
Multiplying through by 3 -m , using the relation ( 9) and using the fact that α < 3, we get

|( √ 5) -1 α n 3 -m -1| < 3 max α n 1 3 m , 3 m 1 -m < max{α n 1 -n+7 , 3 m 1 -m+1 }. ( 12
)
For the left-hand side, we apply the result of Matveev, Theorem 2 with the following data:

t = 3, γ 1 = √ 5, γ 2 = α, γ 3 = 3, b 1 = -1, b 2 = n, b 3 = -m.
Throughout we work with the field K := Q( √ 5) with D = 2. Since max{1, n, m} ≤ 2n, we take B := 2n. Furthermore, we take

A 1 := 2h(γ 1 ) = log 5, A 2 := 2h(γ 2 ) = log α, A 3 := 2h(γ 1 ) = 2 log 3. We put = ( √ 5) -1 α n 3 -m -1.
First we check that = 0, if it were, then α 2n ∈ Q, a contradiction. Thus, = 0. Then, by Matveev's theorem, the left-hand side of ( 12) is bounded as

log | | > -1.4 • 30 6 • 3 4.5 • 2 2 (1 + log 2)(1 + log 2n)(log 5)(log α)(2 log 3).
By comparing with [START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF], we get

min{(n -n 1 -7) log α, (m -m 1 -1) log 3} < 1.
(n -n 1 ) log α, (m -m 1 ) log 3} = (n -n 1 ) log α.
In this case, we rewrite (6) as

α n -α n 1 √ 5 -3 m = β n -β n 1 √ 5 -3 m 1 < 1 + 3 m 1 ≤ 3 m 1 +1 , which implies α n-n 1 -1 √ 5 α n 1 3 -m -1 < 3 m 1 -m+1 . ( 13
)
We put

1 = α n-n 1 -1 √ 5 α n 1 3 -m -1.
To see that 1 = 0, for if 1 = 0, then

α n -α n 1 = √ 5 • 3 m .
By conjugating the above relation in K, we get that

β n -β n 1 = - √ 5 • 3 m .
The absolute value of the left-hand side is at most

|β n -β n 1 | ≤ |β| n + |β| n 1 < 2,
while the absolute value of the right-hand side is at least

| - √ 5 • 3 m | ≥ √ 5 > 2 for all m > 127, which is a contradiction.
We apply Theorem 2 on the left-hand side of [START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF] with the data

t = 3, γ 1 = α n-n 1 -1 √ 5 , γ 2 = α, γ 3 = 3, b 1 = 1, b 2 = n 1 , b 3 = -m.
The minimal polynomial of γ 1 divides

5X 2 -5F n-n 1 X -((-1) n-n 1 + 1 -L n-n 1 ),
where {L k } k≥0 is the Lucas companion sequence of the Fibonacci sequence given by L 0 = 2, L 1 = 1, L k+2 = 2L k+1 + L k for all k ≥ 0, for which the Binet formula for its general term is given by

L k = α k + β k for all k ≥ 0.
Thus, we obtain

h(γ 1 ) ≤ 1 2 log 5 + log α n-n 1 + 1 √ 5 < 1 2 log(2 √ 5α n-n 1 ) < 1 2 (n -n 1 + 2) log α < 8.4 × 10 11 (1 + log 2n). (14)
So, we can take A 1 := 1.67 × 10 12 (1 + log 2n). Furthermore, as before, we take Then,

log | 1 | > -1.72 × 10 24 (1 + log 2n) 2 .
By comparing the above relation with [START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF], we get that

(m -m 1 ) log 3 < 1.80 × 10 24 (1 + log 2n) 2 . ( 15
)
Case 2. min{(nn 1 ) log α, (mm 1 ) log 3} = (mm 1 ) log 3.

In this case, we rewrite (6) as

α n √ 5 -(3 m-m 1 -1) • 3 m 1 = β n + α n 1 -β n 1 √ 5 < α n 1 + 2 √ 5 < α n 1 , which implies that |( √ 5(3 m-m 1 -1)) -1 α n 3 -m 1 -1| < α n 1 3 m -3 m 1 ≤ 3α n 1 3 m < 3α n 1 -n+4 < α n 1 -n+7 . ( 16
)
We put

2 = ( √ 5(3 m-m 1 -1)) -1 α n 3 -m 1 -1.
Clearly, 2 = 0, for if 2 = 0, then α 2n ∈ Q, which is a contradiction. We again apply Theorem 2 with the following data

t = 3, γ 1 = √ 5(3 m-m 1 -1), γ 2 = α, γ 3 = α, b 1 = -1, b 2 = n, b 3 = -m 1 .
The minimal polynomial of γ 1 is X 2 -5(3 1 -1) 2 . Thus,

h(γ 1 ) = log( √ 5(3 m-m 1 -1)) < (m -m 1 + 1) log 3 < 1.25 × 10 12 (1 + log 2n).
So, we can take A 1 := 2.5 × 10 12 (1 + log 2n). Further, as in the previous applications, we take A 2 := log α and A 3 := 2 log 3. Finally, since max{1, n, m 1 } ≤ 2n, we can take B := 2n. Then, we get Finally, we rewrite the equation ( 6) as

log | 2 | > -1.4 • 30 6 • 3 4.5 • 2 2 (1 + log 2)(1 + log 2n)(2.
(α n-n 1 -1) √ 5 α n 1 -(3 m-m 1 -1) • 3 m 1 = β n -β n 1 √ 5 < |β| n 1 < 1.
Dividing through by 3 m -3 m 1 , we get

α n-n 1 -1 √ 5(3 m-m 1 -1) α n 1 3 -m 1 -1 < 1 (3 m -3 m 1 ) ≤ 3 3 m ≤ 3α -(n-4) ≤ α 7-n , ( 19
)
since α < 3 and α < α n 1 . We again apply Theorem 2 on the left-hand side of (19) with the data

t = 3, γ 1 = α n-n 1 -1 √ 5(3 m-m 1 -1) , γ 2 = α, γ 3 = 3, b 1 = 1, b 2 = n 1 , b 3 = -m 1 .
Using the algebraic properties of the logarithmic height function, we get

h(γ 1 ) = h α n-n 1 -1 √ 5(3 m-m 1 -1) ≤ h α n-n 1 -1 √ 5 + h(3 m-m 1 -1) < 1 2 (n -n 1 + 4) log α + (m -m 1 ) log 3 < 2.80 × 10 24 (1 + log 2n) 2 ,
where in the above inequalities, we used the argument from ( 14) as well as the bounds (18). Thus, we can take A 1 := 5.60×10 24 (1+log 2n), and again as before A 2 := log α and

A 3 := 2 log 3. If we put 3 = α n-n 1 -1 √ 5(3 m-m 1 -1) α n 1 3 -m 1 -1,
we need to show that 3 = 0. If not, 3 = 0 leads to

α n -α n 1 = √ 5(3 m -3 m 1 ).
A contradiction is reached upon a conjugation in K and by taking absolute values on both sides. Thus, 3 = 0. Applying Theorem 2 gives

log | 3 | > -1.4 • 30 6 • 3 4.5 • 2 2 (1 + log 2)(1 + log 2n)(5.6 × 10 24 (1 + log 2n) 2 ) ×(log α)(2 log 3), a comparison with (19) gives (n -4) < 3 × 10 36 (1 + log 2n) 3 , or 2n < 6.2 × 10 36 (1 + log 2n) 3 . ( 20
)
Now by applying Lemma 3 on (20) with the data m = 3, T = 6.2 × 10 36 , and x = 2n, leads to n < 2 × 10 40 .

Reducing the bound for n

We need to reduce the above bound for n and to do so we make use of Lemma 2 several times. To begin, we return to [START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF] and put

:= n log α -m log 3 -log( √ 5).
For technical reasons, we assume that min{nn 1 , mm 1 } ≥ 20. We go back to the inequalities for , 1 , and 2 . Since we assume that min{n

-n 1 , m -m 1 } ≥ 20 we get |e -1| = | | < 1 4 . Hence, | | < 1 2
and since the inequality |y| < 2|e y -1| holds for all y ∈ -1 2 , 1 2 , we get

| | < 2 max{α n 1 -n+5 , 3 m 1 -m+1 } ≤ max{α n 1 -n+8 , 3 m 1 -m+2 }.
Assume that > 0. We then have the inequality

0 < n log α log 3 -m + log(1/ √ 5) log 3 < max α 8 (log 3)α n-n 1 , 6 (log 3)3 m-m 1 . < max{45α -(n-n 1 ) , 8 • 3 -(m-m 1 ) }.
We apply Lemma 2 with the data

τ = log α log 3 , μ = log(1/ √ 5) log 3
, (A, B) = (45, α) or [START_REF] Dujella | A generalization of a theorem of Baker and Davenport[END_REF][START_REF] Bravo | On a problem of Pillai with Tribonacci numbers and powers of 2[END_REF].

Let τ = [a 0 ; a 1 , a 2 , . . .] = [0; 2, 3 , 1, 1, 6, 1, 49, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 10, 3,
. . .] be the continued fraction of τ . We choose M := 2 × 10 40 and consider the 91th convergent p q = p 91 q 91 = 487624200385184167130255744232737921512174859336581 1113251817385764505972408650620147577750763395186265 .

It satisfies q = q 91 > 6M. Furthermore, it yields ε > 0.4892 and, therefore, either

n -n 1 ≤ log(45q/ε) log α < 254, or m -m 1 ≤ log(8q/ε) log 3 < 110.
In the case < 0, we consider the inequality

m log 3 log α -n + log( √ 5) log α < max α 8 log α α -(n-n 1 ) , 8 log α • 3 -(m-m 1 ) < max{98α -(n-n 1 ) , 18 • 3 -(m-m 1 ) }.
On a problem of Pillai with Fibonacci numbers and powers of 3

We then apply Lemma 2 with the data

τ = log 3 log α , μ = log √ 5 log α , (A, B) = (98, α), or (18, 3). Let τ = [a 0 ; a 1 , a 2 , . . .] = [2; 3, 1 , 1, 6, 1, 49, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 10, 3, 12,
. . .] be the continued fraction of τ . Again, we choose M = 2 × 10 40 , and in this case we consider the 101th convergent

p q = p 101 q 101 = 106360048375891410642967692492903700137161881169662 56228858848524361385900581302251812795713192394033 ,
which satisfies q = q 101 > 6M. Further, this yields ε > 0.125 and, therefore, either

n -n 1 ≤ log(98q/ε) log α < 254 , or m -m 1 ≤ log(18q/ε) log 3 < 110.
These bounds agree with the bounds obtained in the case > 0. As a conclusion, we have that either nn 1 ≤ 253 or mm 1 ≤ 109 whenever = 0. Now, we distinguish between the cases nn 1 ≤ 253 and mm 1 ≤ 109. First, we assume that nn 1 ≤ 253. In this case, we consider the inequality for 1 , ( 13) and also assume that mm 1 ≤ 20. We put

1 = n 1 log α -m log 3 + log α n-n 1 √ 5 .
Then, the inequality (13) implies that

| 1 | < 6 3 m-m 1 .
If we further assume that 1 > 0, we then get

0 < n 1 log α log 3 -m + log((α n-n 1 -1)/ √ 5) log 3 < 6 (log 3)3 m-m 1 < 6 3 m-m 1 .
Again we apply Lemma 2 with the same τ as in the case > 0. We use the 91th convergent p/q = p 91 /q 91 of τ as before. But in this case, we choose (A, B) := (8, 3) and use

μ l = log((α l -1)/ √ 5)
τ - m n 1 + 2 < 6 3 m-m 1 (n 1 + 2) , or τ - m -3 n 1 + 6 < 6 3 m-m 1 (n 1 + 6) ,
respectively. We assume that mm 1 > 150. Then, 3 m-m 1 > 8 × (4 × 10 40 ) > 8 × (n 1 + 6); therefore

6 3 m-m 1 (n 1 + 2) < 1 3(n 1 + 2) 2 , and 6 3 m-m 1 (n 1 + 6) < 1 3(n 1 + 6) 2 .
By Lemma 1, it follows that m/(n 1 + 2) or (m -3)/(n 1 + 6) are convergents of τ , respectively. So, say one of m/(n 1 + 2) or (m -3)/(n 1 + 6) is of the form p k /q k for some k = 0, 1, 2, . . . , 92. Here, we use that q 92 > 4 × 10 40 > n + 1 + 6. Then

1 (a k + 2)q 2 k < τ - p k q k . Since max{a k : k = 0, 1, 2, . . . , 92} = 140, we get 1 142q 2 k < 6 3 m-m 1 q k
and q k divides one of {n 1 + 2, n 1 + 6}.

Thus, we get

3 m-m 1 ≤ 6 × 142(n 1 + 6) < 6 × 142 × 4 × 10 40 ,
giving mm 1 ≤ 92. Now let us turn to the case mm 1 ≤ 109 and we consider the inequality for 2 , (16). We put

2 = n log α -m 1 log 3 + log(1/( √ 5(3 m-m 1 -1))),
and we also assume that nn 1 ≥ 20. We then have

| 2 | < 2α 8 α n-n 1 .
We assume that 2 , then we get

0 < n log α log 3 -m 1 + log(1/( √ 5(3 m-m 1 -1)) log α < 3α 8 (log 3)α n-n 1 < 130 α n-n 1 .
We apply again Lemma 2 with the same τ, q, M, (A, B) := (130, α) and

μ l = log(1/( √ 5(3 l -1))) log 3 for k = 1,
= n 1 log α -m 1 log 3 + log α n-n 1 -1 √ 5(3 m-m 1 -1) .
Since n ≥ 300, the inequality (19) implies that

| 3 | < 3 α n-4 = 3α 4 α n .
Assuming that 3 > 0, then As before, we have a problem at (k, l) := (4, 1), [START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF][START_REF] Baker | The equations 3x 2 -2 = y 2 and 8x 2 -7 = z[END_REF], [START_REF] Dujella | A generalization of a theorem of Baker and Davenport[END_REF]2). The cases (k, l) := (4, 1), [START_REF] Herschfeld | The equation 2 x -3 y = d[END_REF][START_REF] Baker | The equations 3x 2 -2 = y 2 and 8x 2 -7 = z[END_REF] were treated before in the case of 1 . The case (k, l) := (8, 2) arises because

α 8 -1 √ 5(3 2 -1) = 3 8 α 4 .
We, therefore, discard the cases (k, l) := (4, 1), (12, 1), (8, 2) for some time. For the remaining cases, we get ε > 0.0015, so we obtain n ≤ log(20q/ε) log α < 264.

A similar conclusion is reached when 3 < 0. Hence, n < 300. Now, we look at the cases (k, l) := (4, 1), (12, 1), [START_REF] Dujella | A generalization of a theorem of Baker and Davenport[END_REF]2). The cases (k, l) := (4, 1), (12, 1) can be treated as before when we showed that nn 1 ≤ 263 implies mm 1 ≤ 115. The case when (k, l) = (8, 2) can be dealt with in a similar way. Namely, it gives that

|(n 1 + 4)τ -m 1 | < 20 α n . Therefore τ - m 1 n 1 + 4 < 20 (n 1 + 4)α n . ( 21
)
Since n ≥ 300, we have α n > 2 × 20 × (4 × 10 40 ) > 40(n 1 + 4). This shows that the right-hand side of the above inequality (21) is at most 2/(n 1 + 4) 2 . By Lemma 1, we get that m 1 /(n 1 + 4) = p k /q k for some k = 1, 2, . . . , 92. We then get by a similar argument as before that α n < 20 × 142 × (4 × 10 40 ), which gives n ≤ 211. Therefore, the conclusion is that n < 300 holds also in the case (k, l) = (8, 2). However, this contradicts our working assumption that n > 300. This completes the proof of Theorem 1.

A 2 :

 2 = log α and A 3 := 2 log 3. Finally, since max{1, n 1 , m} ≤ 2n, we can take B := 2n. Then, we get log | 1 | > -1.4 • 30 6 • 3 4.5 • 2 2 (1 + log 2)(1 + log 2n)(16.8 × 10 11 (1 + log 2n))×(log α)(2 log 3).

0 < n 1 log α log 3 -

 3 where (k, l) := (nn 1 , mm 1 ). We again apply Lemma 2 with the same τ, q, M, (A, B) := (20, α) andμ k,l = log((α k -1)/( √ 5(3 l -1)) log 3 for 1 ≤ k ≤ 265, 1 ≤ l ≤ 115.

  log 3 , instead of μ for each possible value of l := nn 1 ∈ [1, 2, . . . , 253]. We have problems at l ∈ {4, 12}. We discard these values for now and we will treat them later. For the remaining values of l, we get ε > 0.0005. Hence by Lemma 2, we get n 1 ≤ 253 implies that mm 1 ≤ 115, unless nn 1 ∈ {4, 12}. A similar conclusion is reached when 1 < 0 with the same two exceptions for n -n 1 ∈ {4, 12}.So, 1 = (n 1 + 2) log αm log 3 , or (n 1 + 6) log α -(m -3) log 3 when l = 4, 12, respectively. Thus we get that

	m -m 1 <	log(8q/0.0005) log 3	< 116.
	Thus, n The reason we have a problem at l ∈ {4, 12} is because
	α 4 -1 √ 5	= α 2 , and	α 12 -1 √ 5	= 2 3 α 6 .

  2, . . . , 109.

	We get ε > 0.004; therefore
	n -n 1 <	log(130q/ε) log α	< 266.

A similar conclusion is reached when 2 < 0. To conclude, we first get that either

nn 1 ≤ 253 or mm 1 ≤ 109. If nn 1 ≤ 253, then mm 1 ≤

115, and if mm 1 ≤ 109 then nn 1 ≤ 265. Thus, we conclude that we always have nn 1 ≤ 265 and mm 1 ≤ 115.

Finally, we go to the inequality of 3 , (19). We put 3

Acknowledgements Open access funding provided by Graz University of Technology. The author thanks the referee like to thank referee for the careful reading of the manuscript and the useful comments that greatly improved the quality of this paper. The research was supported by the Austrian Science Fund (FWF) Projects: F5510-N26-Part of the special research program (SFB), "Quasi-Monte Carlo Methods: Theory and Applications", P26114-N26-"Diophantine Problems: Analytic, geometric and computational aspects", and W1230-"Doctoral Program Discrete Mathematics". The author also thanks his supervisor Professor Robert Tichy for support and encouragement.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

The author was supported by the Austrian Science Fund (FWF) Projects: F5510-N26-Part of the special research program (SFB), "Quasi-Monte Carlo Methods: Theory and Applications", P26114-N26-"Diophantine Problems: Analytic, geometric and computational aspects", and W1230-"Doctoral Program Discrete Mathematics".

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Milliam Maxime

Zekeng Ndadji

Maurice Tchoupé Tchendji
email: maurice.tchoupe@univ-dschang.org

A Projection-Stable Grammatical Model to Specify Workflows for their P2P and Artifact-Centric Execution

Keywords: Workflow Management Systems, Workflow Specification Language, Artifacts, Accreditation, Projection, Partial replica, Peer to Peer Système de Gestion des Workflows, Langage de Spécification des Workflows, Artefacts, Accréditation, Projection, Réplique Partielle, Pair à Pair

In this manuscript, we are interested in the specification and decentralized execution of administrative workflows. We present a grammatical model to specify such processes by indicating, in addition to their fundamental elements, the permissions (reading, writing and execution) of each actor in relation to each of the tasks that compose them. We then present a decentralized and artifact-centric execution model of these processes, on a Peer to Peer (P2P) Workflow Management System (WfMS). Our execution model allows the confidential execution of certain tasks by ensuring that, each actor potentially has only a partial perception of the overall process execution status. In our approach, we propose various stable projection algorithms, making it possible to obtain, to verify the coherence and to guarantee the convergence of the various potentially partial perceptions. Our algorithms are then coded and tested using a graphical tool to simulate the decentralized execution of administrative processes.

RÉSUMÉ

. Dans ce manuscrit, nous nous intéressons à la spécification et à l'exécution décentralisée des processus workflows administratifs. Nous présentons un modèle grammatical permettant de spécifier de tels processus en précisant en plus de leurs éléments fondamentaux, les permissions (lecture, écriture et exécution) de chaque acteur relativement à chacune des tâches qui les composent. Nous présentons ensuite un modèle d'exécution décentralisée et centré-artefact de ces processus, sur un Système de Gestion de Workflows (SGWf) Pair à Pair (P2P). Notre modèle d'exécution autorise l'exécution confidentielle de certaines tâches en permettant à chaque acteur, de n'avoir potentiellement qu'une perception partielle de l'état d'exécution global du processus. Dans notre démarche, nous proposons divers algorithmes de projection stables, permettant d'obtenir, de vérifier la cohérence et de garantir la convergence des diverses perceptions potentiellement partielles. Nos algorithmes sont ensuite codés et expérimentés à l'aide d'un outil graphique de simulation d'exécution décentralisée des processus administratifs.

Introduction

To easily automate their business processes and increase their competitiveness, organisations are increasingly interested in workflow technology. Indeed, in its most widespread approach, the latter reduces the automation of a given business process to its formal specification (modeling) using a Workflow Process Specification Language (WfPSL) [START_REF]Wfmc Standards: the Workflow Reference Model[END_REF][START_REF] Amine Chaâbane | Actes des 7èmes journées scientifiques des jeunes chercheurs en Génie Electrique et Informatique (GEI)[END_REF].

To automate a given administrative process 1 , we can conceive it as a set of annotated trees called target artifacts, representing possible execution scenarios of the said process (see fig. 1). From this set of artifacts, we will derive an equivalent grammar called Grammatical Model of Workflow (GMWf). By associating this model with a list (subsequently called set of accreditations) informing on the permissions of each process actor for each task, we obtain a complete (ready to be executed) specification of the studied process.

Instances of an administrative process described using a GMWf and a set of accreditations, can be executed in a decentralized mode, by a P2P-WfMS-View; it is a Workflow Management System (WfMS) whose geographically dispersed instances (peers) communicate in Peer to Peer (P2P) by exchanging artifacts (artifact-centric). As a prerequisite to do so, each peer is configured using the model (GMWf + set of accreditations) of the process. From the (global) GMWf, each peer completes its configuration by deriving by projection, a local GMWf according to the accreditations of the local actor (GMWf projection). During the proper execution, actors coordinate by exchanging locally updated artifacts, to incrementally build one of the target artifacts: that's why the workflow execution process is said to be artifact-centric. On a given peer, when an artifact is received, it is projected (artifact projection) in accordance with the accreditations of the local actor in order to meet any confidentiality requirements. The local actor potentially manipulates only a partial replica of the global artifact under execution; his actions on it are guided by his local GMWf. At the end of its local execution, his updates to the manipulated partial replica must be integrated into the global artifact being executed. We then merge these two artifacts that are conform to two different models (we say that we realize the expansion-pruning of the updated partial replica). When this merging is complete, the process execution continues with the updated global artifact under execution. Contributions: In this paper, we propose 1) a simple Language for Specifying Administrative Workflow Processes (LSAWfP), allowing to model each administrative process in the form of a GMWf to which a set of actors accreditations is added; 2) stable projection operations (GMWf projection, artifacts projection and expansion-pruning) for decentralized execution of processes using a P2P-WfMS-View. Manuscript organization: in section 2 of this manuscript, we give a formal definition of our language (LSAWfP) and then we discuss its expressiveness. Then, in section 3, we present our model for decentralized execution of administrative processes specified using our LSAWfP, as well as its key algorithms. Finally, in appendices, we analyze the stability of our algorithms (sec. A) and we present a prototype of a P2P WfMS integrating our algorithms.

A Language for the Specification of Administrative Workflow Processes (LSAWfP)

Language definition

In the rest of this manuscript, we designate by the expression "a Language for the Specification of Administrative Workflow Processes" (LSAWfP), the language that we propose for processes specification. Also, we refer to Grammatical Model of Administrative Workflow Process (GMAWfP) as any specification of a business process produced using the language LSAWfP. Formally, a GMAWfP is defined as follows :

Définition 1 The Grammatical Model of Administrative Workflow Process (GMAWfP)

W f of a given business process, is a triplet W f = G, L P k , L A k wherein G is the studied process (global) GMWf, L P k is the set of k actors taking part in its execution and L A k represents the set of these actors accreditations.

Concept of GMWf

The set of tasks of an administrative process and their ordering can be described using a set of target artifacts (see fig. 1). In these, nodes represent tasks to be executed and each hierarchical decomposition (a node and its sons) represents an ordering. In the artifacts that we handle, we consider only two types of ordering : (1) sequential ordering, noted X 0 → X s1 . . . X sn for a set {X 0 , X s1 . . . X sn } of tasks, which describes how the task X 0 precedes tasks X s1 . . . X sn that should be executed sequentially and, (2) parallel ordering, noted X 0 → X p1 . . . X pn , which indicates how the task X 0 precedes tasks X p1 , . . . , X pn that can be executed concurrently. The set of target artifacts can therefore be substituted by a grammar G (a GMWf) in which, each symbol refers to a task and, each production p is of one of the following two forms: p : X 0 → X 1 . . . X n or p : X 0 → X 1 . . . X n . In this case, each target artifact t i is conform to G and we note t i ∴ G. More formally, Définition 2 A Grammatical Model of Workflow (GMWf) is defined by G = (S , P , A) where S is a finite set of grammatical symbols or sorts corresponding to various tasks to be executed in the studied business process; A ∈ S is a particular symbol called axiom, and P ⊆ S × S * is a finite set of productions decorated by the operators " " (is sequential to) and " " (is parallel to): they are precedence rules. A production P = X P(0) , X P(1) • • • X P(|P|) is either of the form P : X 0 → X 1 . . . X |P| , or of the form P : X 0 → X 1 . . . X |P| and |P| designates the length of P right-hand side. Each grammatical symbol X ∈ S is associated with an attribute called status, that can be updated when the task X is executed; X.status provides access (read and write) to its content. A production with the symbol X as left-hand side is called a X-production.

For some business processes, there may be particular cases where it is impossible to strictly order all tasks using the two production forms adopted for GMWf 2 . In such case, it is neccessary to insert new symbols, known as (re)structuring symbols, not related to tasks, to achieve a correct ordering that meets the forms of productions. To take into account such cases, we adapt the previously made GMWf definition to obtain the following complete one: Workflow (GMWf) is defined by G = (S , P , A G ) wherein P refers to the same purpose as in definition 2, S = {A G }∪T ∪T Struc is a finite set of grammatical symbols or sorts in which, those of T correspond to tasks of the studied business process, while those of T Struc are (re)structuring symbols. A G is the axiom of grammar.

Définition 3 A Grammatical Model of

Concept of accreditation of an actor

Since business processes are generally carried out collectively, it is necessary for coordination and confidentiality reasons, to clarify the permissions of each actor on each process task. In our case, we propose to do this when specifying the process, using accreditations. The accreditation of a given actor provides information on the rights (permissions) he has on each sort (task) of the GMWf. The nomenclature of the rights we handle is inspired by the one used in Unix-like operating systems. Three types of accreditation are therefore defined: accreditation in reading (r), in writing (w) and in execution (x).

1. Accreditation in reading (r): an actor accredited in reading on sort X, may be not only informed on the execution of the related task, but he may also access its execution status. We call an actor's vue the set of sorts on which he is accredited in reading.

Accreditation in writing (w):

an actor accredited in writing on sort X is in charge of the execution of the related task. Any actor accredited in writing on a symbol must necessarily be accredited in reading on it.

Accreditation in execution (x):

an actor accredited in execution on sort X is authorized to ask to the actor who is accredited in writing on it to execute it.

More formally, an accreditation is defined as follows:

Définition 4 An accreditation A A i defined on the set S of grammatical symbols for an actor A i , is a triplet

A A i = A A i (r) , A A i (w) , A A i (x) such that, A A i (r) ⊆ S also called view of actor A i ,

is the set of symbols on which A i is accredited in reading,

A A i (w) ⊆ A A i (r)
is the set of symbols on which A i is accredited in writing and A A i (x) ⊆ S is the set of symbols on which A i is accredited in execution.

On the expressiveness of our language

Our language (LSAWfP) covers the three workflow conceptual models identified in [START_REF] Divitini | Interorganizational workflows for enterprise coordination[END_REF]. In our case, the organizational model (that expresses and assigns the resources that must execute tasks) is specified using the couple L P k , L A k . As for the information model (it describes the data structure being manipulated), it is given by the type of the attribute status 3 . LSAWfP does not impose any constraints on the type of this attribute and leaves the responsibility to the designer to specify it; by default it is a string type. The process model (that provides information on the tasks and their sequencing) is provided by the GMWf (G). Note that it is possible, in our case, to represent all forms of routing between tasks (sequential, parallel, alternative 4 and iterative 5 ).

Compared to other languages used in the literature (Business Process Model and Notation [START_REF]Notation (BPMN) version 2.0. OMG Specification, Object Management Group[END_REF], Petri Nets [START_REF] Wil | The Application of Petri Nets to Workflow Management[END_REF], Petri Nets with Objects [START_REF] Amine Chaâbane | Actes des 7èmes journées scientifiques des jeunes chercheurs en Génie Electrique et Informatique (GEI)[END_REF] etc.), ours is mainly distinguished by the fact that it allows designers to statically specify the accreditations of various actors with regard to each task.

GMAWfP P2P execution model and corresponding algorithms

Key elements and constraints of the execution model

The execution environment

To execute a given GMAWfP in a decentralized mode, we use a completely decentralized (P2P) WfMS model (which we call P2P-WfMS-View), proposing an artifact-centric execution of business processes specified using grammatical models. For this purpose, instances of a P2P-WfMS-View are installed on the sites of the various actors involved in processes execution. These communicate (sending/receiving requests/responses) by service invocation.

During the process execution, each peer keeps a copy of a (global) artifact said to be under execution, which represents the current process execution status. Such an artifact provides information on already executed tasks, on those ready to be executed and on their executors. Technically, an artifact under execution is materialized by the presence within it of buds. These indicate at a moment, the only places where contributions are expected. A bud can be either unlocked or locked depending on whether the corresponding task is ready to be executed or not. Buds are typed; a bud of type X ∈ S is a leaf node labelled either by X ω or by X ω depending on its state (locked or unlocked) (see fig. 2). The local actions of a given actor will therefore have the effect of extending its local copy of the (global) artifact by developing, through appropriate productions, the different unlocked buds it contains.

Confidential execution of certain tasks

For confidentiality reasons, each actor acts potentially on a partial replica of his local copy of the (global) artifact. Technically, a partial replica t V i of an artifact t is obtained by 4. Such routing is represented by several productions with the same symbol as left-hand side. 5. To do such routing, all you need is to include recursive symbols in the definition of GMWf productions.

projection (using an operator π said of artifact projection) of t according to the view V i of the concerned actor: we note t V i = π V i (t). To ensure the effectiveness of the confidential execution of certain tasks, each actor acts locally through a specialized graphic tool (see fig. 3).

The need of a local GMWf at each site

Since the local actions of a particular actor depend on his perception of the process, it is necessary to control them in order not only to preserve the possible confidentiality of certain tasks, but also to ensure the consistency of local updates. To do this, we must derive a local GMWf on each site, by projecting the global GMWf according to the view of the local actor (GMWf projection). This projection is carried out using Π operator and the GMWf obtained is noted

G V i = Π V i (G).

The expansion operation

Still with the aim of ensuring system convergence, the contributions made by a given actor and contained in an updated partial replica t ma j V i , must be integrated into the local copy of the (global) artifact before any synchronization between peers. It is therefore necessary to be able to merge these two artifacts, which are based on two different models. We find here, a version of the expansion problem as formulated in [START_REF] Badouel | Merging Hierarchically-Structured Documents in Workflow Systems[END_REF].

Execution model and peer activity

Globally then, before the execution of a given process, peers are configured using its GMAWfP W f = G, L P k , L A k . From the global GMWf G and the view V i of the local actor, each peer derives a local GMWf G V i = Π V i (G). Then, the execution of a case of the process is triggered when an artifact t is introduced into the system (on the appropriate peer); this artifact is actually an unlocked bud of the type of the axiom A (initial task) of G. During execution, peers synchronize themselves by exchanging their local copies of the artifact being executed.

After receiving an artifact on a given peer, the latter merges it (see fig. 3) with its local copy (if it exists) and then the result is projected. The partial replica obtained is then completed when needed, using the specialized editor.

At the end of the completion, the expansion-pruning of the obtained updated partial replica is made in order to obtain the updated configuration of the (global) artifact local copy. If the resulting configuration shows that the process should be continued at other sites 6 , then replicas of the artifact are sent to them by invoking the service "forwardTo". Else 7 , a replica is returned to the pair from whom the artifact was previously received by invoking the service "returnTo".

The main algorithms of the execution model

The GMAWfP execution model is mainly based on three algorithms: artifact projection, GMWf projection and expansion. In this section, we propose versions of these algorithms.

6. This is the case when the artifact contains buds created on the current peer and whose actors accredited in writing are on distant peers.

7. The artifact is complete (it no longer contains buds), or semi-complete (it contains buds that were created on other peers and on which, the actor on the current peer is not accredited in writing).

The artifact projection algorithm

Technically, the projection t V i of an artifact t in the view V i = A A i (r) is obtained by deleting in t all nodes whose types do not belong to V i . In our case, the main challenges in this operation are: (1) to preserve the previously existing execution order between nodes of t, (2) to preserve the use of the only two forms of production retained for GMWf and (3) to obtain only one artifact after projection to ensure the continuation of processes execution.

Our operation is noted π. Inspired by the one proposed in [START_REF] Badouel | Merging Hierarchically-Structured Documents in Workflow Systems[END_REF], we project an artifact by preserving the hierarchy (father-son relationship) between nodes of the artifact (we thus meet challenge (1)); but in addition, we insert into the projected artifact when necessary, new additional (re)structuring symbols (accessible in reading and writing by the actor for whom the projection is made). This enables us to meet challenge (2). Figure 4 presents an artifact t as well as two partial replicas t v 1 and t v 2 respectively obtained from the views V 1 = {A, B,C} and V 2 = {A, B, D}. Note the presence in t v 2 of the (re)structuring symbol

S1.

The algorithm: concretely, to project an artifact t according to a given view V (i.e to find pro js t = π V (t)), the following recursive processing is applied to the root node n = X [t 1 , . . . ,t m ] (this notation indicates that n is labelled with the symbol X and has m sub-artifacts t 1 , . . . ,t m ) of t (note by p n , the production of the GMWf that was used to extend node n; note also that the algorithm described here can return several artifacts):

Algorithm for projecting a given artifact according to a given view.

• If symbol X is visible (X ∈ V ) then :

1. n is kept in the artifact; 2. For each sub-artifact t i of n, having node n i = X i t i 1 , . . . ,t i k as root (of which p n i is the production that was used to extend it), the following processing is applied : a. The projection of t i according to V is done. We obtain the list pro js

t i = π V (t i ) = t i V 1 , . . . ,t i V l ; b.
If the type of p n i is the same as the type of p n or the projection of t i has produced no more than one artifact (|pro js t i | ≤ 1), we just replace t i by artifacts

t i V 1 , . . . ,t i V l of the list pro js t i ;
Otherwise, a new (re)structuring symbol S i is introduced and we replace the sub-artifact t i with a new artifact new_t i whose root node is

n t i = S i t i V 1 , . . . ,t i V l ; 3. If the list of new sub-artifacts of n contains only one element t 1 having n 1 = S 1 t 1 V 1 , . . . ,t 1 V l
(with S 1 a newly created (re)structuring symbol) as root node, we replace in this one, t 1 by the sub-artifacts

t 1 V 1 , . . . ,t 1 V l of n 1 .
This removes a nonimportant (re)structuring symbol.

• Else, n is deleted and the result of the projection (pro js t ) is the union of the projections of each of its sub-artifacts:

pro js t = π V (t) = m i=1 π V (t i )
To avoid that the previous algorithm produces a forest in some cases and thus meet challenge (3), we make the following assumption: GMAWfP manipulated in this work are such that all actors are accredited in reading on the GMWf axiom (axiom's visibility assumption). It should be noted that with our language, all administrative processes can be designed in a form that validates this assumption.

GMWf projection algorithm

The goal of this algorithm is to derive by projection of a given GMWf G = (S , P , A G )

according to a view V , a local GMWf G V = S V , P V , A G V (we note G V = Π V (G)).
The algorithm we propose is as follows:

Algorithm for projecting a given GMWf according to a given view.

1.

First of all, it is necessary to generate all the target artifacts denoted by G; we thus obtain a set arts G = {t 1 , . . . ,t n }; 2. Then, each of the target artifacts must be projected according to V . We thus obtain a set arts G V = t V 1 , . . . ,t V m (with m ≤ n because there may be duplicates; in this case, only one copy is kept) of artifacts partial replicas; 3. Then, collect the different (re)structuring symbols appearing in artifacts of arts G V , making sure to remove duplicates and to consequently update the artifacts and the set arts G V . We thus obtain a set S V Struc of symbols and a final set arts G V = t V 1 , . . . ,t V l (with l ≤ m) of artifacts. These are exactly the only ones that must be conform to the searched GMWf G V . So we call them, local target artifacts for the view V ; 4. At this stage, it is time to collect all the productions that made it possible to build each of the local target artifacts for the view V . We obtain a set P V of distinct produc-tions. REMARK. -The GMWf projection algorithm presented here only works for GMWf that do not allow recursive symbols 8 . We therefore assume that, for this execution model, the manipulated GMAWfP are those whose GMWf do not contain recursive symbols (nonrecursive GMWf assumption). Therefore, it is no longer possible to express iterative routing between process tasks (in the general case); except in cases where the exact number of iterations is known in advance.

The searched local GMWf

G V = S V , P V , A G V is

The expansion algorithm

Consider a (global) artifact under execution t and t V = π V (t), its partial replica on the site of an actor A i whose view is V . Consider the partial replica t ma j V ≥ t V obtained by developing some unlocked buds of t V as a result of A i contribution. The expansion problem is to find a (global) artifact under execution t f , which integrates nodes of t and t V . To solve this problem made difficult by the fact that t and t V are conform to two different models (G and G V = Π V (G)), we perform a three-way merge [START_REF] Mens | A State-of-the-Art Survey on Software Merging[END_REF]. We merge the artifacts t and t V using a (global) target artifact t g such that: (a) t is a prefix of t g 8. It is only in this context that all the target artifacts can be enumerated.

(t ≤ t g ) and (b) t ma j

V is a prefix of the partial replica of t g according to V (t ma j V ≤ π V (t g )).

Our algorithm proceeds in two steps.

Step 1 -Search for the merging guide t g : the algorithm is as follows :

Algorithm searching for a merging guide.

1. First of all, we have to generate the set arts G = {t 1 , . . . ,t n } of target artifacts denoted by G; 2. Then, we must filter this set to retain only the artifacts t i admitting t as a prefix (criterion (a)) and whose projections according to V (t i V j ) admit t ma j V as a prefix (criterion (b)). We obtain the set guides = t g 1 , . . . ,t g k of artifacts that can guide the merging; 3. Finally, we randomly select an element t g from the set guides.

Step 2 -Merging t, t ma j V and t g : we want to find an artifact t f that includes all the contributions already made during the workflow execution. The structure of the searched artifact t f is the same as that of t g : hence the interest to use t g as a guide. The merging is carried out as follows :

Three-way merging algorithm.

A prefixed depth path of the three artifacts is made simultaneously until there is no longer a node to visit in t g . Let n t i (resp. n t ma j V j and n t g k ) be the node located at the address w i (resp. w j and w k ) of t (resp. t ma j V and t g ) and currently being visited. If nodes n t i , n t ma j V j and n t g k are such that (processing):

1. n t ma j V j is associated with a (re)structuring symbol (fig. 6(d)) then: we take a step forward in the depth path of t ma j V and we resume processing; 2. n t i , n t ma j V j and n t g k exist and are all associated with the same symbol X (fig. 6(a) and 6(b)) then: we insert n t ma j V j (it is the most up-to-date node) into t f at the address w k ; if n t ma j V j is a bud then we prune (delete sub-artifacts) t g at the address w k ; we take a step forward in the depth path of the three artifacts and we resume processing.

n t i , n t ma j

V j and n t g k exist and are respectively associated with symbols X i , X j and X k such that X k = X i and X k = X j (fig. 6(e)) then: we add n t g k in t f at address w k . This is an upstair bud; we take a step forward in the depth path of t g and we resume processing. 4. n t i (resp. n t ma j V j ) and n t g k exist and are associated with the same symbol X (fig. 6(c) and 6(f)) then: we insert n t i (resp. n t ma j V j ) into t f at the address w k ; if n t i (resp. n t ma j V j

) is a bud, we prune t g at the address w k ; we take a step forward in the depth path of the artifacts t (resp. t ma j V ) and t g , then we resume processing.

Note that an upstair bud is a node associated with a process task that has not yet been executed, and which is revealed during expansion rather than during the execution phase via the specialized editor (this is due to the fact that it is associated with a symbol that does not belong to the view of the considered site). Such a bud appears above other buds in the expansion artifact (hence its name).

Conclusion

In this work, we proposed LSAWfP, a Workflow Specification Language using grammatical models. We then presented a decentralized and artifact-centric model of workflow processes execution (P2P-WfMS-View). Based on the principles of this model, we proposed versions of its key algorithms (artifact projection, GMWf projection and expansionpruning). These algorithms are perfectly usable (proof of their stability is given in appendix A). We implemented them in several languages and tested them with highly satisfactory results (see appendix B). However, in order to ensure that our algorithms produce the expected results, we have made some assumptions. In particular, the assumption of non-recursivity of GMWf, which had the direct effect of slightly limiting the expressiveness of our language. Therefore, one perspective of this work, is to propose other versions of the algorithms presented here, which would incorporate the same key principles while addressing the assumption of non-recursivity of GMWf in order to offer more comfort to GMAWfP's designers.

Figure 1 .

 1 Figure 1. Example of target artifacts for a given process (peer-review process)

2 .

 2 For example, this is true for a process with four tasks A,B,C et D such as: A precedes all the others, B and C can be concurrently executed and they precede D. Here, the insertion of a new symbol S solves the issue and allows to obtain the following productions: p 1 : A → S D, p 2 : S → B C, p 3 : B → ε, p 4 : C → ε and p 5 : D → ε.

3 .

 3 Reminder: each task is represented by a grammatical symbol with an attribute named status (see definition 2)

Figure 2 .

 2 Figure 2. An intentional representation of an annotated artifact containing buds.

Figure 3 .

 3 Figure 3. Activity of a peer in the system.

Figure 4 .

 4 Figure 4. Example of projections made on an artifact and partial replicas obtained.

  such as: a. its set of symbols isS V = V ∪ S V Struc ; b. its set of productions is P V ; c. its axiom is A G V = A G Illustration: Figure 5 illustrates the research of a local model G V EC such as G V EC = Π V EC (G) with V EC = A EC(r) = {A,B,C, D, H1, H2, I1, I2, F}. Target artifacts generated from G (fig. 5(b)) are projected to obtain two local target artifacts for the view V EC (fig. 5(c)). From the local target artifacts thus obtained, the searched GMWf is produced (fig. 5(d)).

Figure 5 .

 5 Figure 5. Example of projection of a GMWf according to a given view.

Figure 6 .

 6 Figure 6. Some scenarios to be managed during the expansion.

These are variable processes of which we know all the cases; that is, the tasks are predictable and the sequence are simple and clearly defined[START_REF] Wil | The Application of Petri Nets to Workflow Management[END_REF].

A. Some properties of our projection algorithms

Proposition 5 For all GMAWfP W f = G, L P k , L A k verifying the axiom's visibility as- sumption, the projection of an artifact t which is conform to its GMWf (t ∴ G) according to a given view V , results in a single artifact t V = π V (t) (stability property of π).

Proof. Note that the only case in which the projection of an artifact t according to a view V produces a forest, is when the root node of t is associated with an invisible symbol X (X / ∈ V ). Knowing that t ∴ G and that W f validates the axiom's visibility assumption, it is deduced that the root node of t is labelled by the axiom A G of G and that A G ∈ V (hence the uniqueness of the produced tree). Since the projection operation preserves the form of productions, it is concluded that t V = π V (t) is an artifact. ✷ Proposition 6 For all GMAWfP W f = G, L P k , L A k verifying the axiom's visibility and the non-recursivity of GMWf assumptions, the projection of its GMWf G = (S , P , A G )

the assumptions of axiom visibility and non-recursivity of GMWf (stability property of Π).

Proof. As W f = G, L P k , L A k validates the non-recursivity of GMWf assumption, the set of target artifacts (arts G = {t 1 , . . . ,t n }) that it denotes is finished and can therefore be fully enumerated. Knowing further that W f validates the axiom's visibility assumption, we deduce that the set arts

) is finite and the root node of each artifact t V i is associated with the axiom A G of G (see proposition 5). G V being built from the set arts G V , its axiom A G V = A G is visible to all actors and its productions are only of the two forms retained for GMWf. In addition, each new (re)structuring symbol (S ∈ S V Struc )) is created and used only once to replace a symbol that is not visible and not recursive (by assumption) when projecting artifacts of arts G . The new symbols are therefore not recursive. By replacing in

verifying the assumptions of axiom visibility and non-recursivity of GMWf. ✷ Proposition 7 For all GMAWfP W f = G, L P k , L A k verifying the axiom's visibility and the non-recursivity of GMWf assumptions, the projection of an artifact t which is conform to the GMWf G according to a given view V , is an artifact which is conform to the projection of G according to

through the renaming of some potential (re)structuring symbols, is part of the set arts G V of artifacts that have generated

B. Experimentation with P2PTinyWfMS

The algorithms presented in this manuscript, have been coded in Haskell and Java then tested with satisfactory results. In java, we introduced these algorithms in the tool P2PTinyWfMS. It is a tool developed under Eclipse 9 and dedicated to the simulation of the completely decentralized execution of administrative workflows specified using LSAWfP. P2PTinyWfMS has a front-end for the display and the graphical editing of artifacts handled during the execution of a business process (see fig. 7), as well as a Service-Oriented communication module built using SON (Shared-data Overlay Network) [START_REF] Ait | A Lightweight Middleware for Developing P2P Applications with Component and Service-Based Principles[END_REF].