Ana Sofia Dias

Marianna Pirone

Gianfranco Urciuoli

Review on Types of Root Failures in Shallow Landslides

Keywords: Root reinforcement, Root-soil interaction, Root failure mechanisms, Shallow landslides

Nowadays the interest of geotechnical engineers for green solutions is being developed and the use of vegetation as a reinforcement to improve slope stability is growing. The sliding surface of shallow landslides tends to not exceed 1.5-2 m depth, and as a consequence it can be crossed by roots that, in this case, work as a stabilizing measure. Therefore, the study of the soil-roots interaction is necessary to quantify the contribution of vegetation to the stability of shallow landslides. The goal of this paper is to provide an overview of the root failure mechanisms that can occur along the sliding surface and of the forces applied by roots, in order to evaluate the safety factor of a reinforced slope. Several prevailing stress states occur along a shallow landslide failure surface: tension stress at the slide crest, shear stresses along the base of the unstable soil layer and passive earth pressures at the slope toe. Some considerations are also made regarding acceptable simplifications, in terms of root geometry and soil-root friction strength, that are currently assumed in the literature.

Introduction

Introducing vegetation in a slope can be considered a stabilization measure, as a result of the beneficial hydro-mechanical effects of plants on preventing shallow landslides. In that case, the sliding surface occurs within the first 1.5-2 m from the ground surface where the plants' roots can penetrate [START_REF] Wu | Investigation of landslides on Prince of Wales Island, Alaska[END_REF]Schwarz et al. 2010b;[START_REF] Burylo | Soil reinforcement by the roots of six dominant species on eroded mountainous marly slopes (Southern Alps, France)[END_REF]. For this reason the assessment of the efficiency of such measure is an interesting task for geotechnical engineers, but it is possible only if the root failure mechanics are clearly understood.

The current review is mainly devoted to investigate the mechanical response of the roots' fibres to different types of stresses acting in a slope, focusing on those occurring along a shallow landslide (Fig. 1). Roots may be stressed by (a) pull out forces, (b) shear ones or (c) compression ones [START_REF] Schwarz | Root reinforcement of soils under compression[END_REF]. In general, the root in the situations (a) and (b) may slip out or break depending on the soil and root types, moisture and root-soil friction [START_REF] Ennos | The anchorage of leek seedlings: the effect of root length and soil strength[END_REF][START_REF] Mickovski | Material stiffness, branching pattern and soil matric potential affect the pullout resistance of model root systems[END_REF][START_REF] Pollen | Temporal and spatial variability in root reinforcement of streambanks: accounting for soil shear strength and moisture[END_REF]. In situation (c), [START_REF] Wu | Study of soil-root interaction[END_REF] and [START_REF] Schwarz | Root reinforcement of soils under compression[END_REF] showed that the most common mechanism is the occurrence of buckling, even though root bending or root tension for larger displacements may be observed. However, the mechanisms previously appointed are complex and depend on the root interaction with the surrounding soil, as well as on root features. Much investigation is still required on this topic, [START_REF] Mao | Evaluation of root reinforcement models using numerical modelling approaches[END_REF] stated that the mechanisms by which vegetation roots mechanically interact with soil are still not well described. In this paper, first the description of the root-soil interaction, root resistance and stiffness will be introduced, then a description of the main root failure modes is presented.

Root-Soil Coupling

Soil-Root Frictional Strength

The root-soil friction strength depends on the type of soil, soil moisture and confining pressure around the root [START_REF] Pollen | Temporal and spatial variability in root reinforcement of streambanks: accounting for soil shear strength and moisture[END_REF][START_REF] Mickovski | Material stiffness, branching pattern and soil matric potential affect the pullout resistance of model root systems[END_REF][START_REF] Bourrier | Discrete modelling of granular soils reinforcement by plant roots[END_REF][START_REF] Veylon | Quantification of mechanical and hydric components of soil reinforcement by plant roots[END_REF] and can be calculated following different approaches.

Referring to an ideal straight smooth fibre, as a basic ingredient for the analysis, the soil-root friction 𝜏 𝑏 , or so-called bond or interface friction stress, was suggested to be given by (1) in the works of [START_REF] Gray | Mechanics of fiber reinforcement in sand[END_REF] and [START_REF] Cazzuffi | Geosynthetic engineering and vegetation growth in soil reinforcement applications[END_REF], as it had been assessed in the previous work of Potyondy (1961). In Eq. ( 1), 𝑧 is the depth of the root from ground surface, 𝛾 is the soil unit weight, 𝑓 is a reductive coefficient (0.7-0.9) modelling the root-soil interface and 𝜙′ is the internal friction angle of the soil. [START_REF] Ennos | The anchorage of leek seedlings: the effect of root length and soil strength[END_REF], on the other hand, considered that 𝜏 𝑏 was constant in depth, given by ( 2), where 𝜏 is the soil shear strength near the root and 𝛼 is the relative strength of the root-soil bond (0 ≤ 𝛼 ≤ 1). This was a reasonable approximation because the roots studied by Ennos presented lengths up to 5 cm.

𝜏 𝑏 = 𝑧 𝛾 (1 -sin 𝜙 ′ ) 𝑓 tan 𝜙′ (1) 𝜏 𝑏 = 𝛼𝜏
(2) In the research of [START_REF] Naaman | Fiber pullout and bond slip. I: analytical study[END_REF] on fibres subjected to pull out load, a mathematical model that describes the response of straight smooth fibres was developed. A distinction between two types of friction, static and dynamic, is made (Fig. 2). The static friction, or bonded, is mobilized when the root is being stretched and is essentially regulated by the peak strength of the soil. Once the root starts to slip out, friction is essentially regulated by the ultimate shear strength of the soil (critical or residual one). It becomes constant along the fibre taking the value of dynamic friction, or debonded friction. The debonded friction is determined by the following phenomena: the pull out force applied to the root strongly modifies the stresses in the shallow subsoil, causing the detachment of the root from the soil and a decrease of the root diameter (Poisson's effect), as well as a decrease of the confining stress around the root [START_REF] Naaman | Fiber pullout and bond slip. I: analytical study[END_REF]) that reduces the bonded effect, supporting the assumption of Schwarz et al. (2010a). [START_REF] Cazzuffi | Geosynthetic engineering and vegetation growth in soil reinforcement applications[END_REF] always consider the debonded friction even if the root-soil bond is still intact.

Fig. 2 Mobilized root-soil friction along the depth of the root

The static friction was ignored by Schwarz et al. (2010a) after showing that it represents a small percentage of the total friction. The friction values calculated using (1) and ( 2) are not a bonded friction, but an alternative way of calculating debonded friction for shallow root systems; in particular Eq. ( 1) provides constant values of 𝜏 𝑏 for roots developing in horizontal direction.

Even in Schwarz et al. (2010a), the debonded friction was simply calculated using Mohr-Coulomb equation. However, the variation of the normal stress with depth (increase of overburden pressure) cannot be ignored in case long vertical roots are considered.

A positive correlation between the root additional resistance and the vertical effective confining stress is reported in the work of [START_REF] Liang | Modelling the seismic performance of rooted slopes from individual root-soil interaction to global slope behaviour[END_REF]. As mentioned by this author, for deeper rooting systems, the normal effective stress varies significantly due to the deeper embedding depth.

Moreover, normal stress may also vary along the periphery of a root when the applied force presents an horizontal component (e.g. when the root intersects the sliding surface, Fig. 3). For this reason the friction will be greater on one side of the root and may be smaller (or null) in the other, as a consequence of the root deformation [START_REF] Nghiem | Analysis of root reinforcement at slip surface[END_REF]Belfiore and Uriciuoli 2004). Even detachment of the root from the soil may occur in cohesive soils, [START_REF] Wu | Study of soil-root interaction[END_REF]. Evidence is presented in the numerical analysis of [START_REF] Dupuy | A numerical investigation into factors affecting the anchorage of roots in tension[END_REF], [START_REF] Vergani | Effects of root tensile force and diameter distribution variability on root reinforcement in the Swiss and Italian Alps[END_REF], [START_REF] Wu | Root reinforcement of soil: review of analytical models, test results, and applications to design[END_REF] and [START_REF] Bourrier | Discrete modelling of granular soils reinforcement by plant roots[END_REF] that show the stresses in roots' surrounding soil. In Fig. 3 the deformation of a root crossing a soil layer subjected to shear and to a relative displacement along a sliding surface is presented; the load applied normally to the axis of the root and the tensile axial force are also indicated.

Tortuosity Effect

A typical response to a pull out force applied to a root embedded in the soil is presented in Fig. 4, in which four phases can be identified. Displacements of the initial phase are a consequence of the root straightening corresponding to a small force. Then, the root material strength is effectively mobilized. However, at the initial rearrangement of the root geometry, generally strains up to 5% can take place (Schwarz et al. 2010b), representing a critical issue in slope stability problems. In Fig. 4, the Young's modulus of the root in the phase I is apparent and lower than in phase II, because the root is being straightened. This is also known as the tortuosity effect, described by [START_REF] Commandeur | Modulus of elasticity and tensile strength of Douglas-fir roots[END_REF], as an influence of the root geometry on the macroscopic elastic behaviour of a pulled root.

For Schwarz et al. (2010b), the effect is considered the form of apparent Young's modulus (𝐸 𝑎𝑝𝑝 ) by multiplying the Young's modulus of straight roots, which is equal to Young's modulus of the root material (𝐸 𝑚 ), by an empirical reductive coefficient (𝑟) that can vary from 0 to 1, as presented in (3). This relation is only applied in case the strain is less than the threshold value corresponding to the end of straightening. Both reductive coefficient and threshold value depend on root tortuosity and soil conditions, such as soil type, confining pressure and water content.

𝐸 𝑎𝑝𝑝 = 𝑟𝐸 𝑚

(3) After the root is straightened, the increase of load tensioning the root is transferred gradually to the soil through friction (phase II of Fig. 4). During the mobilization of this frictional resistance there is no appreciable extension of the root and the Young's modulus increases (Schwarz et al. 2010b). To describe the variation of the Young's modulus with strain (𝜀), Eq. ( 4) is used, where 𝐸 𝑎𝑝𝑝 is the apparent Young's modulus of progressively activated root length, 𝐸(𝑑) is the Young's modulus of the root of diameter 𝑑, and 𝛽 and 𝜔 are fitting parameters derived from laboratory experiments.

𝐸 𝑎𝑝𝑝 = 𝐸(𝑑)𝛽𝜀 -𝜔

(4) Simplifying the previous approaches, the secant Young's modulus is used in [START_REF] Schwarz | Modelling root reinforcement using a root-failure Weibull survival function[END_REF], as the ratio of root strength over strain at failure. Nevertheless, this operation does not affect the estimation of the maximum tensile load of a root, only the value of strain at which the maximum force is observed, as highlighted by the author.

Root Failure Modes

Pull Out Gray and Leiser (1996) highlighted that if the fibres are very short, unconstrained and subjected to low confining stresses, they will tend to slip out when the root is tensioned. In fact, when a root is pulled, shear friction between the root and soil takes place, and the failure occurs either by the rootsoil bond or by the soil. Nevertheless, roots will continue to contribute to the soil reinforcement, as observed in phase IV of Fig. 4, depending on the contact area between root and soil that decreases with the removal of the root. [START_REF] Gray | Mechanics of fiber reinforcement in sand[END_REF] studied the mechanics of fibre-reinforced soil and showed the importance of the mechanisms of root stretching and root pulling out. During the direct shear test carried out by these authors, none of the fibres immersed in the tested sand broke in tension. Indeed, they either pulled out or stretched depending upon the confining stress and length. Back calculations revealed that in those tests less than 25% of the tensile strength of the fibres was mobilized. In those direct shear tests, [START_REF] Gray | Mechanics of fiber reinforcement in sand[END_REF] observed that the fibre-reinforced soil tended to increase the ultimate shear strength of the composite material, and to limit reductions in post peak shearing resistance (the material becomes more stable and ductile), as in Fig. 5. As a consequence of these results, [START_REF] Gray | Mechanics of fiber reinforcement in sand[END_REF] classified the roots as ideally extensible inclusions, i.e. inclusions characterized by failure strains larger than the maximum strains in the soil without inclusions, allowing a greater ductility of the composite material [START_REF] Dupuy | A numerical investigation into factors affecting the anchorage of roots in tension[END_REF].

Fig. 5 Stress strain curve of fibre-reinforced soil (1 ksf = 47.9 kN/m 2 ) [START_REF] Gray | Mechanics of fiber reinforcement in sand[END_REF] The force, 𝐹 𝑃 , necessary to break the root-soil bond (also known as pull out resistance) is given by ( 5), where 𝐿 is the root length and 𝜏 𝑝 is the root-soil frictional resistance [START_REF] Cazzuffi | Geosynthetic engineering and vegetation growth in soil reinforcement applications[END_REF].

𝐹 𝑃 = 𝜋𝑑𝐿𝜏 𝑝

(5) To be noticed that in the work of Schwarz et al. (2010a), a great advance was achieved by considering the evolution of root diameter (𝑑) with length and root tortuosity. This has been ignored by many authors, such as [START_REF] Ennos | The anchorage of leek seedlings: the effect of root length and soil strength[END_REF] and [START_REF] Gray | Mechanics of fiber reinforcement in sand[END_REF]. The use of constant diameter could become a rough approximation because the transmission of stress from root to friction in the soil varies, not only with the normal stress and soil frictional properties, but also with the available contact area between soil and roots. Therefore, with decreasing root diameter, there is less and less capability of transferring stresses to the soil, and consequently the root will pull out faster.

In the work of Schwarz et al. (2010a) a root of tortuous length 𝑙 (length along root axis) was discretised using elements of length 𝑏. Each of these elements presents a different diameter, which is function of the diameter of the root tip (𝑑 0 ) and of a scaling factor (𝑠) called root diameter proportionality factor. The total soil-root friction force in a root segment is dependent on its lateral surface area.

As reported in [START_REF] Dupuy | A numerical investigation into factors affecting the anchorage of roots in tension[END_REF] and [START_REF] Mickovski | Material stiffness, branching pattern and soil matric potential affect the pullout resistance of model root systems[END_REF], the additional resistance due to the presence of branches along the studied root ranges from 0.1 to 5 N per branching point, depending on soil moisture and branching geometry. Once that the increase of resistance in presence of branches may not be despicable, Schwarz et al. (2010a) proposed that the maximum pull out force of a root (𝐹 𝑝𝑙𝑏 ) is given by ( 6), where 𝑌 is an empirical branching coefficient (0.1 < 𝑌 < 1) and 𝑑 𝑖 are the diameters of each segment of the discretised root's length (𝑖 segments).

𝐹 𝑝𝑙𝑏 = 𝑏𝜏 𝑝 𝑑 0 𝜋 + (𝜋𝑏𝜏 𝑝 + 𝑌) ∑ 𝑑 𝑖 (6)
Nevertheless, roots interact among them in a root system leading to a superposition of shear stresses within the soil. To take into account this interaction Giadrossich et al. ( 2013) proposed ( 7) to obtain the global pull-out resistance of a bundle of 𝑛 roots, 𝑓 𝑏𝑙𝑜𝑐 . In this equation, 𝑤 𝑖 is the radial distance from the surface of one of the roots of the bundle to a given point 𝑥 in the surrounding soil. Therefore, the shape of a block of soil permeated by 𝑛 roots that is pulled out is obtained by determining the points 𝑥 where the soil strength is exceeded by 𝑓 𝑏𝑙𝑜𝑐 .

𝑓 𝑏𝑙𝑜𝑐 = 𝜏 𝑝 ∑ 𝑑 𝑖 𝑤 𝑖 𝑛 𝑖=1 (7)

Breakage

A root breaks when the tensile load applied to the cross-section area exceeds its tensile strength, assuming that the root is well anchored in the soil. This way is observed in the phase III of Fig. 4, in which there is an abrupt drop in the root tension, with a null capacity during phase IV.

From experimental works on root tensile strength, 𝑡 𝑟 (force distributed in the area of the cross section of the root, in MPa), a simple empirical relation with root diameter with the form of a power-law was obtained. It is represented by ( 8), where 𝑑 is the root diameter at the section of breakage [mm] and the parameters 𝜁 and 𝜌 are calibration constants.

𝑡 𝑟 = 𝜁𝑑 𝜌 (8) The experimental works showed that 𝜌 takes values always lower than zero because the roots with larger diameters present lower tensile strengths. Some experimental values can be found in the works of [START_REF] Gray | Mechanics of fiber reinforcement in sand[END_REF], [START_REF] Comino | Root reinforcement effect of different grass species: a comparison between experimental and models results[END_REF], [START_REF] Burylo | Soil reinforcement by the roots of six dominant species on eroded mountainous marly slopes (Southern Alps, France)[END_REF] and [START_REF] Gray | Biotechnical and soil bioengineering, slope stabilization[END_REF]. Obviously, the root tensile load 𝐹 𝐵 is given by ( 9). It may be of interest to mention that the strength of roots decrease when the vegetation is removed. The computation of the root strength decay can be found in more detail in [START_REF] Cazzuffi | Geosynthetic engineering and vegetation growth in soil reinforcement applications[END_REF].

𝐹 𝐵 = 𝜋(𝑑/2) 2 𝑡 𝑟 (9)

Critical Length

According to Fig. 2, the increase of pull out force (𝐹) mobilizes friction on increasing root length. If that force reaches the root tensile resistance (𝐹 𝐵 ), the root breaks. If all the root-soil interaction resistance is mobilized, then 𝐹 takes the value of the pull out resistance 𝐹 𝑃 . The critical length is obtained by equalling ( 5)-( 9) and it represents the separation of the occurrence of breakage from slippage. Therefore, in order to break, a root should have a length greater than the critical length 𝐿 𝑐𝑟𝑖𝑡 , given by ( 10). Consequently, the critical length also represents the maximum length along which friction can be mobilized.

𝐿 𝑐𝑟𝑖𝑡 = 𝑡 𝑟 𝑑 4𝜏 𝑝 (10)
This threshold value depends on the frictional strength between roots and soil and on the tensile strength of the roots of the given species. Roots with small diameters tend to slip out and roots with larger diameters tend to break [START_REF] Pollen | Temporal and spatial variability in root reinforcement of streambanks: accounting for soil shear strength and moisture[END_REF].

Buckling [START_REF] Schwarz | Root reinforcement of soils under compression[END_REF] investigated the influence of the roots on soil resistance under compression focusing on failure, due to passive earth pressures at the toe of a landslide. Moreover roots are also subjected to compression when root orientation is opposite to the movement of the unstable layer [START_REF] Gray | Mechanics of fiber reinforcement in sand[END_REF][START_REF] Wu | Study of soil-root interaction[END_REF]).

However, the reinforcement provided by roots has been neglected, based on the assumption that the compressive strength of roots is low compared with that of soil [START_REF] Cazzuffi | Geosynthetic engineering and vegetation growth in soil reinforcement applications[END_REF]). Nevertheless, it was found that the resistance of some roots in compression was of the same order of magnitude of the tensile one. For example, roots having diameters of 4 and 8 mm, failing at tensile forces of approximately 200 and 400 N could stand compression loads of 50 and 150 N, respectively, before buckling occurs [START_REF] Wu | Study of soil-root interaction[END_REF]).

In [START_REF] Wu | Study of soil-root interaction[END_REF], the measured values of critical load fell between the values predicted by the solutions of Toakley (1965) and Euler, solution 1 and 2, respectively (Fig. 6). [START_REF] Schwarz | Root reinforcement of soils under compression[END_REF] suggested the use of the solution by [START_REF] Timoshenko | Buckling loads for elastically supported struts[END_REF] in order to calculate the critical buckling load. Euler's solution is represented by ( 11), provided that a more conservative estimation was obtained (Fig. 6), where 𝐿 𝑏 is the length of the buckled root (between 3 and 10 cm for increasing root diameter in [START_REF] Wu | Study of soil-root interaction[END_REF], 𝑛 is the mode number, 𝐸 is the Young's modulus considered by the author, 𝐼 is the inertia of the root cross-section and 𝐹 𝑐 is the critical load.

𝐹 𝑐 = 𝑛 2 𝜋 2 𝐸𝐼 𝐿 𝑏 2 (11)
Fig. 6 Measured critical load compared with the theoretical solutions (adapted from [START_REF] Wu | Study of soil-root interaction[END_REF])

Modelling

Physical Model

In the first models proposed in the literature, roots were assumed to be fibrous elements without shear and bending strength, presenting only tensile resistance, [START_REF] Wu | Investigation of landslides on Prince of Wales Island, Alaska[END_REF] and [START_REF] Gray | Mechanics of fiber reinforcement in sand[END_REF].

It is generally accepted that roots present an elastic behaviour, as [START_REF] Comino | Root reinforcement effect of different grass species: a comparison between experimental and models results[END_REF], [START_REF] Burylo | Soil reinforcement by the roots of six dominant species on eroded mountainous marly slopes (Southern Alps, France)[END_REF] and [START_REF] Vergani | Effects of root tensile force and diameter distribution variability on root reinforcement in the Swiss and Italian Alps[END_REF], among others, did.

In order to describe the behaviour of the roots subjected to shear, [START_REF] Wu | Investigation of landslides on Prince of Wales Island, Alaska[END_REF] proposed an analytical model. The shearing of the soil causes the root to distort, as presented in Fig. 7, originating elongation of the roots that result into tensile stresses, which act on the unstable soil volume as stabilizing external forces [START_REF] Wu | Investigation of landslides on Prince of Wales Island, Alaska[END_REF][START_REF] Gray | Mechanics of fiber reinforcement in sand[END_REF]. 12) represents this decomposition assuming that the root out of the shear zone stays perpendicular to it [START_REF] Wu | Investigation of landslides on Prince of Wales Island, Alaska[END_REF], where 𝜃 is the shear distortion of the root and 𝑡 𝑟 is the tensile stress (Fig. 7).

{ 𝜎 𝑟 = 𝑡 𝑟 cos 𝜃 𝜏 𝑟 = 𝑡 𝑟 sin 𝜃 (12)

Beam on Elastic Foundation Model

Even though, many works assume that roots are cable elements whose only resistance is the tensile one, other authors investigated the bending and shear resistance of these elements. This is important not only for the case of large-diameter roots, where bending resistance can be mobilized, but also in order to describe the effect of soil movement on roots deformation, and the constrain of roots to the soil [START_REF] Nghiem | Analysis of root reinforcement at slip surface[END_REF]. [START_REF] Nghiem | Analysis of root reinforcement at slip surface[END_REF] compared the behaviour of a root to an inclined pile, which solution in terms of stresses and strains is derived from a problem of beam on elastic foundation, however ignoring the bending resistance of the root. [START_REF] Wu | Study of soil-root interaction[END_REF] and [START_REF] Wu | Root reinforcement of soil: review of analytical models, test results, and applications to design[END_REF] used cable and beam solutions for the same beam on elastic foundation problems. The comparison of the beam and cable models was produced by [START_REF] Mao | Evaluation of root reinforcement models using numerical modelling approaches[END_REF], differentiating the behaviour of different types of roots. [START_REF] Wu | Root reinforcement of soil: review of analytical models, test results, and applications to design[END_REF] suggested that the cable solution should be adopted when the tensile resistance of the root 𝑡 𝑟 is much greater than the flexure stiffness, 𝐸𝐼, i.e. when the product 𝜂𝐿 > 2.5. Conversely, the beam solution is the most adequate when 𝜂𝐿 < 1.5, where the parameter 𝜂 is given by ( 13).

𝜂 = √𝑡 𝑟 𝐸𝐼 ⁄ (13) The generic beam on elastic foundation problem is described by ( 14), where 𝑢 is the displacement in the direction normal to the root axis, 𝑧 is the coordinate along the axis and 𝑝 is the soil reaction, which is the product between the displacement 𝑢 and the reaction constant.

𝐸𝐼 𝑑 4 𝑢 𝑑𝑧 4 -𝑡 𝑟 𝑑 2 𝑢 𝑑𝑧 2 = 𝑝𝑑 (14)
The beam solution can be found in [START_REF] Wu | Study of soil-root interaction[END_REF]. The ultimate load 𝑝 𝑦 is given by ( 15), where 𝑐 is the soil cohesion, 𝑁 𝑐 , 𝑁 𝑞 and 𝑁 𝛾 are bearing capacity numbers, 𝑞 is the lateral surcharge and 𝑝 𝑦 is the yielding reaction pressure of the soil on the root [limit of 𝑝 of ( 14)].

𝑝 𝑦 = 𝑐𝑁 𝑐 + 𝑞𝑁 𝑞 + 1 2 ⁄ 𝛾𝑑𝑁 𝛾 (15) At large displacements, the depth of the plastic zone increases, which makes the flexible cable solution more adequate to describe the behaviour of the root, because the coupling between horizontal and vertical component of the tensile strength of a root should be considered [START_REF] Wu | Study of soil-root interaction[END_REF]. Due to the large deformations, the ultimate soil resistance 𝑝 𝑦 given by ( 15) is mobilized along all the root. This way, [START_REF] Wu | Study of soil-root interaction[END_REF] described the problem using ( 14) and adopted the flexible cable solution described by ( 16), where 𝑇(0) is the tensile force in the root at the shear surface and 𝑇(𝐿) is the tensile force at the tip of the root. For further details it is recommended to consult the work of [START_REF] Wu | Root reinforcement of soil: review of analytical models, test results, and applications to design[END_REF].

Investigation on the root-soil interaction when the soil is sheared was also made using numerical models. [START_REF] Mickovski | Simulation of direct shear tests on rooted and non-rooted soil using finite element analysis[END_REF] and [START_REF] Mao | Evaluation of root reinforcement models using numerical modelling approaches[END_REF] used FEM and DEM to simulate the response of roots in a direct shear test, observing that the root reinforcement varied as a function of soil strain, and was closely related to root geometry, position in the soil and mechanical traits. On the other hand, [START_REF] Liang | Modelling the seismic performance of rooted slopes from individual root-soil interaction to global slope behaviour[END_REF] considered the interaction between roots and soil using existing p-y curves for piles under lateral loading in a FEM model of a root, validated using experimental observations. It was concluded that the plastic deformations dominates over elastic deformations, because large relative soil-root deformations were found even for relatively small global slip of the rooted soil (Fig. 8). This means that the ultimate resistance of the soil surrounding the root is reached before the critical state of the soil. This also justifies the use of 𝑝 𝑦 by [START_REF] Wu | Study of soil-root interaction[END_REF] in the beam solution. In fact, at shallow depths, the soil was pressed against the root forming an upward moving wedge due to the low confining pressures. When the confining pressure increases, soil tends to contour the root.

Conclusions

This review regards the mechanical effects of roots on shallow landslides, by identifying and describing the types of root failure that can occur in a slope. It was intended to provide basic knowledge on the loads that lead to root failure, so that geotechnical engineers can better consider root resistance for the improvement of slope stability analysis, in particular, with regards to shallow landslides.

Three main forms of root failure were identified: pull out, breakage and buckling. Nevertheless, soilroot interaction under shear actions and tortuosity effect are also considered.

The main observations made in this work are that:

-a better estimation of the pull out resistance may be obtained if the variation of root diameter with depth is considered, because the available contact area between soil and root that mobilizes friction varies; -there is more than one way of calculating soil-root frictional resistance. However it depends on the normal stresses which are considered in a very simplified way, ignoring its variation along the root length. Importance is also given to the effect of root deformation under shear; -the comparison of roots to structural elements, such as beams and cables, provides an improvement on the description of root response; -the tortuosity effect is responsible for a great variation of the Young's modulus of the root, which causes the root to mobilize its frictional resistance only after straightening; -root under compression is still a less investigated topic, however it is generally accepted that a root tends to buckle.

Fig. 1

 1 Fig. 1 Stresses acting on root reinforcement along the sliding surface of a shallow landslide: a tensile, b shear and c compression stresses

Fig. 3

 3 Fig. 3 Scheme of the root deformation due to the shear of the soil layer (the upper layer moves towards the right on the lower one). The normal load acting on the root's axis, compression in the soil and tensile axial force in the root are represented

Fig. 4

 4 Fig. 4 Scheme of the tensile stress acting in the root during pull out with the indication of different mechanisms of failure (III and IV)

Fig. 7

 7 Fig. 7 Scheme of the stresses in the root during shearThis new representation of the tensile force in the fibre can be divided into a normal 𝜎 𝑟 and tangential 𝜏 𝑟 component to the shear plane, acting in the soil. The normal component 𝜎 𝑟 increases the confining stress on the failure plane, thereby mobilizing additional shear resistance in the soil, whereas the tangential component 𝜏 𝑟 directly decreases the acting shear stress. The system of equations (12) represents this decomposition assuming that the root out of the shear zone stays perpendicular to it[START_REF] Wu | Investigation of landslides on Prince of Wales Island, Alaska[END_REF], where 𝜃 is the shear distortion of the root and 𝑡 𝑟 is the tensile stress (Fig.7).

  cos 𝜃 𝐿 -𝑝 𝑦 𝑑(2𝐿 -𝑧)𝑧 2𝑇(𝐿)

Fig. 8

 8 Fig. 8 Comparison of the displacements observed in the soil and in roots of different diameter subjected to shear loading (adapted from Liang et al. 2015)

Acknowledgements

The authors wish to acknowledge the support of the European Commission via the Marie Skłodowska-Curie Innovative Training Networks (ITN-ETN) project TERRE 'Training Engineers and Researchers to Rethink geotechnical Engineering for a low carbon future' (H2020-MSCA-ITN-2015-675762).


Lutz Straßburger

On the Decision Problem for MELL

Keywords: linear logic, MELL, decidability, relevant logic logique linéaire, MELL, décidabilité

.

Introduction

Since the beginning of linear logic [START_REF] Girard | Linear logic[END_REF], the complexity of the decision problem of its various fragments has been studied by many researchers. For example, its multiplicative fragment (MLL) is NP-complete [START_REF] Kanovich | Horn programming in linear logic is NP-complete[END_REF], and its multiplicative-additive fragment (MALL) is PSPACE-complete [START_REF] Lincoln | Decision problems for propositional linear logic[END_REF]. But the complexity of its multiplicativeexponential fragment (MELL) is still unknown. In fact, it is an open problem whether that logic is decidable. Finally, the multiplicative-additive-exponential fragment, i.e., full propositional linear logic (LL) is undecidable [START_REF] Lincoln | Decision problems for propositional linear logic[END_REF]. However, if we add second-order propositional quantifiers, already the multiplicative fragment (MLL2) is undecidable [START_REF] Lafont | The undecidability of second order multiplicative linear logic[END_REF]. On the other hand, if we add a self-dual non-commutative multiplicative connective, the multiplicative fragment stays NP-complete [START_REF] Kahramanogulları | System BV is NP-complete[END_REF] (this logic is called pomset logic [START_REF] Retoré | Pomset logic: A non-commutative extension of classical linear logic[END_REF] or BV [START_REF] Guglielmi | Non-commutativity and MELL in the calculus of structures[END_REF][START_REF] Guglielmi | A system of interaction and structure[END_REF]), but the multiplicative exponential fragment is undecidable [START_REF] Straßburger | System NEL is undecidable[END_REF] (this logic is called NEL [START_REF] Guglielmi | A non-commutative extension of MELL[END_REF][START_REF] Straßburger | A system of interaction and structure IV: The exponentials and decomposition[END_REF][START_REF] Guglielmi | A system of interaction and structure V: the exponentials and splitting[END_REF]).

It was observed early on that the reachability problem for Petri nets [START_REF] Petri | [END_REF] can be encoded into MELL [START_REF] Mogbil | Sémantique des phases, réseaux de preuve et divers problèmes de décision en logique linéaire[END_REF]. That problem has been shown to be decidable [START_REF] Mayr | An algorithm for the general petri net reachability problem[END_REF][START_REF] Kosaraju | Decidability of reachability in vector addition systems (preliminary version)[END_REF][START_REF] Mayr | An algorithm for the general petri net reachability problem[END_REF], but its precise complexity is still unknown. It is EXPSPACE-hard [START_REF] Lipton | The reachability problem requires exponential space[END_REF] and the known algorithms have runtimes that are not primitive recursive [START_REF] Leroux | Demystifying reachability in vector addition systems[END_REF]. It has been known for a long time that the reachability problem for Petri nets is equivalent to the reachability problem of vector addition systems with states (VASS) [START_REF] Reutenauer | Aspects mathématiques des réseaux de Petri[END_REF]. Furthermore, it has been shown recently that the decidability problem of MELL is equivalent to the reachability problem for branching VASS [START_REF] De Groote | Vector addition tree automata[END_REF], for which very recently a non-elementary lower bound has been found [START_REF] Lazic | Nonelementary complexities for branching VASS, MELL, and extensions[END_REF]. More precisely, if MELL turns out to be decidable it will be at least TOWER-hard [START_REF] Lazic | Nonelementary complexities for branching VASS, MELL, and extensions[END_REF]Corollary 22].

Since all known proofs [START_REF] Mayr | An algorithm for the general petri net reachability problem[END_REF][START_REF] Kosaraju | Decidability of reachability in vector addition systems (preliminary version)[END_REF][START_REF] Mayr | An algorithm for the general petri net reachability problem[END_REF][START_REF] Reutenauer | Aspects mathématiques des réseaux de Petri[END_REF][START_REF] Leroux | Vector addition systems reachability problem (A simpler solution)[END_REF] of the decidability of the reachability for Petri nets and VASS are very involved-in fact, the complete proof fills a textbook [START_REF] Reutenauer | Aspects mathématiques des réseaux de Petri[END_REF]-there is interest in the community in an alternative proof, which could be provided by a proof-theoretical proof of the decidability of MELL. The recent proposal by Bimbó [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] gives such a proof which is surprisingly simple. All the details could be given in less than ten pages. 1However, an inspection of the proof by Bimbó [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] shows immediately that the same argument also works for the subexponential variant of MELL, denoted by MSELL, that is obtained by allowing not one but several pairs of the modalities ? and !, which are subject to a partial order relation. But that logic has recently been shown to be undecidable for the case of three pairs of ? and ! by Chaudhuri in [START_REF] Chaudhuri | Undecidability of multiplicative subexponential logic[END_REF]. Furthermore, a minor variation of the argument in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] (using results from [START_REF] Okada | The finite model property for various fragments of intuitionistic linear logic[END_REF]) would also allow to prove decidability of LL, which also known to be undecidable [START_REF] Lincoln | Decision problems for propositional linear logic[END_REF], as mentioned above.

This, of course, is enough to dismiss [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] as being erroneous. However, that is not helpful, neither for the author of [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF], nor anyone else who would like to understand what is going on. For this reason, I will in Section 4 of this paper explain in more detail the technical mistakes of [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF]. After all, the main gap in the proof is a rather subtle mistake that could easily be repeated in other contexts.

Before coming to that, let me finish this introduction with the observation that the decision problem for MELL is-no matter how it will turn out-very close to the border between the decidable and the undecidable: adding just a little bit, i.e., a third self-dual (non-commutative) multiplicative connective or the additive connectives or subexponentials, renders the problem undecidable, and removing just a little bit, i.e., stepping down from branching VASS to non-branching VASS, puts the problem in the realm of the decidable.

This might be one of the reasons why the problem receives so much attention and is, after more than 3 decades, still open.

MELL and RMELL

To make this paper self-contained, I give the sequent calculus for MELL (called CLL int in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF]) and its relevant version RMELL (called RLL int in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF]) below. Formulas (denoted by capital Roman letters A, B, C, . . .) are generated from propositional variables {a, b, c, . . .} and their duals {a ⊥ , b ⊥ , c ⊥ , . . .} via the grammar:

A, B ::= a | a ⊥ | A B | A B | ?A | !A
As in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF], I work here in the unit-free fragment. Sequents (denoted by capital Greek letters Γ, ∆, . . .) are (possibly empty) finite multisets of formulas, written as list separated by commas, with a preceding turnstile:

A 1 , A 2 , . . . , A n . The inference rules for MELL are the following2 :

id --------- a, a ⊥ Γ, A, B -------------- Γ, A B Γ, A B, ∆ --------------------- Γ, A B, ∆ A, ?∆ ! ----------- !A, ?∆ Γ, A ?d ---------- Γ, ?A Γ ?w ---------- Γ, ?A Γ, ?A, ?A ?c -------------- Γ, ?A (1)
where ?∆ stands for a sequent in which every formula is of shape ?B for some B.

The relevant version RMELL, is obtained from MELL by adding a general contraction rule

Γ, A, A c ------------ Γ, A (2)
Of course, the standard linear logic contraction rule ?c is a special case of this and can therefore be omitted from the system. Note that in order to save paper, ink, and the patience of the reader, I use here the one-sided presentation of the sequent calculus, whereas [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] uses a two-sided presentation. Everything I present here also works in the two-sided systems, we only have to replace the one-sided axiom by a negation rule and a two-sided identity axiom:

id --------- a, a ⊥ → id ------- a a (•) ⊥ --------- a, a ⊥

MSELL and RMSELL

For making some arguments in this paper clearer, I will also introduce the subexponential version of MELL, also called multiplicative subexponential linear logic [START_REF] Nigam | Exploiting non-canonicity in the sequent calculus[END_REF][START_REF] Nigam | Algorithmic specifications in linear logic with subexponentials[END_REF], or MSELL. However, for the main points of this paper, subexponentials are not needed, and the reader can safely skip over this section on first reading. Formulas of MSELL are are generated from propositional variables and their duals (see previous section) via the grammar:

A, B ::= a | a ⊥ | A B | A B | ? v A | ! v A
where (as before) we omit the units, and where exponentials are indexed by elements from a (countable) set V of labels which comes equipped with a partial order ≤ ⊆ V × V and a subset U ⊆ V of unbounded labels. The inference rules for MSELL are the ones in the first line of (1) above together with ! v A, ? w1 B 1 , . . . ? wn B n ! v -----------------------------v ≤ w i for all i ∈ {1, . . . , n} ! v A, ? w1 B 1 , . . .

? wn B n Γ, A ? v d ----------- Γ, ? v A Γ ? u w -----------u ∈ U Γ, ? u A Γ, ? u A, ? u A ? u c -----------------u ∈ U Γ, ? u A (3)
Note that contraction and weakening are only allowed if the label of the ? is among the unbounded labels, and promotion on a ! is only allowed if its label is smaller or equal than all the labels of the ? in the context. Only the dereliction rule ? v d can be applied without restriction.

In [START_REF] Chaudhuri | Undecidability of multiplicative subexponential logic[END_REF] it has been shown that provability in MSELL is undecidable for the case V = {α, β, γ} with ≤ being the reflexive closure of ≤ 0 with α ≤ 0 γ and β ≤ 0 γ and U = {γ}, by a straightforward encoding of two-counter machines [START_REF] Minsky | Recursive unsolvability of Post's problem of "tag" and other topics in theory of Turing machines[END_REF][START_REF] Lambek | How to program an infinite abacus[END_REF].

Clearly, we can also define a relevant version of MSELL, that I call here RMSELL, and that is obtained from MSELL by adding the general contraction rule c in [START_REF] Bimbó | On the decidability of implicational ticket entailment[END_REF]. As before in that case, the restricted contraction rule ? u c can be omitted.

Technical flaws in Bimbó's decidability proof

There are three main step in Bimbó's proof [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] of the decidability of MELL:

1. Cut admissibility for MELL and RMELL and modified systems that restrict the application of contraction.

2. Decidability for RMELL by showing that proof search in the modified system terminates.

3. Decidability of MELL by deriving an upper bound for the proof search trees from the decision procedure for RMELL.

In the paper [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] all three steps have technical flaws. For the first and the second, these are easily fixable, but for the third this is not the case, and therefore the decision problem for MELL remains open. Below I explain the mistakes in each of the three steps.

Cut admissibility

The cut rule for all logical systems presented so far can be given as follows:

Γ, A A ⊥ , ∆ cut ---------------------------- Γ, ∆ (4)
where A ⊥ is the De Morgan dual of A. The cut admissibility result for a logical system S says that if a formula or sequent can be derived in S+cut then it can also be derived in S without the cut-rule. All four logical systems presented here have this property. For MELL and MSELL this has been proved in [START_REF] Girard | Linear logic[END_REF] and [START_REF] Nigam | Exploiting non-canonicity in the sequent calculus[END_REF], respectively. For their relevant version, it can be shown by similar methods. But since we defined the logics without cut, the cut admissibility theorem is technically not needed for the decidability proof (for RMELL and RMSELL).

Nonetheless, [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] provides a cut admissibility proof for MELL and claims that this is a new proof. However, the proof method in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] is standard: permute the cut upwards, start with the topmost one, and make sure that some measure decreases at each step. The measure used in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] consists of the tuple ρ, µ, δ where ρ is the the number of rule instances above the cut having the cut formula in the conclusion, µ is the number of ?c-instances that are applied to ancestors of the cut formula, and δ is the size of the cut formula, and where the order is lexicographic. 3 But one case in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] is problematic, namely when ------------------------------- ?∆, Γ is reduced to ------------------------------- ?∆, ?A ⊥ , Γ cut ------------------------------------------- ?∆, ?∆, Γ ?c .......................... ?∆, Γ where the dotted line stands for several instances of the ?c-rule. On first sight one might be tempted to agree with the argument that the two new cuts have a lower µ than the original one, and therefore the induction hypothesis can be applied to both cuts. However, the elimination process for the first cut can increase the µ-value for the second cut, and thus, the induction hypothesis cannot be applied. In order to make this kind of argument work, it is not enough to just count how often the cut formula is duplicated in a contraction. One also has to find a way to take into account the instances of ?c that are newly created in the cut elimination process. This is best achieved through a notion "flow graph", as done for classical logic in [START_REF] Carbone | Turning cycles into spirals[END_REF] using Buss' logical flowgraphs or in [START_REF] Guglielmi | Normalisation control in deep inference via atomic flows[END_REF][START_REF] Guglielmi | Breaking paths in atomic flows for classical logic[END_REF] using atomic flows. For MELL, these flow graphs are studied in [START_REF] Straßburger | MELL in the Calculus of Structures[END_REF] and [START_REF] Straßburger | A system of interaction and structure IV: The exponentials and decomposition[END_REF] in the setting of the calculus of structures. Then, for ensuring termination, it has to be shown that there no cycle in the flow graph, or that the cycles can be eliminated. In any case, cut elimination for MELL is a well-established result with several different published proofs, so that there is no need to go into further details here.

?∆, A ! ----------- ?∆, !A ?A ⊥ , ?A ⊥ , Γ ?c ------------------ ?A ⊥ , Γ cut -
?∆, A ! ----------- ?∆, !A ?∆, A ! ----------- ?∆, !A ?A ⊥ , ?A ⊥ , Γ cut -

Decidability of RMELL

Okada and Terui have shown in [START_REF] Okada | The finite model property for various fragments of intuitionistic linear logic[END_REF] via a semantic argument that the relevant version of full propositional linear logic, denoted by RLL4 , is decidable. Therefore, RMELL is also decidable. 5On the other hand, for a reader familiar with the syntactic proof of the the decidability of various fragments of relevant logic, attributed to Kripke [START_REF] Kripke | The problem of entailment (abstract)[END_REF] and first written up in all detail by by Belnap and Wallace in [START_REF] Nuel | A decision procedure for the system eī of entailmengt with negation[END_REF] for the logic of entailment with negation 6 , it should be clear that the decidability of RMELL can also be shown by almost literally the same proof. This proof is presented in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF], but contains a mistake.

More precisely, Kripke's theorem (Theorem 18) is stated wrongly in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF]. That theorem is needed for cognate sequents in general, and not just modally cognate sequents. I will explain this in further detail now, because for understanding the problem with the decidability proof for MELL, one needs to understand how the decidability proof for RMELL works. So, in order to make this paper self-contained, I will give this proof.

For this, we need some definitions: Two sequents Γ and ∆ are cognate if they contain the same formulas, i.e., they only differ in the number of occurrences of the formulas in the sequent. They are modally cognate, if additionally every formula that is not of the shape ?A has the same number of occurrences in both sequents.

A set of sequents that are cognate to each other are called a cognation class. In the example below, all four sequents are in the same cognation class, but only the first two are modally cognate: a, b, b, ?a a, b, b, ?a, ?a, ?a a, a, a, b, ?a, ?a a, a, a, a, a, b, b, b, ?a, ?a, ?a

In the following, we use the notation Γ 1 Γ 2 if there is a derivation with premise Γ 1 and conclusion Γ 2 , using only the c-rule, and we write Γ 1 Γ 2 (or equivalently Γ 2 Γ 1 ) iff Γ 1 Γ 2 or Γ 1 = Γ 2 (where = stands for multiset equality).

A (finite or infinite) sequence Γ 1 , Γ 2 , . . . of sequents is irredundant if for all i < j, we have Γ i Γ j . We can now state Kripke's lemma: Theorem 4.1 (Kripke [START_REF] Kripke | The problem of entailment (abstract)[END_REF]). If a sequence of cognate sequents is irredundant, then it is finite.

A proof can be found in [14, p.289]. Note that for this theorem it is irrelevant what the inference rules are and what the language of the formulas is. 7 We can now use Theorem 4.1 to exhibit a terminating complete proof search procedure for RMELL, from which decidability follows. This is done by using the following variant of the proof system, called RMELL :

8 id --------- a, a ⊥ Γ, A, B ---------------- Γ, A B Γ, A B, ∆ --------------------- Γ, A B, ∆ A, ?∆ ! ----------- !A, ?∆ Γ, A ?d ------------ Γ, ?A Γ ?w ---------- Γ, ?A (6)
where the rules , , and ?d are variants of the rules , , and ?d, respectively, which have the contraction rule built in, i.e., a sequent Σ is obtained from Σ by application of the c-rule in (2), such that the following additional conditions are satisfied: ?d : If the formula ?A occurs n times in Γ, then it occurs at least max(n, 1) times in Γ, ?A . Any other formula occurs as often in Γ as in Γ, ?A .

We can now prove the following two theorems about RMELL . The first one states the height-preserving admissibility of contraction, and is in the literature often (e.g. in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF]) attributed to Curry. With these ingredients, we can give the full proof of decidability for RMELL.

Theorem 4.4. Provability in RMELL is decidable.

Proof. By Theorem 4.3 we can restrict proof search to RMELL . By Theorem 4.2, we can stop the search when we reach a sequent Γ such that on the current branch of the proof search tree there is an ancestor Γ with Γ Γ. Then, we observe that any formula that occurs in an RMELL is a subformula of the endsequent and there are only finitely many such formulas. There are infinitely many sequents that can be formed from these formulas, 7 In fact, it is equivalent to Dickson's Lemma in number theory which states that every set of n-tuples of natural numbers has finitely many minimal elements (see also [START_REF] Riche | Urquhart and Relevant Decidability & Complexity: "Das ist nicht Mathematik, das ist Theologie[END_REF]). 8 The • notation is taken from [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF]. 9 In [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF], the formulation of these conditions is slightly ambiguous. For this reason, I took here the formulation of [START_REF] Nuel | A decision procedure for the system eī of entailmengt with negation[END_REF], adapted to the case of RMELL. 10 In [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] it is stated that also cut admissibility is necessary for obtaining this theorem, but as we have shown here, it is not needed.

Inria

but there are only finitely many cognation classes. Furthermore, from the previous paragraph and Theorem 4.1 it follows that from each cognation class only finitely many sequents need to be visited in a single branch in the proof search tree. Since the proof search tree is finitely branching (each inference rule has only finitely many premises and at each step there are only finitely many choices for applying an inference rule), we can conclude by König's lemma that the proof search tree is finite.

It is important to observe that the whole argument breaks down if we assume that Theorem 4.1 only holds for sequences of modally cognate sequents. In that case it is still true that in each branch only finitely many sequents of the same cognation class are visited, but now there are infinitely many cognation classes (see example in ( 5)), i.e., the sequents visited in a single branch of the proof search tree can become arbitrarily large. And this is exactly the reason why the decidability proof for MELL in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] is not correct, as we will see in the next section.

Finally, it is easy to see that we can define a system RMSELL in the same way as RMELL , and that we can prove the decidability of provability in RMSELL with almost literally the same proof as for RMELL.

Theorem 4.5. Provability in RMSELL is decidable.

Concerning the Decidability Proof for MELL

The proof of the decidability of MELL in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] is based on the observation that every MELL-proof is also a RMELLproof of the same endsequent. Thus, given a sequent Γ, if the RMELL proof search for Γ comes back with a failure, then we know that Γ is not provable in MELL. On the other hand, if there is an RMELL-proof of Γ, then we can count for each subformula occurrence ?A of Γ, how often the contraction rule is applied to it in any RMELL-proof of Γ.

Even though there is no bound on the proof search for RMELL, we have that RMELL proof search is bounded. The argument of [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] is now that it is enough to count for each occurrence of a subformula ?A of a sequent Γ, how often the subformula occurrence A and its ancestors11 are contracted in any RMELL proof of Γ. This is called the heap number of ?A [1, Def. 22], which is then used to restrict proof search in the system [MELL] shown below:

id --------- a, a ⊥ Γ, A, B -------------- Γ, A B Γ, A B, ∆ [ ] --------------------- [Γ, A B, ∆] A, ?∆ ! ----------- !A, ?∆ Γ, A [?d] ----------- [Γ, ?A] Γ ?w ---------- Γ, ?A (7)
where [ ] and [?d] are variants of the rules and ?d, respectively, which have the ?c-rule built in, i.e., a sequent [Σ] is obtained from Σ by application of the ?c-rule, such that the following additional conditions are satisfied: We have then for [MELL] analogous results as stated in Theorems 4.2 and 4.3 for RMELL , namely the ?c is height-preserving admissible for [MELL], and a sequent is provable in [MELL] if and only if it is provable in MELL.

The decidability proof for MELL in [1, Theorem 23] now simply bounds the number of application of the [?d]-rule to (the ancestors of) an occurrence of a ?A-formula by its heap-number.

However, the same argument would apply to prove decidability of MSELL from the decidability of RMSELL, by going via the system [MSELL] that is defined in the same way as [MELL]. But MSELL has been shown undecidable in [START_REF] Chaudhuri | Undecidability of multiplicative subexponential logic[END_REF].

Therefore, something must be wrong with Bimbó's argument in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF]. First, it is obvious that Bimbó's decision procedure terminates. This follows from the termination argument for RMELL . However, there is no argument explaining why her decision procedure should be complete.

To understand the problem, consider for example the following MELL sequent:

(a ⊥ a ⊥ ) (a ⊥ a ⊥ ), a, ?(a ⊥ (a a)) (8)
The obvious proof in RMELL removes the ?-formula with a weakening, applies the -rule three times, and puts a copy of a into each branch:

id ----- a ⊥ , a id ----- a ⊥ , a ----------------- a ⊥ a ⊥ , a id ----- a ⊥ , a id ----- a ⊥ , a ----------------- a ⊥ a ⊥ , a ----------------------------------------- (a ⊥ a ⊥ ) (a ⊥ a ⊥
), a ?w --------------------------------------------------- (a ⊥ a ⊥ ) (a ⊥ a ⊥ ), a, ?(a ⊥ (a a))

For proving the sequent [START_REF] Guglielmi | A system of interaction and structure[END_REF] in [MELL] we need to apply the [?d]-rule three times in order to create the necessary four copies of a: --------------------------------- a ⊥ a ⊥ , a, ?(a ⊥ (a a)) --------------------------------- a ⊥ a ⊥ , a, ?(a ⊥ (a a)) [ ] ----------------------------------------------------------------------------- (a ⊥ a ⊥ ) (a ⊥ a ⊥ ), a, a, ?(a ⊥ (a a)) ------------------------------------------------------------------------------- (a ⊥ a ⊥ ) (a ⊥ a ⊥ ), a, a ⊥ (a a), ?(a ⊥ (a a))

id ----- a, a ⊥ id ----- a, a ⊥ id ----- a ⊥ , a id ----- a ⊥ , a [ ] ------------------ a ⊥ a ⊥ , a, a -------------------- a ⊥ a ⊥ , a a [ ] -------------------------------- a ⊥ a ⊥ , a, a ⊥ (a a) [?d] -
id ----- a, a ⊥ id ----- a ⊥ , a id ----- a ⊥ , a [ ] ------------------ a ⊥ a ⊥ , a, a -------------------- a ⊥ a ⊥ , a a [ ] -------------------------------- a ⊥ a ⊥ , a, a ⊥ (a a) [?d] -
* ------------------------------------------------------- (a ⊥ a ⊥ ) (a ⊥ a ⊥ ), a a, ?(a ⊥ (a a)) [ ] -
[?d] ------------------------------------------------------------------ (a ⊥ a ⊥ ) (a ⊥ a ⊥ ), a, ?(a ⊥ (a a)) (10)
This proof is at the same time a correct RMELL proof, but it is not visited by the RMELL proof search described in the proof of Theorem 4.4, because the sequent marked with a * in ( 10) is in relation with the conclusion, and therefore the RMELL proof search is aborted at that point. For this reason, we cannot assume, a priori, that the heap numbers determined by the RMELL decision procedure are high enough to ensure a complete [MELL] proof search, which leaves a large gap in the decidability proof for MELL in [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF].

However, if we consider again the derivation in [START_REF] Guglielmi | Breaking paths in atomic flows for classical logic[END_REF], we can see that there is a rule permutation variant that does not visit the sequent * : we can permute one instance of [?d] down below the [ ] and the -rule instance and keep a copy of ?(a ⊥ (a a)). Unfortunately, it is not at all clear whether such a rule permutation always exists. More generally, we can formulate the following conjecture: Conjecture 4.6. Let π be a proof in [MELL] of a sequent Γ. Then there is a proof π in [MELL] with the same endsequent Γ, such that for any two sequents Γ 1 and Γ 2 occurring in π , such that Γ 1 is an ancestor of Γ 2 in the tree of π , we have Γ 1 Γ 2 .

It follows immediately from [START_REF] Chaudhuri | Undecidability of multiplicative subexponential logic[END_REF] that Conjecture 4.6 does not hold if we replace [MELL] by [MSELL], but it might well be that it holds for the special case where the label set V is a singleton. A proof of Conjecture 4.6 would indeed close the gap in Bimbó's proof of the decidability of MELL, and would provide an alternative proof for the decidability of the reachability problem for VASS and Petri nets.

Note that a counterexample to Conjecture 4.6 would not show that MELL is undecidable. But it would show that the proof idea of [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] cannot work.

Conclusion

Whenever a paper is published whose main proof is faulty, it is always easy to blame the author or the editor or the referee or some other victim. However, in this case this would be too short-sighted. I think that the publication of [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] is the consequence of a systemic problem in the field of structural proof theory. Namely, that it is studied by two different communities: one coming from a computer science background and the other coming from a philosophical background. These two communities use different notation and terminology and do not talk to each other. Furthermore, they also consider each other to be less skilled and take this as a justification for not taking each other's papers seriously.

On the one hand, the philosophers rightfully accuse the computer scientists of ignoring the vast amount of literature on substructural logic that existed before linear logic, and the computer scientists rightfully accuse the philosophers of considering linear logic as just another substructural logic, ignoring the semantic consideration that gave rise to it and the vast amount of literature that came after linear logic, exhibiting its enormous influence in many areas of theoretical computer science that makes linear logic very special among the zoo of substructural logics.

The decision problem for MELL is open for three decades now, and many people in the computer science community worked on it, but nobody has observed its containment in RMELL, even though decision problems for relevant logics have been studied since the 1960's, and the relation between MELL and RMELL is obvious to anyone with a background in philosophical logic. This observation leads naturally to Bimbó's proof idea: Can we bound the proof search for MELL by using information that we can extract from the RMELL proofs?

On the one hand, it is quite embarrassing for the computer science community that nobody has explored this idea before Bimbó. 12 On the other hand, the gap in Bimbó's reasoning could be spotted immediately by anybody familiar with the peculiarities of MELL. However, Bimbó's error is very easy to overlook for someone not familiar with linear logic, and this led to the unfortunate publication of [START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF].

Interestingly, there is a recent proof by Bimbó and Dunn (presented in [START_REF] Bimbó | On the decidability of implicational ticket entailment[END_REF]) of the decidability of the logic of ticket entailment 13 , using similar methods as discussed here. In [START_REF] Dawson | Issues in machine-checking the decidability of implicational ticket entailment[END_REF] Dawson and Goré tried to formally verify that proof using the interactive theorem prover Isabelle/HOL. Their work uncovered a gap in the proof of [START_REF] Bimbó | On the decidability of implicational ticket entailment[END_REF] which seems to be of similar nature as the gap uncovered here for MELL.

We should take this as a lesson to take each other more seriously in the future. It seems that with the decidability of MELL we have a problem that needs for its solution both communities, philosophy and computer science.

9 :

 9 If the formula A B occurs n times in Γ, then it occurs at least max(n, 1) times in Γ, A B . Any other formula occurs as often in Γ as in Γ, A B . : If the formula A B occurs n times in Γ and m times in ∆, then it occurs at least max(n, m, 1) times in Γ, A B, ∆ . If another formula occurs n times in Γ and m times in ∆, then it occurs at least max(n, m) times in Γ, A B, ∆ .

  [ ]: If a formula ?C occurs n times in Γ and m times in ∆, then it occurs at least max(n, m) times in [Γ, A B, ∆]. [?d]: If the formula ?A occurs n times in Γ, then it occurs at least max(n, 1) times in [Γ, ?A]. Any other formula occurs as often in Γ as in [Γ, ?A].

  Proof. This is proved by a straightforward induction on h.Theorem 4.3. A sequent Γ is provable in RMELL if and only if it is provable in RMELL.Proof. Any proof in RMELL can be expanded to a proof in RMELL by adding the necessary instances of the c-rule. Conversely, every rule in RMELL, except for c, is an instance of a rule in RMELL . Hence, a proof in in RMELL is already a proof in RMELL + c, and the result follows from Theorem 4.2. 10

Theorem 4.2 (Curry). If

Γ has a RMELL proof π with height h, and Γ Γ , then Γ has a RMELL proof π with height h ≤ h.

Bimbó[START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF] does not speak about complexity, but previous work by Urquhart[START_REF] Urquhart | The complexity of decision procedures in relevance logic[END_REF][START_REF] Urquhart | The complexity of decision procedures in relevance logic II[END_REF] gives an EXPSPACE lower bound, and more recently Weiermann and Bunder[START_REF] Weiermann | Ackermannian lower bounds for a combinatorial problem related to the decidability of certain relevant logics[END_REF] have shown an ACKERMANN lower bound.

I am using here the standard names from the linear logic literature.RR n°9203

In[START_REF] Bimbó | The decidability of the intensional fragment of classical linear logic[END_REF], the value ρ is called the rank of the cut, whereas usually in the literature the value δ is called the rank of the cut.

That is LL extended with the general contraction rule (2). This logic is called CLL in[START_REF] Okada | The finite model property for various fragments of intuitionistic linear logic[END_REF], but the C is more often used for "classical". For this reason I use the R for "relevant" throughout this paper.

More recently, it has been shown that RLL is ACKERMANN-complete[START_REF] Lazic | Nonelementary complexities for branching VASS, MELL, and extensions[END_REF] Corollary 25], and that RMELL is in 2EXP[START_REF] Schmitz | Implicational relevance logic is 2-exptime-complete[END_REF] Theorem

[START_REF] De Groote | Vector addition tree automata[END_REF].1].[START_REF] De Groote | Vector addition tree automata[END_REF] This logic is equivalent to the one obtained from MLL (the three rules in the first line of (1)) together with the contraction rule (2).RR n°9203

I.e., the ancestors of the auxiliary formulas of the instances of the ?d -rule in which an ancestors of that ?A are principal.RR n°9203

At this point it is worth mentioning that the Karp-Miller trees[START_REF] Karp | Parallel program schemata[END_REF], which are finite structures that are crucial for the decidability proof for VASS, play in that proof a similar role as the RMELL derivations in Bimbó's proof attempt for MELL. Given that it is still a big leap from Karp-Miller trees to the full decidability of the reachability problem for VASS, a proof of Conjecture 4.6 will very likely be much harder than it seems on first sight, if it exists at all.

Another published proof of the decidability of ticket entailment can be found in[START_REF] Padovani | Ticket entailment is decidable[END_REF].RR n°9203

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Acknowledgments

Many thanks to Sylvain Schmitz, Dale Miller, Willem Heijltjes, and Revantha Ramanayake for helpful comments and discussions improving the readability and comprehensiveness of this short paper.