Moving Objects Detection by Conflict Analysis in Evidential Grids

Julien Moras, Véronique Cherfaoui, Philippe Bonnifait Abstract-Advanced Driving Assistance Systems exploit exteroceptive sensors to help the driver in perceiving the dynamic environment, like other vehicles or pedestrians. This paper proposes an original approach to deal with this perception challenge in urban environments. The method detects mobile objects motions using grids elaborated thanks to a lidar range scanner and an enhanced map of the drivable space. The data fusion is performed using the Dempster-Shafer theory which provides an interesting framework particularly well adapted to manage the uncertainties of the sensors. By analyzing conflicting information, objects movements can be efficiently characterized. This formalism provides also the interesting possibility to introduce decay factors that are useful for forgetting old information. Experimental results obtained with an IBEO Alasca and an Applanix positioning system show that such a perception strategy can be effective compared to deterministic accumulation strategies.

I. INTRODUCTION

Nowadays, Advanced Driving Assistance Systems (ADAS) are promising technologies which will appear on many vehicles, like Electronic Stability Programs or Speed Regulators that are commonly used in curren vehicles. This next generation of ADAS is coming out and will be the basis of auto-braking systems or overtaking assistance systems. In such kind of application, reliable perception of the environment is crucial to avoid collisions and eliminate false alarms while reducing missed detections. An essential part of the perception problem is the detection of mobile objects because they constitute the more difficult part of the problem when using lidar technology. The velocity of the objects hit by the laser beam is not measured in opposition to a radar exploiting the Doppler effect.

Mobile Object Tracking (MOT) has been studied for a long time and many approaches have been proposed [START_REF] Blackman | Design and Analysis of Modern Tracking Systems[END_REF]. Classical MOT systems are based on features and rely on two principal steps: the clustering of the sensor raw data for object detection and a simultaneous tracking stage. The clustering step can be done for instance by image processing [START_REF] Hoffmann | Fusing multiple 2d visual features for vehicle detection[END_REF] or by clustering a set of lidar scan points [START_REF] Fayad | Tracking objects using a laser scanner in driving situation based on modeling target shape[END_REF]. This process provides a set of features referenced in the sensor frame. Single Hypothesis Tracking (SHT) systems [START_REF] Rodriguez Florez | Visual Confirmation of Mobile Objects Tracked by a Multi-layer Lidar[END_REF] associate the detected objects with the tracks previously registered, using a specific strategy like JPDA (Joint Probabilistic Data Association, see e.g. [START_REF] Bar-Shalom | Multitarget-Multisensor tracking : Applications and Andvances[END_REF]), or create new tracks. The update of the tracks is then done with a filtering stage based on the Kalman Filter. Multi-hypothesis tracking systems keep in memory several hypotheses which are updated using the detected features while unlikely hypotheses are rejected.

Feature-based approaches are some times coupled with grid approaches in order to manage the mobile objects in complement to the grid framework that is dedicated to the static part [START_REF] Leonard | A perception driven autonomous urban vehicle[END_REF]. Indeed, occupancy grids have been used for mobile robotics in static environments. New approaches have appeared in the last years, like the Bayesian Occupancy Filter (BOF) [START_REF] Coue | Bayesian occupancy filteing for multitarget tracking : an automotive application[END_REF] which is able to estimate what is evolving in the grid. This approach uses a grid built in the sensor frame and estimates simultaneously the occupancy and the relative speed of each cell. This is performed trough the use of a probabilist association of the cells along the time.

In this paper, we focus on the detection of moving obstacles using only a lidar sensor. This kind of range sensor provides two types of information: the presence of echoes in the different scanned lines-of-sight, and their relative positions. The problem we are addressing is therefore to infer the dynamics of the scene by using lidar echoes. In this study, we consider the case of an equipped vehicle which evolves on a planar surface, like a road. We suppose that the pose of the vehicle is precisely provided by an external localization system. In the experiments, we used an Applanix system. The main contribution of this paper is to present a new way to detect mobile objects in the surrounding of the vehicle using a Belief Function (also called Evidential in this paper) framework coupled with a grid representation. The main benefit of this approach is that there is no assumption on the shape of the objects which is suitable for a wide range of urban obstacles like pedestrians, vehicles, or bicycles. In this strategy there is no feature detection and association step. Moreover, decay factors are introduced to model the fact that perception memory fades due to the passage of time. Finally, we present an interesting approach to determine the direction of mobile objects.

The paper is organized as follows. In the first section, we present the evidential occupancy grid framework that is used in the data fusion process. Then, a lidar model is proposed to obtained evidential measurements from the raw sensor data. In section III-A, the fusion processing core is detailed. We present how the detection of the mobile objects is performed and how a decay strategy is applied on the perceived grid. Finally, real experimental results are given and compared to a classical accumulation strategy.

II. EVIDENTIAL GRIDS

A. Evidential grid framework

An occupancy grid is a discrete representation of the environment which splits the space into a set of cells that may be free or occupied. Usual approaches attempt to determine the probability that some cell is occupied from sensor data. The grid framework was proposed by Elfes [START_REF] Elfes | Using occupancy grids for mobile robot perception and navigation[END_REF], but it remained little used at that time because this approach is time, memory and computation consuming. During the recent years, this representation has become more and more popular because it is an efficient framework for vehicle navigation and because embedded computers have more powerful calculation capabilities (memory, multi-core, GPU, etc...). Bayesian inference is the common background used to cope with errors and uncertainty and many extensions have been published in the literature [START_REF] Coue | Bayesian occupancy filteing for multitarget tracking : an automotive application[END_REF], [START_REF] Gindele | Bayesian occupancy grid filter for dynamic environments using prior map knowledge[END_REF].

In this work, we propose to use an evidential approach based on Dempster-Shafer (DS) theory because this framework allows making the difference between unknown (there is no information) and doubt caused by conflicting information gathered incrementally in the fusion process. Similar ideas have been developed in [START_REF] Pagac | An evidential approach to map-building for autonomous vehicles[END_REF], [START_REF] Yang | Evidential mapping for mobile robots with range sensors[END_REF] to implement evidential occupancy grids intended to manage uncertainty arising from range sensors in the representation of the environment.

Let us study now how these concepts are handled. In a way similar to probabilistic grids, the frame of discernment includes two possible states: Free(F ) and Occupied(O). In DS theory, this set is augmented by considering the power-set which is defined as 2 Ω = {∅, {F } , {O} , {F, O}}, where Ω = {F, O} represents the ignorance in the same way as in [START_REF] Yang | Evidential mapping for mobile robots with range sensors[END_REF]. For each cell, a mass function m is calculated and provides a quadruplet on the state of the cell m

(F ) m (O) m (Ω) m (∅) ′ . For an element A, m (A) can represent the piece of evidence that the space is free (A = F ), occupied (A = O) or unknown (A = Ω).
The piece of evidence on the empty set, m (∅), is generally null, unless it results of a conflicting combination. m verifies property: A⊆Ω m (A) = 1.

The Transferable Belief Model (TBM) framework proposed by Smets [START_REF] Smets | The transferable belief model[END_REF] provides also powerful tools (plausibility, belief, mass, pignistic probability) and many fusion operators have been developed in order to cope with different kinds of problems.

The strategy used in this paper manages two grids: an instantaneous grid computed from one scan called ScanGrid (SG) that will be fused into a global grid called MapGrid (MG).

B. Sensor Model

In this section, we propose a SG sensor model that attributes an initial mass m S (denoted in the following "BBA", for "Basic Belief Assignment") for each cell, depending on the reliability of the sensor. The SG captures the information coming from one lidar scan at time k.

Sensors provide data elaborated from physical phenomenas. To interpret such an information, one has to model the measurement method by making use of some simplifying assumptions. This paper focusses on lidar sensors and we propose here an evidential model for them. At every time step, a lidar provides a scan which is a set of laser telemetric points measured during one rotation. Even if the scanning process takes time [START_REF] Bezet | Time error correction for laser range scanner data[END_REF], we consider it as instantaneous, e.g. all points have the same date.

Let us assume that the different laser beams are above the road i.e. the echoes are coming from potential obstacles. In order to be as close as possible to the sensor acquisition process, we make a 2D polar grid interpretation of the data. The space is divided in several angular sectors, each sector being divided in several cells along the range and each cell containing a mass function computed from the sensor model. The SG can be represented by four classical grids, using one for the mass of each state. Figure 1 illustrates each component of the grid for one echo. Since a lidar scan captures a sparse representation of the real world, one has to compute the probability for an object to be hit by the laser beam in one scan. Let λ M D and λ F A be two coefficients which correspond respectively to the probability of missed detection and the probability of false alarm. λ F A depends essentially on sensor noise. λ M D is mainly related to the reflexivity of the target and to the blind area between two consecutive laser beams, when the beam divergence ϕ is smaller than the angular step width β (see Fig. 2). Since the reflective and geometrical properties of the objects in the scene are often unknown, we have developed a model that is only based on the geometrical coverage of the laser scans.

Blind Angle Laser Beam

Fig. 2: Laser scanning coverage and blind angles Because of the angular resolution of the map and the multi echoes and multi layers new lidar capabilities (multi echoes and multi layers), several points being in different cells can be located in the same angular sector, as illustrated by Figure 3. Classical lidar grid approaches doesn't handle this kind of configuration [START_REF] Thrun | Probabilistic Robotics (Intelligent Robotics and Autonomous Agents)[END_REF].

Let suppose that the angular precision of the lidar is very good compared to the grid angular resolution. Under this assumption, the sectors of the SG can be evaluated independently. Basically, if the laser beam returns an echo in a cell in one angular sector, then this cell is occupied, the cells crossed before are free and the cells behind are unknown. Now, if there are several laser beams in the same angular sector, they are handled together for quantifying the BBA of the SG sector (see Fig. 3). Let Z Θ be the set of n scan points in the angular sector Θ:

Z Θ = {z i = [r i θ i ] | θ i ∈ Θ, i ∈ [0, n]} For every column R of the row Θ, the mass function quadruplet [m S (O) m S (F ) m S (Ω) m S (∅)] ′ is given by ∀i ∈ [0, n]: mS {Θ, R} (O) = 1 -λF A if ∃ri ∈ R 0 otherwise mS {Θ, R} (F ) = 1 -λMD if r + < min(ri) 0 otherwise mS {Θ, R} (Ω) =      λF A if ∃ri ∈ R λMD if r + < min(ri) 1 otherwise mS {Θ, R} (∅) = 0 (1)

III. MOBILE OBJECTS DETECTION BY GRID FUSION

A. Fusion process Several successive scans are needed to create a map in order to detect mobile objects in the scene [START_REF] Moras | Credibilist occupancy grids for vehicle perception in dynamic environments[END_REF]. In this part we detail how a grid in a global world frame (MapGrid) is built to map the navigable space and the environment (static and mobile). The MG is a Cartesian map referenced in a world frame and composed of L-length squared cells. Each cell refers to an occupancy mass function defined on 2 Ω as described in part II-A. m M {i, j} () is the mass function assigned to the cell (i, j) of MG(k). The MapGrid can be initialized using no prior information, i.e. with vacuous mass m Ω on every cell (m (Ω) = 1). As the fusion is done cell by cell in the full grid, we omit the indices of the cells of grid. So, m M refers to ∀i, j, m M {i, j}.

The updating mechanism is sequential. At a given time sample, the previous MapGrid, MG(k-1) is updated with the current ScanGrid SG(k) built from the sensor measurements. The result is a new MapGrid MG(k). The updating step allows the classification of the cells containing moving objects. The proposed scheme is composed of five phases as described in Figure 4. Grid transformation: To be able to merge the information contained in the SG, we have to transform the grid to be spatially compatible with the MapGrid. This is achieved in two stages. First, the SG is projected from polar to Cartesian coordinates which induces approximations for cells that are far away from the vehicle due to size of the polar cell that increases with respect to the range. Then, the SG is moved from the sensor frame into the reference frame of the MapGrid. This requires the knowledge of the absolute pose of the vehicle at time k. Both transformations are achieved by using bi-linear interpolation methods on each piece of evidence of mass functions. The resulting grid S ′ is called Global ScanGrid (GSG) and each cell contains a new mass function m S ′ (see [START_REF] Moras | A lidar Perception Scheme for Intelligent Vehicle Navigation[END_REF] for details).

Decay factors:

In dynamic environment, information becomes aging as time passes. Since no grid evolution model is considered, we cannot predict the MG. To handle this, we propose to determine a decay factor α resulting from the time difference ∆t = t St M and the time constant τ . t M is the time of the last update of the MG, t S refers to the time of the current SG. τ represents the validity time of the cell information. The decay factor is an exponential decay function as described by α = e -∆t τ . This aging process consists in computing α m M = decay (m M ) with equation 2:

α mM (A) = α • mM (A) A ⊂ Ω α mM (Ω) = 1 -α + α • mM (Ω) (2)
Fusion: When a new Global ScanGrid GSG(k) is computed, it is fused with the previous MapGrid MG(k -1) by the mean of the Dempster's rule (Eq. 3). Each cell is considered independent from the others ones. Fusion is performed using first the conjunctive combination rule and then the Dempster normalization. The conjunctive rule is detailed in Eq. 4. Before the Dempster's normalization (Eq. 5), we store the terms of the empty set mass m ∩M,t (∅) . This step is crucial for the mobile cells detection process.

A = ∅ m M,k (∅) = 0 else (5)

B. Mobile Object detection and Motion Detection by analyzing the empty mass

Let us focus on the behavior of the Dempster combination comparatively to the Bayesian rule. Transfers of belief are illustrated by Fig. 5. Each link is numbered and described here after. Initially the cells are assigned with vacuous mass m Ω . When a cell is observed, the belief is transferred to F or O (1, 2). If conflicting information is observed, masses are transferred to the empty set [START_REF] Fayad | Tracking objects using a laser scanner in driving situation based on modeling target shape[END_REF][START_REF] Rodriguez Florez | Visual Confirmation of Mobile Objects Tracked by a Multi-layer Lidar[END_REF]. The normalization process distributes proportionally the conflicting mass on the other subsets [START_REF] Bar-Shalom | Multitarget-Multisensor tracking : Applications and Andvances[END_REF][START_REF] Leonard | A perception driven autonomous urban vehicle[END_REF][START_REF] Coue | Bayesian occupancy filteing for multitarget tracking : an automotive application[END_REF]. With probabilities, two belief transfer are possible : from F to O (1') or from O to F (2') during the change of status of the cell. So, ignorance is defined by an equal probability for F and O.

These remarks lead us to study the dynamic of the grid using the evidential framework. Three main advantages can be noticed:

• A high value for mass on empty set (∅) highlights a change of state of the cell [START_REF] Fayad | Tracking objects using a laser scanner in driving situation based on modeling target shape[END_REF][START_REF] Rodriguez Florez | Visual Confirmation of Mobile Objects Tracked by a Multi-layer Lidar[END_REF]. In a dynamic environment, it is a mean of detecting moving objects on the grid. • Thanks to the conflicting mass redistribution [START_REF] Bar-Shalom | Multitarget-Multisensor tracking : Applications and Andvances[END_REF][START_REF] Leonard | A perception driven autonomous urban vehicle[END_REF][START_REF] Coue | Bayesian occupancy filteing for multitarget tracking : an automotive application[END_REF], the fusion rule acts as a filter: the state of the cell doesn't change immediately. This provides robustnes against outliers. • The unknown state (e.g. a not-observed cell) is managed independently from the distribution of the beliefs between F and O which corresponds explicitly to contradictory information. The approach uses the fact that a moving object occupies different cells along the time The motion of an object is represented by the conflict resulting from the conjunctive fusion, as detailed in section III-A. The empty mass is computed by equation 6. One can notice two different terms. The first term C 1 refers to the case of a free cell becoming occupied (link 3 on Fig. 5) and represents the mobile object.

F O F O O O (1) (2) (4) (3) (5) (7) (6) (1') (2')
The second term C 2 corresponds to the case of an occupied cell becoming free (link 4 on fig. 5) and represents past the position of the mobile object.

m ∩M,k (∅) = C1 + C2 C1 = m S ′ ,k (O) . α m M,k-1 (F ) F becomes O C2 = m S ′ ,k (F ) . α m M,k-1 (O) O becomes F (6)
Figure 6 presents a simulation of the evolution of the mass functions for one cell. Top figures display the ground truth (1 means that the cell is occupied, 0 means it's free) and a simulated ScanGrid detection. The central plot shows the result of the conjunctive combination, illustrating the conflict generation process and the bottom figure shows the final result of the DS combination. In the first step, the MG cell, initialized with the vacuous mass, converges to a free state according to the SG. Then, since no change occurs, it remains stable. At scan 11 the state changes from F to O. Contradictory information creates a part of conflict C 1 (first term in Eq. 6), the other part of the mass remains on F . Thanks to the normalization, the final state of the MG cell doesn't change, whereas the conflict points out an oncoming object. When the SG remains on the same state during such a time, it forces the MG to change from F to O as shown between time 15 and 18. The following step does the inverse process. Figure 7 shows the result of the detection based on this method. The top figure shows the camera image of the scene, the bottom figure represents the grid containing mobile object (red cells, C 1 ) and mobile object past positions (blue cells, C 2 ) in a local frame. The ego vehicle is represented by the green square at the bottom of the grid and is heading up. The detected objects are filtered using a precise road map in order to keep only mobile objects on the road (white cells). One can see that the direction of the travel has been correctly detected thanks to the management of the conflict (from the blue to the red). The presented algorithm was tested on a set of data acquired with a real vehicle in urban conditions and was compared with an approach that uses accumulation [START_REF] Moras | A lidar Perception Scheme for Intelligent Vehicle Navigation[END_REF]. It is currently implemented in Matlab and runs offline. The data set is a 20-minutes long sequence, acquired in collaboration with the French Geographic Institute (IGN) in Paris. The vehicle follows a reference track repeated 3 times. The results presented were obtained during the first loop.

The data used are provided by three sensors. Two of them are really used by the algorithm: Lidar ALASCA XT and Applanix positioning system. To synchronize the data between the different sensors, time stamps in time GPS have been used. The lidar has been arranged to have its lowest layer horizontally. This means that, if we assume a low pitch, we do not have to consider the floor. The acquisition frequency is 15 Hz with a maximum angular resolution of 0.25 °in the vehicle front and 1°on the sides. The Applanix positioning system consists of two GPS, an IMU and an odometry sensor. A camera has been installed just below the lidar located on the front bumper and was triggered by the lidar. So far, the camera is not used for treatment and is just used for validation and visualization of the scene.

The SG covers a distance of 100 meters and the angular field of 180°with a resolution of 0.5 m wide and 1°angle. MG used covers an area of 800m × 700m with a resolution of 0.5m × 0.5m. The discounting factor α was set such that the time constant τ = 1.3s . The λ M D was computed considering that the angle ϕ = 0.25°and β = 0.5°and the λ F A was set to 0.5.

One sequence of the results is presented through Figures 8a, 8b and8c. At each time step, the front-view camera image is displayed and two grids are plotted (Left: the accumulation approach and Right: the evidential approach). Each grid contains the detected mobile cells on a MG replaced in a local frame They represent an area of 50m by 50m in front of the vehicle, with a resolution of 0.5m. We use a prior accurate map provided by IGN to extract the drivable space. The white cells denote the cells that belong to road map, the black ones denote the cells which are out of the road. The red cells indicate the presence of a mobile object. Again, the ego vehicle is represented by the green square at the bottom of the grid and is heading up. The detected moving objects are circled in the camera view and a link indicates the cluster of cells that are represented. During the sequence, the ego vehicle is moving at the speed of 35 km/h and crosses two cars and one van.

On Figure 8a, both accumulation and evidential algorithm detect correctly the van. The left side of the car is more recognizable with the evidential approach.

Figure 8b shows an interesting phenomenon. The car in the background is correctly detected by both methods. The van is properly detected by the evidential method but poses serious problems for the accumulation method. The van is seen by the lidar (the data used are the same) but most cells are classified as static, this is because the van presents a large surface and produces a high number of laser measurements which increment rapidly the accumulation of the cells that exceeds the threshold between mobile and static.

Figure 8c shows similar results with a better detection in the accumulation approach, but the evidential one remains more comprehensive. We can see the shape of vehicles and also see the scanning effect. Indeed, the front of vehicles form a spike, like a "V". Because the sensor scans clockwise, the left side is scanned before the right side. If the vehicle moves, it produces this effect.

A global comparison of the two methods during the complete sequence exhibits that mobile objects are detected with their global shape when using the evidential approach. Moreover, a nice quality of this approach is that its tuning is easier to do compared to the accumulation strategy that needs the specification of several thresholds. The tuning parameters necessary for the evidential detections are high level with a physical interpretation. These results are promising and this qualitative study shows the added value of the use of evidential grids for moving object detection. Next experimentation with ground truth will be considered in order to give quantitative results on this method.

V. CONCLUSION

This paper has presented a vehicle perception scheme based on both local and global grid interactions able to detect motion in the surrounding of the vehicle by using an evidential framework. The main contribution of this paper is the presentation of a smart way for detection of moving objects by the means of a grid framework. The DS theory provides interesting tools to manage evidentials in occupancy grids which makes the detection simpler and more robust. The determination of the motion direction using the two kinds of conflict generated by the fusion process is the main perspective of this research. We plan also to study different decision strategies able to provide robustness to false alarms. In its current implementation, the main drawback of this method is that it needs a very precise pose, provided in this experimentation by an Applanix system. One perspective is so to analyze how a dead-reckoning localization method, using for instance lidar odometry, will degrade the method.

Fig. 1 :

 1 Fig. 1: Polar evidential representation, colors intensity refers to the value of the mass

10 Fig. 3 :

 103 Fig. 3: Example of an angular sector of the SG with three lidar points ; colors refers to the most likely state of the cell (green F , yellow O, blue Ω)

Fig. 4 :

 4 Fig. 4: Fusion architecture

Fig. 5 :

 5 Fig. 5: Transfer of belief during the fusion process. Left: probabilistic framework, right: evidential framework

Fig. 6 :

 6 Fig. 6: Simulation of the evolution of the mass function (λ M D = λ F A = 0.7)

Fig. 7 :

 7 Fig. 7: Result of the detection of a vehicle crossing the street

Fig. 8 :

 8 Fig. 8: Result of mobile object detection: top: camera view, left grid: accumulation detection results and right grid: evidential detection results.

ACKNOWLEDGMENTS

This work has been supported by ANR (French National Agency) CityVIP project under grant ANR-07_TSFA-013-01. The system has been tested on sequences recorded by the test platform "STEREOPOLIS". The authors would like to thank the CityVIP group.


Benoît Sagot
email: benoit.sagot@inria.fr

Géraldine Walther
email: geraldine.walther@linguist.jussieu.fr

Non-Canonical Inflection: Data, Formalisation and Complexity Measures

Keywords: Inflectional Morphology, Description Complexity, MDL, Paradigm Shape, Canonicity, Inflection Zone, Stem Zone, Inflection Pattern, Stem Pattern

 enable reducing the complexity of morphological descriptions w.r.t. both traditional or more recent models.

Introduction

Automatically generating all forms of a language's inflectional paradigms is often considered a rather unchallenging task, since it has long been solved for most languages of interest to the area of natural language processing (NLP).

On the other hand, there is much ongoing work in theoretical morphology within lexicalist approaches, and especially within Word and Paradigm related frameworks [START_REF] Matthews | Morphology[END_REF][START_REF] Zwicky | How to describe inflection[END_REF][START_REF] Anderson | A-morphous Morphology[END_REF][START_REF] Stump | Inflectional Morphology. A Theory of Paradigm Structure[END_REF][START_REF] Corbett | Network Morphology: a DATR account of Russian nominal inflection[END_REF], on describing, modeling and explaining inflection, and in particular non-canonical inflection phenomena. For example, the Surrey Morphology Group has been working on projects on Syncretism (1999)(2000)(2001)(2002), Suppletion (2000Suppletion ( -2003)), Deponency (2004Deponency ( -2006) ) and Defectiveness (2006Defectiveness ( -2009)). In 2003 G. G. Corbett publishes his first article on Canonical Typology [START_REF] Corbett | Agreement: the range of the phenomenon and the principles of the surrey database of agreement[END_REF], thereby laying the foundations for a theoretical approach aiming at capturing the discrepancy between regularity and irregularity in inflectional paradigms.

However, studies in theoretical morphology are sometimes limited by the lack of complete formalisations and large-scale implementations of the concepts they manipulate, both in terms of morphological and lexical coverage. Still, such resources are required for achieving qualitative assessments of the validity of a given approach or to compare the relevance of several morphological models describing a given language or a specific part of a language's morphology, including the information encoded in the lexicon. This direction of research points towards recent work aiming at measuring linguistic and more specifically morphological complexity [START_REF] Bane | Quantifying and measuring morphological complexity[END_REF]. Indeed, this provides valuable insights into typological phenomena and properties of linguistic structures, and allows to compare various linguistic descriptions with objective metrics (see Section 5).

In this paper, we follow a Word and Paradigm based view of morphology. We introduce metrics allowing for measuring the complexity of a morphological description. The underlying idea is that inflectional complexity lies in the amount and distribution of inflectional irregularities. Irregularities can be represented as specific rules within the morphological grammar or as additional information within the lexicon. Hence our complexity metrics apply to both the morphological grammar and the information stored in the lexicon, thus allowing for comparing different competing descriptions in terms of descriptive complexity.

After a brief summary of the related work in both computational and theoretical inflectional morphology (Section 2), we first recall the definition of a large variety of non-canonical inflectional phenomena likely to cause increased descriptive complexity. These definitions are illustrated with data from French, Latin, Italian, Sorani Kurdish, Persian, Croatian and Slovak (Section 3). In Section 4, we then present our formal model of inflectional morphology. 1 We show how it covers all those non-canonical phenomena and allows for a formal representation of inflectional irregularity. Finally, in Section 5, we implement our model, putting a particular emphasis on French verbal inflection, which exhibits several of these phenomena and received much renewed attention in the last few years. We show how the implementation of this formal model within the Alexina lexical framework [START_REF] Sagot | The Lefff , a freely available, accurate and large-coverage lexicon for French[END_REF] makes it possible to define an information-theoretic notion of complexity for morphological descriptions that includes both the model and the corresponding lexicon, based on description length. We assess the complexity of four different accounts of French verbal inflection. As a side-result of this experiment, we also show that our formal model is not only able to encode previously proposed morphological descriptions [START_REF] Bonami | Quels verbes sont réguliers en français?[END_REF][START_REF] Sagot | The Lefff , a freely available, accurate and large-coverage lexicon for French[END_REF] but also provides a way to write a description of French verbal inflection that has a lower complexity.

Related Work

Related Work in Computational Morphology

Within contemporary computational morphology, inflection is treated in rulebased, supervised and unsupervised methods, sometimes combined [START_REF] Tepper | Inducing morphemes using light knowledge[END_REF]. 1 The reader will find a complete formal presentation in [START_REF] Walther | parsli: an inferential-realisational model of inflectional morphology for the Paradigm Shape Lexicon Interface[END_REF].

Among the first are : (i) stemming methods like the desuffixation algorithm by Porter [START_REF] Porter | An algorithm for suffix stripping[END_REF]; (ii) bi-directional analysis and generation methods like Koskeniemmi's Two-level morphology [START_REF] Koskenniemi | A general computational model for word-form recognition and production[END_REF] that uses transducers linking a deep (lexical) level to surface forms by applying systematic phonotactic transformations, as well as other finite state approaches [START_REF] Beesley | Finite State Morphology[END_REF]; (iii) morphosemantic approaches, mostly for specialised corpora [START_REF] Pratt | Automated processing of medical english[END_REF][START_REF] Lovis | Word segmentation processing: A way to exponentially extend medical dictionaries[END_REF][START_REF] Namer | Morphologie, lexique et tal : l'analyseur dérif[END_REF][START_REF] Cartoni | Lexical morphology in machine translation : A feasibility study[END_REF].

The second type of approach, namely supervised learning methods, rely on annotated learning corpora that define the expected output [START_REF] Van Den Bosch | Memory-based morphological analysis[END_REF][START_REF] Stroppa | Du quatrième de proportion comme principe inductif : une proposition et son application à l'apprentissage de la morphologie[END_REF].

Finally, there are the unsupervised approaches which can be used even for languages for which no preliminary description is available. These approaches rely on at least four different methods.

-Acquisition of morphological information can be achieved through direct comparison of graphemes distributed over a given corpus. This has be done through edit distance measures [START_REF] Baroni | Unsupervised discovery of morphologically related words based on orthographic and semantic similarity[END_REF], maximum affix recognition [START_REF] Jacquemin | Guessing morphology from terms and corpora[END_REF][START_REF] Gaussier | Unsupervised learning of derivational morphology from inflectional lexicons[END_REF][START_REF] Zweigenbaum | Liens morphologiques et structuration de terminologie[END_REF], word insertion tests [START_REF] Keshava | A simpler, intuitive approach to morpheme induction[END_REF][START_REF] Demberg | A language-independent unsupervised model for morphological segmentation[END_REF][START_REF] Bernhard | Apprentissage non supervisée de familles morphologiques par classification ascendante hiérarchique[END_REF], and analogies [START_REF] Lepage | Solving analogies on words : an algorithm[END_REF][START_REF] Hathout | From wordnet to celex : acquiring morphological links from dictionaries of synonyms[END_REF][START_REF] Moreau | Automatic morphological query expansion using analogy-based machine learning[END_REF]; -Another method relies on entropy models based on Harris' hypothesis [START_REF] Harris | From phoneme to morpheme[END_REF].

They are mainly used for automatically detecting morph(eme)-boundaries through entropy measures [START_REF] Keshava | A simpler, intuitive approach to morpheme induction[END_REF][START_REF] Bernhard | Apprentissage non supervisée de familles morphologiques par classification ascendante hiérarchique[END_REF][START_REF] Spiegler | Promodes : A probabilistic generative model for word decomposition[END_REF]; -Probabilistic methods are of various types: bayesian inference models [START_REF] Creutz | Unsupervised discovery of morphemes[END_REF], bayesian hierarchy models [START_REF] Snyder | Unsupervised multilingual learning for morphological segmentation[END_REF] or probabilistic generative models [START_REF] Spiegler | Promodes : A probabilistic generative model for word decomposition[END_REF].

-Segmentation methods relying on data compression [START_REF] Goldsmith | Unsupervised learning of the morphology of a natural language[END_REF][START_REF] Creutz | Unsupervised discovery of morphemes[END_REF] using the Minimum Description Length (MDL) principle [START_REF] Rissanen | Universal coding, information, prediction, and estimation[END_REF]. The underlying idea is that morphology always tends to use the most compact encoding by relying on inflectional regularity (see Section 5 for more details). Such regularity is stated to show in the stem vs. exponent [START_REF] Matthews | Morphology[END_REF] distribution, which should allow for a segmentation of corpora into the smallest possible morph(eme)-units;

All these unsupervised methods can also be combined with rule-based models, as in Tepper and Xia [START_REF] Tepper | Inducing morphemes using light knowledge[END_REF] who define contextual re-writing rules which they apply to the results of an unsupervised analysis in order to account for allomorphy in English and Turkish.

Although there appear to be many different methods that can be used in Computational Morphology, none of these explicitly tackle the question of complexity, regularity or canonicity per se.

Related Work in Theoretical Morphology

Morphological complexity however plays an important role in modern theoretical approaches to morphology. Within formal approaches to morphology, there are those who accept the existence of morphology and those who refuse it. The latter approaches are quite widespread and represented by Chomsky and Halle [START_REF] Chomsky | The sound pattern of English[END_REF], Lieber [START_REF] Lieber | Deconstructing Morphology : Word Formation in Syntactic Theory[END_REF] and Distributed Morphology, Halle and Marantz [START_REF] Halle | Distributed morphology and the pieces of inflection[END_REF], while the former are illustrated by what is called the Word and Paradigm approach, e.g. Matthews, Aronoff and Stump [START_REF] Matthews | Morphology[END_REF][START_REF] Aronoff | Morphology by Itself[END_REF][START_REF] Stump | Inflectional Morphology. A Theory of Paradigm Structure[END_REF]. Only in the Word and Paradigm approach, which are lexeme-based, does the question whether there is regularity within paradigms really matter. In this work, we consequently adopt a lexemebased approach to morphology that vastly relies on Matthews' view of stems and exponents [START_REF] Matthews | Morphology[END_REF].

The question of regularity in morphology is not always specifically addressed. Even in lexeme-based approaches, some works do not give the notion of regularity any theoretical status [START_REF] Stump | Inflectional Morphology. A Theory of Paradigm Structure[END_REF]. Yet, based on psycholinguistic evidence, such as presented by Pinker [START_REF] Pinker | Words and Rules[END_REF], there are modern approaches, such as illustrated in Deriving Inflectional irregularity by Bonami and Boyé [START_REF] Bonami | Deriving inflectional irregularity[END_REF], that treat irregularity as "a real grammatical phenomenon, that is manifest not only in psycholinguistic behaviour but also in language change and in synchronic grammar".

Work on (ir-)regularity in lexeme-based approaches has also been done in the subfield of Canonical Typology, such as presented by Corbett [START_REF] Corbett | Agreement: the range of the phenomenon and the principles of the surrey database of agreement[END_REF][START_REF] Corbett | Deponency, Syncretism, and What Lies Between[END_REF]. Noncanonical/irregular phenomena such as suppletion [START_REF] Boyé | Suppletion[END_REF][START_REF] Bonami | Deriving inflectional irregularity[END_REF], deponency [START_REF]Deponency and Morphological Mismatches[END_REF], heteroclisis [START_REF] Stump | Heteroclisis and paradigm linkage[END_REF], defectiveness [START_REF] Baerman | Defective Paradigms: Missing forms and what they tell us[END_REF] and more recently overabundance [START_REF] Thornton | Towards a typology of overabundance[END_REF] have been studied within this approach, giving rise to quite a series of publications. 2However, these works have seldom explicitly targeted the development of descriptions that optimise their compacity. Indeed, in order to be able to evaluate a description's compacity, a large scale implementation is required. This implementation has to rely on large-scale lexical resources covering (almost) all the described language's relevant lexical items. It must also be able to implement the measured descriptions. A short state-of-the-art presentation of existing compacity measures is given in the introduction to Section 5.

Data on Non-Canonical Inflection

Couching our work in a Word and Paradigm approach to morphology, we define a morphological description as the combination of a set of inflectable lexical entries and corresponding realisation rules realising specific morphosyntactic features. The result of all applied realisation rules are the paradigms of a language's lexemes. In order to assess the complexity of specific morphological descriptions, we start with identifying those phenomena that tend to be paradigm-complexityincreasing. These phenomena are the irregular, non-default cases. In terms of Canonical Typology, they are the non-canonical inflectional phenomena.

Canonical Inflection

The concept of canonical typology can be traced back to Corbett [START_REF] Corbett | Agreement: the range of the phenomenon and the principles of the surrey database of agreement[END_REF] in an attempt to better understand what exactly differs from a hypothetic ideal canonical stage in the different occurrences of non-canonical phenomena. In this approach, canonical inflection must not to be mistaken for prototypical inflection. Canonical inflection is rare. It corresponds to an ideal stage, seldom met, but that constitutes a purely theoretical space from which deviant phenomena can be formally distinguished [START_REF] Corbett | Canonical typology, suppletion and possible words[END_REF].

Canonical inflection is a notion that affects both the relation between the cells of a given lexeme's paradigm and the corresponding cells belonging to two different lexemes' paradigms. Canonical inflection is thus defined through the comparison of both the cells of one given lexeme and the lexemes themselves.

We preliminarily consider an inflectional paradigm canonical if it satisfies the following criteria given in Table 1 [START_REF] Corbett | Canonical typology, suppletion and possible words[END_REF]. To these criteria we add the ones in Table 2 that further define canonical paradigm shape. 3 Deviation from these criteria leads to non-canonical paradigmatic properties. [START_REF] Corbett | Canonical typology, suppletion and possible words[END_REF].

canonical inflection 1 feature expression There is no "mismatch between form and function" [START_REF] Baerman | Morphological Typology of Deponency[END_REF].

stem

Each lexeme has exactly one stem that combines with a series of exponents.

completeness

There exists exactly one form corresponding to the expression of a specific morphosyntactic feature structure.

inflection class

All forms of a lexeme are built from one single inflection class.

Table 2. Additional criteria for Canonical Inflection.

In this work, we present a representation of five types of non-canonical inflection phenomena, namely suppletion, deponency, heteroclisis, defectiveness and overabundance within the inferential realisational model for inflectional morphology developed by Walther in [START_REF] Walther | parsli: an inferential-realisational model of inflectional morphology for the Paradigm Shape Lexicon Interface[END_REF]. 4 In section 5 we show the impact these phenomena can have on the complexity of morphological descriptions.

Stem alternations/Suppletion

Suppletion comes in two types: stem suppletion and form suppletion [START_REF] Boyé | Suppletion[END_REF]. Stem suppletion occurs whenever, inside a paradigm, the forms' exponents remain regular, but their stems vary. This is for example the case for the French verb aller 'to come' which, according to most descriptions, shows as much as four different stems, all-, v-, i-and aill-. Form suppletion corresponds to cases where a whole form is inserted in a paradigm cell that should canonically be filled by a certain stem and the exponent corresponding to this specific cell. Form suppletion is described in [START_REF] Bonami | Suppletion and dependency in inflectional morphology[END_REF] for the French verbe être 'to be' in the present indicative. For this verb, the 1 st person plural form sommes, for example, does not show the regular 1 st person plural exponent -ons that canonically appears with corresponding forms of other verbs (see Table 3). Suppletion can be more or less transparent in the sense that it can be regularly associated with variation in the feature structure of a given word. Thus, Iranian languages such as Persian, show a stem alternation mainly related to tense-alternation: Persian uses a stem1 for the present tenses and a stem2 for the past tenses. The stem2 is also used for the infinitives and the participle, while stem1 serves as a stem for imperative forms.

According to traditional descriptions [START_REF] Ernout | Syntaxe Latine[END_REF] Latin verbs also display three distinct stems that are linked to specific morphosyntactic features and subparts of the inflectional paradigms, namely present, past and supine: amo 'I love', amāv ī 'I loved', amātum 'loved'. Yet the distribution of these stems does not follow strictly transparent feature-form associations. The third stem, for example, is associated with the passive past participle, but also with the active future participle and the finite passive perfective forms. There is no explicit morphosyntactic feature that appears to trigger the use of the third stem. Yet the distribution of the third stem is regular over all regular Latin verbs. Thus they are morphomic in the sense of Aronoff [START_REF] Aronoff | Morphology by Itself[END_REF].

Moreover, suppletion can be more or less massive a phenomenon. While the Latin data only concerns three different stems, French verbs show stem suppletion that extends to twelve different stems [START_REF] Bonami | Supplétion et classes flexionnelles dans la conjugaison du français[END_REF]. Bonami and Boyé [START_REF] Bonami | Supplétion et classes flexionnelles dans la conjugaison du français[END_REF] show that there are up to twelve different feature combinations that can trigger stem suppletion. They call these twelve combinations stem spaces. The stems belonging to the stem spaces are linked through stem dependency.

Yet, among those languages for which stem selection seems to be an expression of morphosyntactic features, such as the Iranian languages, further irregularity can still occur. Thus, Sorani Kurdish displays specific stem selection irregularities: As Persian, Sorani Kurdish has distinct stems for present and past tense forms (respectively stem1 and stem2). Usually passive stems are built from stem1; yet for some verbs, the passive uses stem2, while for a third type of verbs a specific passive stem is required [START_REF] Mccarus | A Kurdish Grammar: descriptive analysis of the Kurdish of Sulaimaniya, Iraq[END_REF][START_REF] Thackston | Sorani Kurdish: A reference grammar with selected readings[END_REF][START_REF] Walther | A derivational account for Sorani Kurdish passives[END_REF] (see Table 5). Such addi-tional irregularities need to be captured before a corresponding morphological description's complexity can be measured.

Deponency

Croatian nouns sometimes use singular forms to express plural [START_REF] Baerman | Deponency in serbo-croatian[END_REF]. This "mismatch between form and function" is what, following Baerman [START_REF] Baerman | Morphological Typology of Deponency[END_REF], we name deponency. Nouns are inflected according to a number of different declension classes. Some classes that are relevant for our discussion are shown in Table 6: the nouns dete 'child' and tele 'calf' inflect according to the singular pattern of respectively the a-stem and i-stem inflection classes. Using a singular inflection to express plural results in this mismatch between form and function.

(feminine) a-stem (feminine) i-stem žena 'woman' stvar 'thing' singular plural singular plural NOM žen-a žen-e stvar stvar-i ACC žen-u žen-e stvar stvar-i GEN žen-e žen-a stvar-i stvar-i DAT žen-i žen-ama stvar-i stvar-ima INS žen-om žen-ama stvar-i stvar-im The Surrey Morphology Group has collected a whole range of data on deponency phenomena in a large database. 5 Even though we will see in Section 4 that our model would not retain all these examples as instances of deponency, this database constitutes an excellent general overview of deponency phenomena. The most often discussed example of deponency probably are the Latin deponent verbs, where active meaning is considered to be conveyed through passive morphology [START_REF] Kiparsky | Blocking and periphrasis in inflectional paradigms[END_REF][START_REF] Hippisley | Declarative Deponency: A Network Morphology Account of Morphological Mismatches[END_REF][START_REF] Baerman | Morphological Typology of Deponency[END_REF][START_REF] Corbett | Deponency, Syncretism, and What Lies Between[END_REF]. However, we shall give an alternate analysis of this particular data in Section 4, showing that these verbs actually are not instances of deponency but rather constitute a textbook example of heteroclisis [START_REF] Walther | Latin passive morphology revisited[END_REF].

Heteroclisis

Heteroclisis refers to the phenomenon where a lexeme's paradigm is built out of (at least) two, otherwise separate, inflection classes.

Examples of heteroclisis are (some) Slovak animal nouns. Indeed, in Slovak, most masculine animal nouns are inflected as masculine animate nouns in the singular, whereas they may (and for some lexemes, must) inflect as masculine inanimate nouns in the plural (except in specific cases, such as personification, which triggers the animate inflection even for plural forms) [START_REF] Zauner | Praktická príručka slovenského pravopisu. Vydava ťe ľstvo Osveta[END_REF]. Compare for example the inflection of chlap 'boy', dub 'oak' and orol 'eagle' in Table 8

Defectiveness

Defectiveness [START_REF] Baerman | Defective Paradigms: Missing forms and what they tell us[END_REF] refers to lexemes which display empty (missing) cells in their paradigm. Sometimes languages contain lexemes for which expected forms are simply non-existing; native speakers are not capable of building the corresponding forms. Whenever such forms are needed, they must be conveyed through forms belonging to a synonymous lexeme. This is for example what we can observe with activa tantum: transitive verbs that do not possess passive forms and must therefore borrow the forms from synonyms. Examples thereof are the Latin verbs facere 'make' and perdire 'destroy' with no passive morphology in the present tense. The missing passive forms are supplied by (different) active verbs, namely fieri 'become' and perire 'perish' [START_REF] Kiparsky | Blocking and periphrasis in inflectional paradigms[END_REF]. These supply verbs are not just passives for the former ones, but also normal intransitives. Hence, they canot be counted as part of the defective verbs' paradigms. They possess their own independent paradigm and constitute independent lexical entries. Another example are the nouns called pluralia tantum which only exist in the plural, cf. English trousers, French vivres 'food supplies' or Slovak Vianoce 'Christmas'.

Overabundance

The obvious counterpart to defectiveness is the concept of overabundance. Overabundance occurs when cells of a paradigm contain more than one form. The notion has been introduced by Thornton and is discussed in [START_REF] Thornton | Towards a typology of overabundance[END_REF] for Italian. Canonical overabundance characterises the case where cell mates of one given cell compete, without any morphological feature permitting to choose one over the other. Table 9 shows examples thereof for Italian verbs. In French, an example is given by the verb asseoir 'to sit' that has two different forms in most cells as shown in Table 10. 7 All French verbs in -ayer also exhibit systematic overabundance (see Table 11). Indeed, for some cells, these verbs may use two competing stems (in -ay-and in -ai-) and therefore have two different inflected forms, morphologically equivalent (although semantic, pragmatic, sociolinguistic and other constraints may interfere). Since the non-canonical phenomena described in section 3 are precisely the irregularities that add complexity to the description of a lexeme's paradigm, we need a model capable of completely formalising the relevant irregularities. Only then can we use the formalised descriptions to measure their complexity with appropriate complexity metrics. We use the formal inferential realisational model for inflectional morphology described in [START_REF] Walther | parsli: an inferential-realisational model of inflectional morphology for the Paradigm Shape Lexicon Interface[END_REF]. In this model a lexeme is considered w.r.t. its formal participation in the inflectional process. Thus, we do not consider any specific semantics or possible derivational properties. In other words, we are here interested in the behaviour of what Fradin and Kerleroux refer to as inflectemes [START_REF] Fradin | Troubles with lexemes[END_REF], as opposed to lexemes, and for which a (very) simplified definition could be "a lexeme minus its semantic and argument-structural information".

This model represents an inflecteme I as the association of five defining elements : (1) the set of morphosyntactic feature structures I can express, (2) the lexeme's morphosyntactic category, (3) a stem formation rule, (4) an inflection rule, (5) a transfer rule.

Defectiveness and Overabundance In our model, categories are assigned to sets of inflectemes that canonically share sets of morpho-syntactic features. Belonging to a specific category creates morphological expectation in the sense of Brown et al. [START_REF] Brown | Defining 'periphrasis': key notions[END_REF] as to which features should be realised by independent forms. If these expectations are not met by an inflecteme's forms this inflecteme is considered defective. Thus, defectiveness is defined for an inflecteme I as the property of not fulfilling its category driven expectations: there is at least one morphosyntactic feature structure that should be expressed by the I's categorie's members for which no form is produced for I.

Whenever more forms are generated than what is expected of a given inflecteme (given its membership of a certain category), this inflecteme is considered overabundant. Thus, defectiveness and overabundance occur whenever the inputs and outputs of an inflection rule f are not in a 1 to 1 correspondance.

Let us consider the French nominal inflecteme I of vivres 'food supplies' as an example. Concerning the feature number, French nouns are expected to express the set of feature-value pairs {number singular, number plural}. However, vivres produces a realisation for the feature structure {number plural} only. It is hence defective.

Conversely, the Italian data in 9 shows instances of overabundance. For example, the inflecteme of languire is such that the realisation associated with the feature structure {number plural, person 3, tense present, mode subj} produces two forms, languano and languiscano.

Stem selection and suppletion: stem zones The stem formation rule and the inflection rule are used for expressing the morphomic dimension of inflectional paradigms belonging to a given lexeme.

Hence, stem alternation in Latin can be represented through the existence of three different stem zones that are sets of cells in which the stem realisation rules associated with expressible morphosyntactic features always produce one type of stem, as shown in Tables 12 and13 for the features listed in Table 14. Suppletion can hence be associated with specific stem zones. Moreover, the model allows for expressing that a given inflecteme I is associated with specific stem zones through the notion of the inflecteme's stem pattern. In Table 14, the active subparadigm's stem pattern comprises stem1, stem2 and stem3, while the passive subparadigm's stem pattern comprises only stem1 and stem3.

Form realisation: inflection zones In [START_REF] Walther | parsli: an inferential-realisational model of inflectional morphology for the Paradigm Shape Lexicon Interface[END_REF], an inflection class is defined as the default association of morpho-syntactic features with form realisation rules that apply to stem zones of a given inflecteme.

Just as we have defined stem zones, we then define an inflection zone as denoting the behaviour of a particular inflection class for a given set of cells. More precisely, each inflection class can be partitioned into inflection zones. In combination with stem zones, inflection zones allow for modeling situtations in which, for example, a given set of exponents is applied to two different stems of the same inflecteme for expressing different morphosyntactic features: the same inflection zone will thus be involved twice in the same paradigm. As sketched above and shown in more details in section 4.1, inflection zones and stem zones allow for a novel analysis [START_REF] Walther | Latin passive morphology revisited[END_REF] of so called Latin deponent verbs [START_REF] Kiparsky | Blocking and periphrasis in inflectional paradigms[END_REF][START_REF] Stump | Heteroclisis and paradigm linkage[END_REF][START_REF] Hippisley | Declarative Deponency: A Network Morphology Account of Morphological Mismatches[END_REF].

Deponency Another non-canonical phenomenon that may occur is deponency. As said above, we follow Baerman [START_REF] Baerman | Morphological Typology of Deponency[END_REF] in defining deponency as a "mismatch between form and function". This mismatch occurs whenever the features to be expressed by an inflecteme do not match the features usually expressed by a specifc realisation rule. This fact is captured by the notion of transfer rule, which takes as input a set of features to be expressed and outputs the set of features corresponding to the appropriate realisation rule. Canonically, the transfer rule is the identity function. An inflecteme is considered deponent whenever an inflecteme's transfer rule differs from the identity function.

In order to model the Croatian data from Table 6, we can thus use the transfer rule. Recall that Croatian sometimes uses singular forms to express plural [START_REF] Baerman | Deponency in serbo-croatian[END_REF]. In our model, an inflecteme I functioning this way has a transfer rule T I such that T I ({number plural})={number singular}.

Heteroclisis Moreover, for Croatian deponent nouns, the inflection rule f outputs the zones in Table 15 for the irregular nouns in Table 7. 8 A, B and C correspond to the three different inflection classes illustrated in Table 7. The nouns dete 'child' and tele 'calf' use exponents from two different inflection classes each to buid their paradigm. It is thus heteroclite. A similar analysis can be made of Latin "deponent verbs". Latin "deponent verbs" show morphological passive ("m-passive") forms, but express active syntax ("s-active"). Therefore, they are usually considered instances of deponency in the sense of [START_REF] Baerman | Morphological Typology of Deponency[END_REF]. On the basis that applying passive morphology to Latin verbs does not necessarily entail applying passive value, as shown in [START_REF] Kiparsky | Blocking and periphrasis in inflectional paradigms[END_REF], 9 we consider that there are distinct inflection classes applying mainly to active vs. passive morphological forms ("m-passive"): changing a verb's inflection class is seen as a derivational process. Since there are distinct endings for m-active and m-passive, we claim that there must be distinct inflection rules, i.e., for every inflecteme, distinct pairings between specific morphosyntactic feature structures and inflection zones belonging to specific inflection classes, see Figures 16 and17. Based on our definition of inflection zones, deponent verbs can be analysed as heteroclite [START_REF] Walther | Latin passive morphology revisited[END_REF], most of their endings being retrieved through inflection zones belonging to a Class B while the additional forms are retrieved from zones in a Class A (namely A3 for the active participles and A4 for the gerunds). Given an inflecteme I, a pair formed by an inflection zone and a corresponding stem zone is called a subpattern. The complete inflection pattern of I, which consists of a set of subpatterns, allows for building all of I's inflected forms. For example, the inflection of a passive Latin verb is fully defined by the following set of subpatterns: B1+stem1, B2+stem3, B3+stem3. They constitute the inflection pattern of all such verbs. Slovak animal nouns also show an instance of heteroclisis. As shown in Table 8, the zones for building the singular forms of the noun orol 'eagle' are partitions of the animate inflection class like those used for inflecting chlap 'boy', whereas the zones for the plural are retrieved in the inanimate inflection class, like for dub 'oak'.

Canonical Inflection It follows from the above-described irregularities that Canonical inflection corresponds to the case where an inflecteme's I inflection pattern and stem pattern consist of only one (inflection resp. stem) zone each, the inflection rules produce exactly one possible realisation for each morphosyntactic feature structure expressable by I's category, there is no mismatch between form and function, i.e. each exponent realised by a given realisation rule for a given morphosyntactic feature structure exactly corresponds to the morphosyntactic feature structure usually expressed in combination with this exponent.

Measuring the complexity of various descriptions of French verbal inflection

We have shown how non-canonical inflectional phenomena can be encoded in the model of inflectional morphology described in [START_REF] Walther | parsli: an inferential-realisational model of inflectional morphology for the Paradigm Shape Lexicon Interface[END_REF], using new notions such as inflection and stem zones. They can be viewed as generalisations of Bonami and Boyé's [START_REF] Bonami | Supplétion et classes flexionnelles dans la conjugaison du français[END_REF] stem spaces (and before them [START_REF] Pirelli | The paradigmatic dimension of stem allomorphy in italian verb inflection[END_REF]), which, in turn, are correspond to stem pattern in this model. With such a formalism, various competing analyses for the same data can be designed, implemented, and therefore quantitatively evaluated with a suitable complexity measure. Not only does this provide a way to compare such analyses w.r.t. their complexity, but it is also a way to get insights into the relevance of these new notions, by examining whether they are used in analyses that have a lower complexity.

For answering these questions, we have developed and implemented a formalism capable of representing the model described in section 4. The basis for our formalism is the morphological layer of the Alexina lexical formalism [START_REF] Sagot | The Lefff , a freely available, accurate and large-coverage lexicon for French[END_REF] used by several morphological (and, for some, syntactic) lexica. We have extended this formalism in order to allow it to deal with inflection zones, transfer rules, patterns and stem patterns.

Next, we have encoded various competing morphological descriptions of French verbal inflection in this formalism, in order to assess the relevance of these newly introduced notions and to quantatively compare these descriptions by means of the notion of complexity.

Descriptions of French verbal inflection

French verbal inflection is interesting in many well-known aspects, some of which have been described above. First, it is a rich system that generates forms corresponding to up to 40 different morphosyntactic features. Second, and this is of particular relevance when trying to assess the complexity of morphological descriptions, it is traditionally described as having one regular and productive inflection class, the class of so-called first-group verbs (verbs in -er), one irregular inflection class, that of third-group verbs, and the inflection class of second-group verbs (verbs like finir), which is sometimes considered as regular, as in traditional grammars, and sometimes as irregular. Analyses differ about the real productivity and regularity of this class [START_REF] Kilani-Schoch | Morphologie naturelle et flexion du verbe français[END_REF][START_REF] Boyé | Problèmes de morpho-phonologie verbale en français, espagnol et italien[END_REF][START_REF] Bonami | Quels verbes sont réguliers en français?[END_REF], which is one first possible source of discrepancy between different accounts of French verbal inflection.

Among first-group verbs, as described in Section 3.6, verbs in -ayer exhibit (regular) overabundance. In [START_REF] Bonami | La morphologie flexionnelle est-elle une fonction?[END_REF], the authors consider them as polyparadigmatic. This is not fully satisfactory given the fact that both supposed paradigms would share the same forms for half of the cells. Another way to represent this situation is to define two stems, one in -ay-and one in -ai-, two inflection zones: one, ζ 1 , that will be used only by the -ay-stem, and one, ζ 2 , that will be used by both stems. Therefore, there would be three subpatterns within the (specific) inflection pattern for -ayer verbs:

ζ 1 +-ay-, ζ 2 +-ai-and ζ 2 +-ay-.
Modeling second-group verbs can also be achieved in different ways. Using Bonami and Boyé's [START_REF] Bonami | Supplétion et classes flexionnelles dans la conjugaison du français[END_REF] twelve-stem approach, these verbs can explicitely specify a secondary stem in -iss in the lexicon, along with the base -i stem (finivs. finiss-for finir 'finish'). The traditional (and widespread) way to represent this inflection class is to consider that it uses suffixes that begin in -ss-in certain cells. Obviously, this is not very satisfying. But as it happens, the cells for which second-group verbs use their secondary stem are exactly those which are covered by the zone ζ 1 defined in the previous paragraph. Therefore, if one defines a unique inflection class for first-and second-group verbs, the same ζ 1 and ζ 2 can be used here as well, together with the following stem pattern: starting from the base stem in -i, the secondary stem can be obtained through the addition of ss, while the inflection pattern is defined by two subpatterns, namely ζ 1 +-iss-and ζ 2 +-i-. Note that this corroborates the empirically grounded analysis in [START_REF] Tribout | Les conversions de nom à verbe et de verbe à nom en français[END_REF].

As for third-group verbs, the only two approaches that we have considered are the traditional one, using many inflection classes, and the twelve-stem approach by Bonami and Boyé [START_REF] Bonami | Supplétion et classes flexionnelles dans la conjugaison du français[END_REF]. Representing the latter approach in our model can be easily achieved, by modeling (default) stem dependencies within a stem pattern, and specifying for each verb (only) those stems that differ from what can be regularily obtained using the defined stem pattern.

Starting from these considerations, we have developed four different descriptions of French inflection in the new version of Alexina that implements our morphological model, in order to try and measure their respective complexity.

Quantifying and measuring morphological complexity 10

In recent years, finding appropriate means to measure language complexity has become an active area of research. In increasing order of specificity, work has been done in that direction by considering languages globally [START_REF] Mcwhorter | The world's simplest grammars are creole grammars[END_REF][START_REF] Juola | Assessing linguistic complexity[END_REF], by restricting the study to one particular level such as morphology [START_REF] Bane | Quantifying and measuring morphological complexity[END_REF], and by measuring the complexity of particular morphological descriptions, most notably in the context of unsupervised or weakly supervised learning of morphology [START_REF] Goldsmith | Unsupervised learning of the morphology of a natural language[END_REF][START_REF] Xanthos | Apprentissage automatique de la morphologie -Le cas des structures racine-schème[END_REF].

Various metrics for measuring linguistic complexity can be found in the litterature. The simplest ones simply count the occurrences of a handcrafted set of linguistic properties: size of various inventories (e.g., phonemes, categories, morph(eme) types. . . ) [START_REF] Mcwhorter | The world's simplest grammars are creole grammars[END_REF]. However, such approaches are intrinsically arbitrary: both the set of properties which are chosen and the criteria underlying the way these properties are described are very hard to define in a principled way (e.g., what would be a suitable objective and language-independant way to build an inventory of categories for any language?). Alternative ways of measuring complexity rely on definitions of complexity that come from information-theoretic considerations. Two distinct definitions have been used in recent work, which apply on any kind of message and not only on linguistic descriptions or models: information entropy (or Shannon complexity), whose main drawback is that it requires encoding the message as a sequence of independent and identicallydistributed random variables according to a certain probabilistic model, which is difficult in practice; and algorithmic entropy (or Kolmogorov complexity) which is a more general and objective measure of the amount of information in a message, but which is not directly computable and has to be approximated.

The Kolmogorov complexity, because it is more general, is more appealing. It relies on the following intuitive idea: a model is more complex than another if it requires a longer message to be described. However, since its computation is not directly possible, one often reduces the problem to computing some kind of entropy within a particular space of possible models, by using an approximation of the Kolmogorov complexity that is defined over this model space: the result is called the description length w.r.t. the model. This is the basis of the paradigm called Minimum Description Length [START_REF] Rissanen | Universal coding, information, prediction, and estimation[END_REF]. Therefore, computing an approximation of the Kolmogorov complexity of a linguistic description requires to define as optimal as possible a way to encode this description as a string (the "code"), and then a means of computing an approximation of the Kolmogorov complexity of that (coded) string [START_REF] Bane | Quantifying and measuring morphological complexity[END_REF]. Moreover, a linguistic model is often structured, contrarily to what studies involving morphological complexity sometimes assume. In particular, assessing the complexity of a representation of a morphological lexicon cannot be reduced to measuring the complexity of a corpus whose forms have been segmented into morphs -which is however the basis of pioneering work in automatic acquisition of morphological information [START_REF] Goldsmith | Unsupervised learning of the morphology of a natural language[END_REF].

In our case, we want to measure the complexity of a given description of (a given part of) a morphological description of a particular language. This is to be contrasted with cross-linguistic comparative studies on morphological (or linguistic) complexity in general [START_REF] Mcwhorter | The world's simplest grammars are creole grammars[END_REF][START_REF] Juola | Assessing linguistic complexity[END_REF][START_REF] Bane | Quantifying and measuring morphological complexity[END_REF]: we do not want to estimate the complexity of a language, but that of particular descriptions of its morphological component, and, more specifically, of its lexical inflectional system.

The description length DL(m) of an unstructured message m within a model that decomposes it as a sequence of N symbols taken from an alphabet W = {w 1 , . . . , w n } can be computed as: DL(m) =i o(w i ) log 2 o(w i )/N , where o(w i ) is the number of occurrences of w i in m. This description length is equal to N times the entropy of the message.

In our case, the code can not be that simple, as a morphological description is structured. First, as explained earlier, it is decomposed into the morphological lexicon and the morphological model. In our formalism, we define a lexical entry as a citation form, an inflection pattern, an optional non-default stem pattern and an optional list of non-predictable stems (predictable stems need not be specified). As for the morphological model, it involves patterns and subpatterns, tables, zones and form formation rules, sandhi rules11 and other factorisation devices (see below for examples). We have designed a code that encodes all this structure in a bijective way (it can be non-ambiguously decoded) using symbols from 16 different alphabets (one for letters in citation forms, one for morphosyntactic tags, one for pattern ids, one for structural information within tables, and so on). As shown by preliminary experiments, the use of various alphabets leads to shorter descriptions as measured by the following generalisation of the above-mentioned formula: if a message m is decomposed loselessly in a sequence of symbols taken from the union of p alphabets W 1 , . . . , W p , W 1 = {w 1 1 , . . . , w 1 n1 }, . . . , W p = {w p 1 , . . . , w p np } (i.e., the alphabet from which a given symbol is taken can be inferred deterministically from its left context), then we define its description length as:

DL(m) = - p j=1 np i=1 o(w p i ) log 2 o(w p i ) N p ,
where N p is the number of symbols from alphabet W p in m. Such a metric allows for approximating the complexity of a structured model, and to measure the contribution of each alphabet to that complexity. This is the way we computed the complexity of various morphological descriptions of French verbal inflection in our model, both for evaluating the relevance of the newly introduced concepts (e.g., inflection zone, inflection pattern) and for comparing these competing morphological descriptions.

Measuring the complexity of various morphological descriptions of French verbal inflection

We have described above a spectrum of possible descriptions that correspond to various ways to balance richer morphological grammars and richer lexical specifications. We used the lexical information in the lexicon Lefff [START_REF] Sagot | The Lefff , a freely available, accurate and large-coverage lexicon for French[END_REF] for our experiments, limiting ourselves to verbs and ignoring multiple entries for the same lexeme (a given lexeme may have several sub-categorisation patterns and several meanings, and therefore have several entries in the Lefff ). The current version of the Lefff contains 7,820 verbs, among which 6,966 first-group verbs, 315 second-group verbs and 539 third-group verbs.

In our new version of Alexina's morphological layer, the morphological information associated with a lexical entry contains the following elements, illustrated by the example below:

a citation form, typically the infinitive for French verbs; an inflection pattern followed by an optional pattern variant: if two patterns only differ on a few slots, they can be merged, and alternate realisation rules are specified for these slots and are lexically triggered by these infection pattern variants; optionally, a stem pattern (a non-specified stem pattern means that the default stem pattern associated with the pattern should be used); optionally, a list of stems (a non-specified stem means that the default stem should be used, as defined by stem formation rules associated with the stem pattern)

For example, an entry such as "bouillir v 23r / bouill,,bou " corresponds to an inflecteme with the citation form bouillir, the pattern v (with the pattern variant 23r), the default stem pattern associated with pattern v in the morphological grammar, as well as bouill as stem 1 and bou as stem 3 (all other stems following the stem pattern). Let us now briefly describe our four competing descriptions of French verbal inflection. The lexical entries for a small set of inflectemes in each of these descriptions is shown in Table 19 for illustration purposes. At one end of the spectrum, we automatically generated a "flat" morphological description, called Flat, that uses no stems, no sandhi and no zones, in the following way. The longest common substring shared by all inflected forms of each lexeme has been identified, and the remainder of each form has been considered a "suffix"; then the list of all suffixes has been ordered w.r.t. the corresponding morphosyntactic tags, thus creating a signature. Finally, all lexemes that share the same signature have been considered as belonging to the same inflection class which is trivially built from the signature and the ordered list of tags. The resulting description has 139 inflection classes. Its description length, measured as explained above, is around 131,400 bits (9,200 bits in the lexicon, 12122,200 bits in the morphological grammar). At the other end of the spectrum lies Bonami and Boyé's [START_REF] Bonami | Supplétion et classes flexionnelles dans la conjugaison du français[END_REF][START_REF] Bonami | Quels verbes sont réguliers en français?[END_REF] analysis, which uses only one inflection class and twelve stems. We started from a preliminary DATR implementation of this model (Bonami, p.c.). Because this analysis was designed on phonemes, we had to apply certain transformations to enable the encoding of graphemic inflection, including by introducing sandhi operations. In order to correctly generate all overabundant forms, we extended it in several ways. The result is a description, called BoBo, that contains only one inflection class, several patterns (4 for non-defective verbs including "v" and "v-ayer" found in Table 19, and a few more for defective ones) and 61 sandhi rules. 13 This description strongly relies on an important feature of our Alexina implementation of the model described in this paper and already mentioned above: any underspecified piece of information is filled by defaults (not specifying a given stem in a lexical entry leads to using the stem formation table for generating it; not specifying a stem generation table for a given lexical entry leads to using the default stem generation table associated with its inflection pattern, and, if not specified, to consider that there is only one stem that applies to all forms, and so on). BoBo's description length is around 52,000 bits (46,600 bits in the lexicon, which is particularily high and is caused by all explicitely specified stems, and 5,400 bits in the morphological grammar, which is very low as expected).

Between these two extremes, the original description Orig used by the Lefff , which heavily relies on sandhis but uses a lot of inflection classes for third-group forms, has a description length of 83,000 bits (8,100 bits in the lexicon, 74,800 bits in the grammar). More interestingly, as mentioned above, using the notion of inflection zone and relying on a reasonable amount of sandhi rules, we were able to develop a more satisfactory morphological description for French verbs, named New, which uses 20 inflection classes (including one for first-group verbs without overabundance, one for first-group verbs in -ayer and one for secondgroup verbs). The corresponding description length, 35,800 bits, is lower than that of BoBo. It corresponds to 20,100 bits in the lexicon (twice more than in Flat, but twice less than in BoBo) and 15,700 bits in the morphological grammar (three times more than in BoBo, but eight times less than in Flat). All these figures for our four descriptions, ordered according to the abovementioned continuum, are summarized in Table 20 and displayed graphically in Figure 1. They make it visible that using the notion of inflection zone, thus generalising the notion of stem space (which in our model corresponds to the notion of stem pattern), leads to accounts of French verbal morphology that constitute a shorter coding of the same information than the three other descriptions, both traditional ones and more original and recent ones [START_REF] Bonami | Quels verbes sont réguliers en français?[END_REF]. Note that this conclusion would have been different if the description length of the lexicon had not been taken into account. However, as the balance between including more information in the lexicon and more in the morphological model depends on the morphological description, it would make little sense to evaluate the description length of the morphological model only.

Conclusion

In this paper, we have addressed the question of measuring the complexity of morphological descriptions using the information-theoretic concept of description length. We have applied our method on four competing descriptions of French verbal inflection. Since descriptive complexity arises with inflectional irregularities, we have couched our descriptions in the formal inferential realisational model developed in [START_REF] Walther | parsli: an inferential-realisational model of inflectional morphology for the Paradigm Shape Lexicon Interface[END_REF]. This model, which relies on new notions such as the one of inflection zone and stem zone, allows for modeling a wide range of non-canonical inflectional phenomena, such as suppletion, deponency, heteroclisis, defectiveness and overabundance. We have developed four descriptions of French verbal inflection in this model and implemented them in the Alexina [START_REF] Sagot | The Lefff , a freely available, accurate and large-coverage lexicon for French[END_REF] morphological framework.

We have also designed an information-theoretic way to assess the complexity of a morphological description in this model.

Our work shows that using information-theoretic concepts to assess description complexity is indeed feasable and relevant as a comparison between competing descriptions. Moreover, quantitative results on our four different descriptions have shown that the traditional way of describing French verbal inflection using many inflection classes, as well as a more recent and radically different proposal [START_REF] Bonami | Quels verbes sont réguliers en français?[END_REF], can both be outperformed in terms of low complexity by using notions such as inflection zones, stem patterns and inflection patterns, in order to find a better balance between the amount of morphological information that is encoded in the lexicon and in the morphological rules.

Fig. 1 .

 1 Fig. 1. Visualisation of the description length of various accounts of French verbal morphology .

Table 1 .

 1 Criteria for Canonical Inflection according to Corbett in

	comparison across comparison
	cells of a lexeme across lexemes

Table 3 .

 3 Form suppletion in the present indicative paradigm of French être 'to be'

		singular plural	lexeme translation stem1 stem2
	p1 suis	sommes	ârâstan 'to adorn'	ârâst ârâ
	p2	es	êtes	âmuxtan 'to learn	âmuxt âmuz
	p3	est	sont	raqsidan 'to dance	raqsid raqs

Table 4 .

 4 Persian present and past stems

Table 5 .

 5 Sorani Kurdish Irregular Passive Stems

Table 6 .

 6 Croatian noun declension

		neut. -et∼a-stem neut. -et∼i-stem
		dete 'child'		tele 'calf'
		singular plural singular plural
	NOM dete	deca	tele	telad
	ACC dete	decu	tele	telad
	GEN deteta	dece	teleta	telad
	DAT detetu	deci	teletu	teladi(ma)
	INS	detetom decom	teletom teladi(ma)

Table 7 .

 7 Croatian deponent noun declension

Table 8 .

 8 .6 Heteroclisis in Slovak masculine animal names inflection

		masculine animate	masculine inanimate	masculine heteroclite
		chlap 'boy'		dub 'oak'		orol 'eagle'
		singular plural	singular plural	singular plural
	NOM chlap	chlap-i	dub	dub-y	orol	orl-y
	GEN chlap-a	chlap-ov	dub-a	dub-ov	orl-a	orl-ov
	DAT chlap-ovi chlap-om	dub-u	dub-om	orl-ovi	orl-om
	ACC chlap-a	chlap-ov	dub	dub-y	orl-a	orl-y
	LOC chlap-ovi chlap-och	dub-e	dub-och	orl-ovi	orl-och
	INS	chlap-om chlap-mi	dub-om	dub-mi	orl-om	orl-ami

Table 9 .

 9 Overabundance in Italian[START_REF] Thornton | Towards a typology of overabundance[END_REF]

		cell-mate 1 cell-mate 2
	'languish' 3pl.prs.subj languano	languiscano
	'possess' 3pl.prs.subj	possiedano	posseggano
	'possess' 3sg.prs.subj	possieda	possegga
	'possess' 1sg.prs.subj	possiedo	posseggo

Table 10 .

 10 Overabundance inFrench asseoir 'to sit'

	ind.pres singular plural	ind.pres singular plural
	p1	assois assieds	assoyons asseyons	p1	balaye balaie	balayons
	p2	assois assieds	assoyez asseyez	p2	balayes balaies	balayez
	p3	assoit assied	assoient asseyent	p3	balaye balaie	balayent balaient

Table 11 .

 11 Overabundance inFrench balayer 'to sweep'

	4 A formal model for inflectional morphology
	4.1 Defining the relevant notions

Table 12 .

 12 A:

		stem1	stem	active subpara. passive subpara.
		... s s s	stem1 imperf. finite stem2 perf. finite stem3 active future part. passive past part. imperf.finite
		stem3		perf. finite (periphr.)
		Table 13. B:
	Stem zones in	Stem zones in
	the Latin ac-	the Latin pas-
	tive (sub-)pa-	sive (sub-)pa-
	radigm	radigm

Table 14 .

 14 Morphomic feature association with Latin verb stems

Table 15 .

 15 Croatian noun inflection zones for deponent lexemes.

	inflection class A: neuter	B: (feminine) C: (feminine)
		-et-stem	a-stem	i-stem
	dete 'child'	sg: zoneA,sg	pl: zoneB,sg
	tele 'calf'	sg: zoneA,sg		pl: zoneC,sg

Table 16

 16

					Lexeme Type	m-active	m-passive
			B1		actives	A1, A2, A3, A4
				B3	passives deponents	A3, A4	B1, B2, B3 B1, B2, B3
					semi-dep. t1	A1, A3, A4	B2, B3
			B2		semi-dep. t2	A2, A3, A4	B1, B3
		.	Table	17.
	Zones	in	Zones	in
	Class A		Class B

Table 18 .

 18 Zone distribution in Latin verb inflection

Table 19 .

 19 Lexical entries for a small set of inflectemes in each of our four competing descriptions of French verbal inflection

	Citation form Flat Orig	New	BoBo
	aimer	v1	v-erstd	v-er	v1
	acheter	v18 v-erstd	v-er	v1
	jeter	v8	v-erdbl	v-eter	v1/,jett,,,,,,,,jett@
	balayer	v12 v-ayer	v-ayer	v-ayer1
	finir	v2	v-ir2	v-ir2	v23r
	requérir	v42 v-ir3	v-ir3	v23r/requér,requier,,,,,,,,requer,requi,requis
	cueillir	v51 v-assaillir v23r/cueill,,,,,,,,,cueill@ v23r/cueill,,,,,,,,,cueill@
	prendre	v24 v-prendre v-prendre	v3re
	mettre	v17 v-mettre v-mettre	v3re/,,met,,,,,,,,mi,mis

Table 20 .

 20 Description nameFlat Orig New BoBo Length of the morphological grammar (bits) 122,200 74,800 15,700 5,400 Length of the lexicon (bits) 9,200 8,100 20,100 46,600 Total length of the morphological description (bits) 131,400 83,000 35,800 52,000 Description length of various accounts of French verbal morphology .

Existing work is mostly done on phonological data. Our work focuses on written data for now. We plan on doing some future work on comparing phonological and graphemic morphological complexity.

Among the additional criteria, criterion 2 derives directly from criterion 2 in[START_REF] Corbett | Canonical typology, suppletion and possible words[END_REF] and criterion

can be seen as derived from criterion 3 in[START_REF] Corbett | Canonical typology, suppletion and possible words[END_REF].[START_REF] Baerman | Morphological Typology of Deponency[END_REF] This model does not include a separate formalisation of syncretism. Syncretism is modeled as a combination of heteroclisis and deponency. For a complete discussion thereof, see[START_REF] Walther | parsli: an inferential-realisational model of inflectional morphology for the Paradigm Shape Lexicon Interface[END_REF].

http://www.smg.surrey.ac.uk/deponency.

Both chlap and dub have a regular inflection: chlap belongs to the standard inflection class for masculine animate stems ending with a consonant, whereas dub belongs to the standard inflection class for masculine inanimate stems ending with what is called a hard or neutral consonant in the Slavic linguistic tradition.

See for example[START_REF] Bonami | La morphologie flexionnelle est-elle une fonction?[END_REF] for a longer discussion thereof.

The representation shows that, in addition to being deponent, Croatian nouns are also heteroclite. Non-canonical behaviours can sometimes combine.

Indeed, Kiparsky[START_REF] Kiparsky | Blocking and periphrasis in inflectional paradigms[END_REF] shows that passive morphology can trigger many kinds of, partly unpredictable, semantic changes. This property is one of derivational morphology -and not inflectional morphology which is usually considered as being semantically predictable[START_REF] Boyé | Régularité et classes flexionnelles dans la conjugaison du français[END_REF].

This title is borrowed from[START_REF] Bane | Quantifying and measuring morphological complexity[END_REF], whose first sections provide a brief but complete and detailed account of recent work on this topic.

We define sandhi rules as morphographemic and/or morphophonemic rules, already implemented in the Alexina formalism. They are local transformations that apply at the boundary between two morph(eme)s. Hence, in French verbal inflection, a stem ending in -g followed by a suffix in [aou]-is associated with a surface form in which an e is appended to the stem: mang ons ↔ mange ons.

Here and in all subsequent figures, the description length of citation forms is not taken into account, as it is the same for all descriptions.

For example, one of these rules handles the @ at the end of some of the stems, depending on its environment.