Mehdi Mirzapour

Jean-Philippe Prost

Christian Retoré

Measuring Linguistic Complexity: Introducing a New Categorial Metric

This paper provides a computable quantitative measure which accounts for the difficulty in human processing of sentences: why is a sentence harder to parse than another one? Why is some reading of a sentence easier than another one? We take for granted psycholinguistic results on human processing complexity like the ones by Gibson. We define a new metric which uses Categorial Proof Nets to correctly model Gibson's account in his Dependency Locality Theory. The proposed metric correctly predicts some performance phenomena such as structures with embedded pronouns, garden paths, unacceptable center embeddings, preference for lower attachment and passive paraphrases acceptability. Our proposal gets closer to the modern computational psycholinguistic theories, while it opens the door to include semantic complexity, because of the straightforward syntax-semantics interface in categorial grammars.

Introduction

Linguistics -especially generative grammar à la Chomsky, makes a distinction between competence and performance in the human processing of natural language [START_REF] Chomsky | Aspects of the Theory of Syntax[END_REF]. The competence is, roughly speaking, our ideal ability without time and resource constraints to parse a sentence, i.e. to decide whether it is grammatical or not. Competence is described by a grammar, a lexicon, assuming an unlimited working memory and time. Performance is how we actually parse a sentence; whether we Mehdi Mirzapour LIRMM, Univ Montpellier, CNRS, Montpellier, France. e-mail: mehdi.mirzapour@gmail.com Jean-Philippe Prost LPL, CNRS & Univ Aix-Marseille, Aix-en-Provence, France. LIRMM, Univ Montpellier, CNRS, Montpellier, France. e-mail: jean-philippe.prost@univ-amu.fr Christian Retoré LIRMM, Univ Montpellier, CNRS, Montpellier, France. e-mail: christian.retore@lirmm.fr succeed in achieving that and how much the sentence resists to our attempt to analyze it, in a limited amount of time for the conversation to work, with our present knowledge, and our memory limits. Computing the space and time algorithmic complexity is a false solution because no one knows the algorithm being used by human beings, whether it depends on the individual or on the kind of conversation. Even if it were so, nothing guarantees that space and time algorithmic complexity matches the degree of difficulty we experience when processing sentences. And finally there are other factors of complexity that are hard to take into account, like the size of the grammar and short time memory limitation [START_REF] Moot | Natural language semantic and computability[END_REF].

This paper, following an earlier body of work [START_REF] Johnson | Special Issue on Recent Advances in Logical and Algebraic Approaches to Grammar[END_REF][START_REF] Morrill | Incremental processing and acceptability[END_REF], tries to provide a formal and computable account of the results of psycholinguistic theories with regard to linguistic complexity. We focus on syntactic complexity of structures with embedded pronouns [START_REF] Gibson | The dependency locality theory: A distance-based theory of linguistic complexity[END_REF], garden pathing [START_REF] Bever | The cognitive basis for linguistic structures[END_REF], unacceptability of center embedded relatives [START_REF] Gibson | The processing complexity of english center-embedded and selfembedded structures[END_REF], preference for lower attachment [START_REF] Kimball | Seven principles of surface structure parsing in natural language[END_REF] and passive paraphrases acceptability [START_REF] Morrill | Incremental processing and acceptability[END_REF]1.

Regarding the psycholinguistic aspects, we mainly follow the studies by Gibson of linguistic complexity of human parsing. Gibson first studied the notion of linguistic difficulty [START_REF] Gibson | A computational theory of human linguistic processing: Memory limitations and processing breakdown[END_REF] through the maximum number of incomplete syntactic dependencies that the processor has to keep track of during the course of processing a sentence. We refer to this theory as Incomplete Dependence Theory (IDT), as coined by Gibson. IDT had some limitations for referent-sensitive linguistic phenomena, which justified the later introduction of the Syntactic Prediction Locality Theory [START_REF] Gibson | Linguistic complexity: Locality of syntactic dependencies[END_REF]. A variant of this theory, namely Dependency Locality Theory (DLT), was introduced later [START_REF] Gibson | The dependency locality theory: A distance-based theory of linguistic complexity[END_REF] to overcome the limitations with respect to the new linguistic performance phenomena. In the original works, both IDT and DLT use properties of linguistic representations provided by Government-Binding Theory [START_REF] Chomsky | Some concepts and consequences of the theory of government and binding[END_REF].

On the formal side, in order to compute the complexity of a sentence -in a way that matches Gibson's results -we use Lambek Categorial Grammar [START_REF] Lambek | The mathematics of sentence structure[END_REF] by means of proof nets construction [START_REF] Moot | The logic of categorial grammars: a deductive account of natural language syntax and semantics[END_REF]Chap 6]. Proof nets were originally introduced by Girard [START_REF] Girard | Linear logic[END_REF] to describe proofs in linear logic, up to some equivalences. Categorial proof nets are introduced by Roorda and Retoré [START_REF] Roorda | Proof nets for Lambek calculus[END_REF][START_REF] Retoré | Calcul de Lambek et logique linéaire[END_REF] to describe proofs in the Lambek calculus and related formalisms. They are to categorial grammar what parse trees are to phrase structure grammar. This kind of approach was initiated by Johnson [START_REF] Johnson | Special Issue on Recent Advances in Logical and Algebraic Approaches to Grammar[END_REF], who defines a measure of the instantaneous complexity when moving from a word to the next one (in particular for center embedded relative clauses) in a way that matches Gibson's and Thomas' analysis [START_REF] Gibson | The processing complexity of english center-embedded and selfembedded structures[END_REF]. To define the complexity of a sentence, Johnson considers the maximum complexity between the words in a given sentence. This approach was refined by Morrill [START_REF] Morrill | Incremental processing and acceptability[END_REF], who re-interprets axiom links in categorial proof nets as incomplete (or unresolved) dependencies. We rename this technique as IDT-based complexity profiling since it clearly inherits many aspects of Gibson's IDT, plus the new notion of profiling that exists in some psycholinguistic theories. This technique is quite successful at predicting linguistic performance phenomena such as garden paths, unacceptability of center embedding and preference for lower attachment. Nevertheless, there is some predictive limitation for referent-1 These linguistic phenomena are described in detail in sections (4) and [START_REF] Chatzikyriakidis | From logical and linguistic generics to hilbert's tau and epsilon quantifiers[END_REF].

sensitive phenomena such as structures with embedded pronouns. Our strategy to overcome this issue is to apply DLT instead of IDT on proof net constructions. This strategy leads us to introduce DLT-based complexity profiling. We will show how this reformulation can improve the predictive power of the existing models in favor of the referent-sensitive linguistic phenomena.

The purpose of developing our computational psycholinguistic model is not solely limited to measuring linguistic complexity. It is potentially applicable to some specific tasks in the domain of formal compositional semantics. For instance, ranking different possible readings of a given ambiguous utterance, or more generally translating natural language sentences into weighted logical formulas. The rest of the paper is organized as follows: Section 2 summarizes Gibson's ideas on modeling the linguistic complexity of human sentence comprehension, namely IDT and DLT. In section 3 we then define proof nets, and recall the success and limitation of IDT-based complexity profiling. In section 4 we define our DLT-inspired measure; we show how it fixes some problems in previous work and how it gives a correct account of those phenomena. In section 5 we provide more linguistic evidence to support our claim in favor of DLT-based complexity profiling over IDT-based approach. In section 6 we discuss limitations of our approach and possible future works for solving them. In the last section we conclude our paper and discuss possible future works.

Gibson's Theories on Linguistic Complexity

We provide a very quick review of Gibson's IDT and DLT in order to make the readers familiar with their underlying concepts. The question of how to automatically compute linguistic complexity based on both theories with categorial proof nets will be covered in the sections (3.2) and [START_REF] Catta | Quantifier scoping and semantic preferences[END_REF].

Incomplete dependency theory relies on counting incomplete dependencies during the incremental processing of a sentence, whenever a new input word must be connected to the current and pending linguistic structure. The main parameter in IDT is the number of incomplete dependencies when the new word integrates the existing structure. It explains the increasing complexity of examples (1a)-(1c) with nested relative clauses. In (1a), the reporter has 1 incomplete dependency; in (1b), the senator has 3 incomplete dependencies; in (1c) John has 5 incomplete dependencies at the point of processing. For room sake we only explain the most complex case, i.e. (1c), in which the incomplete dependencies at the moment of processing John are: (i) the NP the reporter depends on a verb that should follow it; (ii) the NP the senator depends on a different verb to follow; and (iii) the pronoun who (before the senator) depends on a verb to follow; (iv) the NP John depends on another verb to follow; and (v) the pronoun who (before John ) depends on a verb to follow. These are five unsaturated or incomplete or unresolved dependencies. IDT in its original form suggests to calculate the maximum number of incomplete dependencies in a sentence. One can observe that the complexity is proportional to the number of incomplete dependencies. Dependency Locality Theory is a distance-based referent-sensitive linguistic complexity measure put forward by Gibson to supersede the predictive limitations of the incomplete dependency theory. DLT posits two different costs for integration and storage. In this paper, we have only focused on the integration cost. The linguistic complexity is interpreted as the locality-based cost of the integration of a new word to the dependent word in the current linguistic structure which is relied on the number of the intervened new discourse-referents. Through the count of these referents, we can predict the relative complexity, such as structures with embedded pronouns, illustrated in example (1d). The experiments [START_REF] Warren | The effects of discourse status on intuitive complexity: Implications for quantifying distance in a locality-based theory of linguistic complexity[END_REF] support the acceptability of (1d) over (1c). According to the discourse-based DLT structural integration cost hypothesis, since referents for the first-person pronoun I is already present in the current discourse, integrating across them consumes fewer cognitive resources than integrating across the new discourse referents before John. By means of just two aspects of DLT, namely the structural integration and the discourse processing cost, we can predict a number of linguistic phenomena, as we will see in details with some examples.

Complexity Profiling in Categorial Grammars

Proof-nets as parse structures

Our exposition of the henceforth classical material on proof nets for categorial grammar follows [START_REF] Moot | The logic of categorial grammars: a deductive account of natural language syntax and semantics[END_REF] -the main original papers on this topic are [START_REF] Lambek | The mathematics of sentence structure[END_REF][START_REF] Girard | Linear logic[END_REF][START_REF] Retoré | Calcul de Lambek et logique linéaire[END_REF][START_REF] Roorda | Proof nets for Lambek calculus[END_REF]. Categorial grammars are defined from a set C of grammatical categories, defined from base categories (for instance B = {np, n, S}) including a special symbol S (for sentence) and operators, for instance :

C ::= B | C\C | C/C
The symbols \ and / can be viewed as logical connectives, namely implication(s) of a logic, namely intuitionistic non-commutative multiplicative linear logic better known as the Lambek calculus. Such formulas can be viewed as formulas of linear logic, with conjunction ⊗ disjunction `and negation (_) ⊥ because implications can be defined from negation and disjunction:

Definition of \ and /: A \ B ≡ A ⊥ `B B / A ≡ B `A⊥ De Morgan equivalences (A ⊥ ) ⊥ ≡ A (A `B) ⊥ ≡ B ⊥ ⊗ A ⊥ (A ⊗ B) ⊥ ≡ B ⊥ `A⊥
Some formulas have a polarity. Formulas are said to be positive (output) • or negative (input) • as follows:2

a : •, a ⊥ : • ⊗ • • • undefined • • • undefined `• • • undefined • • • undefined
So a `b has no polarity; a ⊥ `b is positive which is equal to a \ b; while b ⊥ ⊗ a is negative and it is equal to the negation of a \ b. Categories are, roughly speaking, analogous to non-terminal categories in phrase structure grammars. But observe that they are endowed with an internal structure, i.e. (np \ S) / np is a compound category and the rules make use of this internal structure, connectors \ and / and subcategories n, np and S. The rules (of the logic) do not depend on the language generated (or analyzed) by the grammar. They are the same for every language, and the lexicon makes the difference. The lexicon maps every word to a finite set of categories. A parse structure in a categorial grammar defined by a lexicon L for a sequence of words w 1 , . . . , w n simply is a proof of c 1 , . . . , c n S with c i ∈ L(w i ) in some variant of the Lambek calculus. The rules for the basic (associative) Lambek calculus are:

A A H 1 , H 2 . . . , H n-1 , H n C H 1 , H 2 . . . , H n-1 C / H n H 1 , H 2 . . . , H n-1 , H n C H 2 . . . , H n-1 , H n H 1 \ C H 1 , . . . , H i , . . . , H n C G 1 , . . . , G n A H 1 , . . . , H i-1 , G 1 , . . . , G n , A \ H i , H i+1 , . . . , H n C H 1 , . . . , H i-1 , H i , H i+1 . . . , H n C G 1 , . . . , G n A H 1 , . . . , H i-1 , H i / A, G 1 , . . . , G n , H i+1 , . . . , H n C
Since the Lambek sequent calculus enjoys the cut-elimination property, whereby a sequent is provable if and only if it is provable without the cut rule, we do not mention the cut rule. Categorial grammars are known for providing a transparent and computable interface between syntax and semantics. The reason is that the categorial parse structure is a proof in some variant of the Lambek calculus, and that this proof gives a way to combine semantic lambda terms from the lexicon into a lambda term which encodes a formula expressing the meaning of the sentence.

We cannot provide more details herein, but the reader is referred to [START_REF] Moot | The logic of categorial grammars: a deductive account of natural language syntax and semantics[END_REF]Chapter 3]. For instance, the categorial analysis of Every barber shaves himself with the proper semantic lambda terms for each word in the sentence yields the logical form ∀x.bar ber(x) ⇒ shave(x, x).

It has been known for many years that categorial parse structures, i.e. proof in some substructural logic, are better described as proof nets [START_REF] Roorda | Proof nets for Lambek calculus[END_REF][START_REF] Retoré | Calcul de Lambek et logique linéaire[END_REF][START_REF] Moot | Proof nets for linguistic analysis[END_REF][START_REF] Moot | The logic of categorial grammars: a deductive account of natural language syntax and semantics[END_REF]. Since categorial grammars follow the parsing-as-deduction paradigm, an analysis of a c phrase w 1 , , . . . , w n is a proof of c under the hypotheses c 1 , ..., c n , where c i is a possible category for the word w i ; and proofs in those systems are better denoted by graphs called proof nets. The reason is that different proofs in the Lambek calculus may represent the same syntactic structure (constituents and dependencies), but these essentially similar sequent calculus proofs correspond to a unique proof net. A proof net is a graph, whose nodes are formulas, and it consists of two parts: subformula trees of the conclusions, in the right order, whose leaves are the base categories and branching are two connectives `and ⊗ -as we saw earlier formulas with \ and / can be expressed from base categories and their negations with `and ⊗ -for nodes that are not leaves the label can be limited to the main connective of the subformula instead of the whole formula, without loss of information; axioms that are a set of edges, pairwise disjoint, and connecting a leaf z to a leaf z ⊥ in such a way that every leaf is incident to some axiom link.

However not all such graphs are proof nets, only those satisfying:3

Acyclicity every cycle contains the two edges of the same `branching.

Connectedness There is a path not involving the two edges of the same `branching between any two vertices.

Intuitionism Every conclusion can be assigned some polarity.

Non commutativity

The axioms do not cross (are well bracketed).

The advantage of proof-nets over sequent calculus is that they avoid the phenomenon known as spurious ambiguities-that is, when different parse structures correspond to the same syntactic structure (same constituent and dependencies). Indeed proofs (i.e. parse structures) with unessential differences are mapped to the same proof net. A (normal) deduction of c 1 , ..., c n c (i.e. a syntactic analysis of a sequence of words as a constituent of category (c) maps to a (normal) proof net with conclusions (c n ) ⊥ , ..., (c 1 ) ⊥ , c [START_REF] Roorda | Proof nets for Lambek calculus[END_REF][START_REF] Moot | The logic of categorial grammars: a deductive account of natural language syntax and semantics[END_REF]. Conversely, every normal proof net corresponds to at least one normal sequent calculus proof [START_REF] Retoré | Calcul de Lambek et logique linéaire[END_REF][START_REF] Moot | The logic of categorial grammars: a deductive account of natural language syntax and semantics[END_REF].

Incomplete Dependency-Based Complexity Profiling and its Limitation

In this subsection we recall the IDT-based measure of linguistic complexity by Morrill [START_REF] Morrill | Incremental processing and acceptability[END_REF], which itself improves over a first attempt by Johnson [START_REF] Johnson | Special Issue on Recent Advances in Logical and Algebraic Approaches to Grammar[END_REF]. Both measures are based on the categorial proof nets. The general idea is simple: to re-interpret the axiom links as dependencies and to calculate the incomplete dependencies during the incremental processing by counting the incomplete axiom links for each word in a given sentence. This is almost the same as Gibson's idea in his IDT, except the fact that he uses some principles of Chomsky Government-Binding theory [START_REF] Chomsky | Some concepts and consequences of the theory of government and binding[END_REF] instead of the categorial proof nets. The notion of counting incomplete dependencies for each node, called complexity profiling, is more effective in terms of prediction than approaches that only measure the maximum number of incomplete dependencies or the maximum cuts [START_REF] Johnson | Special Issue on Recent Advances in Logical and Algebraic Approaches to Grammar[END_REF]. We can rewrite IDT-based complexity profiling [START_REF] Morrill | Incremental processing and acceptability[END_REF] with the following definitions:

Definition 1 Let π be a a syntactic analysis of w 1 , • • • , w n with categories C 1 , . . . , C n -that is a categorial proof net with conclusions (C n ) ⊥ , ..., (C 1 ) ⊥ , S. Let C i 0 be one of the C i (i ∈ [1, n]).
The incomplete dependency number of C i 0 in π, written as

I D π (C i 0 ), is the count of axioms c -c in π such that c ∈ (C i 0 -m ∪ S) (m ≥ 0) and c ∈ C i 0 +n+1 (n ≥ 0).
Definition 2 Let π be a a syntactic analysis of w 1 , • • • , w n with categories C 1 , . . . , C n -that is a categorial proof net with conclusions (C n ) ⊥ , ..., (C 1 ) ⊥ , S. We define the IDT-based linguistic complexity of π, written f idt (π) by (1

+ n i=1 I D π (C i )) -1
. Definition 3 Given two syntactic analyses π i and π j , not necessarily of the same words and categories, we say that π i is IDT-preferred to π j whenever f idt (π i ) > f idt (π j ).

Example 2. 2a. The reporter who sent the photographer to the editor hoped for a good story. 2b. The reporter who the photographer sent to the editor hoped for a good story. Figures [START_REF] Bever | The cognitive basis for linguistic structures[END_REF] and [START_REF] Blache | A computational model for linguistic complexity[END_REF] show the two relevant proof nets for examples (2a) with subjectextracted relative clause and (2b) with object-extracted relative clause (examples from [START_REF] Gibson | The dependency locality theory: A distance-based theory of linguistic complexity[END_REF]). The relevant complexity profiles for (2a) and (2b) are illustrated in the figure [START_REF] Blache | Evaluating language complexity in context: New parameters for a constraint-based model[END_REF]. As shown in the figures, the total sum of the complexity for (2b) is greater than (2a). Therefore it can predict correctly that (2a) is preferred over (2b). That preference is confirmed by reading time experiments [START_REF] Gibson | An integration-based theory of computational resources in sentence comprehension[END_REF]. [START_REF] Catta | Quantifier scoping and semantic preferences[END_REF] Obviously, IDT-based account does not use DLT as its underlying theory. Not surprisingly, the linguistic phenomena that can only be supported by DLT would not be supported by IDT-based complexity profiling. Figure [START_REF] Catta | Quantifier scoping and semantic preferences[END_REF] shows this failure. We can verify this by applying the definitions on the relevant proof nets as it is illustrated in the figures ( 5) and ( 6). As one may notice, the corresponding proof nets for the examples (1c) and (1d) are almost the same. Consequently, IDT-based complexity profiling cannot discriminate both examples, i.e. it generates the same number for both sentences in contrast to the experiments [START_REF] Warren | The effects of discourse status on intuitive complexity: Implications for quantifying distance in a locality-based theory of linguistic complexity[END_REF] as it is shown in the figure (4). This shows the importance of introducing DLT-based complexity profiling for proof nets in order to make more predictive coverage-as we will do so.

A New Proposal: Distance Locality-Based Complexity Profiling

As we discussed, IDT-based complexity profiling is a distance-based measure. However, it does not have a referent-sensitive criterion and due to this fact, it cannot support some of the linguistic phenomena such as structures with embedded pronouns. One plausible strategy to overcome this lack is introducing DLT-based complexity profiling. This will allow us to have a referent-sensitive measure. In this section, we provide the precise definitions of our DLT-based proposal on the basis of the categorial proof nets. Here they are: Definition 4 A word w is said to be a discourse referent whenever it is a proper noun, common noun or verb.

Definition 5 Let π be a a syntactic analysis of w 1 , • • • , w n with categories C 1 , . . . , C n -that is a categorial proof net with conclusions (C n ) ⊥ , ..., (C 1 ) ⊥ , S. Let cc be an axiom in π such that c ∈ C i and c ∈ C j (i, j ∈ [1, n]). We define the length of axiom cc as the integer i + 1j.

Definition 6

Let π be a a syntactic analysis of w 1 , • • • , w n with categories C 1 , . . . , C n -that is, a categorial proof net with conclusions (C n ) ⊥ , ..., (C 1 ) ⊥ , S. Let C i 0 be one of the C i , and let consider axioms cc with c in C i 0 and c in some C i 0 -k . Let us consider the largest k for which such an axiom exists -this is the longest axiom starting from C i 0 with the previous definition. The dependency locality number of C i 0 in π, written DL π (C i 0 ) is the number of discourse referent words between w i 0 : C i 0 and w i 0 -k : C i 0 -k . The boundary words, i.e. w i 0 : C i 0 and w i 0 -k : C i 0 -k should also be counted. Alternatively, it may be viewed as k + 1 minus the number of non-discourse references among those k + 1 words.

Definition 7

Let π be a a syntactic analysis of w 1 , • • • , w n with categories C 1 , . . . , C n -that is, a categorial proof net with conclusions (C n ) ⊥ , ..., (C 1 ) ⊥ , S. We define the DLT-based linguistic complexity of π, written f dlt (π) as (1

+ n i=1 DL π (C i )) -1 .
Definition 8 Given two syntactic analyses π i and π j , not necessarily of the same words and categories, we say that π i is DLT-preferred to π j whenever f dlt (π i ) > f dlt (π j ).

Examples:

We apply our new metric on examples (1c) and (1d). Figures ( 5) and ( 6) show the relevant proof nets for (1c) and (1d). The proof nets for both examples are the same except a difference in one of the lexicons in each example, i.e. John and I.5 Figure [START_REF] Chomsky | Some concepts and consequences of the theory of government and binding[END_REF] shows the accumulative chart-based representation of our measure for each example. The axis Y shows the accumulative sum of the dependency locality function applied to each category on axis X. The quick analysis of the profiles shows the total complexity numbers 14 and 11 for (1c) and (1d) respectively. This correctly predicts the preference of example (1d) over (1c), which was not possible in the IDT-based approaches.

The measure for dependency locality number is quite straightforward. As an example, we calculate the dependency locality number for the word attacked in figure [START_REF] Chomsky | Aspects of the Theory of Syntax[END_REF] for (1d). We can find the longest axiom link starting from attacked and ended to its uppermost category, namely who. This axiom link is highlighted in figure [START_REF] Chomsky | Aspects of the Theory of Syntax[END_REF] for the reader's convenience. Now, we can count the number of discourse referents intervened in the axiom link, which is actually three: attacked, met and senator.

More Linguistic Support

We can evaluate our proposal for measuring the linguistic complexity against other linguistic phenomena. We basically illustrate the examples based on a survey provided in [START_REF] Morrill | Incremental processing and acceptability[END_REF]. As we will see, the new metric supports both referent-sensitive and Fig. 5: Proof net analysis for example (1c). Although (3b) is grammatical, it is perceived ungrammatical due to strong tendency to interpret the initial segments as in (3a). The sentence (3b) leads the reader toward a seemingly familiar meaning, namely (3a), that is actually not the one intended. This often requires re-reading so that the meaning may be fully grasped after careful parsing. The difficulty in correctly parsing the sentence (3b) results from the fact that race can be interpreted transitively or intransitively. The figure [START_REF] Gibson | An integration-based theory of computational resources in sentence comprehension[END_REF] illustrates the proof net analyses for (3a) and (3b). Figure [START_REF] Chomsky | Aspects of the Theory of Syntax[END_REF] shows the complexity profiles in which the total complexity numbers are 2 and 7 for (3a) and (3b), respectively. This correctly predicts the high level of complexity in (3b) and the preferred reading of (3a) compared to (3b) as it happens in the real sentence comprehension. Example 4. 4a. ?The cheese that the rat that the cat saw ate stank. 4b. The dog that chased the cat that saw the rat barked. The unacceptability of center embedding phenomena is illustrated by the examples (4a) and (4b). The example (4a) shows object relativization, which exhibits deterioration in acceptability, while the sentence (4b) exhibits little variation in acceptability since it carries subject relativization [START_REF] Chomsky | Aspects of the Theory of Syntax[END_REF]Chap. 1]. This linguistic phenomenon can be captured in our model. The figures ( 12) and ( 13) illustrate the proof net analyses for (4a) and (4b) respectively. Figure [START_REF] Gibson | Linguistic complexity: Locality of syntactic dependencies[END_REF] shows the complexity profiles in which the total complexity numbers are 14 and 10 for (4a) and (4b) respectively. attachment [START_REF] Kimball | Seven principles of surface structure parsing in natural language[END_REF]. The three profiles in figures ( 14), ( 15) and ( 16) show the highest, middle and lowest attachment of the adverb today to the verbs said, believes and fell, respectively. As it is illustrated in Figure [START_REF] Gibson | The dependency locality theory: A distance-based theory of linguistic complexity[END_REF], our model can predict the preference of (5a) over (5b), and (5b) over (5c). In other words, the lower the attachment of the adverb, the higher the preference [START_REF] Kimball | Seven principles of surface structure parsing in natural language[END_REF].

As for Example [START_REF] Chomsky | Aspects of the Theory of Syntax[END_REF], two proof nets in Figure [START_REF] Kimball | Seven principles of surface structure parsing in natural language[END_REF] show the lowest and highest attachment of the phrase 's title to the sentence the book that shocked Mary. As illustrated in Figure [START_REF] Lafourcade | Collecting weighted coercions from crowd-sourced lexical data for compositional semantic analysis[END_REF], our model can predict the preference of the right interpretation in Figure [START_REF] Kimball | Seven principles of surface structure parsing in natural language[END_REF] over the left one, despite the fact that the right one is non-sensical [START_REF] Morrill | Incremental processing and acceptability[END_REF].

Example 7. 7a. Ingrid was astonished that Jack was surprised that two plus two equals four. 7b. ?That that two plus two equals four surprised Jack astonished Ingrid.

The last linguistic performance phenomena that we want to discuss is Passive Paraphrases Acceptability [START_REF] Morrill | Incremental processing and acceptability[END_REF], exemplified in (7a) and (7b). Notice that the DLTbased complexity profile of (7a) is lower, even though the number of sentences and axiom links are more than (7b). The real preference is on the syntactic forms, namely, the one which is less complex. In our case the example (7a) is preferred to (7b) since it is simpler. The relevant proof nets and the accumulated complexity profiles are illustrated in the figures ( 19)-( 20) and ( 21), respectively.

Limitation

As discussed in this paper, our DLT-based complexity profiling is quite successful at predicting the referent-sensitive phenomenon, i.e., structures with embedded pronouns, that cannot be treated by the IDT-based approach. Moreover, it can also predict almost all the linguistic phenomena that the IDT-based approach can predict. To be more specific, the covered phenomena are: garden pathing, unacceptability of center embedding, preference for lower attachment, and passive paraphrases acceptability. The only exception-in terms of prediction-is Heavy Noun Phrase Shift, a phenomenon that our proposed approach cannot predict. This limitation is reported in [START_REF] Mirzapour | Modélisation de préférences pour l'interprétation d'énoncés ambigus)[END_REF]Chapter 7] and consequently, some treatments are considered in the aggregation phase when different metrics are collected, compared and preferred. Nevertheless, we admit this problem as one of the limitations of the DLT-based complexity approach, which can not supersede the IDT-based complexity profiling on this point.

The second limitation is the problem of ranking valid semantic meanings of a given multiple-quantifier sentence, which cannot be supported by our proposal. A study [START_REF] Mirzapour | Modélisation de préférences pour l'interprétation d'énoncés ambigus)[END_REF]Chapter 3] shows that the same problem exist in the IDT-based approach when dealing with some type of the expressions such as sentence-modifier adverbials and nested sentences. Our experiment shows that the same limitations hold in our DLT-based approach. Thus, both IDT-based and DLT-based complexity profiling cannot correctly predict ranking the quantifier scoping problem. Hopefully, this can be treated with the hybrid models (discussed in [START_REF] Mirzapour | Modélisation de préférences pour l'interprétation d'énoncés ambigus)[END_REF]Chapter 3]) in which Hilbert's epsilon and tau [START_REF] Hilbert | Die logischen grundlagen der mathematik[END_REF][START_REF] Chatzikyriakidis | From logical and linguistic generics to hilbert's tau and epsilon quantifiers[END_REF] are exploited in order to neutralize the quantifier effect, and after that, the complexity is measured on the basis of the penalty of the quantifiers re-ordering.

The third drawback of the DLT-based motivated approaches is that they are not applicable cross-linguistically for human parsing processes. One study [START_REF] Vasishth | Quantifying processing difficulty in human language processing[END_REF] has shown the failure of DLT-based approaches in modeling some human performances in the Hindi language. DLT uses the distance hypothesis: the increasing distance between dependents and heads results in increased processing difficulty. This hypothesis is not a universal linguistic feature as it is claimed [START_REF] Vasishth | Quantifying processing difficulty in human language processing[END_REF]. This cross-linguistic problem in DLT-based approaches has motivated the activation-based models [START_REF] Richard | An activation-based model of sentence processing as skilled memory retrieval[END_REF], which are formalized in computational form in the Adaptive Control of Thought-Rational (ACT-R) architecture [START_REF] John R Anderson | An integrated theory of the mind[END_REF]. This theory construes parsing as an efficient series of the guided memory retrievals, as opposed to a proof net construction in a way we have practiced in this chapter. Nevertheless, activation-based approaches cannot properly treat the structures with embedded pronouns [START_REF] Richard | An activation-based model of sentence processing as skilled memory retrieval[END_REF]. Moreover, it has no systematic approach for automatically deriving meaning representations in logical formulas as one is generally capable to do in the Categorical Grammar frameworks and specifically in our proposed DLT-based complexity profiling.

Conclusion and Possible Extensions

In this paper we explored how our DLT-based complexity profiling on proof nets can give a proper account of the complexity of a wide range of linguistic phenomena. We have also shown that the IDT-based method could not support phenomena of referentsensitive linguistic performance. It was one of the main reasons for introducing the DLT-based complexity profiling technique within the framework of Lambek calculus. There are some extensions for our study and research:

-As we mentioned it is possible to bridge our model with other study [START_REF] Catta | Quantifier scoping and semantic preferences[END_REF] to overcome the problem of ranking quantifier scoping. As we discussed, both IDTbased and DLT-based complexity profiling fail to predict quantifier scoping. This bridging strategy can help us to solve this issue by exploiting Hilbert's epsilon and tau operators [START_REF] Hilbert | Die logischen grundlagen der mathematik[END_REF][START_REF] Chatzikyriakidis | From logical and linguistic generics to hilbert's tau and epsilon quantifiers[END_REF] for neutralizing the quantifier effect and making possible the complexity measure by a penalty cost of the quantifiers re-ordering.

-Another important direction is to take into account not only the axioms of the proofnets but also the logical structure, i.e., par-links, tensor-links and the correctness criterion. This is important indeed, because this structure is needed to compute the logical form (semantics) from the syntactic structure given by proof nets. For instance, nesting Lambek slashes (that are linear implications, and therefore par-links in the proof net) corresponds to higher order semantic constructions (e.g. predicates of predicates) and consequently this nesting of par-links increases the complexity of the syntactic and semantic human processing.

-Our method can be combined with studies in other directions, e.g. the task of sentence correction/completion in Lambek Calculus [START_REF] Mirzapour | Finding missing categories in incomplete utterances[END_REF][START_REF] Mirzapour | Modélisation de préférences pour l'interprétation d'énoncés ambigus)[END_REF]. The other task is measuring semantic gradience in natural language. Some line of research suggests this feature within lexical/compositional frameworks by creating and enrichment of the wide-coverage weighted lexical resources from crowd-sourced data [START_REF] Lafourcade | Collecting weighted coercions from crowd-sourced lexical data for compositional semantic analysis[END_REF].

-We have solely relied on Gibson's theories in this paper. However, we can enrich our grounds with some of the existing tracks. Our work can be extended by adapting the best-fitting model introduced in [START_REF] Cooper-Martin | Measures of cognitive effort[END_REF], which has three constructs of total cognitive effort, cognitive strain, and time. Also, our future model can potentially use the outcomes of an experiment [START_REF] Engonopulos | Language and cognitive load in a dual task environment[END_REF] that shows how different linguistic structures may cause measurable changes in cognitive workload. There are a couple of works in linguistic complexity surveyed in [START_REF] Perfetti | Reviewed work: Linguistic complexity and text comprehension: Readability issues reconsidered by alice davison, georgia m. green[END_REF] and [START_REF] Kramsch | Why is everyone so excited about complexity theory in applied linguistics[END_REF], which can be potentially insightful for our future study.

  The reporter disliked the editor. 1b. The reporter [who the senator attacked] disliked the editor. 1c. The reporter [who the senator [who John met] attacked ] disliked the editor]. 1d. The reporter [who the senator [who I met] attacked ] disliked the editor].

Fig. 1 :

 1 Fig. 1: Proof net analyses for (2a) subject-extracted relative clause.

Fig. 2 :

 2 Fig. 2: Proof net analyses for (2b) object-extracted relative clause.

Fig. 3 :

 3 Fig. 3: IDT-based Complexity Profiles for (2a) and (2b).

Fig. 4 :

 4 Fig. 4: IDT-based Complexity Profiles for (1c) and (1d).

Fig. 6 :

 6 Fig. 6: Proof net analysis for example (1d).

Fig. 7 :

 7 Fig. 7: DLT-based Complexity Profiles for (1c) and (1d).some of the non-referent-sensitive phenomena6. We start with the Garden pathing[START_REF] Bever | The cognitive basis for linguistic structures[END_REF], which can be illustrated by the following examples:Example 3. 3a. The horse raced past the barn. 3b. ?The horse raced past the barn fell.

Fig. 8 :

 8 Fig. 8: Accumulative DLT-based Complexity Profiles for (3a) and (3b).

Fig. 9 :

 9 Fig. 9: Proof net analyses for (3a) located in top (first attempt reading) and (3b) in bottom (full garden path sentence).

Fig. 10 :Example 5 .

 105 Fig. 10: Accumulative DLT-based Complexity Profiles for 4a and 4b Example 5. Joe said that Marthus believes that Ingrid fell today. Example 6. The book that shocked Mary's title. Examples (5) and (6) carry different readings due to the syntactical ambiguity that exists in each of them. As for Example (5), the preference goes to the lowest

Fig. 12 :

 12 Fig. 12: Proof net analyses for 4a (object relativization).

Fig. 13 :

 13 Fig. 13: Proof net analyses for 4b (subject relativization).

Fig. 14 :

 14 Fig. 14: Proof net analyses for 5 with highest adverbal attachments.

Fig. 15 :

 15 Fig. 15: Proof net analyses for 5 with middle adverbal attachments.

Fig. 16 :

 16 Fig. 16: Proof net analyses for 5 with lowest adverbal attachments.

Fig. 17 :

 17 Fig. 17: Proof net analyses for 6 with sensical (left) and nonsensical (right) interpretations.

Fig. 18 :Fig. 19 :

 1819 Fig. 18: Accumulative DLT-based complexity profiles for two readings of 6

Fig. 20 :

 20 Fig. 20: Proof net analyses for 7b.

Fig. 21 :

 21 Fig. 21: Accumulative DLT-based complexity profiles for 7a and 7b

Here we are stricter than in other articles, i.e. we neither allow ⊗ of positive formulas nor òf negative formulas, because we only use the \ and / symbols in categories (and not ⊗): only combining heterogeneous polarities guarantees that a positive formula is a category, and that a negative formula is the negation of a category.

The same procedure can show the increasing complexity of the examples (1a)-(1c) by drawing the relevant proof-nets. This practice is avoided in this paper due the space limitation and its simplicity comparing to the running examples.

Following Lambek[START_REF] Lambek | The mathematics of sentence structure[END_REF], we have assigned the category S/(np\S) to relative pronoun I. Note that even assigning np, which is not a type-shifted category, would not change our numeric analysis at all.

It is worth mentioning that DLT-based complexity profiling can not support two linguistic phenomena: Multiple Sentences and Heavy Noun-Phrase Shift. For more details on the problem related to the Multiple-Quantifier Sentences and a possible solution consider[START_REF] Mirzapour | Modélisation de préférences pour l'interprétation d'énoncés ambigus)[END_REF] Chap 5,[START_REF] Chomsky | Some concepts and consequences of the theory of government and binding[END_REF].

Acknowledgement We would like to show our gratitude to Philippe Blache for his insightful discussion at our lab and also for inspiration that we got from his papers [2,3]. We would like to thank our colleague Richard Moot as well for his numerous valuable comments on this work.


Jérôme Mathieu

Gaël Caro

Lise Dupont

Methods for studying earthworm dispersal

Keywords: Movements, landscape structure, connectivity, behavior, capture mark recapture methods

Dispersal is a key driver of species composition and functional traits in earthworm communities. However, it has been largely overlooked in ecological literature on earthworms because it is particularly difficult to study. In this publication, we review recent developments that have been made in this field of research. We present methods to assess dispersal distance, such as Capture-Mark-Recapture and molecular tools, and methods using dispersal corridors or X-ray imagery aiming at identifying the mechanisms triggering dispersal in earthworms communities.

Introduction

Dispersal plays a major role in shaping biodiversity, evolution, and ecosystem functioning. It connects localities together through fluxes of individuals and alleles. The direct consequence is that species abundance and genetic composition in different places of a landscape are not independent. In other words, local population abundance, genetic structure and community structure not only depend on local factors and processes such as habitat features, demography, genetic drift or species sorting and competition, but are also dependent on the properties of neighboring populations and communities [START_REF] Leibold | The metacommunity concept: a framework for multi-scale community ecology[END_REF]. In this perspective, we need to study local community and genetic structure at both local and regional scales in order to understand the structure of local populations or communities, as well as their functional role.

The magnitude of the dependence between local and regional scales directly results from dispersal rate. Theoretically, when dispersal is very high, local sites are well interconnected and tend to behave like a unique population or community [START_REF] Economo | Species diversity in neutral metacommunities: a network approach[END_REF][START_REF] Mouquet | Community Patterns in Source-Sink Metacommunities[END_REF]. Local interactions are then a major driver of species composition and a low genetic differentiation among populations is expected. When dispersal is very low, local sites are isolated and behave like islands. Local populations and communities are well differentiated and severe genetic drift occurs. Extinction risk is then high for small populations. In nature, dispersal is generally between these two extremes, and complex behaviors such as source -sink dynamics may occur. In such case, certain sites can act as sources of dispersers (source), while others behave like sinks. Such dynamics can prevent extinction or speciation in certain sites. Local populations connected by dispersal are called "metapopulation" [START_REF] Hanski | Metapopulation Biology: Ecology, Genetics, and Evolution[END_REF], local communities connected by dispersal are called "metacommunity" [START_REF] Leibold | The metacommunity concept: a framework for multi-scale community ecology[END_REF]. In both cases, we need to understand the drivers and the magnitude of dispersal in order to understand the behavior and the properties of the system, either at local and regional scales.

Dispersal is usually defined as the movement of individuals away from their natal habitat, or from their usual home range, to a new habitat [START_REF] Clobert | Dispersal Ecology and Evolution[END_REF]. It is usually decomposed in three successive steps. First step is departure from the usual home range. Second is the transfer between the departure site and the arrival site. Third is the establishment in a new habitat (Fig. 1).

Figure 1 All these steps can originate from two very distinct processes: it can come from individuals' own willing, a process called "active dispersal", or from movements driven by an external force such as wind, water runoff, displacement by another animal, or by human activities [START_REF] Matthysen | Multicausality of dispersal: a review, Dispersal Ecology and Evolution[END_REF]. Dispersal direction is generally controlled in active dispersal but not in passive dispersal.

Studying dispersal is challenging for all organisms [START_REF] Nathan | The challenges of studying dispersal[END_REF], because it is hard to track individuals. This is particularly true for earthworms, because they are subterranean and cannot be seen from surface. Several approaches have been developed to address these difficulties, and can be classified in two groups. The first ones focus on dispersal patterns. They intend to measure typical dispersal distances of organisms during life span, or during a precise period of time. This approach aims at producing a histogram of distances travelled over a period of time, the so called "dispersal kernel" [START_REF] Nathan | Dispersal kernels: review[END_REF]. It is thus centered on the spatiotemporal aspects of the dispersal. The second group of methods focuses on the factors that drive dispersal, such as habitat quality or conspecific density. The products of this approach are dispersal rules, like positive density dependent dispersal. This approach is often framed in game theory and evolutionary ecology. It aims at predicting dispersal behavior and understanding the reasons why different dispersal behaviors evolved.

In this work, we present techniques that are currently available to study earthworms' dispersal from these two angles: 1) dispersal distances and 2) the factors that drive dispersal.

Methods for studying earthworm dispersal distance

One of the most basic question regarding dispersal of organisms is to determine how far they can move. In order to address this point, we need to estimate the distribution of the distances travelled for a given period of time. For this, two kinds of movements are often defined : the usual and most frequent movements, related to foraging, and the rare long distances movements [START_REF] Nathan | A movement ecology paradigm for unifying organismal movement research[END_REF]. In this framework, true dispersal usually refers to the rare and long distances dispersal events (LDD), located in the tail of the dispersal distribution [START_REF] Nathan | Dispersal kernels: review[END_REF]. These rare events play a critical role in colonization processes and invasion processes [START_REF] Trakhtenbrot | The importance of longdistance dispersal in biodiversity conservation[END_REF].

The estimation of traveling distances is challenging for earthworms, but can be done in two broad ways: by tagging and by studying the genetic profiles of individuals or populations.

Tags

Methods based on tags usually share the same principle: a group of individuals is captured, marked with a tag, then released at a known location in the field or in an experimental device.

After a period of time, the marked individuals are searched at proximity from the release point. The distribution of the recapture distances from the release point (Fig. 3A) gives the "dispersal kernel" of the individuals over the respective period of time. These methods are generally referred to as Capture Mark Recapture (CMR) methods [START_REF] Seber | The Estimation of Animal Abundance and Related Parameters[END_REF]. They are better suited for short distances estimation than for LDD. In theory, individuals can be released and recaptured several times, but this is difficult to achieve in the field on earthworms. Variations in CMR methods include the type of tags, the capacity to use tags specific to individuals, and the number of recapture events. In earthworms, several types of tags have been used so far. Early attempts used food colorants such as E102 + E132 (green) or E124 (red) and red biological stains such as Safranin and Phloxine [START_REF] Mazeaud | Evaluation de methodes de marquage permettant le repérage des lombriciens; premières application[END_REF]. More recently, 15 NH4+ and U-13 C glucose [START_REF] Dyckmans | A simple and rapid method for labelling earthworms with 15 N and 13 C[END_REF], Rubidium [START_REF] Hamou | Marking earthworms for release-recapture studies using the trace element rubidium[END_REF], or 60 Co [START_REF] Capowiez | Three-dimensional trajectories of 60Colabelled earthworms in artificial cores of soil[END_REF] have been used with success. However, all these methods are difficult to apply because they require a significant equipment or lab work to detect the tags, because they are not visible with naked eyes. We now present in detail two recent methods that circumvent these caveats.

Visual Tags for earthworm: VIE

Visible Implant Elastomers (VIE, Fig. 2A&B and3A) are colored tags that are injected below the skin [START_REF] Butt | A viable technique for tagging earthworms using visible implant elastomer[END_REF][START_REF] Gonzalez | A Fluorescent Marking and Re-count Technique Using the Invasive Earthworm, Pontoscolex corethrurus (Annelida: Oligochaeta)[END_REF]. They are injected in a liquid state but soon become solid through a polymerization process. They are available from Northwest Marine Technology, Inc (http://www.nmt.us/) and are relatively affordable.

Figure 2 This kind of tags has been used with success on a variety of organisms such as fish, frogs, turtles and seeds, in order to assess population size or dispersal distance. They are visible through the skin of the animals, particularly with a UV light. The tags are well supported by earthworms [START_REF] Butt | Is tagging with visual implant elastomer a reliable technique for marking earthworms?[END_REF]. They can stay up to 27 months in their body, which is much longer than the previously mentioned tags [START_REF] Butt | Is tagging with visual implant elastomer a reliable technique for marking earthworms?[END_REF].

Figure 3 The main drawback of this method is the impossibility of monitoring individuals in a continuous manner. It is not possible to track individuals between two capture events.

Marking and recapturing require a lot of time, and tags are hardly specific to individuals, even though combining tags of different colors is possible for large individuals (>1.5g, Fig. 2B).

This method requires releasing a high number of marked animals for statistical reasons, which can trigger density dependence dispersal [START_REF] Mccrea | Analysis of Capture-Recapture Data[END_REF]. Indeed, in many species, high level of conspecific density leads to active dispersal [START_REF] Caro | Are dispersal behaviours of earthworms related to their functional group?[END_REF][START_REF] Mathieu | Habitat quality, conspecific density, and habitat pre-use affect the dispersal behaviour of two earthworm species, Aporrectodea icterica and Dendrobaena veneta, in a mesocosm experiment[END_REF]. Finally, statistical methods for capture recapture data are complex [START_REF] Amstrup | Handbook of Capture-Recapture Analysis[END_REF][START_REF] Lee | Estimating Population Size via Sample Coverage for Closed Capture -Recapture Models[END_REF]. Despite these difficulties, this approach is very useful to get an estimation of the dispersal kernel of individuals.

Electronic tags: RFID tags

A new promising technique is the use of miniaturized RFID tags (Radio Frequency IDentification. These tags (Fig. 4) offer the possibility to mark each individual specifically, with a unique barcode ID. A scanner (Fig. 4B) retrieves the ID of an individual when the tag is close enough from the receptor, typically less than 0.5 cm.

Figure 4

Detecting individuals in a continuous fashion can be achieved by installing antennas at the surface of the ground, which detect any individual with a tag that comes close to the antenna. This offers the possibility to track individuals continuously and at different sites.

This method suffers from several limitations that still impede its use on a regular base. First, RIFD tags are more harmful than VIE tags. They are bigger (1.4 x 8mm) and are frequently ejected out of the body by earthworms. During preliminary tests, only larger individuals, typically >1g, supported them. Second, detection can only be done at very short distance from the antenna (<0.5cm), and cannot be done through the soil. Even if the antenna can be buried, it is much more efficient to detect individuals on the surface. Thus, this technique is better suited for large epigeic and anecic species, which crawl on surface, but not for endogeic ones.

Third, the tags are completely invisible once injected, which makes it difficult to find their location in the body during manual scanning, or to determine if an individual is tagged or not.

A VIE can be injected in addition, close to the RIFD tag, however this increases the risk of mortality. At the moment, miniaturized RFID can only be detected by the scanners provided by the RFID manufacturers, which are expensive and have a limited efficiency. However, in the future it should be possible to build its own system of scanners or antenna and data logger.

Molecular approach

Recent progresses in molecular biology offer new opportunities for the estimation of dispersal distance of earthworms from genetic data [START_REF] Dupont | Perspectives on the application of molecular genetics to earthworm ecology[END_REF][START_REF] Torres-Leguizamon | Isolation of polymorphic microsatellite markers in Aporrectodea icterica (Savigny 1826)[END_REF][START_REF] Zeller | Estimating landscape resistance to movement: a review[END_REF]. As dispersal leads to gene flow, genetic information can be used to infer dispersal patterns. They can be assessed in two ways: first by comparing observed populations' genetic structure to theoretical ones under no dispersal, and second by statistical assignment methods that allow identifying the parents or the population of origin of individuals, based on their genetic profile [START_REF] Broquet | Molecular estimation of dispersal for ecology and population genetics[END_REF]. In addition, the effect of landscape structure on dispersal patterns can be assessed with the tools developed within the framework of landscape genetics [START_REF] Manel | Landscape genetics: combining landscape ecology and population genetics[END_REF]. The principle is to determine if there is any relationship between the spatial patterns of populations' genetic structure, or individuals' genetic profile, and landscape features. This approach was applied at the plot scale (100x50m) on the earthworm Allolobophora chlorotica [START_REF] Dupont | Dispersal constraints and fine-scale spatial genetic structure in two earthworm species[END_REF]. It suggested that soil properties influence dispersal in the field, and estimated that gene flow occurs at approximately 7 m.year-1. At the landscape level, it showed the role of landscape features on the dispersal of two earthworm species, A. chlorotica and Aporrectodea icterica [START_REF] Dupont | Landscape features impact connectivity between soil populations: a comparative study of gene flow in earthworms[END_REF].

Figure 5 The genetic approach is based on the variation of highly variable DNA sequences (i.e. molecular markers) which are selectively neutral (i.e. the variation is not selected by the environment). Microsatellites, Amplified fragment length polymorphisms (AFLPs), and single nucleotide polymorphisms (SNPs) are typical markers used for this kind of study [START_REF] Dupont | New insight into the genetic structure of the Allolobophora chlorotica aggregate in Europe using microsatellite and mitochondrial data[END_REF][START_REF] Torres-Leguizamon | Isolation of polymorphic microsatellite markers in Aporrectodea icterica (Savigny 1826)[END_REF]. Because population genetic models are dedicated to diploid (2N) species, this kind of approach can hardly be used for studying dispersal of earthworm species with a ploidy level different to 2N. Once the molecular data are acquired, they can be matched with landscape data in two ways. First, they can be compared to landscape structure in the neighborhood of the sampling sites (Fig. 5). For this, an area (a "buffer") is defined around each sampling site, and the relevant features of landscape structure are described with the help of a GIS. For instance, the distance to the nearest forest, the percentage of crops or the length of roads can be computed [START_REF] Mcgarigal | FRAGSTATS: spatial pattern analysis program for quantigying landscape structure[END_REF]. The result of this process is a table describing different landscape features for each sampling site. These features can then be matched with genetic descriptors of populations, such as allelic richness and heterozygote frequency, using, for instance, traditional regression techniques. The second way to proceed is through connectivity analysis.

In this case, the landscape is considered in its totality, not only around a certain radius of each sampling site. In this approach, all sites are compared by pairs, both in terms of genetic difference and geographical distance (Fig. 5). This is done by correlating the geographical distance between pairs of sites to their genetic distance. If there is any significant relationship between the two matrixes, it suggests that landscape feature, and thus dispersal, has an influence on spatial genetic structure. Geographical distance is calculated by estimating the cumulated costs of movements between each pairs of sites. For this, specific costs are attributed to the different elements of the landscape. The product of this step is a "resistance map". This map is used to calculate the cumulative traveling costs between each pair of sites, using different algorithms of movements. Finally, either the cost of the shortest path, or the weighted mean of the different paths, is used to estimate the distance between all pairs of sites [START_REF] Mcrae | Using circuit theory to model connectivity in ecology and conservation[END_REF]. The final product of this step is a pairwise squared distance matrices giving the traveling costs between pairs of sites. This process can be realized with different scenario of costs. By comparing different scenarios of costs, this technique allows identifying which elements are most likely to play a role on dispersal, either by facilitating it (corridors) or by impeding it (barriers) [START_REF] Dupont | Landscape features impact connectivity between soil populations: a comparative study of gene flow in earthworms[END_REF]. In order to match the distance matrix with genetic features, the genetic differentiation between all sampling sites must also be estimated.

This can be done through different indices, but the most widespread is the fixation index Fst [START_REF] Wright | The genetical structure of populations[END_REF]. The result of this step is a pairwise squared dissimilarity matrix between sites.

In order to correlate the matrix of genetic differentiation to the matrixes of traveling costs, specific statistical techniques must be used, because the same sites are involved in several comparisons, and there is non independency in the residuals. The historical technique used for this was Mantel test, but it has been criticized and the recommend test is now MLPE (Maximum Likelihood Population Effect). This technique identifies the scenario of costs that match with differences in genetic structure [START_REF] Van Strien | A new analytical approach to landscape genetic modelling: least-cost transect analysis and linear mixed models[END_REF]

Methods for studying mechanisms of earthworm dispersal

Another important aspect in the study of dispersal is to determine the conditions that drive individuals to disperse. These drivers can be internal, such as body size, hormonal status and age, or external, such as habitat quality or biotic interactions [START_REF] Clobert | Dispersal Ecology and Evolution[END_REF]. In earthworms, only external drivers of dispersal have been studied so far. Two techniques are now well established: the dispersal corridors [START_REF] Mathieu | Habitat quality, conspecific density, and habitat pre-use affect the dispersal behaviour of two earthworm species, Aporrectodea icterica and Dendrobaena veneta, in a mesocosm experiment[END_REF], and the X ray imagery [START_REF] Caro | Is earthworms' dispersal facilitated by the ecosystem engineering activities of conspecifics?[END_REF].

Dispersal corridors

The principle of dispersal corridors is to expose individuals to a stimuli, to wait a certain amount of time, and then to compare the movements of individuals in the stimuli treatment to the control [START_REF] Clobert | Dispersal Ecology and Evolution[END_REF]. In this kind of experiment, it is necessary to be able to make the difference between active dispersal and random diffusion. Active dispersal results from a decision to leave a patch, without knowing in advance where to go, and what type of environment will be available after leaving the initial patch. Random diffusion results from usual movements, such as foraging, that do not imply a decision to leave. To be able to make the distinction between active dispersal and random diffusion, a specific experimental setup must be used (Fig. 6).

Figure 6

The experimental device is elongated and separated in three parts: 1) the "departure section";

2) the "crossing section"; 3) the "settlement section". The "departure section" is the section where the stimuli or the control treatment is implemented, and where the earthworms are inoculated. The treatment can consist for example in soil quality, litter quality, conspecific density, or interspecific density. The crossing section must be an unhospitable soil, which individuals would cross only if the conditions in the departure section are really unhospitable. This is the key point to make the distinction between random diffusion and active dispersal.

The "settlement section" must contain a very hospitable soil, which individuals will not leave once they reached it. In top of that, it is required to put litter on the surface of the departure and of the settlement zones, even for endogeic species. After a fixed period of time after inoculation, the soil and the litter of the tree sections are separated and the number of earthworms in each section is retrieved. The percentage of dispersal is compared between control and stimuli treatment with a logistic model [START_REF] Caro | Are dispersal behaviours of earthworms related to their functional group?[END_REF].

X ray imagery

Another promising tool is X ray imagery. This method takes advantage of recent progress in imagery machines. It consists in using X ray imaging device such as Philips Diagnostic C generator to scan the soil (Fig. 7). This kind of device is capable of differentiating earthworms and their gallery in the soil, if the soil is less than 6 cm thick. The soil must not be too mineral otherwise X ray cannot penetrate. Adding soft material, such as peat, greatly increase image quality. Introducing by incision a small metal tag (1mm) below the skin of individuals can also help to localize them on the X ray shots. With this kind of setup, it is possible to determine in real time the location of individuals in the ground [START_REF] Caro | Is earthworms' dispersal facilitated by the ecosystem engineering activities of conspecifics?[END_REF], to tell if it is located inside or outside a gallery, and to measure its movement speed [VIDEO].

Conclusion

The several techniques that now exist to study earthworm dispersal (Table 1) can give us insights about dispersal distances and mechanisms leading to active dispersal, either in the field or in experimental devices. They offer new opportunities to quantify this process, and to estimate its contribution to population dynamics, community assembly, and finally ecosystem functioning.

Figure captions

Figure 7

 7 Figure 7

Fig. 1 -

 1 Fig. 1 -Dispersal is usually decomposed in three steps: Emigration from the source, transfer, and immigration to a new site.

Fig. 2 -

 2 Fig. 2 -A/ Tagging of a small Allolobophora chlorotica B/ An individual of Aporrectodea longa tagged with two yellow bands.

Fig. 3 -

 3 Fig. 3 -Summary of the steps required to study earthworm dispersal with tags.

Fig. 4 -

 4 Fig. 4 -A/ Picture of a RFID tag ready to be injected. B/ The scanner recognizes a tagged Allolobophora chlorotica individual.

Fig. 5 -

 5 Fig. 5 -Overview of the workflow to perform landscape genetics on earthworm.

Fig. 6 -

 6 Fig. 6 -Example of a dispersal corridors with its three sections. The triangle indicates the inoculation point of earthworms.

Fig. 7 -

 7 Fig. 7 -Experimental setup to take X ray shots of earthworms in the soil.

Acknowledgments

We would like to thank Augusto Zanella for his commitment in this Special Issue of Applied

Soil Ecology

Table 1 Comparison of the methods currently available for studying earthworm dispersal. †: + = small extent, +++ = large extent, ‡: + = least, +++ = most, § References : 1) Dupont, L. and al ., 2017. Mol Ecol 26, 3128-3140. 2) Butt, K.R., Lowe, C.N., 2007. Appl. Soil Eco. 35, 454-457, 3) unpublished, 4) Mathieu J. and al., 2010. Soil Biol. Biochem. 42, 203-209. 5) Caro and al. , 2012. Biol. Fert


Keywords: Entropy, Biodiversity, Phylogenetic diversity, Functional diversity

We unify the definitions of phylogenetic and functional entropy and diversity as a generalization of HCDT entropy when an ultrametric tree is considered. We derive the decomposition of phylodiversity and its estimation bias correction to allow its estimation from real, often undersampled data.Phyloentropy can be transformed into phylodiversity to provide a measure of true diversity, i.e. an effective number of species or communities.

Introduction

Diversity partitioning using conventional species-neutral approach (based on Hill numbers, i.e. the effective number of species) is now being replaced by far more interesting frameworks taking into account the species relatedness. This is what we call 'phylodiversity' [START_REF] Webb | Integrating phylogenies into community ecology[END_REF], i.e. how species assemblages are distributed in a phylogenic tree. When bothrelative abundance and relatednessdegree between species (or individuals) are quantified, [START_REF] Pielou | Ecological diversity[END_REF]suggested that diversity measures should be generalized integrating taxonomic differences between species. A little later, [START_REF] Rao | Diversity and dissimilarity coefficients: a unified approach[END_REF] proposed that the average of the species differences can be used as a measure of biodiversity. Despite some attempts to take into account phylogenic diversity into a generalized species richness [START_REF] Vane-Wright | What to protect: systematics and the agony of choice[END_REF], this 'avant-garde' ideahas been hardly applied in ecology (e.g. [START_REF] Warwick | New 'biodiversity' measures reveal a decrease in taxonomic distinctness with increasing stress[END_REF][START_REF] Crozier | Preserving the information content of species: Genetic diversity, phylogeny, and conservation worth[END_REF].During the last decade, increasing interests into the evolutionary history of communities [START_REF] Webb | Exploring the phylogenetic structure of ecological communities: An example for rain forest trees[END_REF] as well as the need for conservation strategies taking phylogenetic risks into account [START_REF] Faith | Threatened Species and the Potential Loss of Phylogenetic Diversity: Conservation Scenarios Based on Estimated Extinction Probabilities and Phylogenetic Risk Analysis[END_REF] revived the interest in phylodiversity partitioning.

Phylogenetic trees are built upon thegenetic similarities and differencesamong various biological individuals or other superior taxa.In a given local assemblage, phylogenetic diversity aims to quantify the evolutionary history shared among individuals since the time of the most recent common ancestor [START_REF] Chao | Phylogenetic diversity measures based on Hill numbers[END_REF]. All else being equal, an assemblage of phylogenetically divergent species is often seen as more diverse than a local assemblage of closely related species [START_REF] Vellend | Measuring phylogenetic biodiversity[END_REF]. There is increasing interest to partition this phylogenetic diversity not only between local communities but also between time periods in order to elucidate community assembly rules [START_REF] Pavoine | Hierarchical partitioning of evolutionary and ecological patterns in the organization of phylogenetically-structured species assemblages: Application to rockfish (genus: Sebastes) in the Southern California Bight[END_REF] and investigate what is commonly called the phylogenetic structure of communities (e.g. Cavender- [START_REF] Cavender-Bares | Phylogenetic overdispersion in Floridian oak communities[END_REF]. For instance, [START_REF] Hardy | Characterizing the phylogenetic structure of communities by an additive partitioning of phylogenetic diversity[END_REF]argued that a proper partitioning of phylodiversity is a necessary step prior to deciphering phylogenetic clustering (either due to local speciation of allopatric clades or habitat filtering of phylogenetically conserved traits) fromphylogenetic overdispersion (allopatric speciation of two ancestral sympatric species, habitat filtering of phylogenetically convergent traits, competitive exclusion of related species).

Functional trees differ from phylogenetic trees as phylogenetic trees reflect evolutionary constraints while functional trees also take into account ecological convergence [START_REF] Herault | Reconciling niche and neutrality through the Emergent Group approach[END_REF].

Each time a proper functional tree can be constructed from a functional trait-based distance matrix [START_REF] Podani | How should a dendrogram-based measure of functional diversity function? A rejoinder to Petchey and Gaston[END_REF], it should be possible to estimate and partition functional diversity in a mannersimilar to phylogenetic diversity [START_REF] Petchey | Functional diversity (FD), species richness and community composition[END_REF]. Functional diversity was often defined as the extent of functional differencesamong individuals or species in a local community [START_REF] Tilman | Functional diversity. Encyclopedia of Biodiversity[END_REF], animportant determinant of ecosystem processes [START_REF] Loreau | Ecology -Biodiversity and ecosystem functioning: Current knowledge and future challenges[END_REF]. Functional diversity based on functional trees is a great tool to estimate the complementarity among individuals' or species' trait values by estimating their dispersion in trait space at all hierarchical scales simultaneously, avoiding discretization of continuous trait variation into functional groups [START_REF] Petchey | Functional diversity (FD), species richness and community composition[END_REF]. However, functional differences among species or individuals being assessed by a number of a priori important functional traits, the degree of similarity among species and, in fine, the functional diversity value itself will both depend strongly on thisa priori choice of functional traits [START_REF] Weiher | Challenging Theophrastus: A common core list of plant traits for functional ecology[END_REF].

In this paper, we consider that all individuals or species of a local community take place in an ultrametric phylogenetic or functional tree. The distance between two species is measured as the length of the branches between them and their first common node. [START_REF] Chao | Phylogenetic diversity measures based on Hill numbers[END_REF] generalized Hill numbers to measure phylogenetic or functional diversity (we will often write phylodiversity and phyloentropy for short) while [START_REF] Pavoine | Hierarchical partitioning of evolutionary and ecological patterns in the organization of phylogenetically-structured species assemblages: Application to rockfish (genus: Sebastes) in the Southern California Bight[END_REF] generalized HCDT entropy [START_REF] Shimatani | On the measurement of species diversity incorporating species differences[END_REF][START_REF] Ricotta | Additive partitioning of Rao's quadratic diversity: a hierarchical approach[END_REF] had already done it, but for Rao's quadratic entropy only). We first show here their equivalence: phyloentropy is transformed into phylodiversity the same way HCDT entropy is transformed into diversity sensustricto. Then, we derive phylodiversity partitioning as a straightforward generalization of that of HCDT diversity.Finally, we provide estimation-bias corrections for phyloentropy in order to obtain biascorrected measures of phylodiversity.

Partitioning phylodiversity

Tsallis entropy

Tsallis entropy, also known as HCDT entropy [START_REF] Havrda | Quantification method of classification processes. Concept of structural -entropy[END_REF][START_REF] Daróczy | Generalized information functions[END_REF][START_REF] Tsallis | Possible generalization of Boltzmann-Gibbs statistics[END_REF]) has proven to be a powerful tool to measure diversity, generalizing the classical indices of diversity, including the number of species, Shannon and Simpson indices. The order of diversity q gives more or less importance to rare species. Entropy can be converted into diversity sensustricto [START_REF] Hill | Diversity and Evenness: A Unifying Notation and Its Consequences[END_REF][START_REF] Jost | Entropy and diversity[END_REF], easy to interpret and compare. Estimators of diversity are intrinsically biased because of unseen species and also because they are not linear functions of probabilities (Marcon et al. submitted). This is a serious issue [START_REF] Dauby | Sampled-based estimation of diversity sensu stricto by transforming Hurlbert diversities into effective number of species[END_REF][START_REF] Beck | Undersampling and the measurement of beta diversity[END_REF], even if some bias corrections are available for HCDT entropy estimators [START_REF] Grassberger | Finite sample corrections to entropy and dimension estimates[END_REF][START_REF] Chao | Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample[END_REF]Marcon et al. in press).

Neutral diversity

We first recall some features of HCDT diversity partitioning (Marcon et al. in press).

Consider a metacommunity made of several local communities. Abundances of species in each local community is denoted , ( = 1, 2, … , is the index of species,the index of communities). is the number of individuals of species in the metacommunity, the number of individuals sampled in local community and the total number. The same notations are used for probabilities of occurrence , which are unknown but estimated with , = , .

Community weights are : they may be equal to but any positive values summing to 1 are allowed. Probabilities in the metacommunity depend on these weights: = , . Diversity of the metacommunity is diversity. Diversity of local communities is diversity. The formalism of deformed logarithms is appropriated: it allows elegant and intuitive algebra. The logarithm of order is defined as:

ln q = 1--1 1 - (1)
Its inverse function is the deformed exponential:

= 1 + 1 - 1 1- (2)
Note that:

+ = 1+ 1- (3)
Tsallis entropy of the metacommunity, , can be written as:

= 1 - -1 = - ln q (4)
Last, diversity is the deformed exponential of entropy, = , and entropy is the deformed logarithm of diversity: = ln q . Consider a phylogenetic or functional ultrametric tree (Figure 1).The first slice starts at the bottom of the tree and ends at the first node. In slice , leaves are found. The probabilities of occurrence of the species belonging to branches that were below leaf in the original tree are summed to give the grouped probability , . We follow [START_REF] Pavoine | Hierarchical partitioning of evolutionary and ecological patterns in the organization of phylogenetically-structured species assemblages: Application to rockfish (genus: Sebastes) in the Southern California Bight[END_REF] to define phyloentropy as the sum of the entropies in each tree slice, weighted by the slice height. However, we choose to normalize it by the total tree height, = =1 . We denote it :

Phyloentropy and Phylodiversity

= =1 (5)
is HCDT entropy inslice . It is calculated as = -, ln q , .

Chao, Chiu and Jost (2010) generalized Hill numbers to phylogenetic diversity, defined as:

= , =1 =1 1 1- (6)
Simple algebra shows that:

= (7)
This relation is exactly the same as the relation between HCDT entropy and diversity. In other words, phyloentropy is the weighted average of entropy along the tree, and phylodiversity is its Hill number. Entropy is linear, it can be summed over slices, but diversity is not: phylodiversity is not the weighted average of diversity along the tree. Marcon et al. (in press) derived the decomposition of HCDT entropy, generalizing Shannon entropy partitioning [START_REF] Rao | Cross entropy, dissimilarity measures, and characterizations of quadratic entropy[END_REF][START_REF] Marcon | The Decomposition of Shannon's Entropy and a Confidence Interval for Beta Diversity[END_REF], based on Patil and Taillie's concept of diversity of a mixture [START_REF] Patil | Diversity as a concept and its measurement[END_REF]. Note that it differs from Jost's The deformed exponential of equation ( 8) is the decomposition of phylodiversity:

Decomposition

= (11) = ; = ; = 1+ 1-
and phylodiversities can be interpreted as an equivalent number of neutral species, that is to say the number of species equally different from each other (i.e. in an ultrametric tree made of a single slice) with the same probability of occurrence that would give the same measure of diversity. phylodiversity is an equivalent number of communities, that is to say the number of completely distinct neutral communities with the same weight as actual ones, that would yield the same diversity as the actual metacommunity.

Bias correction

and HCDT entropies can be corrected following Marcon et al. (in press): when q is low, unsampled species are the main issue that can be corrected according to [START_REF] Chao | Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample[END_REF]; when q is high, the contribution of rare species to entropy is small so the bias they cause is little but entropy is less linear with respect to probabilities, requiring the correction of [START_REF] Grassberger | Finite sample corrections to entropy and dimension estimates[END_REF]. Bias correction relies on the number of sampled individuals (probabilities are not enough) and can be computed for positive values of . The unbiased estimators are denoted instead of . Their formulas are in Marcon et al. (in press) and are not repeated here.

Phyloentropy can be corrected by summing the bias-corrected estimators of HCDT entropy in each slice of the tree.Bias-corrected entropy, , relies on values of Since the number of individuals in each leaf , increases in slices close to the root of the tree, the bias decreases with . = , is calculated the same way. phyloentropy is obtained as the difference between and because Grassberger's correction is not available to allow direct calculation.

Example

We used the tropical forest dataset alreadyinvestigated by [START_REF] Marcon | The Decomposition of Shannon's Entropy and a Confidence Interval for Beta Diversity[END_REF] and Marcon et al. (in press). Two 1-ha plots were fully inventoried in the Paracou field station in French Guiana. 1124 individual trees (diameter at breast height over 10 cm) have been sampled

among 229 species. The phylogenetic tree was built introducing a rough taxonomy of the 229 species in the analysis: distance between species of the same genus is set to 1, 2 for different genera of the same family, and 3 for different families. The functional tree was based onspecies relatedness using four key functional traits related to one axis of the leaf-height-seed-stem economic spectra of tropical trees (Baraloto et al. 2010b): seed mass and tree maximum height [START_REF] Herault | Functional traits shape ontogenetic growth trajectories of rain forest tree species[END_REF])plusspecific leaf area and wood specific gravity (Baraloto et al. 2010a).The functional tree was built from a Gower's similarity matrix agglomerated using Ward's method We first calculated the neutral, phylogenic and functional diversity of order 1 of the metacommunity (the two plots) and partitioned it (each plot is considered as a local community, weights are proportional to the numbers of individuals). The neutral diversity (Hill number of Shannon entropy) is 134 effective species, partitioned into diversity equal to 92 effective species (82 and 107 in each plot) and diversity equal to 1.46 equivalent communities.

Phylogenetic and functional diversity values respectively are:

1 =55 and 5.9, =42 and 5.5 with =1.29 and 1.06. Considering the taxonomy of Paracou species, phylodiversity is around 2.5 times smaller than neutral diversity. Functional diversity is only 6 equivalent species,

showing an extreme redundancy according to the functional tree (FD is estimated equal to 18 for 297 estimated species). Since diversity if the product of by , they can be represented as nested rectangles (Figure 2). The rectangle of size by has the same area as that of size by 1.

Plotting neutral and phylodiversity together summarizes the essential information: the reduction of diversity due to the consideration of species phylogenetic or functional proximity. Profiles (Figure 3) can be drawn for neutral, phylogenetic and functional diversities.

Discussion and conclusion

Unification of measures of diversity

Phyloentropy generalizes manyprevious indices of diversity. [START_REF] Rao | Diversity and dissimilarity coefficients: a unified approach[END_REF] quadratic entropy is phyloentropy of order 2 multiplied by , the tree height. It has been explored in depth and several results obtained here were already known in this special case. It has been partitioned early by Rao himself, weighting communities according to number of individuals, as [START_REF] Villeger | Additive partitioning of diversity including species differences: a comment on Hardy & Senterre (2007)[END_REF] while [START_REF] Hardy | Characterizing the phylogenetic structure of communities by an additive partitioning of phylogenetic diversity[END_REF] or [START_REF] Pavoine | A New Technique for Analysing Interacting Factors Affecting Biodiversity Patterns: Crossed-DPCoA[END_REF] used equal weights. [START_REF] Hardy | Interpreting and estimating measures of community phylogenetic structuring[END_REF] validated both weightings but a general framework allowing additive partitioning of Rao's entropy was missing [START_REF] Guiasu | The weighted quadratic index of biodiversity for pairs of species: a generalization of Rao's index[END_REF]. We showed that arbitrary weights are acceptable.

Other indices of diversity can be considered as special cases of phyloentropy. We summarize them in Table 1. [START_REF] Faith | Conservation evaluation and phylogenetic diversity[END_REF]) and FD [START_REF] Petchey | Functional diversity (FD), species richness and community composition[END_REF].

1 equals , the phylogenetic generalization of Shannon's index [START_REF] Allen | A New Phylogenetic Diversity Measure Generalizing the Shannon Index and Its Application to Phyllostomid Bats[END_REF].

Neutral diversity

Entropy:

Diversity:

+ 1 0 is species richness. 1 is Shannon entropy. 2 is Simpson entropy.

Conclusion

In this paper, we provide a general, consistent and operational framework to decompose measures of neutral, phylogenetic or even functional diversity into α (within local communities) and β (between local communities) components. We show that entropy can be calculated and its estimation bias corrected in each slice of the phylogenetic or functional tree, summed over slices and finally transformed into diversity. In fact, phylogenetic diversity could be analyzed without using any species concept (i.e. without categorizing individuals into a set of species) provided that phylogenetic or functional distance between individuals can be assessed, for example using molecular data or functional trait measured for each member of a meta community [START_REF] Paine | Functional traits of individual trees reveal ecological constraints on community assembly in tropical rain forests[END_REF]. Being able to properly partition phylodiversity is a necessary step towards deciphering the ecological and evolutionary mechanisms that underlie the structure and assembly of communities. Moreover, diversity partitioning will improve our assessment of human-driven modifications of ecosystem functioning in conservation studies.

Figure 1 :

 1 Figure 1: Hypothetical ultrametric tree. (a) The whole tree contains three slices, delimited by two nodes. The length of slices is . b) Focus on slice 2. The tree without slice 1 is reduced to 3 leaves. Frequencies of collapsed species are , . c) Slice 3 only.

(

  [START_REF] Jost | Partitioning diversity into independent alpha and beta components[END_REF] non-additive partitioning when community weights are unequal; seeMarcon et al. (in press) for a full discussion.Since phylogenetic entropy is a linear transformation of generalized entropy, its decomposition is identical and follows equation (8). Inslice , HCDT entropy is denoted , and the contributions of local community to and entropy are ,

(

  full details in Herault & Honnay 2007). Phylogenicdiversity was calculated with the entropart package (Marcon & Hérault 2013)under R (R Development Core Team 2014): bias-corrected entropy was calculated first, summed and finally transformed into diversity. Necessary R codes are in Appendix 1.

Figure 2 :

 2 Figure 2: Graphical representation of the diversity of order 1 in Paracou plots. Transparent rectangles represent neutral diversity, hatched rectangles phylogenetic diversity and shaded rectangles functional diversity. In each case, the horizontal rectangle of height 1 represents diversity (respectively 134, 55 and 6 effective species). The other rectangle has the same area, but its size is diversity by diversity.

Figure 3 :

 3 Figure 3: Diversity profile of Paracou plots. Neutral diversity (solid line), phylogenetic diversity (dashed line) and functional diversity (dotted line) are plotted against the order of diversity, between 0 (number of species, PD and FD) and 2 (Simpson diversity and Rao's quadratic entropy transformed into diversity), with estimation-bias correction. Diversity scale is logarithmic for readability.

Table 1 : Many usual measures of diversity are special cases of phyloentropy, either reducing it to neutral diversity or limiting it to values of equal to 0, 1 or 2. Diversity of order Special values of

 1

	Phylogenetic or	Entropy:
	functional entropy / diversity	Diverstiy:

0 + 1 equals PD

ACKNOWLEDGEMENTS

This work has benefited from an "Investissementd'Avenir" grant managed by AgenceNationale de la Recherche (CEBA, ref. ANR-10-LABX-0025).

Appendix 1: R code of the example

Load the package and the dataset:

Calculate diversity

# Shannon diversity DivPart(q = 1, Paracou618.MC, Biased = FALSE) # Phylodiversity of order 1 based on the taxonomy DivPart(q = 1, Paracou618.MC, Biased = FALSE, Tree=Paracou618.Taxonomy) # Functional diversity of order 1 DivPart(q = 1, Paracou618.MC, Biased = FALSE, Tree=Paracou618.Functional) # FD, with bias correction DivPart(q = 0, Paracou618.MC, Biased = FALSE, Tree=Paracou618.Functional) # Number of species DivPart(q = 0, Paracou618.MC, Biased = FALSE) Plotting Fig. 2: data(Paracou618) q <-1 # Neutral diversity (Neutral <-DivPart(q, Paracou618.MC, Biased = FALSE)) par(mar = c(5, 5, 2, 2)) plot(c(0, Neutral$GammaDiversity), c(0, Paracou618.MC$Ncommunities), type = "n", xlab = expression(paste(alpha, " / ", gamma, " diversity")), ylab = expression(paste(beta, " diversity")), cex.lab=1.8) rect(0, 0, Neutral$GammaDiversity, 1, lty=2) rect(0, 0, Neutral$TotalAlphaDiversity, Neutral$TotalBetaDiversity, lty=2) # Phylogenetic diversity (Phylo <-DivPart(q, Paracou618.MC, Biased = FALSE, Tree=Paracou618.Taxonomy)) rect(0, 0, Phylo$GammaDiversity, 1, angle=135, density=5) rect(0, 0, Phylo$TotalAlphaDiversity, Phylo$TotalBetaDiversity, density=5) # Functional diversity (Functional <-DivPart(q, Paracou618.MC, Biased = FALSE, Tree=Paracou618.Functional)) rect(0, 0, Functional$GammaDiversity, 1, angle=135, density=20) rect(0, 0, Functional$TotalAlphaDiversity, Functional$TotalBetaDiversity, density=20)

Plotting Fig. 3: (5,5,2,2)) plot(y = Neutral, x = seq(0, 2, 0.1), type = "l", col = "black", lwd = 2, lty=1, xlab = "Order of Diversity", ylab = "Gamma Diversity", ylim = c(1, max(Neutral)), log = "y", cex.lab=1.8, main = "") lines(y = Phylo, x = seq(0, 2, 0.1), lwd = 2, lty=2) lines(y = Functional, x = seq(0, 2, 0.1), lwd = 2, lty=3)