Maxime Amblard
email: amblard@loria.fr

Minimalist Grammars and Minimalist Categorial Grammars, definitions toward inclusion of generated languages

. In this article, we give full definitions of MG with algebraic tree descriptions and of MCG, and take the first steps towards giving a proof of inclusion of their generated languages.

, unified more than fifty years of linguistic research in a theoretical way. MP postulates that a logical form and a sound could be derived from syntactic relations. Stabler, [2], proposes a framework for this program in a computational perspective with Minimalist Grammars -MG. These grammars inherit a long tradition of generative linguistics. The most interesting contribution of these grammars is certainly that the derivation system is defined with only two rules: merge and move. The word Minimalist is introduced in this perspective of simplicity of the definitions of the framework. If the merge rule seems to be classic for this kind of treatment, the second rule, move, accounts for the main concepts of this theory and makes it possible to modify relations between elements in the derived structure.

Even if the phonological calculus is already defined, the logical one is more complex to express. Recently, solutions were explored that exploited Curry's distinction between tectogrammatical and phenogrammatical levels; for example, Lambda Grammars, [4], Abstract Categorial Grammars, [5], and Convergent Grammars [6]. First steps for a convergence between the Generative Theory and Categorial Grammars are due to S. Epstein, [7]. A full volume of Language and Computation proposes several articles in this perspective, [8], in particular [9], and Cornell's works on links between Lambek calculus and Transformational Grammars, [10]. Formulations of Minimalist Grammars in a Type-Theoretic way have also been proposed in [11], [12], [13]. These frameworks were evolved in [14], [3], [15] for the syntax-semantics interface.

Defining a syntax-semantics interface is complex. In his works, Stabler proposes to include this treatment directly in MG. But interactions between syntax

and semantic properties occur at different levels of representation. One solution is to suppose that these two levels should be synchronized. Then, the Curry-Howard isomorphism could be invoked to build a logical representation of utterances. The Minimalist Categorial Grammars have been defined from this perspective: capture the same properties as MG and propose a synchronized semantic calculus. We will propose definitions of these grammars in this article. But do MG and MCG genrate the same language? In this article we take the first steps towrds showing that they do.

The first section proposes new definitions of Minimalist Grammars based on an algebraic description of trees which allows to check properties of this framework, [START_REF] Amblard | Calcul de représentations sémantiques et suntaxe générative: les grammaires minimalistes catégorielles[END_REF]. In the second section, we will focus on full definitions of Minimalist Categorial Grammars (especially the phonological calculus). We will give a short motivation for the syntax-semantics interface, but the complete presentation is delayed to a specific article with a complete example. These two parts should be viewed as a first step of the proof of mutual inclusion of languages between MG and MCG. This property is important because it enables us to reduce MG's to MCG, and we have a well-defined syntax-semantics interface for MCG.

Minimalist Grammars

Minimalist Grammars were introduced by Stabler [START_REF] Stabler | Derivational minimalism[END_REF] to encode the Minimalist Program of Chomsky, [START_REF] Chomsky | The Minimalist Program[END_REF]. They capture linguistic relations between constituents and build trees close to classical Generative Analyses. These grammars are fully lexicalized, that is to say they are specified by their lexicon. They are quite different from the traditional definition of lexicalized because they allow the use of specific items which do not carry any phonological form. The use of theses items implies that MG represent more than syntactic relations and must be seen as a meta-calculus lead by the syntax.

These grammars build trees with two rules: merge and move which are trigged by features. This section presents all the definitions of MG in a formal way, using algebraic descriptions of trees.

Minimalist Tree Structures

To provide formal descriptions of Minimalist Grammars, we differ from traditional definitions by using an algebraic description of trees: a sub-tree is defined by its context, as in [START_REF] Huet | The zipper[END_REF] and [START_REF] Levy | Algorithmes et Programmation[END_REF]. For example, the figure on the left of the figure 1 shows two subtrees in a tree (t 1 and t 2 ) and their context (C 1 and C 2 ). Before we explain the relations in minimalist trees, we give the formal material used to define a tree by its context.

Graded alphabets and trees: Trees are defined from a graded set. A graded set is made up of a support set, noted Σ, the alphabet of the tree, and a rank function, noted σ, which defines node arity (the graded terminology results from the rank function). In the following, we will use Σ to denote a graded (Σ, σ).

The set of trees built on Σ, written T Σ , is the smallest set of strings (Σ ∪ {(; ); , }) * . A leaf of a tree is a node of arity 0, denoted by α instead of α(). For a tree t, if t = σ(t 1 , • • • , t k ), the root node of t is written σ .

Moreover, a set of variables X = {x 1 , x 2 , • • •} is added for these trees. X k is the set of k variables. These variables mark positions in trees. By using variables, we define a substitution rule: given a tree t ∈ T Σ(X k ) (i.e. a tree which contains instances of k variables x 1 , • • • , x k ) and t 1 , • • • , t k , k trees in T Σ , the tree obtained by simultaneous substitution of each instance of x 1 by t 1 , . . . ,

x k by t k is denoted by t[t 1 , • • • , t k ]. The set of all subtrees of t is noted S t .
Thus, for a given tree t and a given node n of t, the subtree for which n is the root is denoted by t with this subtree replaced by a variable.

Minimalist trees are produced by Minimalist Grammars and they are built on the graded alphabet {<, >, Σ}, whose ranks of < and > are 2 and 0 for strings of Σ. Minimalist Trees are binary ones whose nodes are labelled with < or >, and whose leaves contain strings of Σ.

Relations between sub-trees We formalise relations for different positions of elements in S t . Intuitively, these define the concept of be above, be on the right or on the left. A specific relation on minimalist trees is also defined: projection that introduces the concept of be the main element in a tree.

In the following, we assume a given graded alphabet Σ. Proofs of principal properties and closure properties are all detailed in [START_REF] Amblard | Calcul de représentations sémantiques et suntaxe générative: les grammaires minimalistes catégorielles[END_REF]. The first relation is the dominance which informally is the concept of be above.

Definition 1 Let t ∈ T Σ , and C 1 , C 2 ∈ S t , C 1 dominates C 2 (written C 1 ⊳ * C 2 ) if there exists C ′ ∈ S t such that C 1 [C ′ ] = C 2 .
Figure 1 shows an example of dominance in a tree. One interesting property of this algebraic description of trees is that properties in sub-trees pass to tree. For example, in a given tree t, if there exists C 1 and C 2 such that C 1 ⊳ * C 2 , using a 1-context C, we could build a new tree t ′ = C[t] (substitution in the position marked by the variable xx 1 of t). Then,

C[C 1 ] and C[C 2 ] exist (they are part of t ′ ) such that C[C 1 ] ⊳ C[C 2 ]. Definition 2 Let t ∈ T Σ , C 1 , C 2 ∈ S t , C 1 immediately precedes C 2 (written C 1 ≺ C 2 ) if there exists C ∈ S t such that: 1. C 1 = C[σ(t 1 , . . . , t j , x 1 , t j+2 , . . . , t k )] and 2. C 2 = C[σ(t 1 , . . . , t j , t j+1 , x 1 , . . . , t k )].
Precedence, written ≺ ∼ , is the smallest relation defined by the following rules (transitivity rule, closure rule and relation between dominance and precedence relation):

C 1 ≺ ∼ C 2 C 2 ≺ ∼ C 3 [trans] C 1 ≺ ∼ C 3 C 1 ≺ C 2 [ * ] C 1 ≺ ∼ C 2 C 1 ⊳ * C 2 [dom] C 2 ≺ ∼ C 1 C 1 C 2 t 1 t 2 C 1 C 2 t 1 t 2
-C1 is the context of the sub-tree t1 -C2 is the context of the sub-tree t2 -C1 ⊳ * C2 means that the root node of t1 is higher than the root node of t2 in the full tree -C1 is the context of the sub-tree t1 -C2 is the context of the sub-tree t2 -C1 < * C2 means that the root node of t1 is on the left side of the root node of t2 in the tree Precedence encodes the relation be on the left (and then be on the right) or be above another element (using the dominance). These two relations stay true for substitution (as mentioned above).

The next relation does not define a tree relation. It realises a linguistic property by leading the concept of be the main element in a structure (or a substructure).

Definition 3 Let t ∈ T Σ M G (A), and C 1 , C 2 ∈ S t , C 1 immediately projects on C 2 (written C 1 < C 2 ) if
there exists C ∈ S t such that one of the two following properties holds:

1. C 1 = C[<(x 1 , t 2 )] and C 2 = C[<(t 1 , x 1 )], 2. C 1 = C[>(t 2 , x 1 )] and C 2 = C[>(x 1 , t 1 )], in this case C ⊳ C 1 and C ⊳ C 2 . If C 1 <C 2 or C 2 <C 1 , then there exists C such that C ⊳ C 1 and C ⊳ C 2 .
< ∼ is the smallest relation defined by the following system of rules:

C ∈ S t [0] C< ∼ C C 1 < ∼ C 2 C 2 < ∼ C 3 [trans] C 1 < ∼ C 3 C 1 <C 2 [∼] C 1 < ∼ C 2 C 1 ⊳ * C 2 C 3 ⊳ * C 4 C 2 <C 3 [A] C 1 < ∼ C 4 C 1 ⊳ C 2 C 2 <C 3 [B] C 2 < ∼ C 1
Note that the projection relation is transitive. All the properties of these three relations are proven in [START_REF] Amblard | Calcul de représentations sémantiques et suntaxe générative: les grammaires minimalistes catégorielles[END_REF]. The figure 2 presents three minimalist trees where in t the main element is the verb walks (which is accessible by following the projection relation).

These three relations could seem quite complicated for a reader who is not familiar with these notations or the zipper theory. But their expressiveness allows to prove the structural properties assumed for MG and moreover to give the proof of languages inclusion with MCG. Finally, in this section, we have defined the concept of parent and child relations in trees plus the projection relation which defines constituents in linguistic descriptions.

Linguistic Structures in Trees

From the linguistic perspective, trees represent relationships between grammatical elements of an utterance. Linguistic concepts are associated with minimalist tree structures. These relationships have been proposed for the analysis of structural analogies between verbal and nominal groups. Thus, groups of words in a coherent statement (phrases), whatever their nature, have a similar structure. This is supposed to be the same for all languages, regardless of the order of subterms. This assumption is one of the basic ideas of the X-bar theory introduced in the seventies, [18] and in the MP, [START_REF] Chomsky | The Minimalist Program[END_REF].

The head is the element around which a group is composed. An easy way to find the head of a minimalist tree is to follow the projection relation of the nodes. Definition 4 Let t ∈ T M G , if for all C ′ ∈ S t , C< ∼ C ′ then C is called the head of t. For a given tree t ∈ T M G , we write H t [x] ∈ S t a sub-tree of t of which x is the head, and head(t) is a leaf which is the head of t. Then t = H t [head(t)].

For a minimalist tree, there always exists a unique minimal element for the projection relation and it is a leaf (which is the head of the tree) [START_REF] Amblard | Calcul de représentations sémantiques et suntaxe générative: les grammaires minimalistes catégorielles[END_REF].

For example, the head of the minimalist tree in figure 2 is the leaf walks (follow the direction of the projection relation in nodes and stop in a leaf). Subtrees have their own head, for example the leaf a is the head of the subtree t 1 (in figure 2) and the preposition in is the head ot t 3 .

Maximal Projection is, for a leaf l, the largest subtree for which l is the head. This is the inverse notion of head. In the minimalist tree of figure 2, the maximal projection of the leaf walks is the full tree t. To describe other maximal projections in this example, the maximal projection of a is the subtree which contains a man and the maximal projection of the man is the leaf man. In a more formal way, the maximal projection is defined as follows:

✟ ✟ ✟ ❍ ❍ ❍ < ✟ ✟❍ ❍ a man < ✟ ✟ ❍ ❍ walks < ✟ ✟ ❍ ❍ in < ✟ ✟❍ ❍ the street < ✟ ✟❍ ❍ a man < ✟ ✟ ❍ ❍ in < ✟ ✟❍ ❍ the street t t1 t2
Fig. 2. a minimalist tree t and two of its sub-tree

Definition 5 Let t ∈ T M G , C ∈ S t .
The maximal projection of C (denoted by proj max (C)) is the subtree defined by:

-if C = x 1 , proj max (C) = x 1 -if C = C ′ [< (x 1 , t)] or C = C ′ [> (t, x 1 )], proj max (C) = proj max (C ′ ) -if C = C ′ [< (t, x 1 )] or C = C ′ [> (x 1 , t)], proj max (C) = C
Then proj max (walks) = t. This logical characterization of minimalist trees and structural relations allows to prove different properties of MG (for example that the projection is anti-symmetric), [START_REF] Amblard | Calcul de représentations sémantiques et suntaxe générative: les grammaires minimalistes catégorielles[END_REF].

Complement and Specifier are relations on subtrees with respect to the head.

Elements coming after the head provide information and they are in the

complement relation. Let t ∈ S M G , C 1 is a complement of head(t) = C, if proj max (C) ⊳ * C 1 and C ≺ + C 1 , denoted by C 1 comp C.
In the tree t of figure 2, the subtree t 2 is in a complement relation with the head walks. It adds information to the verb.

By contrast, elements placed before the head determine who

(or what) is in the relationship. Let t ∈ S M G , C 1 is a specifier of head(t) = C, if proj max (C) ⊳ * C 1 and C 1 ≺ + C, denoted by C 1 spec C.
In the tree t of figure 2, the subtree t 1 is in a specifier relation with the head walks. It specifies interpretation of the verb.

Minimalist Grammars

The computational system of MG is entirely based on features which represent linguistic properties of constituents. Rules are trigged by these features and they build minimalist trees. A Minimalist Grammar is defined by a quintuplet V, F eatures, Lex, Φ, c where:

-V is a finite set of non-syntactic features, which contains two sets: P (phonological forms, marked with / /), and I (logical forms, marked with ()). -Features= {B ∪ S ∪ L a ∪ L e } is a finite set of syntactic features, -Lex is a set of complex expressions from P and Features (lexical items), -Φ = {merge, move} is the set of generative rules, -c ∈Features is the feature which allows to accept derivations.

The final tree of a derivation which ends with acceptance is called a derivational tree, which corresponds to a classical generative analysis. Phonological forms are used as lexical items (and they could be seen as the grammar's terminal symbols). A left-to-right reading of phonological forms in derived and accepted structures provides the recognized string. But intermediate trees in a derivation do not stand for this. Only the derivational tree allows to recognize a string. This results from the move rule which modifies the tree structure. For a MG G , the language L G recognized by G is the closure of the lexicon by the generation rules.

Features

A MG is defined by its lexicon which stores its resources. Lexical items consist of a phonological form and a list of syntactic features. The syntactic set of features is divided in two subsets: one for basic categories, denoted B, and one for move features, denoted D. Different types of features are: Vermaat, [START_REF] Vermaat | Controlling movement: Minimalism in a deductive perspective[END_REF], proposes an automata which recognises the acceptable sequences, proposed in figure 3. This structure could be divided in two parts: the first containing a sequence of selectors and licensors (features which trigger rules, as we shall see), and the second which contains only one basic feature (the grammatical category associated to the expression) and a sequence of licensees. The first part corresponds to stat I and II and the second to stat III and transitions to this state. In the following, e will denote any feature and E a sequence of features (possibly empty).

-B = {v, dp, c, • • •} the set
For example, the sequence associated with an intransitive verb will be: =d +case v which means that this verb must be jointed with a determinal phrase (determinal comes from the Generative Theory), a complex expression with feature d. Then it must be combined with a -case, we will see how in the next section, an then there is a structure associated with verb (feature v).

Transitive verbs will extend the intransitive ones wth the list:

=d +case =d +case v

The two =d correspond to the subject and the object of the verb. The first case will be accusative and the second nominative.

Another example is determiners: they are combined with a noun to build a determiner phrase and need to be unified in the structure (see the next section).

Here is an example of lexicon which contains a verb, a noun and a determiner:

walks : =d +case v a : =n d -case man : n 1.5 MG Rules
Φ, the set of generating rules, contains only: merge and move. A derivation is a succession of rule applications which build trees. These trees are partial results: the structural order of phonological forms does not need to correspond to the final one. In the MP, a specific point, called Spell-Out is the border between the calculus of derivations and the final result. Rules are trigged by the feature occurring as the first element of list of features of the head.

Merge is the process which connects different parts. It is an operation which joins two trees to build a new one:

merge : T M G × T M G → T M G
It is triggered by a selector (=x) at the top of the list of features of the head and it is realised with a corresponding basic feature (x) at the top of the list of features of the head of a second tree. Merge adds a new root which dominates both trees and cancels the two features. The specifier/complement relation is implied by the lexical status of the tree which carried the selector. The new root node points to this tree.

Let For example, to derive a man walks, we first need to combine a with man, and then to combine the result with the verb:

t,t ′ ∈ T M G be such that t = H t [l : =h E] and t ′ = H t ′ [l ′ : h E ′ ] with h ∈ B: merge(t, t ′ ) = < (l : E, H t ′ [l ′ : E ′ ]) if t ∈ Lex, > (H t ′ [l ′ : E ′ ], H t [l : E]) otherwise.
< ✟ ✟ ✟ ❍ ❍ ❍ a ✟ ✟ =n d -case man ✚ n and < ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ walks ✟ ✟ =d +case v < ✟ ✟ ✟ ❍ ❍ ❍ a ✟ ✟ =n ✁ d -case man ✚ n
Obtained trees do not verify the word order (only the final tree will check the right word order). In this example, the selectors are carried by lexical items, then projection relations point to the left in both cases.

Move encodes the main idea of the Minimalist Program. It corresponds to the movement of a constituent at the top position in a derivation. Move is trigged by a licensor (+x) at the top of the list of features of the head of a tree. Then, it looks for a corresponding licensee (-x) at the top of the list of features of the head inside the tree. If these conditions are met, the maximal projection of the node which carries the licensee is moved to the left of a new root. This node points to the right (the subtree which carries the former head). Both licensor and licensee are cancelled. The root of the moved maximal projection is substituted by an empty leaf (ǫ). This new leaf is called the trace of the move.

Figure 4 shows a graphical representation of the move rule where the head of C carries a +g in its top features list. Then we look for a leaf with -g in its top features list and then find its maximal projection (C 2 ) which contains all the elements which depend on it. Finally this sub-tree is moved to the left position of a new root node. Intuitively, a linguistic property is checked and the consequence is a move in first position in the tree. And strictly:

move :

T M G → T M G For all tree t = C[l : +g E, l ′ : -g E ′ ], such that t = H t [l : +g E], there exists C 1 , C 2 ∈ S t such that: C 2 is the maximal projection of the leaf l ′ and C 1 is t deprived of C 2 . Then, t = C 1 [l : +g E, C 2 [l ′ : -g E ′ ]
] where:

-C 2 [l ′ : -g E ′ ] = proj max (C[l ′ : -g E]) -C 1 [l : +g E, x 1 ] = proj max (C[l : +g E, x 1 ]) move(t) = >(C 2 [l ′ : E ′ ], C 1 [l : E, ǫ])
Figure 4 presents the graphical representation of move. Stabler introduces some refinements to these grammars. Let us mention them. He introduces a second move: weak move, which does not move the phonological forms. The precedent move is then called strong move, which is trigged with capital features. The weak move is, like strong move:

move(t) = >(C 2 [ǫ : E ′ ], C 1 [l : E, l ′ ])
Variations on strong/weak values achieve variations on phonological order. This is an instance of the use of parameters of the Minimalist Program.

Moreover, restrictions can be introduced on MG derivations. An important one is the Shortest Move Condition (SMC) which blocks move in case of ambiguity on licensees. Then, the move operation of MG with SMC is deterministic.

A locality condition could also be introduced: Specifier Island Condition -SPIC. "Islands" define areas which prohibit extractions. With SPIC, a subtree cannot be moved if it is in a specifier relation within a subtree. This condition was introduced by Stabler, in [START_REF] Stabler | Remnant movement and structural complexity. Constraints and Resources in Natural Language Syntax and Semantics[END_REF] drawing on works of [START_REF] Koopman | A verbal Complex[END_REF] and [START_REF] Kayne | Overt vs covert movment[END_REF], who proposes that moved elements had to be in a complement relation.

In the previous example, the head of the last tree is the leaf walks which contains a +case feature as first element of its list. Then, a move is trigged in the tree with the leaf a which carries a (-case). The resulting tree is the following:

> ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ < ✟ ✟ ✟ ❍ ❍ ❍ a ✟ ✟ =n ✁ d ✘ ✘ ✘ -case man ✚ n < ✟ ✟ ❍ ❍ walks ✟ ✟ =d ✘ ✘ ✘ +case v ǫ
The move operation modifies the position of the maximal projection of the leaf which carries the -case. The old position is substituted by an empty leaf (ǫ). Finally, the tree contains only one feature which is v. In this small example, I did not discuss the validity of the final feature, but in a real derivation, we assume that it is not the verb which carries the +case licensor which corresponds to the nominal case, but it is a specific item. This item corresponds to the morphological mark of the verb. Then each acceptable derivation assumes that a verb has received its time (and other properties). But exhibiting the use of this item needs other refinements of the two rules (Head-movement and Affix-Hopping).

This section did not propose a new framework for computational linguistics. This is a new definition of Stabler proposal. This way, assumed properties of minimalist trees have been fully proved, [START_REF] Amblard | Calcul de représentations sémantiques et suntaxe générative: les grammaires minimalistes catégorielles[END_REF]. Moreover this algebraic definition of MG is a perfect description to compare generated languages with other frameworks. Finally, this modifies the point of view on derivations and shows all steps of the calculus as substitution. One missing point is still the introduction of a semantic calculus. Let us now develop MCG which are defined with a syntax-semantics interface.

Minimalist Categorial Grammars -MCG

In this section, we define a new Type-Theoretic Framework which is provided by the mixed calculus, a formulation of Partially Commutative Linear Logic. It proposes to simulate MG and then keep linguistic properties of the Minimalist Program. MCG are motivated by the syntax-semantics interface, [START_REF] Amblard | Calcul de représentations sémantiques et suntaxe générative: les grammaires minimalistes catégorielles[END_REF]. This interface, as for Lambek calculus, is based on an extension of the Curry-Howard isomorphism, [START_REF] Howard | The formulae-as-types notion of construction[END_REF]. Even though this interface is not the aim of this paper, let us discuss some important points.

The idea of encoding MP with Lambek calculus arises from [START_REF] Lecomte | Towards a logic for minimalist[END_REF] and extended versions of this work. In these propositions, the calculus is always noncommutative, a property needed to model the left-right relation in sentences. But the move operation could not be defined in a proper way with non-commutative relation. In particular, in complex utterances, the non-commutativity implies that a constituent (for example the object DP) must be fully treated before another one is introduced (for example the subject DP). Otherwise, features are mixed and non-commutativity blocks resolutions. It is not acceptable to normalize the framework with such a strong property and it makes the system inconsistent in regard to linguistics.

The solution we propose is to define a new framework which allows to deal with commutative and non-commutative connectors: the mixed calculus. The main consequence on the model of this calculus is that variables in logical formulae are introduced at different places and must be unified later. In [START_REF] Amblard | Calcul de représentations sémantiques et suntaxe générative: les grammaires minimalistes catégorielles[END_REF] we show how the unification is used to capture semantic phenomena which are not easily included. In few words, the idea is to consider proofs of mixed calculus as phases of a verb. Phases have been introduced by Chomsky to detail different modifications which occur on a verb. Several linguists have showed that phases have implications on semantics, for example the theta-roles must be allocated after a specific phase. This is exactly the result of the syntax-semantics interface of MCG. Full explanations need more space to be presented, but the main contribution of MCG is to propose an efficient syntax-semantics interface in the same perspective as MG.

In this section, we will detail MCG and expose their structural link with MG. First we present the mixed calculus, then we give definitions of MCG and show proofs of the mixed calculus produced by MCG (together with their linguistic properties).

Mixed calculus

MCG are provided with mixed calculus, [START_REF] De Groote | Partially commutative linear logic: sequent calculus and phase semantics[END_REF], a formulation of Partially Commutative Linear Logic. Hypotheses are either in a non-commutative order (<; >) or in a commutative one ((, )) The plain calculus contains introduction and elimination rules for:

the non-commutative product ⊙:

∆ ⊢ A ⊙ B Γ, < A; B >, Γ ′ ⊢ C [⊙ e ] Γ, ∆, Γ ′ ⊢ C ∆ ⊢ A Γ ⊢ B [⊙ i ] < ∆; Γ >⊢ A ⊙ B -its residuals (/ and \): Γ ⊢ A ∆ ⊢ A\C [\ e ] < Γ ; ∆ >⊢ C ∆ ⊢ A/C Γ ⊢ A [/ e ] < ∆; Γ >⊢ C < A; Γ >⊢ C [\ i ] Γ ⊢ A\C < Γ ; A >⊢ C [/ i ] Γ ⊢ C/A -the commutative product ⊗: ∆ ⊢ A ⊗ B Γ, (A, B), Γ ′ ⊢ C [⊗ e ] Γ, ∆, Γ ′ ⊢ C ∆ ⊢ A Γ ⊢ B [⊗ i ] (∆, Γ ) ⊢ A ⊗ B -its residual ⊸: Γ ⊢ A ∆ ⊢ A ⊸ C [⊸ e ] (Γ, ∆) ⊢ C (A, Γ ) ⊢ C [⊸ i ] Γ ⊢ A ⊸ C
The product connectors of the mixed calculus use in a first step hypotheses to mark positions in the proof and in a second one substitute the result of an another proof in these positions using a product elimination (the commutative/noncommutative status depends on relations between hypotheses). This is exactly the process we will use to define the move rule of MCGs.

Moreover, the calculus contains an axiom rule and an entropy rule. This last one allows to relax the order between hypotheses. We will use this rule to define merge in MCG as we will see in the following section.

[axiom]

A ⊢ A Γ ⊢ C [entropy -whenever Γ ′ ⊏ Γ ] Γ ′ ⊢ C
This calculus has been shown to be normalizable, [START_REF] Amblard | Natural deduction and normalisation for partially commutative linear logic and lambek calculus with product[END_REF] and derivations of MCG will be proofs of the mixed calculus in normal form.

Minimalist Categorial Grammars

As MG, MCG are lexicalized grammars. Derivations are led by formulae associated with lexical items built with connectors of the mixed logic. They are specific proofs of the mixed logic, labelled to realise the phonological and semantic tiers. Phonological labels on proofs will be presented with definitions of MCG rules.

A MCG is defined by a quintuplet N,p, Lex, Φ, C where :

-N is the union of two finite disjoint sets P h and I which are respectively the set of phonological forms and the one of logical forms. p is the union of two finite disjoint sets p 1 and p 2 which are respectively the set of constituent features (the set B of MG) and the one of move features (the set D of MG). -Lex is a finite subset of E × F × I, the set of lexical items1 .

-Φ = {merge, move} is the set of generative rules, -C ∈ p is the accepting formulae.

As mentioned in the previous section, move is defined using a product elimination. In MG, a constituent is first introduced in a tree using its basic feature and then can be moved using its licensees. In MCG, a constituent will be introduced only when all its positions (which correspond to the basic feature and its licensees) have been marked in the proof by specific hypotheses. But we need to distinguish the type of the basic feature from the licensees features. That is why p is divided in two subsets p 1 and p 2 . This sub-typing of formulae is used to well define lexicons of MCG.

The set E is P h * , and the set F , the set of formulae used to build Lex, is defined with the set p, the commutative product ⊗ and the two non-commutative implications / and \. Formulae of F are recognized by the non-terminal l of the following grammar:

l ::= (b) / p 1 | c b ::= p 1 \ (b) |p 2 \ (b) | c c ::= p 2 ⊗ (c) | c 1 c 1 ::= p 1
In more details, MCG formulae start with a / which is followed by a sequence of \. This sequence contains operators allowing to compose the proof with another one (operators are the translation of selectors and licensors). Lexical formulae are ended by a sequence of ⊗. To sum up, these formulae have the structure (c m \ . . .

\c 1 \(b 1 ⊗ . . . ⊗ b n ⊗ a))/d, with a ∈ p 1 , b i ∈ p 2 , c j ∈ p and d ∈ p 1 .
This structure corresponds to the two parts of the list of features we have mentioned in the previous section.

For the example a man walks, the MCG's lexicon is the following: walks : case\v/d a : (case ⊗ d)/n man : n Licensees, which express the need for an information, are there seen as a specific part of the basic feature (a part of the main sub-type). Licensors will be cancelled with an hypothesis to mark a position in the proof. Distinction between them is not written by an ad hoc marker but by structural relations inside the formula. Before we explain the move and merge rules, let us present the phonological tiers.

Derivations

Labels. Derivations of MCG are labelled proofs of the mixed calculus. Before defining labelling, we define labels and operations on them.

Let V be an uncountable and finite set of variables such that: P h∩V = ∅. T is the union of P h and V . We define the set Σ, called labels set as the set of triplets of elements of T * . Every position in a triplet has a linguistic interpretation: they correspond to specifier/head/complement relations of minimalist trees. A label r will be considered as r = (r spec , r head , r comp ).

For a label in which there is an empty position, we adopt the following notation: r -head = (r spec , ǫ, r comp ), r -spec = (ǫ, r head , r comp ), r -comp = (r spec , r head , ǫ). We introduce variables in the string triplets and a substitution operation. They are used to modify a position inside a triplet by a specific material. Intuitively, this is the counterpart in the phonological calculus of the product elimination. The set of variables with at least one in r is denoted by V ar(r). The number of occurrences of a variable x in a string s ∈ T * is denoted by |s| x , and the number of occurrences of x in r by ϕ x (r). A label is linear if for all x in V , ϕ x (r) 1.

A substitution is a partial function from V to T * . For σ a substitution, s a string of T * and r a label, we note s.σ and r.σ the string and the label obtained by the simultaneous substitution in s and r of the variables by the values associated by σ (variables for which σ is not defined remain the same).

If the domain of definition of a substitution σ is finite and equal to x 1 , . . . , x n and σ(x i ) = t i , then σ is denoted by [t 1 /x 1 , . . . , t n /x n ]. Moreover, for a sequence s and a label r, s.σ and r.σ are respectively denoted s[t 1 /x 1 , . . . , t n /x n ] and r[t 1 /x 1 , . . . , t n /x n ]. Every injective substitution which takes values in V is called renaming. Two labels r 1 and r 2 (respectively two strings s 1 and s 2 ) are equal modulo a renaming of variables if there exists a renaming σ such that r 1 .σ = r 2 (resp. s 1 .σ = s 2 ).

Finally, we need another operation on string triplets which allows to combine them together: the string concatenation of T * is noted •. Let Concat be the operation of concatenation on labels which concatenates the three components in the linear order: for r ∈ Σ, Concat(r) = r spec • r head • r comp .

We then have defined a phonological structure which encodes specifier/complement/head relations and two operations (substitution and concatenation). These two operations will be counterparts in the phonological calculus of merge and move.

Labelled proofs. Before exhibiting the rules of MCG, the concept of labelling on a subset of rules of the mixed logic is introduced. Minimalist logic is the fragment of mixed logic composed by the axiom rule, \ e , / e , ⊗ e and ⊏.

For a given MCG G = N,p, Lex, Φ, C , let a G-background be x : A with x ∈ V and A ∈ F , or G 1 ; G 2 or else (G 1 , G 2 ) with G 1 and G 2 some Gbackgrounds which are defined on two disjoint sets of variables. G-backgrounds are series-parallel orders on subsets of V ×F . They are naturally extended to the entropy rule, noted ⊏. A G-sequent is a sequent of the form: Γ ⊢ G (r s , r t , r c ) : B where Γ is a G-background, B ∈ F and (r s , r t , r c ) ∈ Σ.

A G-labelling is a derivation of a G-sequent obtained with the following rules:

s, A ∈ Lex [Lex] ⊢ G (ǫ, s, ǫ) : A x ∈ V [axiom] x : A ⊢ G (ǫ, x, ǫ) : A Γ ⊢ G r 1 : A / B ∆ ⊢ G r 2 : B V ar(r 1 ) ∩ V ar(r 2 ) = ∅ [/ e ] Γ ; ∆ ⊢ G (r 1s , r 1t , r 1c • Concat(r 2 )) : A ∆ ⊢ G r 2 : B Γ ⊢ G r 1 : B \ A V ar(r 1 ) ∩ V ar(r 2 ) = ∅ [\ e ] Γ ; ∆ ⊢ G (Concat(r 2 ) • r 1s , r 1t , r 1c ) : A Γ ⊢ G r 1 : A ⊗ B ∆[x : A, y : B] ⊢ G r 2 : C V ar(r 1 ) ∩ V ar(r 2 ) = ∅ A ∈ p 2 [⊗ e ] ∆[Γ ] ⊢ G r 2 [Concat(r 1 )/x, ǫ/y] : C Γ ⊢ G r : A Γ ′ ⊏ Γ [⊏] Γ ′ ⊢ G r : A
Note that a G-labelling is a proof tree of the minimalist logic on which sequent hypotheses are decorated with variables and sequent conclusions are decorated with labels. Product elimination is used with a substitution on labels and implication connectors with concatenation (a triplet is introduced in another one by concatenating its three components).

If Γ ⊢ G r : B is a G-sequent derivable, then r is linear, and V ar(r) is exactly the set of variables in Γ . Finally, for all renamings σ, Γ.σ ⊢ G r.σ : B is a G-sequent differentiable.

Merge and Move rules are simulated by combinations of rules of the minimalist logic producing G-labeling.

Merge is the elimination of / (resp. \) immediately followed by an entropy rule. The meaning of this rule is joining two elements in regard to the left-right order (then non-commutative connectors are used) and, as mentioned earlier, all hypotheses must be accessible. To respect this, a commutative order between hypotheses is needed. Then an entropy rule immediately follows each implication elimination.

For the phonological tier, a label is concatenated in the complement (respectively specifier) position in another one. Note that a merge which uses / must be realized with a lexical item, so the context is always empty.

⊢ (r spec , r head , r comp ) : A / B ∆ ⊢ s : B [/ e ] ∆ ⊢ (r spec , r head , r comp • Concat(s)) : A [⊏] ∆ ⊢ (r spec , r head , r comp • Concat(s)) : A ∆ ⊢ s : B Γ ⊢ (r spec , r head , r comp ) : B \ A [\ e ] ∆; Γ ⊢ (Concat(s) • r spec , r head , r comp ) : A [⊏] ∆, Γ ⊢ (Concat(s) • r spec , r head , r comp ) : A
These combinations of rules are noted [mg].

For example, the proof of the utterance a man walks begins with the formulae of walks: case\v/d. The first step of the calculus is to introduce two hypotheses, one for d and the other for case. The result is the following proof:

v : case ⊢ (ǫ, v, ǫ) : case ⊢ (ǫ, walks, ǫ) : case\v/d u : d ⊢ (ǫ, u, ǫ) : d [mg] u : d ⊢ (ǫ, walks, u) : case\v [mg] (v : case, u : d) ⊢ (ǫ, walks, u) : v
In parallel, the derivation joins the determiner a and the noun man:

⊢ (ǫ, a, ǫ) : (case ⊗ d)/n ⊢ (ǫ, man, ǫ) : n [mg] ⊢ (ǫ, a, man) : case ⊗ d
Note that the first proof contains two hypotheses which correspond to the type of the main formula in the second proof. The link between these two proofs will be made by a move, as we will show later.

Move is simulated by an elimination of a commutative product in a proof and, for the phonological calculus, is a substitution. We have structured the lexicons and the merge rule to delay to the move rule only the substitution part of the calculus.

Γ ⊢ r 1 : A ⊗ B ∆[u : A, v : B] ⊢ r 2 : C [⊗ e ] ∆[Γ ] ⊢ r 2 [Concat(r 1 )/u, ǫ/v] : C This rule is applied only if A ∈ p 2 and B is of the form B 1 × . . . B n × D where B i ∈ p 2 and D ∈ p 1 .
This rule is noted [mv]. Move uses hypotheses as resources. The calculus places hypotheses in the proof, and when all hypotheses corresponding to a constituent are introduced, this constituent is substituted. The hypothesis p 1 is the first place of a moved constituent and hypotheses of p 2 mark the different places where the constituent is moved or have a trace.

In recent propositions, Chomsky proposes to delay all moves after the realisation of all merges. MCG could not encode this but contrary to MG where a move blocks all the process, in MCG merge could happen, except in the case of hypotheses of a given constituent shared by two proofs which must be linked by a move.

In our example, we have two proofs:

one for the verb: (v : case, u : d) ⊢ (ǫ, walks, u) : v one for the DP: ⊢ (ǫ, a, man) : case ⊗ d

The first hypothesis corresponds to the entry position of the DP in MG and the second to the moved position. Here, we directly introduce the DP by eliminating the two hypotheses in the same step:

⊢ (ǫ, a, man) : case ⊗ d (v : case, u : d) ⊢ (ǫ, walks, u) : v [mv] ⊢ (a man, walks, ǫ) : v
The phonological result is a man walks. The proof encodes the same structure as the derivational tree of MG (modulo a small transduction on the proof).

For cyclic move (where a constituent is moved several times) all hypotheses inside this move must be linked together upon their introduction in the proof. For this, when a new hypothesis A is introduced, a [mv] is applied with a sequent with hypothesis A⊗B ⊢ A⊗B where A is in p 2 and B is of the form B 1 ⊗. . .⊗B n ⊗D where B i ∈ p 2 and D ∈ p 1 .

x

: A ⊗ B ⊢ (ǫ, x, ǫ) : A ⊗ B ∆[u : A, v : B] ⊢ r : C [⊗ e ] ∆[A ⊗ B] ⊢ r[x/u, ǫ/v] : C
In the definition of merge, the systematic use of entropy comes from the definition of move. As it was presented, move consumes hypotheses of the proof. But, from a linguistic perspective, these hypotheses could not be supposed introduced next to each other. The non-commutative order inferred from \ e and / e blocks the move application. To avoid this, the entropy rule places them in commutative order. In MCG, all hypotheses are in the same relation, then to simplify the reading of proofs, the order is denoted only with ','.

The strong/weak move could be simulated with the localization of the substitution (if hypotheses are in p 1 or p 2 ).

s : Γ ⊢ A ⊗ B r[u, v] : ∆[u : A, v : B] ⊢ C [move strong ] r[Concat(s)/u, ǫ/v] : ∆[Γ ] ⊢ C s : Γ ⊢ A ⊗ B r[u, v] : ∆[u : A, v : B] ⊢ C [move weak ] r[ǫ/u, Concat(s)/v] : ∆[Γ ] ⊢ C
This version of move is quite different from the one presented for MG, but is close to one developed for later MG such as [START_REF] Kobele | Generating Copies: An Investigation into Structural Identity in Language and Grammar[END_REF].

The main difference between MG and MCG comes from move: in MCG, constituents do not move but use hypotheses marking their places. MCG uses commutativity properties of mixed logic and see hypotheses as resources. To sum up, the derivation rules of MCG is the following set of rules:

s, A ∈ Lex [Lex] ⊢ G (ǫ, s, ǫ) : A ⊢ (r spec , r head , r comp ) : A / B ∆ ⊢ s : B [mg] ∆ ⊢ (r spec , r head , r comp • Concat(s)) : A ∆ ⊢ s : B Γ ⊢ (r spec , r head , r comp ) : B \ A [mg] ∆, Γ ⊢ (Concat(s) • r spec , r head , r comp ) : A Γ ⊢ r 1 : A ⊗ B ∆[u : A, v : B] ⊢ r 2 : C [mv] ∆[Γ ] ⊢ r 2 [Concat(r 1 )/u, ǫ/v] : C
The set D G of recognized derivations by a MCG G is the set of proofs obtained with this set of rules and for which the concluding sequent is ⊢ r : C. The language generated by G is L(G) = {Concat(r)| ⊢ r : C ∈ D G }.

These derivations do not formally conserve the projection relation (nor the specifier, head and complement relations). These principles are reintroduced with strings. However, the head of a proof could be seen as the principal formula of mixed logic, and then by extension, the maximal projection is the proof for which a formula is the principal one. Specifier and complement are only elements on the right or left of this formula.

An interesting remark is that rules of MCG do not use the introduction rule of the mixed calculus. This way, they only try to combine together formulae extracted from a lexicon and hypotheses. As in MG where a derivation cancels features, the MCG system only consumes hypotheses and always reduces the size of the main formula (only the size of the context could increase). This corresponds to the cognitive fact that we stress the system in the analysis perspective. Introduction rules could be seen as captured by the given lexicon. But, because of the strong structure of the items, we directly associate formulae and strings.

We have presented all the MCG rules and lexicon, and illustrated them with a tiny example which encodes the main properties of this framework.

Conclusion

In this article, we propose new definitions of MG based on an algebraic description of trees. These definitions allow to check properties of this framework and moreover give a formal account to analyse links with other frameworks. Then, we give the definitions of MCG, a Type-Theoretic framework for MG. In this framework, merge and move are simulated by rules of the mixed logic (an extension of Lambek calculus to product and non-commutative connectors). The phonological calculus is added by labelling proofs of this logic.

The main contribution of MCG is certainly its syntax-semantics interface. This calculus is synchronized on proofs of MCG. But more technical details are needed to present this interface and the linguistic properties which it encodes. We delay the presentation of this interface to a future presentation.

Finally, the syntax-semantics interface of MCG should be used under the condition they keep properties of MG. This is the aim of another future article which will present the proof of inclusion of MG generated languages in MCG generated languages. To prove this property, two alternative representations of MG and MCG derivations are introduced: alternative derived structures and split proofs and the corresponding merge and move. These structures and rules make the gap between the two kinds of derivations. They need technical details and more space to be presented.

Definitions and proofs could be easily extended to refinements of merge: Affix-Hopping and Head-Movement because these operations derived the same strings in both structures. But we have not included these rules in this presentation. On another hand, the proof of inclusion presented here does not include the SMC. The interpretation of SMC in MCG must be better defined before being included in such perspective. The generative power of these grammars with shortest move condition is still open.

This article is a first step to several perspectives which make a strong link between a well defined framework with many linguistic properties and a new one which captures this framework and proposes a syntax-semantics interface.

Fig. 1 .

 1 Fig. 1. Dominance and precedence relations in trees.

Fig. 3 .

 3 Fig. 3. Automata of acceptable sequences of features where b ∈ B and d ∈ D.

Figure 4

 4 Figure 4 presents the graphical representation of merge.

Fig. 4 .

 4 Fig. 4. Tree representation of merge and move.

In the following, Lex is a subset of E × F . The semantic part is used for the syntaxsemantics interface which is not detailed here.

Acknowledgements

The author would like to express his deep gratitude to his supervisors Alain Lecomte and Christian Retoré. In particular, discussions with Alain Lecomte was a source of supports and good advices which turn the author to this research.

The author also wants to thank the associated editors and the anonymous reviewers for their constructive remarks and suggestions, and finally Patrick Blackburn and Mathieu Morey for their careful readings.


Andreas Gumbsch

Giovanni Barcaro

Mike G Ramsey

Svetlozar Surnev

Alessandro Fortunelli
email: fortunelli@ipcf.cnr.it

Falko P Netzer
email: falko.netzer@uni-graz.at

Kondo vs magnetic coupling of cobalt dimers in a Cu-O stripe phase

teaching and research institutions in France or abroad, or from public or private research centers.

Kondo versus magnetic coupling of cobalt dimers in a

CuO (2 x 1) reconstruction

In the last 12 years, modern developments in surface science have allowed researchers to demonstrate the Kondo effect, a previously well-known and widely studied phenomenon in condensed-matter physics [1,2,3], on novel classes of materials and to characterize its features with unprecedented accuracy [4,5,6]. This combined very well with the theme of collective many-body interactions that are at the basis of the surface Kondo effect and make it an ideal test ground for understanding and controlling quantum phenomena that are expected to dominate as the limits of miniaturization are approached in nanotechnological devices. Spinflip scattering processes between the electrons of the magnetic species and the conduction band electrons of the host produce a non-magnetic correlated electronic quantum state, whose signature in scanning tunneling spectroscopy (STS) is a resonance positioned in close proximity of the Fermi level [7]. The resonance in STS has a Fano-like appearance [8], as it arises from the interference of two competing scattering channels. An analysis of its features thus provides access to information on the electronic interactions between the host and the impurity which would be very difficult to obtain otherwise. The length scale of these interactions, the so-called Kondo screening length, in which the impurity spin is screened, is estimated to extend for tens of nanometers (10-100 nm), involving 10 5 -10 8 atoms [9].

Although the basic physics of the Kondo effect has been known since a long time [1], a quantitative understanding is needed if control on and design of real devices is to be achieved.

Finding new, atomically characterized examples is crucial in this respect, as surprises and counterintuitive trends have been shown to occur [10,11], in particular by enlarging the set of well-defined systems from the original magnetic metal adatoms on single-crystal metal surfaces to more exotic substrates, but at the same time closer to real-world applications, such as surface or ultrathin oxides and nitrides [7]. In this perspective, it is especially important that not only the principles governing the impurity/host but also the interaction among single scatterers need to be fully clarified as functions of the chemistry and geometry of adsorption, another important theme in current nanoscience [12].

In the present work, this research plan is pursued by investigating via STM/STS Co adatoms on and Co atoms and dimers embedded in a Cu(110)(2x1)-O surface oxide. Co adatoms on the bare Cu(110) surface have been studied in a previous work [10], showing that they exhibit a Kondo resonance with a peak-like instead of a dip-like Fano shape, as observed for Co adatoms on other Cu surfaces [13]. The main finding here is that pairs of first-neighbour Co atoms incorporated into adjacent parallel rows of the Cu(110)(2x1)-O surface show Kondo resonances essentially decoupled from each other, whereas they are anti-ferromagnetically coupled (and thus do not exhibit any Kondo effect) when they are aligned in the same Cu-O row. In other words, we find that Co first-neighbor dimers can exhibit a completely different magnetic behaviour depending on the geometry of adsorption, despite the fact that they lie at comparable close-contact distances. These results, which are in keeping with the predictions of density-functional (DF) simulations, indicate first of all that -despite the intrinsically longrange character of the Kondo phenomenon -it is possible to decouple single carriers of the Kondo quantum effect from each other via a proper choice of the interaction geometry, with promising implications in terms of technological devices. Additionally, it is shown that an oxidic environment (closer to real-world applications) presents features somewhat different from pure metal systems, not only because the oxygen atoms can mediate the interactions among impurity atoms, in one present case switching from ferromagnetic to antiferromagnetic coupling, but also because they influence appreciably their charge and spin state, and thus ultimately their Kondo response. This is supported by arguments based on theoreticallyderived quantities such as the weighted coordination number, the 3d population, and the local density of states (LDOS) at the Fermi level.

The STM/STS experiments have been carried out in a three-chamber low-temperature STM system (Createc, Germany) in ultrahigh vacuum (UHV) with a base pressure of ~5 x 10 -11 mbar, operated at a measurement temperature of ~5-6 K in the liquid He STM cryostat stage as described previously [10]. The Cu(110)(2x1)-O reconstruction was prepared by exposing the clean Cu(110) surface to 1 L O 2 (1 Langmuir (L) = 1 x 10 -6 torr.s) at a sample temperature of 600 K followed by annealing in UHV. Further details of the experiment and the preparation of the Cu(110)2x1-O surface including the deposition of Co impurity atoms are described in the Supplementary Material. DF calculations were performed using the PWscf (Plane-Wave Self-Consistent Field) computational code [14], employing ultrasoft pseudopotentials [15] and the PBE exchangecorrelation functional [16] within a DF+U approach [17] with a value of U for the Co species equal to 3.0 eV. A DF+U approach is known to produce more accurate results than a pure DF one for Co in oxidic environments [17]. It can be noted that we use a conservative value of U = 3.0 eV, which lies at the lower edge of the range usually considered in the literature [18].

The calculations on the non-oxidized systems were also performed using a standard DF approach, finding completely analogous results. Other computational details can be found in the Supplementary Material. As mentioned above, the form of the Kondo resonance in the STS spectra may be described by a Fano line function [7,8]:

(dI/dV) (V) α (q + ε) 2 / (1 + ε 2 ) + c (1)
with the normalised energy ε = (eV -δ) / Γ and a constant c. The parameters q, δ and Γ are fit parameters for the experimental data and signify the asymmetry parameter q, the energy shift δ of the Kondo resonance from zero bias (i.e. E F ), and the half width of the feature at half maximum Γ, which is related to the characteristic Kondo temperature The geometries of Co atoms and dimers on/in the oxidized Cu(110) surface layer are schematically depicted in Fig. 1(a-d). The distances between the Co atom and its nearestneighbor Cu atoms (r ij ) as derived from DF energy minimizations are reported in Table 1 for all the systems here considered, together with the average number of electrons in the Co 3d and 3s majority (n dα and n sα , respectively) and minority (n dβ and n sβ , respectively) spin bands, evaluated via a Lowdin projection of the wave function, and the experimentally derived values of the Kondo temperature (T K ), the position of the Kondo resonance with respect to the Fermi energy (δ), and the asymmetry parameter of the Kondo resonance (q). A few notable differences among the various systems here considered can be drawn from an inspection of Table 1, which can be analyzed in detail in comparison with experimental data.

T K = Γ/k B (k B is the Boltzmann constant).
We start by analyzing the Co interaction geometry. From Table 1, one can see that both the number and the distances of Cu atoms next to Co vary significantly for the various systems.

This variation can be summarized into a single parameter as the coordination number [13]:

CN = Σ j-th Cu neighbor exp ( -r ij ) (2)
where r ij is the distance of the j-th Cu neighbor from the Co impurity atom, evaluated from the DF local energy minimizations and reported in Table The Co atom is therefore partially oxidized in the latter cases. Second, the spin polarization in the d-space (n dα -n dβ ) is ≈ 2 for the bare systems, whereas (n dα -n dβ ) is ≈ 1.82 for the adatom on the oxidized (110) surface and even smaller (1.75) for a Co atom embedded in the same surface (the fact that the spin polarization for the oxidized systems is reduced will be used below).

The charge state of the impurity immediately reflects on the position δ of the Kondo resonance. This quantity is in fact related to the occupation number n d of the 3d states of the adatom on the surface by the relation [2]:

δ = Γ • tan [π /2 • (1-n d )] (3)
The average occupation number n d can be estimated using atomic populations derived via a Lowdin projection of the DF wave function (see Table 1) as:

n d = n dα + n dβ -7 (4)
In Fig. 3 the values of the ratio δ/Γ for the five systems under consideration are plotted as a function of n d . In qualitative agreement with equation ( 3), the ratio between the Kondo shift δ and the width of the Kondo resonance Γ increases with decreasing occupation number n d , reaching its maximum for the a-Cu(110)(2x1)-O system, even though an accurate interpolation is not possible, also due to uncertainties in the δ values and to the approximate values of n d derived from Lowdin populations.

The Kondo temperature can be discussed along lines similar to those followed in Refs.

[ 13,26,27], where it has been argued that the coordination number of the adsorption site may be taken as a simple scaling measure of the Kondo temperature, and experimental data on a number of systems have been presented in support of this notion. However, it has been also pointed out that this simple scaling behaviour may not be sufficient to describe all the observations on adsorbate cluster and ligand systems [11,28] and that a more refined picture may be necessary involving detailed electronic structure calculations to reveal the local density of states at the Fermi level. We use the equation (derived in the weak coupling limit of the s-d model [7] via the Schrieffer-Wolff transformation [29,2])

k B T K = [(UΔ)/2] 1/2 exp[ -π /(2 UΔ) • | ε d | • | ε d + U | ] ( 5
)
where Δ is the hybridization matrix element that couples the localized state with the continuum of band states as above, U is the on-site Coulomb repulsion and ε d is the position of the impurity d-level with respect to the Fermi energy. By then assuming that U and ε d are roughly constant, linearizing T K as a function of Δ (which is legitimate as Δ varies in a narrow range), and assuming that Δ, being related to the broadening of the impurity state, is roughly proportional to the coordination number CN [13], we can plot T K as a function of the CN, as shown in Fig. 4(a). An inspection of this figure immediately reveals that the oxidized systems, e-Co/Cu(110)-O in particular, do not scale properly with CN, in contrast with the linear relationship found for the other systems. This is not unexpected, as the above analysis of the atomic populations already pointed out a clear difference between non-oxidized and oxidized systems, in particular for e-Co/Cu(110)-O. Indeed, it is precisely for these systems that one finds values of the spin polarization appreciably smaller than the value of ~2 valid for the bare Cu surfaces. One can try to correct for this effect by multiplying CN by a weighting factor that takes spin polarization into account. We derive this factor from the analysis of Ref. [23], where the Kondo response of Fe atoms embedded in fcc Au was studied, and it was found from renormalization group calculations that the Kondo temperature scaled linearly with respect to the spin state of the Fe atom according to the formula:

T K ≈ T K 0 • [ 0.4 + 0.3 • (n dα -n dβ ) ] (6)
being 0.7 for S=1/2, 1.0 for S=1 and 1.3 for S=3/2 (see Fig. 3 of Ref. [23] in proper units). By correcting the coordination number for the spin polarization weighting factor given by equ.

(6), the plot of Fig. 4(b) results, in which a better agreement with a linear behavior can be appreciated, despite the euristic character of the present analysis.

Co 3d-LDOS plots are shown for comparison purposes in Fig. 1 of the Supplementary Material for the systems here considered. In passing, we observe that in analogy with Ref. [10] a correlation between the density of Co 3d states at the Fermi energy and the shape of the Kondo resonance is found: a low Co d-LDOS at the Fermi energy is associated with a dip-like shape of the Kondo resonance, whereas a high Co d-LDOS at the Fermi energy is associated with a peak-like shape.

The previous analysis shows that an oxidic environment entails appreciable differences with respect to the more thoroughly investigated metal supports, specifically in terms of both the charge and spin state of the impurity atom. Even more striking results are obtained for the by an oxygen atom, and, even though the oxidation state of Co is only a little higher than that of Co atoms on bare Cu surfaces, the system in some respects resembles a CoO oxide, which is an antiferromagnet in the bulk [18]. This is consistent with the Goodenough-Kanamori rules [32,33] which also predict antiferromagnetic coupling in the given geometry. The energy scale associated with this coupling (0.18 eV from DF calculations) is much larger than that typical of the Kondo interaction, the total spin of the system is zero and the Kondo resonance is predicted to disappear, in agreement with experiment. Despite the presence of antiferromagnetic rather than ferromagnetic coupling, the latter phenomenon is thus in some sense analogous to the disappearance of the Kondo resonance for Co dimers on a Au(111) surface first observed by Chen et al. [30].

In conclusion, the Kondo response of Co atoms and dimers adsorbed on or embedded into a parameters are given in Table 1. The corrected CN is modulated by the spin polarization (SP) for each system, see equ. (6). Experimental error bars are also shown.

Figure captions

Fig. 1

 1 Fig. 1 contains atom-resolved constant-current topographic STM images of Co atoms on

Fig. 2

 2 displays the Fano fits of the STS data for a Co atom adsorbed on the Cu(110)2x1-O reconstruction (panel a), and for a Co atom embedded into the Cu-O rows (panel b).The data in Fig.2have been background corrected before fitting, that is the corresponding STS spectra of the bare substrates, recorded in the same experimental run with strictly the same tip conditions, have been subtracted directly without normalisation from the raw spectra (such as shown in Fig.1). The average Fano fit parameters obtained from the analysis of a large data set are given in

Cu( 110 )

 110 (2x1)-O stripe phase has been investigated via STM/STS and DF simulations, finding that Co dimers incorporated into adjacent rows of the surface reconstruction show Kondo resonances decoupled from each other, whereas they are anti-ferromagnetically coupled (and do not exhibit Kondo effect) when they are aligned on the same row. This shows that it is possible to couple/uncouple single carriers of the Kondo effect via a proper choice of the adsorption and host geometry, an important finding for the understanding and design of strongly correlated electronic devices. Additionally, it is shown that an oxidic environment (thus closer to real-world systems) presents features remarkably different from pure metal substrates, not only because the oxygen atoms can mediate the interactions among impurity atoms, but also because they influence their charge and spin state, and thus ultimately their Kondo response.

Figure 1 :

 1 Figure 1: Topographic STM images (leftmost side), schematic geometrical models (middle) and differential conductance STS spectra (rightmost side, dI/dV versus V): (a) Single Co adatom adsorbed on Cu(110)2x1-O (a-Co/Cu(110)-O), Cu-O rows along the [001] direction are indicated by lines in the STM images (STM image parameters: 36 x 38 Å 2 ; sample bias: 1V; tunneling current: 0.5 nA); (b) Co monomer embedded in the Cu(110)2x1-O layer (e-Co/Cu(110)-O), (36 x 32 Å 2 ; -10 mV; 1 nA); (c) Co dimer embedded in Cu(110)2x1-O along [110] (e-Co 2 [110]/Cu(110)-O), (36 x 33 Å 2 ; -10 mV; 1 nA); (d) Co dimer embedded in Cu(110)2x1-O along [001] (e-Co 2 [100]/Cu(110)-O), (36 x 36 Å 2 ; -10 mV; 1 nA).

Figure 2 :

 2 Figure 2: Fano lineshape fits to the STS spectra of Co monomers: (a) adsorbed on Cu(110)2x1-O; (b) embedded in Cu(110)2x1-O. The corresponding fit

Figure 3 :

 3 Figure 3: Plot of the position of the Kondo resonance with respect to the Fermi energy divided by its width (δ / Γ) (experimental values from the present work or Ref. [13]) as a function of the occupation number of the 3d states, n d , for the five Co adsorption configurations considered in the present work. n d is calculated according to equ. (4) (see text). Experimental error bars are also shown.

Figure 4 :

 4 Figure 4: Kondo temperature T K (experimental values in Kelvin from the present work or Ref. [13]) vs. standard (a) or corrected (b) coordination number (CN). CN is derived from the present DF calculations for Co atoms on different Cu surfaces.

Table 1 :

 1 Number of spin majority (n dα ) and spin minority (n dβ ) electrons on Co d-orbitals (the corresponding values for s-orbitals: n sα and n sβ , are given in parentheses); spin polarization (SP= n dα -n dβ ) and total population (n d = n dα + n dβ ) in the d-space; distances (r ij , in Å) of Cu close-neighbors from Co atoms; coordination number (CN) calculated according to equ. (2); average Kondo temperature T K (in Kelvin) and average Fano fit parameters δ, Γ, and q. δ and Γ are in meV.

Figure 1 Figure 2 Figure 3 Figure 4 Figure 1 :

 12341 Figure 1

Figure 2 :

 2 Figure 2:

Table 1

 1 first-principles calculations to predictions of the shape and position of the Kondo resonance, see e.g. Refs.[20][21][22][23][24][25][26][27], although many rely in one way or another on qualitative fits or on the use of empirical parameters, whose values are fixed on the basis of physical intuition or to match experimental data or first-principles quantities. A really quantitative and general understanding of the Kondo effect has not yet been achieved. In this context, electronic structure DF calculations can provide a framework in which simplified models operate, and information useful to discuss experimental data.

	, which also contains data of Co adsorbed on

which STS experiments have been conducted

[10,19]

and which will be used for comparison purposes. DF theory being essentially a mean-field approach, it cannot capture the complex many-body physics of the Kondo phenomenon. Several approaches have been proposed in the literature to link

  In between, we find a group of three systems, namely Cu(100) and the oxidized Cu(110) surfaces, with similar CN values, CN = 0.35÷0.38. It can be noted that in the oxidized systems Co has 5-6 neighbors, but the increase in the Co-Cu distances decreases the CN value from the maximum realized for the bare Cu(110) surface.The difference between oxidized and non-oxidized systems results even clearer from an analysis of the atomic populations: n dα , n sα , n dβ and n sβ . The atomic configuration of Co, as it emerges from these values, is 3d 8 4s 1 , realized via atomic populations which -very roughlyread: n dα ≈ 5, n dβ ≈ 3, n sα ≈ n sβ ≈ 0.5. The net spin polarization is therefore around 2,

. From Table 1, one can see that CN varies from a minimum of 0.28 for Co/Cu(111) to a maximum of ≈ 0.44 for Co/Cu(110).

corresponding to two impurity orbitals or a two-channel Kondo system. A more detailed analysis however highlights marked differences. First, (n sα + n sβ ) ≈ 1 and n dα ≈ 5 for for the bare surfaces, whereas (n sα + n sβ ) ≈ 0.66-0.73 and n dα ≈ 4.74-4.78 for the oxidized systems.

  This research has been supported within the Advanced Grants of European Research Council

	System	n dα (n sα )	n dβ (n sβ )	SP n d r ij (Å) CN T K (K)	δ (meV)	Γ (meV)	q
	Co/Cu(110)	4.95 (0.58)	2.87 (0.44)	2.08 0.82	2.24 + 4 x 2.45	0.44	125 ± 30	6.9 ± 2.1	10.8 ± 2.6 peak
	a-Co/Cu(110)-O	4.78 (0.35)	2.96 (0.38)	1.82 0.74	2 x 2.53 +4 x2.89	0.38	93 ± 8	8.9 ± 3.0	8.0 ± 0.7	dip
	Co/Cu(100)	4.94 (0.54)	2.97 (0.46)	1.97 0.91 4 x 2.44 0.35	88 ± 4	-1.3 ± 0.4	7.6 ± 0.3	dip
	Co/Cu(111)	4.96 (0.58)	2.94 (0.44)	2.02 0.90 3 x 2.38 0.28	54 ± 2	1.8 ± 0.6	4.9 ± 0.2	dip
	e-Co/Cu(110)-O	4.74 (0.32)	3.01 (0.34)	1.75 0.75 5 x 2.61 0.36	57 ± 4	3.3 ± 1.3	4.9 ± 0.3	dip

(project "SEPON" Grant No. ERC-2008-AdG-227457) and by the Rektor of the Karl-Franzens University Graz.

dimer configurations -Fig. 1(c-d). First of all, we immediately find a qualitative difference in the magnetic properties of the system between the case in which the two Co atoms are positioned in the same chain, Fig. 1(d), vs. in parallel chains, Fig. 1(c), i.e., between dimers positioned along the [001] or [110] direction, respectively. Total energy calculations show that in the case that the dimers are along [110] the energy difference between spin configurations in which the Co atoms are ferromagnetically or antiferromagnetically coupled is below numerical accuracy (less than 0.01 eV), and the system resembles a pair of essentially decoupled embedded Co atoms. It is therefore to be expected that the Kondo response of the system will be a superposition of two single-atom contributions, in agreement with experimental observations. This occurs despite the fact that the spin waves induced in the substrate by the two Co impurities interfere with each other, as visible in Fig. 2 of the Supplementary Material. On the opposite, in the case that the dimers are along [001] one finds that this energy difference amounts to 0.18 eV, with the Co atoms antiferromagnetically coupled in the ground state. Also the LDOS of a dimer along the [110] direction (see Fig. 1 of the Supplementary Material) is very similar to the single-atom case, whereas it presents appreciable differences for a dimer along the [001] direction. The complete decoupling of close-neighbors in the [110]-oriented dimer is particularly surprising. It is known in fact that the perturbation produced by the impurity decays slowly with distance, in such a way that the cloud of screening electrons extend for tens of nm inside the host [9], and collective phenomena due to long-range interactions among single scatterers have been experimentally demonstrated [12]. Our finding proves that the Kondo phenomenon is extremely sensitive to the geometry of the interaction, an important point when the design of independent scatterers is sought for. As for the [001]-oriented dimer, a Co-Co antiferromagnetic coupling may sound unexpected, as Co is a ferromagnet in the bulk and ferromagnetic coupling has been observed for close pairs of Co atoms [30], whereas antiferromagnetic coupling is only expected for much larger distances [31,28]. However, in the present case the Co-Co interaction is mediated Supporting Information available, describing details of the theoretical and experimental methods.

Supplementary Material for

Kondo vs magnetic coupling of cobalt dimers on a Cu-O stripe phase

1 Andreas Gumbsch, 2 Giovanni Barcaro, 1 Mike G. Ramsey, 1 Svetlozar Surnev, 2 Alessandro Fortunelli * , 1 Falko P. Netzer * 1 Institute of Physics, Surface and Interface Physics, Karl-Franzens University Graz, A-8010 Graz, Austria 2 Molecular Modeling Laboratory, Istituto per i Processi Chimico-Fisici (IPCF), CNR, I-56124 Pisa,

Italy

Theoretical methods

Density-functional (DF) calculations are described in the text. Here we provide more details on the energy cutoffs, the unit cells and the k-sampling of the Brillouin zone, and we discuss LDOS plots.

A value of 40 Ryd as the energy cutoff for the selection of the plane wave basis set for the description of the wave-function and of 240 Ryd for the description of the electronic density were employed. The metal support was described using seven layers and considering a symmetric adsorption of cobalt atom or dimers on both sides of the metallic slab (to cancel the total dipole moment of the system). The atoms of the innermost three layers were kept fixed in the positions of bulk fcc crystal (a lattice constant corresponding to a first-neighbour distance of 2.58 Å was chosen, corresponding to the equilibrium value of bulk Cu predicted by the present DF approach), whereas the positions of the outer two layers on each side, as well as the coordinates of the Co atoms, were fully optimized until the forces were smaller than 0.01 eV/Å per atom. All the calculations were performed spin-unrestricted, and a Gaussian smearing of the energy levels of about 0.03 eV was used. The dimensions of the unit cell in the surface (xy) plane were chosen as to maintain a minimum distance between Co atoms in replicated vicinal cells of at least 7 Å. In more detail, in the case of a Co atom adsorbed on the clean Cu(100) and Cu(111) surface, a square or hexagonal 3x3 unit cells were employed (edge dimension equal to 7.74 Å). In the case of adsorption on the clean Cu(110) surface, a rectangular 4x2 cell was employed (measuring 10.36 Å x 7.32 Å, where in the notation L 1 x L 2 the first number L 1 refers to the cell length along the [110] direction and the second one L 2 to that along the [001] direction, respectively). In all the three cases, the first Brillouin zone [1], we thus find a correlation between the density of Co 3d states at the Fermi energy and the shape of the Kondo resonance: a low Co d-LDOS at the Fermi energy is associated with a dip-like shape of the Kondo resonance, whereas a high Co d-LDOS at the Fermi energy is associated with a peak-like shape (peak-like line shapes have also been observed in other systems [2,3]). We observe that, in this scenario, the e-Co/Cu(110)-O represents a borderline case.

Whereas in all other cases, in fact, the value of Co d-LDOS at the Fermi energy is robust with respect to the use of a DF or DF+U approach, for the e-Co/Cu(110)-O system the introduction of the U term makes a qualitative difference. The reason is that the crystal field for e-Co/Cu(110)-O is normal, but with a not too large splitting, so that at the pure DF level the LDOS pertaining to the lower and upper bands in practice overlap precisely at the Fermi energy. When a U term is introduced, instead, the Co d-orbitals are slightly depopulated and hence stabilized, the width of the corresponding d-LDOS peaks shrinks, and a gap at the Fermi energy in the d-LDOS clearly appears [1]. As the embedded Co experiences an oxidic rather than a metallic situation (see e.g. the increased positive charge and the antiferromagnetic coupling in an embedded Co dimer discussed in the main text), it is appropriate to augment DF theory with a U term as it is necessary for bulk CoO.

It should also be stressed that even a small value of U (1 eV or even 0.6 eV) is sufficient to open a gap at the Fermi energy in the d-LDOS. We finally note in addition that the e-Co/Cu(110)-O system is somewhat different from the other ones also in terms of the interaction geometry: e-Co/Cu(110)-O is in fact the only case of a Co embedded into the surface, i.e., not being an adatom, so that its height with respect to neighboring atoms, defined as the difference in the z-coordinates, is lower by 0.4 Å, whereas it is higher or at least equal in all the other cases. This might contribute to decrease the direct scattering channel into impurity d-orbitals and thus to promote a dip-like Fano line shape.

Finally, for illustrative purposes the spin density of a Co dimer embedded in adjacent rows of the Cu-O stripe phase is shown in Fig. 2. The spin polarization induced on neighboring Cu atoms, and in particular the interference occurring on the Cu atoms equidistant from the Co scatterers the can be clearly appreciated in this figure.

Experimental methods

In


Keith Davidson

Fotoon Sayegh

David Js Montagnes

Oxyrrhis marina based models as a tool to interpret protozoan population dynamics 4 5

Keywords: Oxyrrhis marina, modelling, population dynamics review, dinoflagellate, experimental design

HAL is

Coprophagy is a well-recognised process in planktonic systems that may have considerable impact on 208 food web dynamics. For instance, low chlorophyll:phaeopigment ratios have been proposed to indicate 209 high levels of mesozooplankton coprophagy, but given the importance of microzooplankton in food 210 webs (e.g. Azam et al., 1983;Davidson, 1996), it may be that they too are important in this process.

211

To this end, Strom (1993) developed a mathematical model to test the hypothesis that re-ingestion of 212 faecal material by protozoa could account for the observed variation in the conversion of chlorophyll to 213 phaeopigment. In this model, which was applied to her experiments on Strombidium and 214 Gymnodinium and to the O. marina-based data of Klein et al. (1986), protozoa preyed on made a function of phytoplankton prey (Table 1, Eq. 1), with a selection factor for phytoplankton cells 217 or faecal particles that allowed the determination of an ingestion rate for faecal particles (Table 1, Eq. predator (N eq ) was then determined from

1 - ⋅ = p pred eq V V N
, where V pred and V p are mean predator and 235 prey cell volumes, respectively. This allow the calculation of the encounter distance

236 ( ) 1 - × × α × = p S N δ p eq
, where

(

)

1 2 2 - + ⋅ = pred p pred C C C α . 237
If predation of a particular prey type continues when δ is greater than that of alternative prey, 238 then the predator is deemed to select the former item. This concept was applied to a set of laboratory The role of prey quality was further explored by Mitra et al. (2003) who hypothesized that it could 264 influence predation through modulation of either 1) the rate of ingestion (Table 1, Eq. 11), with "a" in 265 Eq. 11 being a variable rather than a constant or 2) the efficiency of assimilation of this ingested 266 material. The study examined the relative importance of these processes through a model that related 267 both maximum rate of predation and assimilation efficiency to prey quality (defined as its C:N ratio) in 268 a range of different functional forms. By making the maximum predation rate a function of prey 269 quality the model simulated experimentally observed phenomena exhibited by O. marina of "surge 270 feeding" (Öpik and Flynn, 1989) and prey rejection (Flynn et al., 1996). In particular, this study The quantitative importance of nutrient regeneration and cycling 318 While mathematical models are often used in predictive mode, an equally important application is the 319 analysis of processes that cannot be easily understood by simple observation. One of the most 320 important of these for microbial community population dynamics is the regeneration of inorganic 321 nutrients by protozoa, and their subsequent use by phytoplankton (Goldman et al., 1987). This process 322 maintains the stoichiometric balance of nutrients within the predator and fuels further growth of the 323 prey (Caron, 1991). et al., 1995a,b) and that functional form used to simulate this selectivity will influence 518 model results (Mitra et al., 2003;Mitra, 2006;Mitra and Flynn, 2006). In this light, there is a et al., 2006). This is an obvious area for

⋅ - ⋅ ⋅       = , 334 where
+ × = max p k p r r r + × = max ' max v p k p v v v + + × = aT p p k p p r r r × - + - × = ) ' ( ) ' ( max c I b T a p k p I I ) ( max - × + × = c v b T a v p k p v v ) ( ' max - ×       + + × = t t t N a N r N × + × = + 1 1       × + × = SF p D D I Re

  between prey types, with selective grazing, on the basis of prey quantity or 199 quality, now being recognised as a key issue in the functioning of microbial food webs (Montagnes et 200 al., 2008a). A range of factors such as morphology, chemical defence, and nutritional quality may 201 govern the selectivity of prey items by protozoa (Montagnes et al., 2008a). The O. marina-based 202 studies that have experimentally addressed these factors are not germane to our work, but the interested 203 reader is directed to (Roberts et al., this issue); here we specifically review how models have addressed 204

218 9 )

 9 and hence, by subtraction from the functional response, an ingestion rate for phytoplankton cells.219Although the model did not lend support to the notion that re-ingestion governed phaeopigment 220 distributions, it demonstrated that the re-ingestion of faecal material by protozoa was a plausible 221 trophic pathway that may have significant implications for energy flow within the microbial loop.

  s (1993) model-based demonstration of discrimination between live and dead food by 225 protozoa, the selection of alternative live prey by O. marina was addressed by Flynn et al. (1996) who 226 presented a theoretical relationship for the distance travelled by a raptorial predator to encounter a equal its own volume (δ,

324

  To this end,Davidson et al. (1995b) studied nutrient regeneration using a predator-prey model 325 that included O. marina and I. galbana, to assess the response of different nitrogen (N) regeneration 326 equations. The model incorporated the phytoplankton growth model of Davidson et al. (1993) that 327 simulates both carbon (C) and N dynamics of I. galbana during unbalanced growth, allowing 328 simulation of experiments in which prey exhibit active growth. Again, as an example of how such 329 models are developed, we provide the relevant equations. 330 The equation for the rate of change of I. galbana N is give by of change of O. marina N was simulated by

519

  somewhat surprising relative lack of more basic combined experimental-modelling studies based 520 around O. marina that specifically seek to model the response to particular environmental drivers (an 521 Davidson et al. 10. Oxyrrhis marina based models 25 exception being the temperature based study of Kimmance

  , c are constants; I = ingestion rate; I max = maximum ingestion rate; p = prey abundance; k I = half 750 saturation constant of the ingestion curve; r = specific growth rate; r max = maximum specific growth 751 rate; p′ = threshold prey abundance (at which r = 0); k r = a constant (k r -p′ = half saturation constant of 752 the growth curve); T = temperature; v = volume; v max = maximum volume; k v = half saturation constant 753 of the volume curve; v′ = the volume at zero food abundance; N = O. marina concentration; Re = re-754 ingestion of faecal particles; D = concentration of faecal particles; SF = selection factor; N eq = number 755 of prey cells equivalent to one predator in terms of biovolume; α = variable related to the swimming 756 speed of prey and predator; S p = cross section of encounter party of predator and a given prey species; 757 e i = preference for different prey types; C i = capture rate of specific prey species; f i = capture rate 758 parameter; E = nitrogen excretion rate; n = O. marina nitrogen content; θ i = C:N ratio of prey or 759 predator; R = respiration rate, S = gross growth efficiency.

Fig 1 .

 1 Fig 1. An indication of how further population dynamics might be obtained for comparison with model output: abundance of Oxyrrhis marina and the prey flagellate Dunaliella primolecta, grown at 16 °C in 32 PSU seawater enriched with f/2 media (Sigma). Panels a-c are 18 d time-course incubations of predator (○) and prey (•) in triplicate flasks. Panels d and e are phase plots: d is a plot of the three 18-d time course incubations (a-c); e is a plot of a series of short (5-11 d) incubations, indicating a semblance of population cycling. 142x159mm (600 x 600 DPI)

Table 1 ,

 1 Eq. 10). To illustrate how such simulations are 228 developed, below we use this model as an example.The model was developed by first determining that the cross sectional area (S p ) of the encounter and prey are speeds are C pred and C p , respectively, 233 and N p is the number of prey. The number of prey cells equivalent, in terms of biovolume, to one

	230
	231	path for the predator and a given prey (p) species is	S	p	= π	( r	pred	+	r	p	) 2	, where r pred and r p are the radius
	232	of the predator and prey cells, respectively. The rate of prey encounters per predator (E) was then
	given by , where predator 234 2 2 pred p p p C C N S E + ⋅ ⋅ =

229

241 communities may be complex and dependent on both prey size and prey quality, both of which may 242 change with time, rather than simple random encounter. Clearly, in this case, O. marina acted as a

	249	marina study of Flynn and Davidson (1993b) and the multi prey-O. marina experiments of Flynn et al.
	250	(1996).
	251	Flynn and Davidson (1993b) suggested that O. marina initially ingested but then rejected the
	252	flagellate prey, but Davidson et al. (1995a) found that no parameterization of a standard type II
	253	functional response (Table 1, Eq. 1) generated adequate simulations of their data. Instead, qualitatively
	254	better simulations were achieved if the maximum ingestion rate was made to decrease with continued
		prey ingestion (
		F o
		r
		P
	239	e e r
	240 243 244	experiments in which O. marina ingested three differently sized prey species. The analysis demonstrated the occurrence of selection of live cells and suggested that selective grazing in microbial model organism to test general issues associated with selection. R e v i e w
	245
	246	Prey inhibition of grazing
	247	The role of selective grazing in governing the temporal changes of both prey and predator was further
	248	studied by Davidson et al. (1995a) through simulation of two independent data sets: the I. galbana-O.
		13

Table 1 ,

 1 Eq. 11), a factor that was attributed to the build up of an inhibitor within O.255 marina through the ingestion of the prey. The same model structure was also able to simulate the O.Kimmance et al., 2006), it would now seem appropriate to test these model predictions with empirical

	256
	257	marina-multiple prey data of Flynn et al. (1996). Thus, this O. marina-based model quantitatively
	258	demonstrated that the quality of a prey item as well as its abundance or size may govern its suitability
		as a prey item for protozoa. Given the tractability of using O. marina for grazing experiments (e.g.
	260
	261	data.
	262
	263	Prey quality governing predator functional response

259

  Ox, Is, Ox N , and Is N represent the number and N-content of O. marina and I. galbana,The protozoan compartment of the model was parameterized from laboratory experiments on O. Nutrient regeneration was related to the relative C:N ratios of prey and predator using three 388 alternative functions all capable of representing, to some degree, the N regeneration efficiency of O.are, therefore, required to better validate these and future models. As guidance, we present an example 433 of such a data set for model comparison (Fig.1). This represents a subset of numerous time course 434 experiments beginning at many initial predator and prey concentrations. Such an approach, while 435 generating time-course data, may minimize the bottle effects (e.g. accumulation of toxins and fouling 436 on surfaces) that can bias long-term incubations. Population models, independently derived from O.Individual based models (IBMs) provide an alternative modelling strategy to those that seek to 500 represent the ecosystem as a whole, and O. marina is an ideal candidate for these. IBMs calculate 501 biological variables while following individual (or meta-) particles in space. These models may then 502 be of particular use for the study of advective populations and/or species that form only a small fraction 503 of the biomass of a trophic level but are important for other reasons. A number of important biotoxin 504 producing phytoplankton species such as the advective Dinophysis spp.(Hart et al., 2007) or the low-505 biomass high-toxicity dinoflagellate Alexandrium tamarense(Touzet et al., 2010) fit these criteria. As

	499	Individual based models
	389 480	here again using O. marina may be instructive. The use of PFT models is somewhat controversial,
	390 481	marina: a constant nitrogen regeneration efficiency (NRE); a stepwise switching function between low with some authors (e.g. Anderson, 2005) suggesting that their application may be premature. However,
	391 437 482	and high NRE, termed threshold elemental ratio (TER), (Table 1, Eq. 16); and a dynamic nutrient notwithstanding this debate, there is consensus that better model parameterization is required. This is
	438 483	regeneration equation (Caron and Goldman, 1988), (Table 1, Eq. 15). marina functional and numerical response and prey growth data may be tested using the resultant particularly pertinent at the microzooplankton level. For example, the dynamic green ocean model (Le
		F o F o F o F o F o F o Quéré et al., 2005) contains five separate autoptrophic functional types, but only a single composite, phase-plot data. F o
		r r r r r r r
		P P P P P P P
	511	e e r e e r e e r e e r e e r e e r e e r
	401 402 447 448 449 512	Summary of synecological work. Improving model structure, using O. marina SUMMARY	R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R w e i v e
	450	Doney (1999) highlighted the need for succinct but realistic mathematical models capable of simulating
	405 451	the cycling of multiple nutrients within microbial food webs. These are necessary to simulate the
	406 452	modelling studied have drawn on this resource, to date. Clearly, considerable scope exists to develop transfer of production to higher trophic levels and the export flux of C to the ocean floor. Within this
	453	and improve existing models, and to produce alternative formulation for comparison, as illustrated context parameterisation of zooplankton or microzooplankton response in a range of different model
	339

335

respectively; I is the ingestion rate of O. marina; and E Is and E ox are the N-regeneration rates of I. 336 galbana and O. marina, respectively. Within the equations, N-regeneration by O. marina was 337 simulated either as a constant or on the basis of an "optimal" O. marina C:N ratio, using a range of N 338 regeneration models. 387 marina. 392 Simulations demonstrated quantitative differences between the output generated by the different 393 models, particularly between the switching and dynamic models and the constant NRE model in terms 394 of the density of phytoplankton blooms. Differences in C:N ratio of model components were also 395 evident with only the dynamic model predicting a stoichiometrically balanced zooplankton C:N ratio 396 close to 6.6, the Redfield value, similar to the values that experimental estimates suggests that 397 protozoa, including O

. marina, maintain (Goldman and Dennett, 1992; Nakano, 1994; Davidson et al., 398 1995a)

. Such O. marina-derived results have important implications for the formulation of the multiple 399 functional type models that are now being formulated to better understand the global C cycle (e.g. le 400

Quéré et al., 2005)

.

403

Considering the importance of protozoa within marine ecosystems and the relative wealth of data, 404 response relationships and models based on O. marina (Table

1

), it is unclear why so few synecological 407 above (The influence of N regeneration on a food web). Moreover, the now recognised diversity of O.

439

One of the reasons that the O. marina-I. galbana data set of Flynn and Davidson (1993b) has 440 been so often used as a comparison with simulations is the availability of a robust model that predicts 441 prey growth in non-steady state conditions

(Davidson et al., 1993; Davidson and Cunningham, 1996)

.

442

Hence, the experimental study and modelling of O. marina in particular, and protozoa in general must 443 be conducted in parallel with that of their prey, to allow both trophic levels to be simulated to the same 444 level of complexity. This requires experimental studies to measure a sufficient array of parameters 445 including numbers, biomass, ingestion, grazing, respiration, and nutrient cycling rates to allow 446 appropriate model parameterization and testing, to minimise the need to "fit" free model parameters. 484 protozooplankton compartment to represent heterotrophic flagellates and ciliates. While this is 485 understandable in terms of model tractability, the parameterisation of the equations used to represent 486 this "functional group" requires deeper consideration in the light of the wealth of behaviour that 487 different species and genera are capable of (Montagnes et al., 2008) and observations of temporal 488 succession of different micro-zooplankton groups (Davidson et al., 2007). 489 Analysis of the functional response of O. marina in comparison with other heterotrophic marine 490 micro-or dinoflagellates, expanding on initial studies such as those of Jeong et al. (2008), and with 491 multiple prey items (John and Davidson, 2001), would add confidence to single functional group 492 parameterisation, or provide definitive evidence that multiple micro-zooplankton functional groups are 493 required in models. The application of developing techniques such as lectin labelling Wootton et al. 494 (2007), flow cytometric separation of prey and predators (Montagnes

et al., 2008)

, the analysis of 495 stable isotope signatures

(Flynn and Davidson, 1993a)

, or stable isotope probing

(Radajewski et al., 506

O. marina ingests biotoxin producing dinoflagellates

(Jeong et al., 2001; Jeong et al. 2003)

it may be a 507 suitable candidate organism for developing grazing terms within such models. Furthermore, in rare 508 cases it forms large blooms of up to 10 5 cells ml -1

(Begun et al., 2004)

and thus own its own may be 509 important in short-term rapid fluxes of nutrients in some ecosystems. Thus, O. marina-IBMs may too 510 be justified in the future.

513

What has O. marina modelling delivered to the scientific community? Of the autoecological models 514 reviewed, the majority deal with some aspect of prey selectivity; the combined body of work in this 515 area is particularly useful in demonstrating that prey selectivity by micro-heteroptrophs is, indeed, 516 capable of influencing the trophic transfer of phytoplankton biomass (e.g.

Strom, 1993; Flynn et al., 517 1996; Davidson

Table 1 .

 1 522 further fruitful study. The relative lack of synecological studies employing O. marina based model 523 parameterisation perhaps reflects this need, with the more sophisticated models of prey selectivity In conclusion, as O. marina is not often abundant in open water samples, it might not be the 526 organism of first choice to parameterize the protozoan component of such models. However, as 527 indicated above, available evidence suggests its use is appropriate, and the relative wealth of O. marina 528 studies makes it a pragmatic choice. Furthermore, there are ecosystems where O. marina may be 529 abundant, and in these regions using O. marina-derived parameters would be entirely appropriate. An indication of how further population dynamics might be obtained for comparison with model 741 output: abundance of Oxyrrhis marina and the prey flagellate Dunaliella primolecta, grown at 16 °C in Studies that determine or apply O. marina based equations and their functional forms.

	524
		requiring a fundamental underpinning prior to their wider application.
	530
	531	Therefore, we support its continued use as a model organism to parameterize simple and more complex
	532	population models.
	533

525

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phytoplankton and faecal particles (with which phaeogiment was associated). Protozoan growth was

Plankton functional type (PFT) models are increasingly being employed in ocean biogeochemistry, and

ACKNOWLEDGEMENTS

534

The Authors would like to thank Edd Codling and Emily Roberts for their constructive comments on