Nicholas Asher

Pascal Denis

Dynamic Typing for Lexical Semantics A Case Study: the Genitive Construction

Introduction

In recent years, there has been a considerable amount of effort in computational work on lexical semantics that seeks to automate the acquisition of disambiguated senses for words. Similar work is less popular in theoretical lexical semantics but still widely practiced throughout the discipline. We believe that this approach is fundamentally mistaken. The only way to build large scale lexicons that are linguistically and practically useful is to use simultaneously the techniques of underspecification and type theory, informed by ontological analysis. These techniques include not only a rich type lattice of simple types but also complex types whose operations of introduction and exploitation (or elimination) must be understood in the form of logical rules. Types put constraints on underspecified lexical meanings or rather constitute them. In fact, the semantic types of the lexicon reflect in a simplified manner our common sense ontology as it surfaces in our language. We think not only that ontological analysis is crucial for lexical semantics but that a careful study of the meanings of words also reveals important principles of common sense ontology.

Through the process of semantic composition, words with highly underspecified meanings may acquire more determinate meanings; on our view semantic composition typically adds to the content supplied by the basic lexical elements. We countenance not only deductive, monotonic rules for types but also non-monotonic rules whose conclusions can be overridden by discourse context; these rules and their associated types are intended to capture the preferred meanings of words in a given context. Such a project, if it succeeds, provides a novel and general view of the lexicon. It will also allow us to make predictions about word meanings in particular contexts. We also hope that it will be possible to acquire information about types and rules automatically by learning from annotated corpora. We summarize our view in the following principles:

• The lexical entry for a word is underspecified but simple-one particular lambda expression that may even include rhetorical information but which is typically underspecified.

• We offset these simple lexical entries with a rich account of types including simple types, functional types and complex types that resemble restricted conjunctions. All variables in the λ-term of a lexical entry are typed.

• The process of composition and predication are type-driven processes, involving not only type checking but also type merging and various coercion operations which are construed as operations of type introduction or exploitation often of complex types.

• Following the Curry-Howard isomorphism between proofs and types but extending it to complex types, type checking and type merging are understood as rules of introduction and exploitation in a deductive logic. We will also have nonmonotonic rules for type exploitation to capture preferred meanings in a given context. These rules can affect the logical form of the λ-terms.

• The dividing line between lexical and non lexical or encyclopedic knowledge has to do more with how the knowledge is expressed (as types lexically; in a much more expressive formalism encyclopedically). So indeed in terms of content Quine and Fodor ([1], [START_REF] Fodor | The emptiness of the lexicon, critical reflections on J. Pustejovsky's The Generative Lexicon[END_REF]) may be right: there is no distinction in terms of content between lexical and encyclopedic information, though the information may have a privileged epistemic status in virtue of the fact that it is grammaticalized.

This paper illustrates these principles on a case study, namely the genitive construction. We will also relate our work to past efforts, in particular the Generative Lexicon of Pustejovsky. We agree with Pustejovsky that the lexicon should have a generative approach to meaning, but in particular we give this notion of generation a dynamic twist. As already detailed in [START_REF] Asher | The metaphysics of words in context[END_REF], we don't agree with the formalization of GL of [START_REF] Pustejovsky | The Generative Lexicon[END_REF], which frankly leaves a lot to be desired. GL is not clear on what lexical information is and on lexical processes, from our perspective. Finally and specifically, we think that the system of types employed in [START_REF] Pustejovsky | The Generative Lexicon[END_REF] is too restrictive; we will show how in particular the so-called qualia have to be replaced with a much more dynamic notion of typing.

The type composition logic (TCL)

Lexical entries for us contain semantic information of two kinds. The first is a λ-term with a model theoretic interpretation, while the second is a typing context. A typing context for a term t determines an assignment of types to all subterms of t. It is the types that are assigned to the various variables in the λ-term that will affect how it combines with other terms in the composition of a logical form. In earlier semantic theories, the composition process was largely a matter of applying a λ-term to its arguments, and the standard λ-calculus with functional types due to Church sufficed. Our composition logic is more complex. It contains a system of types and rules for introducing and exploiting these types. Our system of types extends the λ-calculus for functional types with complex types like the •-types ("dot-types") of Asher and Pustejovsky ([3]), and types that generalize the qualia structures that Pustejovsky and others have argued for. As one of us argued in [START_REF] Asher | The semantics and pragmatics of metaphor[END_REF], [START_REF] Asher | The metaphysics of words in context[END_REF], and as already outlined in [START_REF] Pustejovsky | The Generative Lexicon[END_REF], •-types are useful to model many copredications, which are constructions where two predications with incompatible typing constraints are made on a single argument as in: The idea is that an object •-type α • β has two aspects, the aspect of being α and the aspect of being β and that the predication of a property to an object may hold in virtue of that aspect. Preferential readings of similar typing conflicts have motivated Pustejovsky's introduction of qualia, for which Asher and Pustejovsky provide a complex type similar to the •type, the so-called qualia constructs (or QCs). Qualia are also an Aristotelian idea; roughly Pustejovsky's qualia for any object correspond to the four metaphysical αιτια or explanatory causes of an object's existence. While we are Aristotelians with respect to restricted predication, we believe that the lexicon is not restricted to encoding Aristotelian causes of objects. We will move away from qualia here, and so we will just introduce this type via its special operator and call it a ⊗-type ("product-type").

The •-type and ⊗-type resemble conjunctive types and our natural deduction rules for exploiting them and introducing them will resemble something like conjunction elimination and introduction. But our rules are quite a bit more complicated than the introduction and elimination rules for simple conjunctive types, as they add material to logical form, as well as revise the types of variables. Besides these rules, we'll assume the presence of a type hierarchy with a subtyping relation that defines a partial order on the set of types and a greatest lower bound operation on the set of types. has the usual properties-e.g., idempotence, commutativity, and α β iff α β = α. We will capture incompatibility between types in terms of their common join, ⊥. Our type language takes as fundamental the notion of a term together with a typing context that our rules can revise or extend. We note by c(t : α) that the type assignment t : α is in the typing context c, by c * (t : α) that the context c is revised in the assignment t : α (which boilds to simply adding this assignment, c + (t : α), if t is not defined in c).

We assume there are simple types and two sorts of complex types, •and ⊗-types as well as their associated functional types (e.g. α --• β) corresponding to λ-expressions:

(2) a. P T: e the general type of entities and t the type of truth values. Below σ, τ range over all simple types, the subtypes of e as well as e and t. Our Type Composition Logic (TCL) combines the usual λ-terms familiar from compositional semantics with a set of type assignments, of the form x : σ, x :σ • τ, or x : (σ ⊗ R 1 ,...R n (τ 1 . . . τ n )). Constraints on types will also be available; for instance, we may need to know that σ is a subtype of τ, something we express as σ τ or that two types are compatible, which we write as σ τ ⊥.

Let us add a little more here regarding the new ⊗-types. Like Asher and Pustejovsky's QCs, but unlike •-types, these new types are "asymmetric". That is, a complex type like x : (σ ⊗ R 1 ,...R n (τ 1 . . . τ n )) combines on its left an intrinsic type σ that defines the nature of the thing denoted by the variable and on its right a set of dependent types τ 1 . . . τ n corresponding to eventualities that are conventionally (but defeasibly) associated to that thing. Each of these dependent types τ i is associated with a particular relation R i that encodes the generic explanatory cause linking τ i to σ. The main point of departure from [START_REF] Pustejovsky | The Generative Lexicon[END_REF] and [START_REF] Asher | The metaphysics of words in context[END_REF] lies in that these relations R i will not be restricted to qualia roles, as was the case with QCs.

Following [START_REF] Asher | The metaphysics of words in context[END_REF], we also make sure that some minimal information about syntactic structure can be exploited in our rules; for instance, we will have a formula head (ψ), where ψ is a term telling us that ψ is derived from some projection of the head or the head itself of the syntactic structure whose meaning we are currently trying to build up. They advance the following principle:

(3) Head Typing Principle: Given a compositional environment X with constituents A and B, and type assignments A : α and B : β (in the type contexts for A and B, respectively) that clash, if A is the syntactic head in the environment, then the typing of A must be preserved in any composition rule for A and B to produce a type for X.

We now introduce the rules for the TCL. As usual a λ-expression denotes a functional type, i.e., a α --• β type. Such rules should be understood as reduction rules, thereby giving rise to equivalent term expressions. The first rule is Application and corresponds in terms of the type calculus to a rule of Modus Ponens for --•. Application is defined in terms of a context, c, which provides typing assignments to both the variable in the applicand and the argument.

(4) Application:

λxφ[t], c(x : α, t : α) φ[t/x], c
The contexts that accompanies the rule of Application and other operations may be updated or combined, as the result of a rule being applied. We will refer to this second rule as Merging Contexts (noted with the symbol " ∧ ").

(

) Merging Contexts: λxφ, c[t, c ] λxφ[t], (c ∧ c ) 5
For applications whose typing contexts are not locally satisfied, but are consistent with the types available in the type semilattice from the lexicon (cf. [START_REF] Pustejovsky | The Generative Lexicon[END_REF], [START_REF] Copestake | Semi-productive polysemy and sense extension[END_REF]), we have the rule of Type Accommodation. This third rule covers what results in type unification, in which a supertype can unify with a subtype yielding the subtype as the result. ( 6) Type Accommodation:

λxφ[t], c(x : α, t : β), α β ⊥ λxφ[t], c * (x, t : α β)
These are the basic rules for computing with types. We will also assume more specific rules for complex ⊗-types which are relevant for the analysis of the genitive construction. We will discuss these additional rules in section (4).

A quick comparison with "classic" GL

In [START_REF] Asher | The metaphysics of words in context[END_REF], it was shown how these rules can be used to model dynamic type shifts in copredication and coercion phenomena. These are phenomena already introduced in classic GL but our approach is quite different. To give a quick overview of the differences, we need to recall that in classical GL, there are four types of information:

1. Argument Structure: Specification of number and type of logical arguments.

2. Event Structure: Definition of the event type of a lexical item. Sorts include , -, and .

Qualia Structure:

A structural differentiation of the predicative force for a lexical item.

4. Lexical Inheritance Structure: Identification of how a lexical structure is related to other structures in the type lattices.

Our approach makes use of argument structure and event structure, but it understands these as types and the relations between types determine the lexical inheritance structure. GL originally only countenanced qualia though in [START_REF] Pustejovsky | The Generative Lexicon[END_REF] dot objects are also introduced.

The qualia structure of a word specifies four aspects of its meaning:

• : the relation between an object and its constituent parts;

• : that which distinguishes it within a larger domain;

• : its purpose and function;

• : factors involved in its origin or "bringing it about".

In way of illustration, below is the qualia structure for the word door:

(7)                       
 : aperture(y)  : physobj(x)  : walk through(P,w,y)  : make(T,z,x)

                      
A quick look at this qualia structure reveals some obvious technical problems of classical GL. Formulas rather than actual types are taken to be the values of the various qualia features. Classic GL's use of AVMs and other tools of unification indicates that the construction of a logical form should proceed by unification, but it's unclear how unification is to proceed on these structures. We can unify AVMs but how are we supposed to use the information in one of the qualia to interpret [START_REF] Vikner | A semantic analysis of the english genitive. interaction of lexical and formal semantics[END_REF] in the sense that enjoy the book is preferably interpreted as enjoy reading the book?

(8) Sheila enjoyed {the book/the sonata/strong coffee} In classic GL, there's no clear answer to our question. In contrast in the current system, the information within these formulas is added to the logical form by using one of the ⊗-Exploitation rules. That is, in our system complex type exploitation results in the addition of information to logical form. Our system also makes clear that what drives the explanation of the phenomena of logical metonymy are type mismatch and then type exploitation or introduction to correct the mismatch, whereas that's not at all clear in classic GL. Our approach also makes clear that qualia are just a special case of complex types, which leads us to the next criticism-that either the qualia are far too restrictive a set of relations to be really useful in determining the meaning of sentences like [START_REF] Vikner | A semantic analysis of the english genitive. interaction of lexical and formal semantics[END_REF] or the relations that are associated with the qualia are too vague to be useful. While problems of overgeneration with qualia have been noticed (see e.g. [START_REF] Godard | Towards a proper treatment of coercion phenomena[END_REF]), our point is that they don't postulate a rich enough system of relations for interpreting many constructions. One type of construction that we help us put our argument to the test is the genitive construction.

The genitive construction

The genitive construction offers us a nice point of comparison to test how our more flexible typing system fares in contrast to the "classic" GL. We will in particular look at the account proposed by [START_REF] Vikner | A semantic analysis of the english genitive. interaction of lexical and formal semantics[END_REF] that explicitly appeals to GL. Let's start with some simple English examples of the genetive constructions: Interpreting phrases like [START_REF] Barker | Possessive Descriptions[END_REF] requires us to compute a relation between the two nominal referents that are introduced, respectively, by the specifier NP (aka genitive NP or possessum) and the head noun (aka the possessor) (see e.g. [START_REF] Barker | Possessive Descriptions[END_REF]). The main problem is that this relation is often not directly specified by the grammar, and as a result these constructions give rise to many interpretations (i.e., many different relations can be inferred). For instance, depending on the context, the girl's car can be the car owned/driven/dreamt about/designed. . . by the girl. This great diversity of meanings has led some authors like [START_REF] Williams | The NP-cycle[END_REF] to the view that any relation was indeed possible.

A more nuanced view emerges from more recent works, especially that of Partee and Borshev (henceforth, P&B, e.g., [START_REF] Borschev | Genitive modifiers, sorts, and metonymy[END_REF][START_REF] Partee | Genitives, relational nouns, and argument-modifier ambiguity[END_REF]) and Jensen and Vikner (henceforth, J&V, e.g., [START_REF] Vikner | A semantic analysis of the english genitive. interaction of lexical and formal semantics[END_REF]). These authors show in particular that the semantic properties of the head noun both play a decisive role in constraining the different available interpretations; they also try to to model how pragmatic information gets into the picture. That the (syntactico-)semantic properties of the head are driving the interpretation is obvious with deverbals (e.g. gift, destruction) and relational nouns in general (e.g., mother, edge, captain). Thus, example (9a) seems to license a single interpretation (i.e., mother o f (x, bill)), one where the relation is indeed intrinsic to the head noun.

These authors actually argue that this claim extends to many other, monadic nominals: these are coerced into relational denotations. For instance, nouns denoting body parts like ear give rise in (9b) to a strongly preferred interpretation, in which the denotation of the head noun is interpreted as a part of the denotation of the genitive NP. Things are a little less clear with nominals like car, but there is still a clear sense that, out of explicit contexts, some relations (like owned or driven) are more preferred than others (say, dreamt about); that is, one would probably want to infer the former relations as defaults.

[8] are probably the most explicit in terms of how lexical semantics of the head noun is exploited during the interpretation of the genitive. Assuming Pustejovsky's GL as their background, they argue that the default relations found in the genitive constructions are provided by the different qualia roles associated with the head noun. More precisely, the idea is that the genitive NP, acting as the functor, is able to type-coerce the (monadic) denotation of head noun into a relation: crucially, this relation corresponds to one of qualia roles in the lexical entry of the noun. This approach has prima facie some empirical appeal, for one can indeed find examples corresponding to most of the qualia roles:

(10) a. : The car's design b. : Mary's book (i.e., the book read by Mary) c. : Bill's cake (i.e., the cake cooked by Bill) However, this approach has a number of problems. The main one is that the qualia are simply not a reliable guide for inferring the genitive relation. For one thing, we can't find any plausible example of a true  relation between a head and the genitive NP; there are lots of genitives that involve material constitution-e.g., the car's metal, the sail's fabric, the computer's motherboard, but in these examples the quale constitution is introduced by the genitive NP not the head. Trying to deal with this problem in the framework of GL has de facto an undesirable consequence, since it supposes either a relaxation of the Head Principle or an ad hoc redefinition of the  role; J&V opt for that latter alternative proposing that the  quale be interpreted as the part-whole relation.

This brings us to another point, namely that J&V are often led to adopt an extremely liberal view of the qualia. For instance, they assume that the same qualia role (namely, the  role) is responsible for explaning both the interpretation of genitive like The girl's nose (clearly a case of part-whole relation) and that of The girl's team (clearly, a set-membership). This makes the  role very unclear. For one thing, there is first a directionality problem: the part-whole relation is between the head N and the genitive NP, while the set-membership relation goes the other around (in potential violation of the head principle). There is a more crucial problem: these two relations part-whole and set-membership are rather different in formal terms (e.g., the former is clearly transitive, while the latter is not). Under our analysis, nouns like nose and team having different intrinsic types (a body-part and a group, respectively) and therefore dependent types; and so, they will naturally give rise to different types of relations.

Another argument against qualia comes from morphologically rich languages that have grammaticalized certain genitive relations. Crucially, none of these languages have grammaticalized qualia roles therein. In Basque for instance, there are two genetive postpositions, namely an unmmarked -(a)ren suffix and a marked -ko suffix that specifically encodes (spatial) localization and which thus requires that the type of the object denoted by the nominal in this genitive case be a location (see [START_REF] Aurnague | Basque genitives and part-whole relations : typical configurations and dependences[END_REF]). Below is a minimal pair from Basque that gives an illustration of the two genitives: Crucially, the relation of localization is not part of the qualia.

Another illustration of the problem of using qualia is that, as it stands, the qualia fail to provide the relation own (i.e., material possession) that one finds (preferred) with most physical objects (e.g., John's car) in many different languages. To avoid this problem, J&V have to resort to a multi-purpose (and therefore rather vague) relation, namely , which encompasses the  role but also the relation of ownership, and basically everything that is not provided by the qualia. This again weakens their position. Even worse is the fact that the authors later on appeal to the  role to account for the modification case with favorite.

These different points seem to make it clear that the qualia roles are in a way arbitrary; they don't seem to have any privileged epistemic status. We suggest that we dispense with the qualia altogether, replacing it with a more dynamic notion of typing.

Before turning to that, a number of points about the genitive constructions are still in order. First, contra previous accounts, it seems that constraints on the interpretation of the genitives not only come from the head: Another way we will move away from previous accounts is in assuming that the relation intrisically provided by relational nouns can also, under certain circumstances, be overridden: [START_REF] Aurnague | Basque genitives and part-whole relations : typical configurations and dependences[END_REF] [Context: Picasso and Raphael both painted a mother with a child.] Picasso's mother is bright and splashy -a typical cubist rendition. It makes Raphael's mother look so somber.

(14) Brancusi's kiss has recently been bought by the Philadelphia Museum of Art.

These data in conjunction with a static view of complex types like that enshrined in the notion of qualia lead us to conclude that genitive NPs always act as modifiers; they are never arguments of the head noun (see also discussion in [START_REF] Partee | Genitives, relational nouns, and argument-modifier ambiguity[END_REF]). However, we reject a static view of complex types. As we'll see below, the Head Principle and the view that genitive NPs are arguments is compatible with a more flexible system of types.

Our account of the genitive construction

Basic story

The previous section led us to conlude that qualia were not a reliable guide for inferring the genitive relation. In our account, qualia will be replaced by ⊗-types of the form

(σ ⊗ R 1 ,...R n (τ 1 • • • τ n )).
As noted, these types are asymmetric: they have the intrinsic type σ associated with the variable on the left and a set of dependent types τ 1 • • • τ n are located on the right. The idea is that these dependent types are exploited when the functor type-clashes with the intrinsic type. Each of the dependent type τ i is associated with a relation R i , that will in turn serve as our genetive relation. The basic ⊗-Exploitation rules are given in ( 15) and (??). These rules will have to be defeasible (this is noted with the prefix "/" on the conclusion), this corresponds to the fact that contextual factors can always override lexical inference (see section 4.4). A key feature of the ⊗-exploitation rules for our concern is that they introduce material that changes the arity of some predicate variable-replacing a property expression with a relational one at logical form. [START_REF] Asher | The metaphysics of words in context[END_REF] argue that the relation links a variable with the complex type together with one of the dependent types that serves as an argument for a predicate. Crucially, however, they claim that the head type does not change, in keeping with the Head Typing Principle. This means that these type shifts are very local.

(15) ⊗-Exploitation:

{λPφ(P(x)), c(P : α ⊗ R i (. . . β i . . .) --• γ)}[ψ, c (ψ : δ --• γ)], δ α = ⊥, δ β i ⊥ /{λPφ[ (R i (y,x)∧P(y)) P(x) ], c * y : β i δ}[ψ, c ] (16) ⊗-Exploitation T S {λPφc(P : (δ --• γ) --• δ)}[λPψ(P(x)), c (P : [α ⊗ R i β] --• γ)], δ α = ⊥, δ β ⊥ /{λPφ[λPψ ∃y(R i (y,x)∧P(y)) P(x) , c * (x : (α ⊗ R i β), y : (β δ))]}
Sometimes, however, changing the types in this way interacts with the compositional semantics derived from the syntax/semantics interface, which may force the existential closure of some of the λ-abstracted variables in the new relational expression. This occurs in examples of logical metonymy where ⊗-Exploitation should convert the complement of enjoy from a simple DP like the book into something like reading the book. The syntactic structure for such a complement of enjoy is some projection of I as argued for in [START_REF] Asher | Reference to Abstract Objects in Discourse[END_REF] -which on a compositional semantics for nominals like that in [START_REF] Asher | Reference to Abstract Objects in Discourse[END_REF] must existentially bind the event variable introduced by the ⊗-Exploitation rule and the gerund reading. In any case we can expect that in many of the qualia motivating constructions there will be existential closure of variables introduced by ⊗-Exploitations within some complete functional complex, typically DP or IP, but also partial closure of other maximal projections like NP as in a quick cigarette, where an ⊗-Exploitation would transform λx(cigarette(x)). But it appears that in the genitive, the closure principle is optional; in fact the empty head determiner will not only serve to bind an extra variable as we would expect in X' theory, but also introduce a relation abstract. The reason for this is simple: genitive constructions, like noun-noun compounds and bridging descriptions require a relation to be determined between the variable introduced by the DP in genitive case and the variable that is introduced by the whole DP.

Consequently, we need a quite particular version of ⊗-Exploitation that takes care of the clash between a monadic property term and a construction that requires a relation term:

(17) ⊗-Exploitation R : {λPφ, c(P : (δ --• ( --• γ)))}[λvψ(v), c (v : α ⊗ β)], α δ ⊥, γ δ ⊥, R β (u, v) → u : α ⊗ β ∧ v : γ {λPφ, c}[λwλv[ψ(v) ∧ R β (v, w)]], c * w : α δ, v : γ δ
We will also need an ⊗-Introduction rule to handle certain interpretations of the genitive construction. It was a contention in [START_REF] Asher | The metaphysics of words in context[END_REF] that such a rule was needed; here we provide empirical evidence for it. We will depart from Asher and Pustejovsky's format for •-introduction rules, and simply write a rule for revising type contexts. Thus, this rule isn't ampliative at all and does not affect the logical form of a construction, making it quite simple to state.

(18) ⊗-Introduction:

(λPφ, c(P :

((δ ⊗ β) --• ( --• γ)))) [λvψ(v)], c (v : α), α δ ⊥ (λPφ, c)[λvψ(v)], c * (v : α ⊗ γ), provided R β (x, y) → R γ (y, x)

A little syntax

As mentionned, the syntax we will assume for the genitive construction follows a DP analysis (cf. [START_REF] Abney | The English Noun Phrase in Its Sentential Aspect[END_REF]; see also [START_REF] Asher | Reference to Abstract Objects in Discourse[END_REF]) in which the construction is headed by an empty functional D head, which assigns (genitive) case to the Spec, DP. This is shown below for the girl's teacher:

Interpreting the genitive

Our basic case involves a deverbal, relational noun in head position; e.g. as in the girl's teacher. This is the simplest case, since the relation is intrinsic to the head N (i.e., teach).

Below are the bits of denotation and the compositional derivation -types are left out for now: We now move to a case involving a monadic nominal like nose in head position, as in:

(21) Every girl's nose had been pierced. This type of cases will be be handled using our ⊗-Exploitation R rule. This rule in effect formally models the meaning shift from a property to a relation which is directly dependent on the complex type of individual denoted by the property: in the case of nose (as for other body parts), the type we exploit is - ⊗ -: We now tackle a case where the nominal head has a simple type as in:

(23) The girl's rock This is an example where we have to use a ⊗-Introduction. But what introduces a complex ⊗-type in this genitive construction? We hypothesize that it is the empty determiner itself. But in contradistinction to the Head Typing Principle, this type can be overridden either by a complex type either in the NP or in the DP in SPEC position. The empty determiner posits a default according to which the first argument of the relation abstract is of the type e ⊗  where  stands for the relation of possession (i.e., ownership). Let's consider the derivation here. We'll only put in the one, complex default type for one variable in the determiner, and we'll assume POSS' to be the converse of POSS; i.e., POS S (x, y) → POS S (y, x) and that POS S (x, y) → (y : e ⊗  ∧ x : e).

(24) a.

[ The rest of the derivation proceeds as before except that the default type for y is overridden by the hard information that our type introduction rule now gives us. And then when we apply the determiner more specific type information will affect the second argument of POSS'. The final result we get is:

[the girl] ] = λP(∃!z(girl(z) ∧ P(z))), z :  b. [ [∅] ] = λRλΦλS (Φ(λy∃!x(R(y)(x) ∧ S (x)))), /y : e ⊗  c. [ [rock] ] = λvrock(v), v : - d. [ [∅] ]([ [rock] ]) = λRλΦλS (Φ(λy∃!x(R(y)(x) ∧ S (x))))[λvrock(v)],
(25) λS (∃!z(girl(z) ∧ ∃!x(rock(x) ∧ ownedby(x, z) ∧ S (x)))), x : - ⊗ , x : .

The question is how do we get more specific interpretations for the relational abstract R when those are given by the DP in SPEC position. It should be noted that some of these are uninterpretable-for instance, the child's woman, except as cases of possession, even though child is relational and arguably woman is not. Other putative examples, however, are the artist's object, which we think can mean the object created by the artist. Object is not plausibly relational; when it occurs in the genitive construction with another noun that introduces a λ-bound variable of simple type, then the only relation that seems to hold between the two variables is POSS. But in this case, we get a different interpretation. How is this possible? What needs to happen, we think, is that somehow the typing information from the DP in SPEC has to affect the interpretation of R in the possessive determiner; and the only way we can do this in our framework is to relax the order in which functional applications are made, as argued for in [START_REF] Asher | Reference to Abstract Objects in Discourse[END_REF]. At this point our type logic becomes polymorphic. We don't go into details but essentially this means that we can "freeze" the application of the determiner meaning to the head NP and go ahead and combine the meaning of the DP in SPEC with that result. This would give us the more specific typing information from the DP onto the relational abstract. And this in turn will allow us to construct a more specific ⊗-type via ⊗-Introduction for the λ-bound variable in the NP. And then we can proceed as above to get a relational meaning for the NP via ⊗-Exploitation R . Our final logical form for ( 26) is (26'):

(26) The artist's object (26') λS (∃!x(artist(x) ∧ ∃!y(ob ject(y) ∧ createdby(y, x) ∧ S (y)))

Contextually Sensitive Typing

The examples we've looked at so far all use complex typing information that we think is plausible to suppose in the lexicon, though it goes far beyond the sort of information stored in the qualia of classic GL. However, it is well-known that discourse can affect typing and that these types can propagate through the discourse. Here's a typical example:

(27) a. All the children were drawing fish. b. Suzie's salmon was blue.

Here we need to draw on a theory of discourse structure like SDRT, which would posit that (27a) and (27b) stand in an Elaboration relation, or even Instance ( [START_REF] Asher | Reference to Abstract Objects in Discourse[END_REF]). But this will be so, only if Suzie's salmon is interpreted in a very particular way; namely, the salmon is the fish that Suzie is drawing. Without this link, we cannot make any clear discourse link between the two sentences, threatening the discourse's coherence. So how does this work? salmon introduces a variable whose subtype is that of the variable introduced fish-i.e.,  . But fish has had it's type changed in the context because it is an argument of draw. draw coerces fish into a complex type, one where we have ⊗-. The discourse context dictates that salmon inherits this complex type, this inheritance being crucial to computing the discourse relation of Instance. This complex type now helps specify the relation between Suzie and her salmon, but by itself it doesn't suffice, because Suzie is not the picture of the salmon. But Suzie is an agent and agents, among other things, by default make things. That is, the variable introduced by Suzie is also an ⊗-type: a ⊗ - type. The discourse context also facilitates the selection of this particular activity associated with agents, since Instance requires that Suzie be one of the children, all of whom are making pictures. The clash between the intrinsic type of the variable associated with Suzie and the typing on the variable associated with the picture induces another ⊗-Exploitation, this time linking Suzie as a maker of the picture of the fish. By performing ⊗-Explotation twice, we get that there is a picture of a fish, which Suzie is making, therefore avoiding type clashes incoherent discourse.

This example shows several things. First, a simple, static typing system like the qualia in classic GL cannot possibly do justice to the phenomena. Second, lexical semantics has to be integrated with a discourse semantics in order to account for the incredible complexity and context sensitivity of predication. We hope that our system of flexible typing will make inroads on these thorny issues.

( 1 )

 1 a. Mary picked up and mastered three books on mathematics. [physical object and informational content] b. Most cities that vote democratic passed anti-smoking legislation last year. [population and legislative entity]

( 9 )

 9 a. Bill's mother b. Mary's ear c. The girl's car

  of the book'

( 12 )

 12 a. The rapist's victims b. The designer's car c. The composer's concerto d. A mother's boy These data suggest that both the type of the complement/modifier can help determine the genitive relation.

teacher

  Previous accounts like J&V and P&B and use a much simpler syntax, but one which fails to predict quantificational facts.

(

  20) a. [ [∅] ] = λRλΦλS (Φ(λy∃!x(R(y)(x) ∧ S (x)))) b. [ [teacher] ] = λyλx(teach(x, y)) c. [ [∅] ]([ [teacher] ]) = λΦλS (Φ(λy∃!x(teach(x, y) ∧ S (x)))) (Application and β-reductions) d. [ [ s] ] = λX.X e. [ [the girl] ] = [ [the girl s] ] = λP(∃!z(girl(z) ∧ P(z))) (Application and β-reduction) f. [ [the girl s teacher] ] = λS (∃!z(girl(z) ∧ ∃!x(teach(x, z) ∧ S (z)))) (Application and β-reductions)

  ] ] = λRλΦλS (Φ(λy∃!x(R(y)(x) ∧ S (x)))), x : e, y : e b. [ [nose] ] = λz(nose(z)), z : - ⊗ - c. [ [∅] ]([ [nose] ]) = λRλΦλS (Φ(λy∃!x(R(y)(x) ∧ S (x)))[λz(nose(z))] x : e, y : e, z : - ⊗ - (Application and Merge) d. λRλΦλS (Φ(λy∃!x(R(y)(x) ∧ S (x)))[λvλy(nose(y) ∧ bodypart(y, v))] x : e, y : - ⊗ -, v :  (⊗-Exploitation R ) e. λΦλS (Φ(λy∃!x(nose(x) ∧ parto f (x, y)) ∧ S (x))) y : - ⊗ -, x :  (β-reductions) f. [ [every girl s] ] = λP(∀z(girl(z) → P(z))) z :  g. [ [every girl s nose] ] = λΦλS (Φ(λy∃!x(nose(x)∧ part -o f (x, y))∧S (x)))[λP(∀z(girl(z) → P(z)))] x : - ⊗ -, y : , z :  (Application and Merge) h. [ [every girl s nose] ] = λS (∀z(girl(z) → ∃!x(nose(x)∧part-o f (x, z))∧S (x))) x : -⊗ -, z :  (β-reductions and Accomodation)

  /y : e ⊗ , z :  (Application and Merge) e. λRλΦλS (Φ(λy∃!x(R(y)(x) ∧ S (x))))[λvrock(v)], /y : e ⊗ , v : - ⊗  (⊗-Introduction) f. λRλΦλS (Φ(λy∃!x(R(y)(x)∧S (x))))[λvλurock(v)∧owns(v, y)], /y : e⊗, v : -⊗ , u : e (⊗-Exploitation R )

  If σ and τ 1 , . . . τ n are types, then so is (σ⊗ R 1 ,...R n (τ 1 . . . τ n )) (where R i is a relation over (σ, τ i )).

	b. F T: If σ and τ are types, then so is (σ --• τ)
	c. D T: If σ and τ are types, then so is (σ • τ)
	d. P T:


Daniel Rouly
email: drouly@laas.fr

Josiane Tasselli

Patrick Austin

Chaymaa Haloui

Karine Isoird

Frédéric Morancho

Design Optimization of a New Nanostructured P-GaN Gate for Normally-off GaN HEMTs

Keywords: HEMT, P-GaN gate, normally-off, TCAD simulations

A new AlGaN/GaN heterostructure is proposed to achieve a normally-off behavior for GaN HEMTs. It relies on multiple P-GaN wells epitaxial regrowth along the gate. Simulation results are presented by focusing on the physical and geometrical parameters of the P-GaN wells. The normally-off behavior of the novel HEMT is demonstrated.

I. INTRODUCTION

AlGaN/GaN high-electron mobility transistors (HEMTs) have attracted worldwide attention in power electronics as candidates for next-generation of high-speed switching devices. Thanks to the large electric field of GaN and the high carrier mobility and density in the two-dimensional electron gas (2DEG), AlGaN/GaN HEMTs can achieve high breakdown voltage and realize ultrahigh power density operation with low power losses.

While most of the demonstrated AlGaN/GaN HEMTs are inherently normally-on with a negative gate threshold voltage, normally-off mode is strongly demanded to fulfill the requirements of power electronics applications; indeed normally-off devices are inherently secure and suitable for energy converters requiring specifically high system reliability. Several approaches, each with its own limitations, have been proposed to convert the inherent depletion mode (normally-on) into an enhancement one (normally-off). The recess of the AlGaN barrier under the gate (or recessed gate) [START_REF] Asubar | Enhancement-Mode AlGaN/GaN MIS-HEMTs with High V TH and High I Dmax Using Recessed-Structure With Regrown AlGaN Barrier[END_REF][START_REF] Huang | Design of novel normally-off AlGaN/GaN HEMTs with combined gate recess and floating charge structures[END_REF], Fluorine plasma ion implantation [START_REF] Chen | Physics of fluorine plasma ion implantation for GaN normally-off HEMT technology[END_REF], oxygen treatment [START_REF] He | Recessed-gate quasi-enhancement-mode AlGaN/GaN high electron mobility transistors with oxygen plasma treatment[END_REF], gate injection transistor (GIT) [START_REF] Okita | Through recessed and regrowth gate technology for realizing process stability of GaN-GITs[END_REF] and P-GaN gate [START_REF] Zhong | Effect of Thermal Cleaning Prior to p-GaN Gate Regrowth for Normally Off High-Electron-Mobility Transistors[END_REF][START_REF] Greco | Review of technology for normally-off HEMTs with p-GaN gate[END_REF] are the most developed ones.

In this paper, we propose a new HEMT structure that combines two of these approaches i.e. recessed and P-GaN regrowth gate. The use of a P-GaN layer on the AlGaN/GaN heterostructure under the gate contact lifts up the conduction band level, inducing the depletion of the two dimensional electron gas (2DEG) channel, even in the absence of an external bias, thus ensuring the normally-off behavior.

The novelty of our structure relies on the gate conception: it is characterized by the implementation of a succession of P-GaN wells spaced by active AlGaN/GaN zones along the gate as shown on Fig. 1. A lateral depletion effect takes place, allowing the decrease in the 2DEG density on each well sidewall. By acting on the well topology, the 2DEG channel can be fully depleted or not.

Design and optimization of P-GaN wells are described with a focus on the influence of their geometrical parameters (width, depth and spacing) and on the influence of the technological parameters of both the AlGaN layer (thickness and aluminum content) and the GaN wells (P doping concentration). The simulated electrical transfer characteristics of the novel AlGaN/GaN HEMT are presented, highlighting the normallyoff behavior. In the last section, we report on the technological developments performed for fabricating the nanostuctured P-GaN gate HEMT.

II. DEVICE STRUCTURE

A. HEMT description

Fig. 1 shows different schematic views of the new AlGaN/GaN HEMT with multiple P-GaN wells along the gate. Fig. 1a is a 3D view of the whole HEMT. Fig. 1b is a schematic cross section along the gate showing the alternation of P-GaN wells and AlGaN/GaN active zones. The schematic cross section along the drain-source axis is drawn across a P-GaN well (Fig. 1c) and between two wells (Fig. 1d).

The epitaxial layers parameters used for the simulations are the following: a 20 nm Al 0.2 Ga 0.8 N barrier, a 1.7 µm GaN layer, a resistive AlGaN buffer layer, a 200 nm AlN transition layer on a silicon substrate. The GaN and AlGaN are considered to be non intentionally doped (NID) at 1 x 10 16 cm -3 and 1 x 10 15 cm -3 respectively. The gate-source spacing, the gatedrain spacing and the gate length are 1 µm, 5 µm and 300 nm respectively. The source and drain contacts are Schottky ones.

B. Simulated elementary cell description

2D and 3D simulations were performed with SENTAURUS TCAD software. In order to first evaluate the impact of the insertion of P-GaN wells through the layers stacking, we present simulations at the thermodynamic equilibrium. The studied structure being symmetrical and in order to minimize calculation times, we considered only a half cell as described on Fig. 2.

As mentioned in the introduction, the insertion of P-GaN wells along the gate leads to the formation of PN junctions formed by the P-regions and the AlGaN and GaN active regions neighboring the different well sides: a space charge region (SCR) or depleted zone is naturally created, which extension depends on the wells design. The on-state or off-state behavior of the HEMT is the consequence of the SCR modulation obtained by applying an electrical potential on the P-GaN The results are reported on Fig. 4: the channel depletion width increases with e T until achieving a maximum value for e T around 100 nm. This depth value will be retained for the following simulations. The depletion width through the channel also increases with Na, and decreases when the aluminum content increases. For example, for a P doping concentration varying from 1 x 10 18 cm -3 to 1 x 10 19 cm -3 and x Al fixed to 15%, the depletion width extends from 30 nm to 45 nm respectively.

B. Influence of the aluminum content x Al

We plotted on Fig. 5 the variations of the lateral depletion width as a function of the polarization charges density at the interface AlGaN/GaN. Indeed, the 2DEG channel is the result of the piezoelectric and spontaneous polarization charges through these two layers: the higher the aluminum content, the higher the charges density and, as a consequence, the higher the 2DEG density. To obtain the normally-off functionality for this HEMT, it will be necessary to maximise the lateral depletion width. For an Al content of 20 %, value commonly used, and a P-doping concnetration of 1 x 10 19 cm -3 e.g., the optimum value of the lateral space charge extension through the 2DEG is 28 nm.

C. Influence of the P-GaN well width L T

We have also analyzed the impact of the well width L T on the width of the depleted zone which extends on each side of the P-GaN regions. The behavior can be fitted by an exponential function from which can extract the minimum value L Tmin of the well width at 95 % of the maximum SCR extension width. In Fig. 6a, it can be seen that the space charge no longer extends from a well width value corresponding to L Tmin . For a 1 x 10 19 cm -3 P-GaN doping concentration, the minimum value is 25 nm whatever the aluminum content. We have determined the L Tmin value necessary to achieve a maximum SCR extension, for various P-GaN doping concentrations and Al contents, with a 100 nm well depth (Fig. 6b). For P-doping concentrations higher than 5 x 10 18 cm -3 , there is a little influence of x Al . For lower doping concentrations, it will be necessary to increase L T in order to obtain an optimum depletion width.

D. Influence of the well spacing D T

The geometrical and physical parameters of the P-GaN wells have been optimized to obtain a maximum depletion extension for each one. We consider now a whole cell, comprising two wells spaced by the length D T . We have determined the spacing necessary to achieve the SCRs overlapping, thus inducing a full channel depletion and obtaining the HEMT's self-blocking.

It is well known that it is necessary to lift the conduction band energy level E C above the Fermi level to achieve the normally-off functionality for the HEMT by depleting the channel at the AlGaN/GaN interface. So we have analyzed the evolution of the minimum E C value extracted at the middle of the channel as a function of two well spacing for different Na doping concentration values (Fig. 7). The minimum energy decreases linearly when D T increases to reach a threshold when the channel is no more depleted, so rending the HEMT normally-on. This spacing threshold increases with the P-GaN doping concentration.

E. On-state behavior of the multi P-GaN well gate HEMT

3D simulations have been performed to observe the on-state behavior of our HEMT structure. The electron mobility in the channel is fixed at 2000 cm 2 /V.s taking into account the models of the mobility degradation of the electrons saturation speed at strong electric field. We also considered a Schottky contact on the P-GaN wells. Due to the symmetry of the structure, a half cell is still studied.

The maximum drain saturation current I Dmax and the threshold voltage V TH are extracted from the simulated transfer characteristics I D (V GS ) plotted for different channel widths D T and a drain voltage V DS of 1 V (Fig. 8). The well parameters are the optimum ones defined on the previous simulations, i.e. L T = 30 nm, e T = 100 nm, x Al = 0.20 and Na = 10 19 cm -3 . The normally-off behavior of the new HEMT structure is validated with positive threshold voltages. When the channel width increases, V TH decreases as it was expected from previous 2D simulations.

With our structure, we can achieve V TH values higher than those obtained for standard P-GaN HEMTs [START_REF] Jiang | Thin-barrier heterostructures enabled normally-OFF GaN high electron mobility transistors[END_REF] and FiNFET AlGaN/GaN HEMT topologies regarding the same channel width [START_REF] Zhu | P-GaN Tri-Gate MOS Structure for Normally-Off GaN Power Transistors[END_REF][START_REF] Alsharef | Theoretical Investigation of Trigate AlGaN/GaN HEMTs[END_REF]. However, we can see that the maximum current value strongly depends on the channel width, due to the area aspect ratio between the P-GaN and the 2DEG channel regions. The maximum drain current value is around 1 kA.cm -2 which is a coherent value similar than the ones found in literature [START_REF] Guo | 3-D Simulation Study of a Normally-OFF GaN Lateral Multi-Channel JFET With Optimized Electrical Field Transfer Terminal Structure[END_REF].

IV. TECHNOLOGICAL DEVELOPMENTS

Most of the P-GaN gate HEMT technologies developed are based on the selective GaN layer etching on top of the AlGaN barrier to define the source and drain access regions. However, the plasma-induced damages may affect the electrical performances of the device due to surface leakage currents. One technology that overcomes these problems and that we have developed in our laboratory consists in fabricating the Mgdoped P-GaN region by selective epitaxy using a SiO 2 growth mask [START_REF] Fu | Selective area regrowth and doping for vertical gallium nitride power devices: Materials challenges and recent progress[END_REF][START_REF] Rolland | High Power Normally-OFF GaN/AlGaN HEMT with Regrown p Type GaN[END_REF].

We present in this part the results obtained for a normallyoff HEMT that we have fabricated by associating both recessgate and P-GaN regrowth technics [START_REF] Charles | Normally-off HEMT with high threshold voltage and a low on-state resistance[END_REF][START_REF] Haloui | Development and technological fabrication of GaN HEMT devices with the normally-off functionality[END_REF][START_REF] Haloui | Recessed and P-GaN regrowth gate development for normally-off AlGaN/GaN HEMTs[END_REF] with the GaN layer regrown on the whole gate area. The validation of this process led us to develop afterwards the new normally-off HEMT structure presented in this paper with multi P-GaN wells along the gate.

A. Normally-off HEMT with a MBE regrown P-GaN gate

The HEMT structure was characterized by an AlGaN barrier of 25 nm thick with an aluminum rate of 20 %. At first, a 100 nm thick silicon oxide SiO 2 layer was deposited by PECVD (Plasma Enhanced Vapor Deposition), acting as a mask for both the AlGaN gate etch and the selectively GaN regrowth. After a CF 4 RIE etch of the SiO 2 layer to define the gate region, a Cl 2 partial etching of the AlGaN layer was performed by ICPECVD with the following conditions: RF power of 60 W, pressure of 5 mTorr and Cl 2 flow rate of 10 sccm. For an etch time of 35 s, 19 nm of AlGaN was removed. A 50 nm GaN layer P-doped with Magnesium (Mg) was then regrown in a MBE (Molecular Beam Epitaxy) reactor with a nominal acceptor concentration of Na-Nd of 4 x 10 18 cm -3 .

Figure 9 shows a TEM cross-section view of the gate region with the regrown P-GaN layer on the etched AlGaN layer. The analysis highlights the perfect nucleation of the GaN layer on the etched AlGaN layer. The next technological steps concerned the mesa isolation and the gate, source and drain metallisations. The measured transfer characteristics I D (V GS ) of the fabricated HEMT for various V DS values are reported on Fig. 10, for a Schottky contact on the gate. The normally-off behavior was demonstrated with positive threshold voltages around 1 V.

B. Normally-off HEMT with a multi P-GaN wells gate

The fabrication of the novel proposed HEMT structure is in progress. The first technological step we have developed is the e-beam lithography of the 200 nm thick HSQ resist at the nanometric scale in order to define the 50 nm wide active regions separating the different P-GaN regions. Fig. 11 shows a MEB view after the lithography of the HSQ lines across the gate, which is delimitated by the SiO 2 layer, and the BCl 3 etching of the AlGaN/GaN layers with both HSQ and SiO 2 as an etch mask, thus resulting in the formation of the different wells. The next step of the process will be the selective area growth of the P-GaN regions inside the etched wells. V. CONCLUSION A new normally-off AlGaN/GaN HEMT structure with multiple P-GaN wells along the gate is proposed in this paper. The P-GaN HEMTs developed up to now necessitate an accurate control of the gate recess with the disadvantage of channel mobility degradation. In our structure, the 2DEG channel properties are maintained between two wells. A lateral space charge region extends on each side of them, thus depleting the adjacent channel created by the AlGaN/GaN heterostructure.

2D simulations have highlighted that with an appropriate design of the well i.e. depth, width, spacing between each one, and P-GaN doping concentration, we can obtain the normallyoff functionality with a degre of freedom in the choice of the AlGaN layer parameters (thickness and aluminum content).The benefit of the proposed HEMT was demonstrated from 3D simulations, positive threshold voltages being achieved. This structure paves the way for the integration of advanced power functions.

The HEMT fabrication is in progress: the critical technological steps such the wells shaping at the nanometric scale and the GaN regions regrowth have been developed.

Fig. 1 .

 1 Fig. 1. 3D and cross sectional views of the multi P-GaN wells gate HEMT

Fig. 2 .

 2 Fig. 2. Schematic cross sectional view of the simulated half cell

Fig. 3 .

 3 Fig. 3. Example of 2D simulations of the space charge region extension for different P-GaN well depths: a) e T = 20 nm, b) e T = 35 nm, c) e T = 100 nm

Fig. 5 .

 5 Fig. 5. Simulated variations of the channel depletion width as a function of the aluminum content x A l and the polarisation charges density for different Pdoping concentrations

Fig. 6 .

 6 Fig. 6. a) Simulated variations of SCR width as a function of L T for a 1x10 19 cm -3 P-GaN doping concentration and different aluminum contents x Al ; b) L Tmin variations as a function of x Al for different P-doping concentrations (e T = 100 nm)

Fig. 7 .

 7 Fig. 7. Simulated variations of the conduction band energy level versus the well spacing D T for different P-GaN doping concentrations Considering the conventional P-GaN normally-off HEMTs studied in the literature, the only way to modify the threshold voltage V TH is acting on the P-GaN doping concentration. The novel HEMT structure presented in this paper offers a better flexibility with the advantage to modulate V TH by varying either the P-GaN wells geometrical parameters or their spacing or their doping concentration.

Fig. 8 .

 8 Fig. 8. Simulated transfer characteristics I D (V GS ) at V DS = 1 V for various D T with L T = 30 nm, e T = 100 nm, x Al = 0.20, Na =1 x 10 19 cm -3 , and a Schottky contact on the gate.

Fig. 9 .

 9 Fig. 9. TEM cross-section view of 50 nm P-GaN layer regrown by MBE on the etched AlGaN layer.

Fig. 10 .

 10 Fig. 10. View of the recessed and regrown P-GaN gate normally-off HEMT developed at LAAS and measured transfer characteristics I D (V GS ) for various V DS , with Na = 4 x 10 18 cm -3 and a Schottky contact on the P-GaN.

Fig. 11 .

 11 Fig. 11. MEB view of the multiwells along the gate after HSQ e-beam lithography and AlGaN/GaN BCl 3 etching.

ACKNOWLEDGMENTS

This work is funded by the IPCEI on Microelectronics research program "Nano2022" . This work was partly supported by LAAS-CNRS Micro and Nanotechnologies Platform, member of the French RENATECH network, and by the calculation platform of LAAS-CNRS.

The authors would like to thank F. VAURETTE from the IEMN Micro and Nano Fabrication Platform for providing HSQ e-beam lithographies and Y. CORDIER and É. FRAYSSINET from CRHEA-CNRS for performing MBE P-GaN epitaxies.

This work is supported by the IPCEI program Nano2022