Philippe Besnard
email: besnard@irit.fr

Marie-Odile Cordier
email: cordier@irisa.fr

Yves Moinard
email: yves.moinard@inria.fr

Arguments using ontological and causal knowledge

We explore an approach to reasoning about causes via argumentation. We consider a causal model for a physical system, and we look for arguments about facts. Some arguments are meant to provide explanations of facts whereas some challenge these explanations and so on. At the root of argumentation here, are causal links ({A1, • • • , An} causes B) and also ontological links (c1 is a c2). We introduce here a logical approach which provides a candidate explanation ({A1, • • • , An} explains {B1, • • • , Bm}) by resorting to an underlying causal link substantiated with appropriate ontological links. Argumentation is then at work from these various explanation links. A case study is developed: a severe storm Xynthia that devastated a county in France in 2010, with an unaccountably high number of casualties.

Introduction and Motivation

Looking for explanations is a frequent operation, in various domains, from judiciary to mechanical fields. We consider the case where we have a (not necessarily exhaustive) description of some mechanism, or situation, and we are looking for explanations of some facts. The description contains logical formulas, together with some causal and ontological formulas (or links). Indeed, it is well-known that, although there are similarities between causation and implication, causation cannot be rendered by a simple logical implication. Moreover, confusing causation and co-occurrence could lead to undesirable relationships. This is why we resort here to a causal formalism such that some causal links and ontological links are added to classical logical formulas. Then, our causal formalism will produce various explanation links [START_REF] Besnard | Ontology-based inference for causal explanation[END_REF].

Each causal link gives rise to explanation links, and each explanation link must appeal to at least one causal link. The ontology gives rise to further explanation links, although only in the case that these come from explanation links previously obtained: no explanation link can come only from ontological information. In fact, the ontology determines a new connective (it can be viewed as a strong implication) which can induce these further explanation links, whereas classical implication cannot. Indeed, given an explanation link, logical implication is not enough to derive from it other explanation links (apart from trivially equivalent ones).

Despite these restrictions, if the situation described is complex enough, there will be a large number of explanation links, i.e., possible explanations, and argumentation is an appealing approach to distinguish between all these explanations.

We introduce in section 2 an explicative model, built from a causal model and an ontological model. Section 4 shows how the explicative model produces explanations. Section 5 deals with argumentation about explanations and we conclude in section 6. Section 3 presents a case study: a severe storm called Xynthia, that resulted in 26 deaths in a single group of houses in La Faute sur Mer, a village in Vendée during a night in 2010.

Explicative Model = Causal Model + Ontological Model

The model that is used to obtain explanations and support argumentation, called the explicative model, is built from a causal model relating literals in causal links and from an ontological model where classes (which denote types of object as usual in the literature about ontology) are related by specialization/generalization links. Data consist of causal links and "is a" relationships (specifying a hierarchy of classes, see the ontological model in section 2.3) and background knowledge (formulae of a sorted logic whose sorts are the classes of the aforementioned hierarchy).

From these data, tentative explanations are inferred according to principles using the so-called ontological deduction links obtained in the explicative model.

Closed literals

By a closed literal, we mean a propositional literal or a formula

∃ x : class ¬ {0,1} P (x) or ∀ x : class ¬ {0,1} P (x)
where x is a variable and P is a unary predicate symbol, preceded or not by negation. Throughout, a closed literal of the form ∃ x : classP (x) is abbreviated as ∃ P (class) and ∀ x : classP (x) is abbreviated as ∀ P (class) and similarly for ¬P instead of P .

From now on, when we write simply literal, we mean a closed literal. Lastly, extension to n-ary predicates is unproblematic except for heavy notation. Henceforth, it is not considered in this paper for the sake of clarity.

The causal model

By a causal model [START_REF] Hugh | The Facts of Causation[END_REF], we mean a representation of a body of causal relationships to be used to generate arguments that display explanations for a given set of facts. Intuitively, a causal link expresses that a bunch of facts causes some effect.

Notation 1 A causal link is of the form {α 1 , α 2 , • • • , α n } causes β where α 1 , α 2 , • • • , α n , β are literals.
These causal links will be used in order to obtain explanation links in section 4.

Example 1 My being a gourmet with a sweet tooth causes me to appreciate some cake can be represented by {sweet tooth gourmet} causes ∃ X : cake Ilike(X) Similarly, my being greedy causes me to appreciate all cakes can be represented by {IamGreedy} causes ∀ X : cake Ilike(X)

In the figures (e.g., part of the causal model for Xynthia in Fig. 2), each plain black arrow represents a causal link. In the figures (e.g., part of the ontological model for Xynthia in Fig. 1), each whiteheaded arrow labelled with is-a denotes an ISA -→ link.

The ontological model

The relation ISA -→ is required to be transitive and reflexive.

(

) 1
Reflexivity is due to technical reasons simplifying various definitions and properties.

The explicative model

The resulting model (causal model + ontological deduction link) is the explicative model, from which explanation links can be inferred. (

) 2
The same principle holds for P replaced by ¬P . That is, from class 1 Summing up, the explicative model consists of the causal links (in the causal model) together with the ontological deduction links (obtained from the ontological model). The explicative model contains all the ingredients needed to derive explanations as is described in section 4.

ISA -→ class 2 , both ∃ ¬P (class 1 ) DEDO -→ ∃ ¬P (class 2

Background knowledge

In addition to the explicative model, background knowledge is used for consistency issues when defining explanation links, as will be seen in Section 4, Notation 5. Background knowledge consists of logical formulas of sorted logic. Part of this knowledge is freely provided by the user. Moreover, we take causal and ontological deduction links to entail classical implication:

{α 1 , • • • , α n } causes β entails ( n i=1 α i ) → β. α DEDO -→ γ entails α → γ. (5)
Consequently, the rightmost logical formulas ( n i=1 α i ) → β and α → γ from (5), are necessarily included in the background knowledge.

The Xynthia Example

In this section, we consider as an example a severe storm, called Xynthia, which made 26 deaths in a single group of houses in La Faute sur Mer, a village in Vendée during a night in February 2010. It was a severe storm, with strong winds, low pressure, but it had been forecast. Since the casualties were excessive with respect to the strength of the meteorological phenomenon, various investigations have been ordered. This showed that various factors combined their effects. The weather had its role, however, other factors had been involved: recent houses and a fire station had been constructed in an area known as being susceptible of getting submerged. Also, the state authorities did not realize that asking people to stay at home was inappropriate in case of flooding given the traditionally low Vendée houses. From various enquiries, including one from the French parliament2 and one from the Cour des Comptes (a national juridiction responsible for monitoring public accounts in France) 3 and many articles on the subject, we have plenty of information about the phenomenon and its dramatic consequences. We have extracted a small part from all this information as an illustration of our approach.

Classes and predicates for the Xynthia example

The classes we consider in the causal model are the following ones: Hurri, SWind, BFPA, House1FPA, HouseFPA, and BFPA have already been introduced in §2.3, together with a few ISA -→ links. Among the buildings in the flood-prone area FPA, there is also a fire station FireSt. Besides Hurri, we consider two other kinds of natural disasters NatDis: tsunami Tsun and flooding Flooding. As far as meteorological phenomena are concerned, we restrict ourselves to very low pressure VLP, together with the aforementioned Hurri and SWind, and add high spring tide HST to our list of classes.

Two kinds of alerts Alert may be given by the authorities, Alert-Evacuate AlertE and Alert-StayAtHome AlertS. Also, PeopleS expresses that people stay at home. There exists an anemometer (able to measure wind strength) with a red light, described by OK Anemo meaning that it is in a normal state and Red Anemo meaning that its light is on, which is caused by strong wind, while during a hurricane the anemometer is in abnormal state.

The following predicates are introduced: Flooded and Vict I, applied to a group of building, respectively meaning that flooding occurs over this group, and that there were victims in this group (I ∈ {1, 2, 3} is a degree of gravity, e.g. Vict 1, Vict 2 and Vict 3 respectively mean, in % of the population of the group: at least a small number, at least a significant number and at least a large number of victims).

Remember that Occ means that some fact has occurred (a strong wind, a disaster, . . . ), similarly a unary predicate Exp means that some fact is expected to occur.

∃ Exp(VLP) causes ∃ Exp(SWind), ∃ Occ(Hurri) causes ¬ OK Anemo, {∃ Occ(SWind), OK Anemo} causes Red Anemo, {¬∃ Occ(SWind), OK Anemo} causes ¬ Red Anemo, ∃ Occ(NatDis) causes ∃ Occ(Alert), ∃ Exp(NatDis) causes ∃ Occ(Alert),    ∃ Occ(VLP), ∃ Occ(SWind), ∃ Occ(HST)    causes ∃ Occ(Flooding), ∃ Occ(Flooding) causes ∀ Flooded(BFPA), ∀ Flooded(BFPA) causes ∀ Vict 1(BFPA), ∃ Occ(AlertS) causes ∃ Occ(PeopleS), ∃ Occ(PeopleS), ∀ Flooded(House1FPA) causes ∀ Vict 2(House1FPA), ∀ Vict 2(House1FPA), ∀ Flooded(FireSt) causes ∀ Vict 3(House1FPA).
Table 1: Part of the causal model for Xynthia

The causal and ontological models for the Xynthia example

The classes and the ontological model are given in Fig. 1.

BFPA (is-a) (is-a)

FireSt HouseFPA Part of the causal model is given in Table 1 and in Fig. 2. It represents causal relations between (sets of) literals. It expresses that an alert occurs when a natural disaster is expected, or when a natural disaster occurs. Also, people stay at home if alerted to stay at home, and then having one level home flooded results in many victims, and even more victims if the fire station itself is flooded,. . .

The explicative model for the Xynthia example

The explicative model can be built, which contains various DEDO -→ links between literals. For instance, from Hurri The explicative model (it consists of causal and ontological links) allows us to infer explanation links. We want to exhibit candidate reasons that can explain a fact by means of at least one causal link. We disregard "explanations" that would involve only links which are either classical implications (→) or DEDO -→ links: some causal information is necessary for an "explanation" to hold. Here is how causal and ontological links are used in order to obtain (tentative) explanations in our formalism.

Notation 5 Let Φ, ∆ and Ψ be sets of literals. An explanation link Φ explains ∆ unless ¬ Ψ is intended to mean that Φ explains ∆ provided that, given Φ, the set Ψ is possible: if adding Φ ∪ Ψ to available data (i.e., background knowledge and formulas from ( 5)) leads to an inconsistency, then the explanation link cannot be used to explain ∆ by Φ.

Ψ is called the provision set of the explanation link.

When the set Ψ is empty, we may omit the " unless ¬ ∅" (i.e., "unless ⊥") part.

Throughout the text, we write as usual Φ for ϕ∈Φ ϕ and ¬Ψ for ¬ Ψ.

We set the following equivalences between explanation links, so that the leftmost link can under any circumstance be substituted for the rightmost link and vice-versa:

Φ explains ∆ is equivalent to Φ explains ∆ unless ¬ Φ. Φ explains ∆ unless ¬ Ψ is equivalent to Φ explains ∆ unless ¬ (Φ ∪ Ψ). (6)
Let us now describe how explanation links are inferred from the explicative model. First is the case that ∆ is a singleton set.

Explaining a singleton from a set of literals

The basic case consists in taking it that a direct causal link Φ causes β between a set of literals Φ and a literal β provides an explanation such that the cause explains the (singleton set of) effect: see (7a).

If β = ∃ P (c) or β = ¬ ∃ P (c), a more interesting case arises. Take β = ∃ P (c) for instance. Since the causal link expresses that the effect of Φ is ∃ P (c), it means that for any subclass c of c, ∃ P (c ) could be caused by Φ (unless a logical inconsistency would indicate that ∃ P (c ) cannot be the case in the presence of Φ and background knowledge and all the formulas from ( 5)). Accordingly, Φ can be viewed as explaining ∃ P (c ). This is the reason for (7b).

Φ causes β yields: Φ explains {β}. (a) Φ causes ∃ β ∃ δ DEDO -→ ∃ β yields: Φ explains {∃ δ}, unless ¬ {∃ δ}. (b) (7)
(5) yields Φ → β (case (7a)) as well as Φ → ∃ β and ∃ δ → ∃ β (case (7b)), hence adding β (case (7a)) or ∃ β (case (7b)) to the provision set makes no difference, thereby justifying the equivalences in [START_REF] Gebser | Answer Set Solving in Practice[END_REF].

If Φ = {ϕ} is a singleton set, we may abbreviate Φ explains {β} as ϕ explains β.

Here are a couple of examples from the Xynthia case. First, that flooding occurred can be explained by the conjunction of very low pressure, strong wind, as well as high spring tide. In symbols,

   ∃ Occ(VLP), ∃ Occ(SWind), ∃ Occ(HST)    causes ∃ Occ(Flooding) yields    ∃ Occ(VLP), ∃ Occ(SWind), ∃ Occ(HST)    explains ∃ Occ(Flooding)
Second, expecting a hurricane can be explained from expecting very low pressure:

∃ Exp(VLP) causes ∃ Exp(SWind) ∃ Exp(Hurri) DEDO -→ ∃ Exp(SWind) yields ∃ Exp(VLP) explains ∃ Exp(Hurri)
Third, that all buildings in the flood-prone area are flooded can be explained by flooding: ∃ Occ(Flooding) causes ∀ Flooded(BFPA) yields ∃ Occ(Flooding) explains ∀ Flooded(BFPA)

In the figures, dotted arrows represent explanation links (to be read explains), these arrows being sometimes labelled with the corresponding provision set. We now introduce explanation links between sets of literals, extending the notion of explanation links from sets of literals to literals presented so far. Since it is an extension, we keep the same name explanation link.

Explaining a set of literals from a set of literals

The patterns (7) inducing an explanation for a single observation (a singleton set) are now extended so that they can be used to obtain an explanation for a set of observations:

Let Φ 1 , Φ 2 , ∆, Ψ 1 and Ψ 2 be sets of literals and β be a literal.

If we have

Φ 1 explains ∆ unless ¬ Ψ 1 , and

Φ 2 explains {δ} unless ¬ Ψ 2 , (8)
then we get

Φ 1 ∪ Φ 2 explains ∆ ∪ {δ} unless ¬ (Ψ 1 ∪ Ψ 2 ).
Notice that the condition in ( 8) is that Ψ 1 ∪ Ψ 2 must be possible (it is not enough that Ψ 1 be possible and that Ψ 2 be possible -the same applies to (10) below).

Still further explanation links can be generated from these, by following DEDO -→ links:

If we have      Φ explains ∆ unless ¬ Ψ Φ 0 DEDO -→ Φ ∆ DEDO -→ ∆ 1      then we get Φ 0 explains ∆ 1 unless ¬ Ψ. (9)
Let us return to our example. Applying (7a), that all the buildings in the flood-prone area are flooded can be explained by flooding (this is shown at the end of section 4.2). This explanation link (Φ is {∃ Occ(Flooding)} and ∆ is {∀ Flooded(BFPA)}) can be exploited through [START_REF] Halpern | Causes and Explanations: A Structural-Model Approach. Part II: Explanations[END_REF], letting Φ 0 = Φ and ∆ 1 = {∀ Flooded(HouseFPA)}. I.e., that all houses in the flood-prone area are flooded can also be explained by flooding:

∃ Occ(Flooding) causes ∀ Flooded(BFPA) ∀ Flooded(BFPA) The last, but not least, way by which explanation links induce further explanation links is transitivity (of a weak kind because provision sets are unioned). arising from the explicative model are those and only those resulting from applications of ( 7), ( 8), ( 9) and [START_REF] Lacave | A review of explanation methods for heuristic expert systems[END_REF].

If Φ explains ∆ unless ¬ Ψ 1 Γ ∪ ∆ explains Θ unless ¬ Ψ 2 then Φ ∪ Γ explains Θ unless ¬ (Ψ 1 ∪ Ψ 2 ).
The reader should keep in mind that Φ must always be included in the set to be checked for consistency, as is mentioned in Notation 5 (cf ( 6)). Definition 6 is such that we can neither explain Φ by Φ itself nor explain Φ by Φ 0 if all we know is Φ 0 DEDO -→ Φ. Intuitively, providing such "explanations" would be cheating, given the nature of an explanation: some causal information is required.

More examples detailed

Let us start with an example from Xynthia illustrating the use of the patterns (7b) and ( 9) depicted in Fig. 4 and5.

In the causal model for Xynthia, we focus on the causal link

∃ Exp(VLP) causes ∃ Exp(SWind)
In the ontological model for Xynthia, we consider the following ontological links We are looking for Exp(NatDis) to be explained by Exp(VLP) hence we consider

∃ Exp(VLP) causes ∃ Exp(SWind) ∃ Exp(Hurri) DEDO -→ ∃ Exp(SWind)
and we apply (7b) in order to obtain, as a first step, ∃ Exp(VLP) explains ∃ Exp(Hurri) unless ¬ ∃ Exp(Hurri) over which we apply (9) using the ontological deduction link obtained above, that is,

∃ Exp(Hurri) DEDO -→ ∃ Exp(NatDis)
in order to arrive at

∃ Exp(VLP) explains ∃ Exp(NatDis) unless ¬ ∃ Exp(Hurri)
Please observe that applying (9) actually requires ∃ Exp(VLP) DEDO -→ ∃ Exp(VLP) which is obtained by using [START_REF] Minh | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF].

That a natural disaster occurs can be explained from the fact that very low pressure is expected. However, if ¬∃ Exp(Hurri) holds (it is impossible that some hurricane be expected), then this explanation no longer stands (because the effect of the causal link underlying it is strong wind and the explanation chain here identifies hurricane as the kind of strong wind expected).

Let us now turn to an example showing how a chain of explanations can be constructed by means of transitivity [START_REF] Lacave | A review of explanation methods for heuristic expert systems[END_REF] applied over explanations already detailed above. The fact that

   ∃ Occ(VLP), ∃ Occ(SWind), ∃ Occ(HST)    causes ∃ Occ(Flooding)
is in the explicative model allowed us to conclude

   ∃ Occ(VLP), ∃ Occ(SWind), ∃ Occ(HST)    explains ∃ Occ(Flooding) (i)
and the fact that

∃ Occ(Flooding) causes ∀ Flooded(BFPA) ∀ Flooded(BFPA) DEDO -→ ∀ Flooded(HouseFPA)
is in the explicative model allowed us to conclude

∃ Occ(Flooding) explains ∀ Flooded(HouseFPA). (ii)
Hence chaining the explanations (i) and (ii) through [START_REF] Lacave | A review of explanation methods for heuristic expert systems[END_REF] by letting Γ = ∅ yields

   ∃ Occ(VLP) ∃ Occ(SWind), ∃ Occ(HST)    explains ∀ Flooded(HouseFPA) (iii)
Let us now suppose that we have multiple observations {∀ Flooded(BFPA), Red Anemo}.

From

{∃ Occ(SWind), OK Anemo} causes Red Anemo, we get {∃ Occ(SWind), OK Anemo} explains Red Anemo.

Then, from (iii), using (8) we get

       ∃ Occ(VLP) ∃ Occ(SWind), ∃ Occ(HST), OK Anemo        explains {∀ Flooded(HouseFPA), Red Anemo} (iv)
Also, from Hurri ISA -→ SWind we get ∃ Occ(Hurri) DEDO -→ ∃ Occ(SWind) Thus from (iii), using (9), we get

   ∃ Occ(VLP) ∃ Occ(Hurri), ∃ Occ(HST)    explains ∀ Flooded(HouseFPA) (v)
However, since ∃ Occ(Hurry) causes ¬ Red Anemo, we do not get

       ∃ Occ(VLP) ∃ Occ(Hurri), ∃ Occ(HST), OK Anemo       
explains {∀ Flooded(HouseFPA), Red Anemo} . Fig. 7 displays another example from Xynthia of various possible explanations (represented by dotted lines) labelled as 1, 1a, . . . The sets of literals, from which the explanation links start, are framed and numbered (1) to (4). These sets are not disjoint, some literals are then duplicated for readability and the copies are annotated with (bis). Transitivity of explanations is again at work, e.g.,

• set 1 explains ∀ Vict 1(BFPA) (label 1+1a+1b) It is obtained by transitivity over explanation links 1, 1a and 1b.

• set 4 explains ∀ Vict 2(House1FPA) (label 1+1a+2) It follows from explanations 1, 1a and 2. The latter results from explanation 1+1a together with the ∀ Flooded(BFPA)

DEDO -→ ∀ Flooded(House1FPA) link. • set 4 explains ∀ Vict 3(House1FPA) (label 1+1a+2+3) Explanation 3 results from the ∀ Flooded(BFPA) DEDO -→ ∀ Flooded(FireSt) link together with explanations 1+1a+2. E E E A A E A A E A E E E E E Occ(AlertS) A A 1+1a (2) (Flooding)

(bis)

Flooded(FireSt) (3) 3 (4)

Argumentation

The explicative causal model allows us to infer explanations for a set of statements and these explanations might be used in an argumentative context [START_REF] Besnard | Elements of Argumentation[END_REF][START_REF] Minh | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]. Let us first provide some motivation from our case study, Xynthia. An explanation for the flooded buildings is the conjunction of the bad weather conditions (very low pressure and strong wind) and high spring tide (see Fig. 2). Let us take this explanation as an argument. It can be attacked by noticing: a strong wind is supposed to trigger the red alarm of the anemometer and no alarm was shown. However, this counter-argument may itself be attacked by remarking that, in the case of a hurricane, that is a kind of strong wind, the anemometer is no longer operating, which explains that a red alarm cannot be observed.

Let us see how to consider formally argumentation when relying on an explicative model and explanations as described in sections 2 and 4. Of course, we begin with introducing arguments.

Arguments

An argument is a tuple (Φ, ∆, Ψ, Θ) such that Θ yields that Φ explains ∆, unless ¬Ψ is an explanation link according to Definition 6. The components of the argument are:

• Φ, the explanation, a set of literals.

• ∆, the statements being explained, a set of literals.

• Ψ, the provision of the explanation (see Section 4), a set of formulas.

• Θ, the evidence, comprised of formulas (e.g., Φ → γ), causal links (e.g., Φ causes β), and ontological deduction links (e.g., ∆ DEDO -→ {β}).

Back to (iii) in the example from Xynthia in section 4.4, that the F P A houses are flooded is explained by the set of literals

Φ =    ∃ Occ(VLP) ∃ Occ(SWind) ∃ Occ(HST)   
on the grounds of the following set consisting of two causal links and one ontological deduction link

Θ =                ∃ Occ(Flooding) causes ∀ Flooded(BFPA),    ∃ Occ(VLP) ∃ Occ(SWind) ∃ Occ(HST)    causes ∃ Occ(Flooding), ∀ Flooded(BFPA) DEDO -→ ∀ Flooded(HouseFPA)               
That is, (iii) gives rise to the argument (Φ, {δ}, Ψ, Θ) where • The explanation is Φ.

• There is a single statement being explained, i.e., δ = ∀ Flooded(HouseFPA).

• The provision of the explanation is empty.

• The evidence is Θ.

As for the argumentation part, our approach is concerned with sense-making. I.e., there is complex information that needs to be made sense of, and our approach is meant to provide a way to organize that information so that the key points are identified. This is a primary task in argumentation, as argumentation (even in computational guise) is much more than evaluating arguments, and in any case, does not begin with evaluating arguments [START_REF] Minh | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]. Accordingly, our approach does not aim at evaluating a collection of arguments and counterarguments (as in the sense of determining extensions or identifying warranted arguments).

Counter-arguments

A counter-argument for an argument (Φ, ∆, Ψ, Θ) is an argument (Φ , ∆ , Ψ , Θ ) which questions 1. either Φ (e.g., an argument exhibiting an explanation for ¬Φ) 2. or ∆ (e.g., an argument exhibiting an explanation for ¬∆) 3. or Ψ (e.g., an argument exhibiting an explanation for ¬Ψ) 4. or any item in Θ (e.g., an argument exhibiting an explanation for the negation of some θ occurring in Θ)

5. or does so by refutation: i.e an argument exhibiting an explanation for a statement known to be false and using any of Φ, Θ, Ψ and ∆. In this case, at least one of Φ , Θ , Ψ intersects one of Φ, ∆, Θ, or Ψ.

Type (5) counter-arguments do not directly oppose an item in the argument being challenged. They rather question such an item by using it to provide an argument whose conclusion is wrong. The presence of such an item is ensured by checking that the challenged argument and the counter-argument indeed share something in common, i.e., that the intersection is not empty. Otherwise, in the case that the intersection is empty, then the two arguments have nothing in common, hence none can be viewed as a counter-argument to the other.

These counter-arguments have the form of an argument. They explain something that contradicts something in the challenged argument.

Dispute.

Let us consider the illustration at the start of this section: The argument (that the houses in the flood-prone area are flooded is partly explained by a strong wind) is under attack on the grounds that the anemometer did not turn red -indicating that no strong wind occurred. The latter is a counter-argument of type 5 in the above list. Indeed, the statement explained by the counter-argument is Red Anemo that has been observed to be false. The explanation uses ∃ Occ(SWind), i.e., an item used by the explanation and then belonging to Φ in the attacked argument.

Taking Red Anemo to be a falsehood, the counter-argument (Φ , ∆ , ∅, Θ ) results from Θ yielding that ∃ Occ(SWind) OK Anemo explains {Red Anemo} where • The explanation is Φ = {∃ Occ(SWind), OK Anemo}

• The statement being explained is ∆ = {Red Anemo} • The evidence is Θ = ∃ Occ(SWind) OK Anemo causes Red Anemo
Notice that Φ does intersect Φ. This is a counter-argument because, taking the anemometer being red as falsity, it is an argument which uses the occurring of a strong wind to conclude the anemometer being red. As explained above in the general case, such a type (5) counter-argument uses an item (a strong wind occurring) from the argument being challenged, in order to conclude a falsity (the anemometer being red).

Dispute (continued)

This counter-argument has in turn a counter-argument (of type 1.). It explains the misbehavior of the anemometer by the occurrence of an hurricane (that is a strong wind), and then explains the negation of an item OK Anemo of the explanation Φ of the counter-argument. The anemoter not getting red, instead of being explained by the absence of a strong wind, is explained by the fact that the wind was so strong (an hurricane) that the anemometer misbehaved.

So, the counter-counter-argument is: (Φ , ∆ , ∅, Θ ) resulting from Θ yielding that:

∃ Occ(Hurri) explains {¬OK Anemo} where • The explanation is Φ = {∃ Occ(Hurri)}

• The statement being explained is

∆ = {¬OK Anemo} • The evidence is Θ = {∃ Occ(Hurri) causes ¬OK Anemo}
The dispute can extend to a counter-counter-counter-argument and so on as the process iterates.

Conclusion

The contribution of our work is firstly to propose a new logic-based formalism where explanations result from both causal and ontological links. It is important to stress that our approach reasons from causal relationships which are given, in contrast to a number of models for causality that aim at finding causal relationships (e.g., [START_REF] Halpern | Causes and Explanations: A Structural-Model Approach. Part I: Causes[END_REF][START_REF] Halpern | Causes and Explanations: A Structural-Model Approach. Part II: Explanations[END_REF]). This causal-based approach for explanations, already defended in [START_REF] Besnard | Deriving explanations from causal information[END_REF], is relatively different from other work on explanations that rely on expert knowledge and are considered as useful functionalities for expert systems and recommender systems (for a synthetic view on explanations in these domains, see [START_REF] Friedrich | A taxonomy for generating explanations in recommender systems[END_REF][START_REF] Lacave | A review of explanation methods for heuristic expert systems[END_REF][START_REF] Leake | The impact of explanation facilities in user acceptance of expert system advice[END_REF]. We then show how these explanation links may be interestingly used as building blocks in an argumentative context [START_REF] Besnard | Elements of Argumentation[END_REF]. It has similarities with the work by [START_REF] Moulin | Explanation and argumentation capabilities: Towards the creation of more persuasive agents[END_REF], who argue that, in the context of knowledge-based decision support systems, integrating explanations and argumentation capabilities is a valuable perspective.

Although explanation and argumentation have long been identified as distinct processes [START_REF] Walton | Explanations and arguments based on practical reasoning[END_REF], it is also recognized that the distinction is a matter of context, hence they both play a role [START_REF] Scott Giboney | User acceptance of knowledge-based system recommendations: Explanations, arguments, and fit[END_REF] when it comes to eliciting an answer to a "why" question. This is exactly what is attempted in this paper, as we are providing "possible" explanations, that thus can be turned into arguments. The argument format has some advantages inasmuch as its uniformity allows us to express objection in an iterated way: "possible" explanations are challenged by counter-arguments that happen to represent rival, or incompatible, "possible" explanations. Some interesting issues remain to be studied. Among others, comparing competing explanations according to minimality, preferences, and generally a host of criteria.

We have designed a system in answer set programming that implements the explicative proposal introduced above. Indeed, answer set programming [START_REF] Gebser | Answer Set Solving in Practice[END_REF] is well fitted for this kind of problem. One obvious reason is that rules such as ( 5), ( 8) or ( 9) can be translated literally and efficiently. Also ASP is known to be good for working with graphs such as the one depicted in the figures of this text. We plan to include our system in an argumentative framework and think it will be a good basis for a really practical system, able to manage with as a rich and tricky example as the Xynthia example.

1 ISA-→ c 2

 12 Our approach assumes an elementary ontology in which specialization/generalization links between classes are denoted c n ISA -→ c m . Notation 2 An ISA -→ link has the form c where c 1 and c 2 are sorts in our logical language that denote classes such that c 1 is a subclass of c 2 in the ontology. Thus, ISA -→ denotes the usual specialization link between classes. E.g., we have Hurri ISA -→ SWind and House1FPA ISA -→ HouseFPA and HouseFPA ISA -→ BFPA 1 : the class Hurri of hurricane is a specialization of the class SWind of strong wing, the class House1FPA of typical Vendée low houses with a single level in flood-prone area is a specialization of the class HouseFPA of houses in this area, which itself is a specialization of the class BFPA of buildings in this area.

Notation 3 1 DEDO-→ Φ 2 1 DEDO-→ ϕ 2 . 1 DEDO-→ ϕ 2 1 ISA-→ c 2 1 ISA-→ class 2 ,

 31212121212 An ontological deduction link has the form Φ where Φ 1 and Φ 2 are two sets of literals. If Φ 1 = {ϕ 1 } and Φ 2 = {ϕ 2 } are singletons, we may actually omit curly brackets in the link {ϕ 1 } DEDO -→ {ϕ 2 } to abbreviate it as ϕ Such a link between literals ϕ actually requires that ϕ 1 and ϕ 2 are two literals built on the same predicate, say P . If ϕ 1 = ∃ P (c 1 ) and ϕ 2 = ∃ P (c 2 ), then ∃ P (c 1 ) DEDO -→ ∃ P (c 2 ) simply means that ∃ P (c 2 ) can be deduced from ∃ P (c 1 ) due to specialization/generalization links (namely here, the c link in the ontological model that relate the classes c 1 and c 2 mentioned in ϕ 1 and ϕ 2 ). Technically, the DEDO -→ links between literals are generated through a single principle: If in the ontology is the link class then, in the explicative model is the link ∃ P (class 1 ) DEDO -→ ∃ P (class 2 ) as well as the link ∀ P (class 2 ) DEDO -→ ∀ P (class 1 ) Also, the following links are added ∀ P (class i ) DEDO -→ ∃ P (class i ).

  ) and ∀ ¬P (class 2 ) DEDO -→ ∀ ¬P (class 1 ) ensue, and the links ∀ ¬P (class i ) DEDO -→ ∃ ¬P (class i ) are added whenever necessary. Let us provide an example from Xynthia, with a predicate Occ so that ∃ Occ(Hurri) intuitively means: some hurricane occurs. between literals introduced in (2) are extended to a relation among sets of literals (as announced in Notation 3), which is done as follows: Let Φ and Ψ be two sets of literals, we add to the explicative model Φ DEDO -→ Ψ if for each ψ ∈ Ψ, there exists ϕ ∈ Φ such that ϕ DEDO -→ ψ and for each ϕ ∈ Φ, there exists ψ ∈ Ψ such that ϕ DEDO -→ ψ.

( 3 )

 3 From[START_REF] Besnard | Deriving explanations from causal information[END_REF], we obtain that ψ DEDO -→ ψ is in the explicative model for each literal ψ. Accordingly, Ψ DEDO -→ Ψ is in the explicative model for each set of literals Ψ.

( 4 )

 4 Back to the hurrican illustration (Hurri ISA -→ SWind), the explicative model contains: {∃ Occ(Hurri), ItRains} DEDO -→ {∃ Occ(SWind), ItRains} but it does not contain {∃ Occ(Hurri), ItRains} DEDO -→ {∃ Occ(SWind) because, in the latter case, ItRains contributes nothing in the consequent. Definition 4 Items (2)-(3) give all and only the ontological deduction links Φ DEDO -→ Ψ (introduced in Notation 3) comprised in the explicative model.

Figure 1 :

 1 Figure 1: Ontological model for Xynthia ( ISA -→ links and constants)

Figure 2 :

 2 Figure 2: Part of the causal model for Xynthia

Fig. 3 FloodedFigure 3 : 4 Explanations 4 . 1

 33441 Figure 3: Part of the explicative model: data used to explain why there were numerous victims in low houses in the flood-prone area House1FPA

Figure 4 :

 4 Figure4: The schema of the explanation link from[START_REF] Scott Giboney | User acceptance of knowledge-based system recommendations: Explanations, arguments, and fit[END_REF]

Figure 5 :

 5 Figure 5: Explanation links follow DEDO -→ links [cf (9)]

Figure 6 :Definition 6

 66 Figure 6: Transitivity of explanations among sets of literals[START_REF] Lacave | A review of explanation methods for heuristic expert systems[END_REF]

OccFigure 7 :

 7 Figure 7: A few explanations for victims

FPA stands for some precise flood-prone area, BFPA for the buildings in this area, HouseFPA for the houses in this area and House1FPA for the one floor low houses in this area.

http://www.assemblee-nationale.fr/13/rap-info/i2697.asp

www.ccomptes.fr/Publications/Publications/Les-enseignements-desinondations-de-2010-sur-le-littoral-atlantique-Xynthia-et-dans-le-Var

Acknowledgements

It is our pleasure to thank the reviewers for their detailed and constructive remarks.


Camille Chane Simon
email: camille.simon@u-bourgogne.fr

Rainer Schutze

Frank Boochs

Franck S Marzani
email: franck.marzani@u-bourgogne.fr

Camille Simon Chane

Rainer Schütze

Fusion of Arbitrary Multi-view 3D Acquisitions

Keywords: 3D imaging, close-range photogrammetry, 3D registration, multi-view registration

come   L'archive ouverte pluridisciplinaire

Introduction

3D imaging is increasingly used for surface inspection. Creating 3D models with contact-less imaging sensors is a greatly useful. However, there are natural trade-offs between the field of view of these devices and their resolution. To scan large objects with a high resolution, multiple acquisitions with complementary fields of view are necessary. These views must then be stitched together to create a complete 3D model. This is the problem of high-accuracy multi-view registration.

In the case of spatially-structured objects, multi-view registration is most often based on the ICP algorithm [START_REF] Besl | A Method for Registration of 3-D Shapes[END_REF] or one of its variations [START_REF] Rusinkiewicz | Efficient variants of the ICP algorithm[END_REF]. This method does not perform well for smooth surfaces and requires a 30 % to 40 % overlap between successive views. On the other hand, basing the registration on feature points requires adequate salient points for each pair of datasets. Targets can be added to the surface if there are no natural salient features, if these features are not well-resolved or if they are insufficient in number. However, it can be tedious and time-consuming to place and remove targets from the multiple objects that we want to digitize. Furthermore, there is always the possibility that a target covers a defect that we want to detect. Also, there may be cases in which we fear damaging the surface with the targets.

Instead of relying on the object data, 3D registration can be based on the known position and orientation of the acquisition system. If a robot is used to control the acquisition system, its' position and orientation is readily available.

But the attained accuracy of 1 mm in untaught mode is not sufficient to base the registration solely on this data. Optical techniques such as photogrammetry can be used to increase robot positioning accuracy via on-line calibration. Two setups are possible: either a camera is fixed to the robot and observes the scene which has been covered with targets [START_REF] Hefele | Real-time photogrammetric algorithms for robot calibration[END_REF][START_REF] Clarke | The control of a robot end-effector using photogrammetry[END_REF] or the robot is itself covered with targets and observed by a number of fixed cameras [START_REF] Schütze | Optopose -a multicamera system for fast and precise determination of position and orientation for moving effector[END_REF][START_REF] Maas | Dynamic photogrammetric calibration of industrial robots[END_REF]. This second setup is the best suited to be adapted to our task. We can measure the precise position and orientation of the acquisition system in a coordinate system defined by set of fixed cameras which observe the acquisition system the camera by fixing targets to it, instead of on the object under study. Our goal is to register 3D datasets with a precision better than half the acquisition system resolution.

Materials and Methods

Principle

Figure 1 illustrates the principle of our technique: a set of photogrammetric cameras observe an acquisition system while it digitizes a surface. A targetframe is fixed to the acquisition system to enable the precise tracking of its' position and orientation during the measurements.

The scope of this experiment is to show that this setup can be used to track a under study is a car door which has been covered with un-coded photogrammetric targets. These targets are not used for the tracking and registration but they enable us to measure the accuracy of our registration procedure.

Materials

All 3D digitizations were performed using a commercial fringe projection digitization system by Gom, the Atos III. The system can be built in different field-of-view/ accuracy setups. For this study we use a 500 mm × 500 mm field of view, yielding an accuracy of 0.24 mm. In this configuration the measuring distance between the Gom Atos III and the surface under study must be 760 mm. This entails that our registration accuracy goal of half the acquisition resolution is 0.12 mm spatially and 0.158 mrad.

The characteristics of the materials used for the tracking procedure were optimized through several simulations, some of which are described in [START_REF] Simon | Asserting the Precise Position of 3D and Multispectral Acquisition Systems for Multisensor Registration Applied to Cultural Heritage Analysis[END_REF]. The four tracking cameras are 5 Mpx AVT Stingray and are used in conjunction with 8 mm Pentax lens. The tracking frame is an aluminum cube of edge length 500 mm and covered with 78 targets (see figure 2). A hexagonal headplate is attached to the bottom of the frame so that it can be fixed to a tripod. A hexagonal plate holder inside the cube on the bottom face is used to fix the acquisition system to the frame. An additional camera, a Nikon D300, is used for certain calibration procedures.

Calibrations

To ensure a precise tracking, all optics and objects in play must be carefully calibrated. We thus introduce the following coordinate systems, linked to the materials in use (illustrated figure 3):

• C S , (O S , x S , y S , z S ) is the coordinate system linked to the acquisition system.

• C F , (O F , x F , y F , z F ) is the coordinate system linked to the tracking frame.

• C Ci , (O Ci , x Ci , y Ci , z Ci ) are the coordinate systems linked to the each tracking camera. O Ci is the optical center of the camera, ( x Ci , y Ci ) define the image plane, z Ci is collinear to the optical axis.

• C 0 , (O 0 , x 0 , y 0 , z 0 ) is the world coordinate system.

We now describe the necessary calibrations and how they are performed.

The following notations will be used throughout this section: A| C U are the homogeneous coordinates (x A , y A , z A , 1) of point A in coordinate system C U .

We define T C U ,C V the transformation matrix between two coordinate system

C U and C V such that T C U ,C V • A| C V = A| C U . ⃗ y F O F ⃗ z F ⃗ y 0 O 0 ⃗ z 0 ⃗ y S O S ⃗ z S A Photogrammetric cameras
Acquisition system Surface under study Tracking frame side and observe the same area so that they can be calibrated together. Previous experience has taught us that the internal orientation can stay stable for over a week if the cameras are handled with care during this time frame. We thus perform the interior orientation a few days before the acquisitions.

⃗ y C1 O C1 ⃗ z C1 ⃗ y C2 O C2 ⃗ z C2
Internal orientation of the acquisition system. It is also necessary to know the distortions introduced by the acquisition system. In the case of the Atos III there is a specific calibration procedure to perform and the output data is corrected by the acquisition software.

Calibration of the tracking frame. The cube calibration is performed by taking over one hundred images of the tracking cube with a scale bar and additional targets. We can then define C F and know the position of each coded target in this coordinate system.

Orientation of the acquisition system with respect to the tracking frame. To know the position of the acquisition system in the system defined by the tracking cube, we proceed in three steps:

1. Fix the acquisition system to the tracking frame.

2. Use the acquisition system to digitize a target-covered 3D object.

3. Take approximately fifty photos of the tracking frame and target 3D object.

We associate a coordinate system C t emp to the 3D object.

Step 2 provides us with the position and orientation of the 3D object in C S , that is T S,temp .

Similarly, step 3 provides us with the position and orientation of the 3D object in C F previously defined, that is T F,temp .

We can thus easily calculate the transformation between C S and C F :

T F,S = T F,temp • (T S,temp ) -1 .
Relative orientation of the photogrammetric cameras. Once the tracking cameras have been positioned to observe the area in front of the surface under study, we can measure their relative position and orientation. This is done by acquiring approximately eighty images of a scalebar in various positions and orientations, simultaneously by the four calibrated cameras.

The position and orientation of the other three cameras are measured with respect to the first.

We choose the coordinate system of the first tracking camera as the world system: C 0 = C C1 .

Data processing

We thus have the necessary data to register the data sets. Each acquisition provides us with the coordinates of a group of surface points in the sensor system, A| C S . The simultaneous tracking provides us with T C0,C F . The known interior orientation of the tracking cameras and their relative orientation ensures that T C0,C F is sufficiently precise.

We can thus calculate A| C0 , the coordinates of the surface points in the world system using:

A| C0 = T C0,C F • T C F ,C S • A| C S . (1)
Tracking and calibration software. We rely on two pieces of software for all photogrammetric image processing: Tritop Deformation Software1 and i3AxOri, a lab-developed software based on the AxOri photogrammetric bundle adjustment library2 . Tritop is used to recognize the coded and uncoded points in the images and to compute a first assessment of the position of the cameras. This data is then exported to i3AxOri in which we have more flexibility and control on what we want to compute given our input parameters.

Evaluating the registration accuracy. The Gom Atos III acquisition software recognizes any coded or uncoded target present in the scene. In our case the full surface is covered with over 60 uncoded targets of diameter 6 mm. More than 10 targets are visible in each mesh. It is possible to export the list of the targets visible in each mesh. We thus have A ij C S the coordinates of target point j from mesh i.

Using equation 1 we calculate A ij C0 for all targets of all meshes. For every target j we now calculate

D (i,k) j = A ij C0 -A kj C0 (2)
for every pair of meshes (i, k) where target j is visible. If the registration were perfect, the result of this subtraction would be a null vector. Since the registration is imperfect, D (i,k) j provides us a measure of the accuracy of the registration.

Results and Discussion

The various acquisitions were performed in the following order: All accuracy values (simulations and measurements) are given at 2 σ.

Individual calibrations

The accuracy of the individual calibrations described in section 2.3 is given in table 1. When available, they are compared with the expected accuracy of the simulations. The simulations were run with two levels of noise: low noise, corresponding to a best-case-scenario and higher noise, corresponding to a realistic situation. The accuracy reached is always in between the results for these two configurations.

The tracking cube calibration is performed with much higher accuracy than expected, though this value does not take into account the deformation that the aluminum cube may undergo due to temperature variations for example.

M4 M2 M5 M17 M11 M1 M12 M15 M14 M16 M6 M0 M10 M7 M9 M13 M8 X Z M0 M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17

Tracking the frame

Eighteen acquisitions are performed with the 3D digitization system. They are numbered M0 to M17. The relative position and orientation of the four tracking cameras and the tracking cube is illustrated figure 4.

This subsection examines the accuracy with which we evaluate T C0,C F for each position. The spatial and angular accuracy achieved are compared to the simulation results of the realistic scenario figure 5. Though the spatial accuracy of the results is not as good as expected, the angular accuracy is always better than the value reached during the simulations.

Compared to the tracking accuracy target fixed by our registration goal (0.12 mm spatially and 0.158 mrad), the results are quite satisfying: the target spatial accuracy is always reached with a comfortable margin while the angular accuracy is insufficient for only three positions: M7, M9 and M10.

Registration

***

Conclusion and Perspectives

• Contactless 3D acquisition and registration

• No need for targets

• Accuracy: ***

  3D acquisition system and to register the individual 3D acquisitions. The object

Figure 1 :

 1 Figure 1: Overview of tracking configuration

Figure 2 :

 2 Figure 2: Tracking cube covered with 78 targets with the acquisition system fixed inside the cube. The car door is visible in the background.

Figure 3 :

 3 Figure 3: Coordinate systems defined for the tracking procedure.

1 . 2 . Tracking frame calibration 3 .

 123 Photogrammetric cameras interior orientation (three days in advance) Acquisition system interior orientation 4. Tracking frame to acquisition system orientation 5. Simultaneous digitization of the car door and photogrammetric tracking of the frame 6. Photogrammetric cameras exterior orientation We first quickly present the accuracy achieved for the individual calibrations (steps 1 to 4 and 6). Then we analyze the accuracy with which we track the target frame (5) and finally we look in to the accuracy of the final registration.

  angular accuracy Green: angular accuracy at 2 σ < 0.176 mrad Red: angular accucarcy at 2 σ > 0.2112 mrad M3

Figure 4 :Figure 5 :

 45 Figure 4: Relative position of the cameras (fixed, blue) and the tracking frame for all acquisitions. *** Redo: colors, orientation, remove text on figure and title

Table 1 :

 1 Individual calibrations accuracy compared to expected accuracy from the simulations.

		Measures	Simulations	Unit
			realistic	best
	Photogrammetric cameras I.O.	0.055	0.1	0.033 pixel
	Acquisitions system I.O.	0.020	***		pixel
	Tracking frame calibration	0.015	0.05		mm
	Tracking frame to acquisition	0.025	-		mm
	system orientation	0.050	-		mrad
	Photogrammetric cameras E.O.	0.017 0.030	0.03 0.02	0.01 0.04	mm mrad

Gom, Tritop Deformation Software, http://www.gom.com/3d-software/ tritop-deformation-software.html

Axios

3D, Axori photogrammetric bundle block adjustment, http://www.axios3d.de

Acknowledgments

We would like to thank the Conseil Régional de Bourgogne, France , as well as i3mainz laboratory, Germany, for their funding and support. GOM lent us with a research license for Tritop which was greatly appreciated. Jens Bingenheimer helped perform the measurements. We thank him for his enthusiasm and his meticulosity.

• Can be extended to 2D projection on 3D model • Aluminum profiles are fine for rapid prototyping but they are heavy and temperature dependent: a real object should be made of carbon


Eric W M Reinheimer

Damian R S Jankowski

Roman S Wietlik •

Marc Fourmigue´

Fourmigue´

Wietlik

Hybrid Material Based on the Lindquist Polyoxometalate [W6O19]22 and the organosulfur donor o-Me2TTF: A Combined Structural and Spectroscopic Study

. and c = 95.6950(10) . Resolution of the solid state structure revealed that the o-Me2TTF radical cations aggregate as isolated face-to-face dimers with intradimer interactions between neighboring sulfur atoms at distances \3.6 A° . Hydrogen bonding was also observed between hydrogen atoms bound to sp 2-hybridized carbon atoms of o-Me2TTF and bridging oxygen atoms of [W6O19]2-. Single crystal IR and Raman spectra were also collected and provide further evidence that the o-Me2TTF donors have been oxidized to their corresponding radical cationic states.

Introduction

A focus of contemporary Materials Science is the preparation of molecule based materials that display active physical properties such as conductivity, ferromagnetism or non-linear optical properties [1]. With the initial discovery of metallic conductivity in TTF-TCNQ as well as the first and second generation of organic superconductors, materials containing oxidized organosulfur (or organoselenium) donors were identified as those most likely to display elevated conductivity (or superconductivity) [2][3][START_REF] Williams | Organic superconductors[END_REF]. Most advantageous to the observation of either non-activated conductivity or superconductivity within these materials was the solid state aggregation of the oxidized donors into long range stacks which created a more facile pathway for the transfer of itinerant electron density.

As the study into electroactive materials persisted, interest in materials that displayed multifunctionality began to manifest. Through the directed, rational synthesis of solid networks in the form of salts, where each ionic network furnished a distinct physical property to the material, the multifunctional materials known as magnetic conductors were prepared [5]. The molecular ferromagnet, (BEDT-TTF)3[MnCr(C2O4)3]. CH2Cl2 (BEDT-TTF = bis(ethylenedithio)tetrathiafulvalene or ET) served as a ''proof of concept'' because conducting layers of partially oxidized ET donors were successfully combined with paramagnetic layers of oxalato-bridged bimetallic complexes to form a material which displayed both metallic conductivity (rRT = 250 S cm -1) and ferromagnetically coupled metal atoms (Tc = 5.5 K) [6].

Oxalato-bridged bimetallic anions are not the only class of inorganic anions studied in the preparation of hybrid 123 J Chem Crystallogr (2013) 43:178-186 salts containing derivatives of TTF. In an effort to further study how novel lattice architectures and physical properties resulting from the association of organic radical cations can be affected by large anions, the use of large polyoxometalate clusters as anions in hybrid salts has been probed. Due to their elevated solubility in organic solvents, ability to possess variable charges and shapes which may alter the corresponding band structure and possibility of incorporating paramagnetic metal atoms into the polyoxoanion core, this class of anions is an ideal candidate for incorporation into hybrid and multifunctional materials [7].

With the Lindquist clusters, with the formula (M6O19)2( M = Mo or W), 2:1 phases involving TTF, TMTTF (tetramethyltetrathiafulvalene), ET, BET-TTF (bis(ethylenethio) tetrathiafulvalene), TPTTF (tetraphenyltetrathiafulvalene) and DMDPTTF (dimethyldiphenyltetrathiafulvalene) have been reported [8][9][10][11][12][13]. Later a 2:1 phase involving [W6O19]2and the donor bis(propylene)dithiotetrathiafulvalene (BPDTTTF) was communicated [14]. In all cases, those salts were found to contain isolated dimers of radical cations and were insulating. Meanwhile, 3:1 and 4:1 phases containing TTF and TMTSF were discovered and found to be semiconducting [15,16]. Finally, a 5:1 hybrid salt involving ET and the substituted Lindquist anion [VW5O19]3-which displayed metallic conductivity, was reported by Ouahab and coworkers in 1993 [17]. While the Lindquist polyoxometalates have yielded a large number of results, other polyoxoanion clusters have also yielded salts when combined with TTF derivatives such as ET. When ET was combined with the polyoxovanadate cluster [20]. Of these salts, the latter two are the most interesting as the Keggin clusters form a chainlike heteropolyanion during the course of the electrocrystallization [21].

Considering the facility by which TTF derivatives have been combined with polyoxoanion clusters, especially the dianionic Lindquist polyoxometalates, we sought to prepare a hybrid salt that combined this cluster type with the TTF derivative o-3,4-dimethyltetrathiafulvalene (o-Me2TTF) (Fig. 1). This member of the TTF family has been used by the groups of Dunbar and Fourmigue´ as a starting material to prepare the redox active chelating diphosphine o-P2 used in various homoleptic transition metal coordination compounds [22][23][24][25][26][27][28]. Prior to 2007, the only salt of o-Me2TTF known was (o-Me2TTF) [ReO4] reported in 1993 where it was oxidized to the ?1 oxidation state [29]. Later, the crystal structure of the neutral donor was published in 1998 [30]. In 2008, Fourmigue´ et al. [31] reported the structures and properties of a series of isostructural, tetragonal 2:1 salts with the halides Cl -,Br -and I which possessed metallic conductivity at room temperature. In 2008 and 2009, Reinheimer et al. [32][33][34] published separate communications describing the syntheses and structural properties of the salts (o-Me2TTF)[I3], (o-Me2TTF) [BF4] and (o-Me2TTF)2[Re2Cl8] where like the [ReO4] -salt, the o-Me2TTF donor exists as its singly oxidized radical cation. Two years later in 2010, Reinheimer and coworkers reported the 1:1 complex o-Me2TTF-TCNB (TCNB = 1,2, 4,5-tetracyanobenzene), which emphasized the complex's hydrogen bonding network [35].

Herein we present the structure of the hybrid salt (o-Me2TTF)2[W6O19] which combines the unsymmetrical TTF derivative o-Me2TTF with the closed shell Lindquist polyoxometalate [W6O19]2-. This structure, together with its single crystal IR and Raman spectroscopy data, are useful for evaluating the o-Me2TTF oxidation state through comparison to theoretical calculations of IR and Raman active modes.

Experimental Section

Preparation of Compounds

For the preparation of the title compound, crystals formed on platinum electrodes of dimensions 1 mm in diameter and 2 cm in length in the anodic compartment of a standard electrochemical H-cell under low constant current [36]. The organosulfur donor o-Me2TTF and the polyoxoanion (TBA)2[W6O19] were prepared via published techniques [37,38]. The solvent CH3CN, used in the electrochemical reactions, was not dried prior to its use. solution in the anodic compartment and exposed to a constant current density of 1.0 lA to yield black block-like crystals on the electrode surface which were harvested after a period of 2 weeks, washed with CH3CN, and dried in air.

X-ray Crystallographic Details and Structure Solution

Diffraction quality crystals were grown on the surface of platinum electrodes by slow anodic oxidation of o-Me2TTF in the presence of electrolytic solutions of (TBA)2[W6O19] dissolved in CH3CN. A black, block-like crystal of dimensions 0.20 9 0.10 9 0.06 mm3 was secured to a glass fiber using paratone oil and placed into the cold gaseous nitrogen stream of a Bruker APEX II diffractometer using Mo-Ka radiation (k = 0.71073 A° )at110 ± 2 K. All data collections were conducted using graphite monochromated radiation and were corrected for Lorentz and polarization effects. Data collection and initial indexing were handled using SMART [START_REF] Smart | Software for the CCD detector system, version 5.01 (NT). Bruker Analytical X-ray Systems, Madison 41. Blessing RH[END_REF]. Frame integration, Lorentz-polarization corrections, and final cell parameter calculations were carried out using SAINT [40]. Multi-scan absorption corrections were performed using SADABS [41]. The space group P 1 was unambiguously assigned by analysis of symmetry and systematic absences using XPREP and further verified by PLATON [START_REF] Sheldrick | XPREP, program for space group determination[END_REF][START_REF] Speck | PLATON, a multipurpose crystallographic tool[END_REF]. The structure was solved using Patterson methods and difference Fourier techniques. The final structural refinement included anisotropic temperature factors on all non-hydrogen atoms, with the exception of atoms C4 and O10 which could not be satisfactorily refined anisotropically. Structure solution, refinement, graphics, and creation of publication material were performed using SHELXS, SHELXL and XSEED [START_REF] Sheldrick | SHELXS-97, program for crystal structure determination[END_REF][45][46]. Hydrogen atoms were inserted at calculated positions and constrained with isotropic thermal parameters. The solid state structure with anisotropic displacement ellipsoids at 50 % probability is shown in Fig. 2. The central C=C bond and C-S bonds of o-Me2TTF are the most susceptible to its oxidation state and have been used by Coppens and coworkers [47] to develop an empirical relationship that can be used to calculate the overall oxidation state of the donor molecules. Information regarding refinement details and bond distances for (o-Me2TTF)2[W6O19] as well calculated average changes for the o-Me2TTF donor in the title salt and other materials are listed in Tables 123. CCDC file 890234 contains the supplementary crystallographic data for this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif.

Single Crystal Spectroscopic Studies

Raman spectra within the wavenumber range 50-3,500 cm -1 were measured on single crystals of (o-Me2TTF)2[W6O19] using backscattering geometry. The measurements at room temperature were performed on a Labram Horiba Jobin Yvon 800 spectrometer equipped with He-Ne laser (632.8 nm) and a liquid-nitrogen cooled CCD detector. To avoid sample overheating and possible decomposition, the power of the exciting laser beam was reduced to *0.1 mW. The electrical vector of the exciting laser beam was nearly parallel and perpendicular to the maximum measured intensity for the C=C stretching vibrations of o-Me2TTF, individual molecules of which were aggregated into dimers, approximately perpendicular (E\a) and parallel (E||a) to the crystallographic a-axis. Polarized reflectance spectra were measured from the bestdeveloped crystal face of single crystals (700-16,000 cm -1). The spectra were measured for various polarizations of the incident IR beam, which corresponded to maximum and minimum measured intensities of the electronic band at 10,800 cm -1 with electrical vector orientations nearly parallel or perpendicular to the (o-Me2TTF)22? stacking axis. The spectra were recorded using a Bruker Equinox 55 FT-IR spectrometer equipped with a Bruker microscope Hyperion 1000. The reflectance data were normalized to the reflectance of a high quality aluminum mirror.

Results and Discussion

X-Crystallography

The solid-state structure for (o-Me2TTF)2[W6O19] represents the first charge-transfer salt between the donor o-Me2TTF and the dianionic, diamagnetic Lindquist polyoxoanion [W6O19]2-. It has been well established by our group and others that in the absence of any intermolecular contacts such as p-p or S S interactions, neutral chalcofulvalene donors are nonplanar, often exhibiting significant bends of up to 30. along dithiole or diseleno bridges [22][23][24][25][26][27][28]. The solid-state structures of neutral tetrathiafulvalenebased molecules are often controlled by weak van der Waals forces. These interactions define the molecular shape, size, and close packing properties in the solid-state and govern the formation of different patterns of association such as the herringbone conformation in a TTF polymorph and the sandwich herringbone pattern of dimers in the structure of neutral BEDT-TTF [48,49].

Upon oxidation of the o-Me2TTF donor, structural changes in the molecule occur, including the adoption of a planar conformation followed by a lengthening of the central C=C bond and a shortening of the C-S bonds in the central TTF core [50][51][52][53][54][55][56][57][58][59][60]. Using the Coppens' formula, the oxidation state for the o-Me2TTF donors was confirmed to be ?1; this was further confirmed upon comparison of the central C=C and mean C-S bond distances from (o-Me2TTF)2[W6O19] to those for other materials known to contain oxidized o-Me2TTF moieties [29][30][31][32][33][34]47].

For the title salt, the o-Me2TTF donors exist as strong face-to-face dimers of their corresponding radical cations (Fig. 2a). While this was surmised on the basis of calculation using the Coppens' formula and direct comparison to other o-Me2TTF-containing materials, this aggregation into dimers within the solid state also furnishes a structural indicator. In previous cases of TTF and its derivatives with the dianionic Lindquist polyoxoanions published to date, most salts contain donors in the ?1 oxidation state [8][9][10][11][12][13]. In those and other cases where radical cations of chalcofulvalene donors [START_REF] Speck | PLATON, a multipurpose crystallographic tool[END_REF] are present, according to a review by Fourmigue´ and Batail [61], the radicals form strong face-to-face dimers with short interatomic distances between their chalcogen atoms being less than the sum of the van der Waals radii. This eclipsed face-to-face interaction between radical cations corresponds to the formation of r-bonding and r*-antibonding combinations of the two SOMOs, a dispersion of molecular orbitals fundamentally equivalent to that seen for diatomic hydrogen. As each SOMO contributes a single electron to the r-bonding MO, a strong r bond occurs between the radicals ultimately resulting in charge localization and insulating properties. A projection in the (111) plane illustrating the formation of the strong, eclipsed face-to-face interactions is presented in Fig. (o-Me2TTF)[ReO4], but is different from the orientation found in (o-Me2TTF)2[Re2Cl8] where a head-to-head orientation of radicals is observed [32][33][34].

Commencing with the unit cell, a dimer containing o-Me2TTF radicals was found to be nearly parallel to the ab plane, others can be generated through translations in all directions. For translations along the b-and c-axes, the dimers are effectively separated by the void containing the [W6O19]2-polyoxometalate. The polyoxoanion itself is centrally located within the ac plane and related to other symmetry generated polyoxoanions through translations in all three dimensions. Structural features of the [W6O19]2polyoxoanion from (o-Me2TTF)2[W6O19] are consistent with those observed previously in other TTF-based salts utilizing this anion [8][9][10][11][12][13][14].

For o-Me2TTF radical dimers related by translations along the a-axis, their stacking is reminiscent of a stair-step pattern and reveals no close interatomic interactions between neighboring sulfur atoms at distances \3.6 A° (Fig. 2c). Despite appearing that there may be long range interactions between sulfur atoms along the a-axis, there is no satisfactory long range stacking between the oxidized 1.340(4) 1.762(3) 0 [30] 1.342(7) 1.755(2) 0 [33] 1.365(5) 1.738(4) ?0.48 [31] 1.365(5) 1.738(4) ?0.48 1.39(1) 1.716(5) ?1.06 [29] 1.401(5) 1.722(4) ?1.14 [32] 1.386(4) 1.721(2) ?0.95 [33] 1.401(3) 1.715(3) ?1.22 [34] 1.373(2) 1.717(4) ?0.82 1.391(5) 1.716(4) ?1.07 This study 1.391(5) 1.716(4) ?1.07 o-Me2TTF moieties which could lead to either activated or non-activated conductivity as short intradimer (\3.6 A° ) and longer interdimer ([3.6 A° ) interactions exist in an alternating pattern involving the o-Me2TTF radicals and successive dimers respectively along a.

Among the solid structure for (o-Me2TTF)2[W6O19], hydrogen bonding interactions, which can be suggested if interatomic distances between oxygen and hydrogen atoms are\2.70 A° (the sum of their corresponding van der Waals radii), were found between hydrogen atoms bound to sp 2-hybridized carbon atoms from o-Me2TTF and the bridging oxygen atoms of the [W6O19]2-polyoxoanion clusters. A 1999 report by Batail and coworkers [62] demonstrated, by the use of by electrostatic potential maps, that bridging oxygen atoms in these clusters are more basic and thus better hydrogen bond acceptors than those in terminal positions. Within our structure (Fig. 3), short Csp2-H O interactions of 2.296 and 2.512 A° exist between H8-O9 and H7-O4 respectively within the (111) plane. In both cases, a bridging oxygen atom from the tungsten cluster acts as a hydrogen bonding acceptor. While other interactions between hydrogen and oxygen atoms at distances \2.70 A° were found, their close approximation (within *0.03-0.07 A° ) to the threshold value of 2.70 A° makes their identification as actual hydrogen bonding interactions problematic. The fact that the observed Csp2-H O interactions are well below the threshold value of ° 2.70 Aand involve bridging oxygen atoms from the [W6O19]2-cluster, provides structural support that those particular oxygen atoms from the cluster, with their elevated basicity, are better hydrogen bond acceptors than their terminal counterparts. Single Crystal Spectroscopic Properties

In previous spectroscopic measurements on o-Me2TTF containing materials, particularly the neutral donor and the 2:1 salts (o-Me2TTF)2[X] (X =Cl -,Br -and I -) where 123

Fig. 3 Detail of the hydrogen bond pattern (see text). Anisotropic displacement ellipsoids have been set to 50 % probability o-Me2TTF bears a ?1/2 charge, IR and Raman spectra were successfully measured [63]. During the course of those studies, the normal vibrational modes of the neutral and the radical cation, as well as their theoretical IR and Raman spectra, were calculated by quantum chemical methods. Utilizing these theoretical calculations, experimentally observed vibrational features of o-Me2TTF were assigned [63]. For (o-Me2TTF)2[W6O19], the solid state structure reveals that each donor molecule bears a ?1 charge and aggregates with other o-Me2TTF radical cations to form isolated dimers. Upon consideration of the structural features of the (o-Me2TTF)2[W6O19] salt, the isolated dimer model can be used to explain the features of the spectra. In this model, one can distinguish two dimeric modes: the in-phase vibration of monomers, which is Raman active, and the out-of-phase mode, which is IR active due to coupling with the electronic transition.

The Raman spectra of the title salt are displayed in Fig. 4 for two polarizations of the incident laser beam which were nearly parallel and perpendicular to the stacking axis of the o-Me2TTF radical cations in the isolated dimers, that being the crystallographic a-axis. For perpendicular polarization, the strongest bands for o-Me2TTF ? are attributed to the C=C stretching modes (m9 = 1,556 cm -1, m10 = 1,483 cm -1, m11 = 1,417 cm -1) and the ring breathing mode (m38 = 518 cm -1) [63]. These experimental data compare closely with those stretches determined theoretically: m9 = 1,603 cm -1, m10 = 1,546 cm -1, m11 = 1,436 cm -1, m38 = 482 cm -1 [63]. In the spectrum we also find some other bands related to the o-Me2TTF radical cations: m20 = 1,101 cm -1 (C-H bend), m21 = 1,036 cm -1 (C-C stretch), m25 = 975 cm -1 (C-S stretch), m26 = 937 cm -1 (C-S stretch, C-C stretch), m31 = 754 cm -1 (rock CH3), m32 = 720 cm -1(C-S stretch), m34 = 627 cm -1 (in-plane deformation), m40 = 473 cm -1 (ring breath), m45 = 274 cm -1 (in-plane deformation), m47 = 234 cm -1 (out-of-plane deformation) [63]. It is well-known that the C=C modes of TTF derivatives are very sensitive to the degree of oxidation and as such are often used for the determination of charge density distribution [47]. Based on the behavior of these modes for charge transfer salts of TMTTF, ET and bis(ethylenedioxo) tetrathiafulvalene (BEDO-TTF), their C=C bond frequencies depend linearly (or nearly linearly) on charge density. When the average oxidation state grows, the wavenumber for the C=C modes decreases considerably by *60-110 cm -1 per ?1 charge. For the title salt, a similar nearly linear charge dependency was observed for the analogous C=C modes of the o-Me2TTF donor and is represented within Fig. 5 where we outline the spectral response for selected Raman active bands for o-Me2TTF0, o-Me2TTF?0.5 and the present data for o-Me2TTF ? . Concurrently, when the donor's oxidation state increases, the ring breathing mode, m38, shifts towards higher wavenumbers (*23 cm -1 per ?1 charge). Such spectroscopic behavior is equivalent to that seen for other TTF derivatives [65][66][67].

The polarized reflectance spectrum for (o-Me2TTF)2

[W6O19] is shown in Fig. 6 for polarizations parallel (E||a) towards lower wavenumbers because of the e-mv coupling effect. Similar shift patterns were also observed for charge transfer salts containing other TTF derivatives, thus pro 1600 viding evidence suggesting strong electronic coupling.

In the IR spectrum we also find two bands related to C-S 0.0 0.5 1.0 .11 .10 .9 stretching modes: the m25 mode at 988 cm - Fig. 6 Reflectance spectrum of (o-Me2TTF)2[W6O19] salt for two perpendicular polarizations, corresponding to maximum (E||a) and minimum (E\a) intensity of the electronic transition at 10,800 cm -1 and perpendicular (E\a) to the dimer axis. For polarization nearly parallel to the dimer axis, we observe an electronic band at *10,800 cm -1, corresponding to an intradimer charge transfer transition. This band is a characteristic feature of the salts containing dimers of TTF-type radical cations; for example, in (ET)2[Mo6O19] and (DMtTTF) [Br], this band was located at 7,000 and 8,300 cm -1 respectively [68]. It is remarkable that in (o-Me2TTF)2 [W6O19], this electronic transition is located at higher frequency, i.e. the interaction between the o-Me2TTF radical cations within the dimer is smaller. The relatively strong vibrational feature at 1,350 cm -1 (E||a) is related to the effect of electron-molecular vibration (e-mv) coupling of the m11 mode (C=C central bond stretching) with the charge-transfer transition. In comparison with the analogous Raman band, this IR feature is shifted by 67 cm -1 structural features are strongly indicative of regions of charge localization which ultimately result in insulating properties. Hydrogen bonds between bridging oxygen atoms of the polyoxometalate cluster and the o-Me2TTF radicals were also observed at distances well below the threshold value of 2.70 A° . As these interactions involve the bridging oxygen atom of the cluster, it supports the computationallydetermined hypothesis that bridging oxygen atoms of polyoxometalate clusters, with their heightened basicity, are better hydrogen bond acceptors than their terminal analogues. Single crystal spectroscopic studies were also completed on the IR and Raman active modes of the oxidized o-Me2TTF donor. Identification and analysis of these modes correlate with theoretical spectra for o-Me2TTF ? determined using quantum chemical methods. The frequency change for the C=C mode in o-Me2TTF with respect to oxidation state follows a linear (or nearly linear) trend as seen for other TTF derivatives. Finally, a strong intradimer charge transfer transition was observed at *10,800 cm -1, which is indicative of the dimerization of TTF-type radical cations in the solid state. The title salt adds another 2:1 phase to the library of salts containing

  [H3V10O28]3-and the Dawson-Wells anion [P2W18O62]6-, the salts (ET)5[H3V10O28] 4H2O and (ET)11[P2W18O62]. 3H2O resulted and were found to possess metallic conductivity [18, 19]. Finally, Keggin polyoxometalate clusters with FeIII the generic formulae [XZM11O39]5-(XZM11 = SiIVCrIII Mo11,SiIVW11,PVCoIIW11,PVNiIIW11,PVCuIIW11, PVZnIIW11,PVMnIIW11 and PVMnIIMo11)were combined with ET to make the salts (ET)8[XZM11O39](XZM11 = FeIIICrIII SiIVMo11,SiIVW11,PVCoIIW11,PVNiIIW11,PVCuII W11 and PVZnIIW11)and (ET)8n[PVMnIIM11O39](M = Mo or W)

  Fig. 1 o-Me2TTF

Fig. 2 (

 2 Fig. 2 (a) View of the unit cell of (o-Me2TTF)2[W6O19] illustrating the tendency of the o-Me2TTF radical cations to form face-to-face dimers in the solid state. (b) Detail of the face-to-face overlap between cation radicals. (c) Chain of dimers running along the a axis. The atoms are draw with van der Waals radii

  Final R indices [I [ 2r] R1 = 0.0222, wR2 = 0.0550 R indices (all data) R1 = 0.0230, wR2 = 0.0555 Goodness-of-fit on F2 1.155 2 22]1/2 R1 = R||Fo| -|Fc||/R|Fo|. wR2 = [R [w(Fo -Fc)2]/R [w(Fo)2. )]1/2 Goodness-of-fit = [Rw(|Fo| -|Fc|)2/(Nobs -Nparameter ° Table 2 Bond distances for (o-Me2TTF)2[W6O19]in A W(1)-O(1) 1.694(4) W(3)-O(8) 1.929(3) W(1)-O(2) 1.917(4) W(3)-O(9) 1.706(4) W(1)-O(3) 1.928(3) W(3)-O(10) 2.3268(2) W(1)-O(6) 1.926(3) S(1)-C(1) 1.721(5) W(1)-O(8) 1.920(3) S(1)-C(2) 1.714(5) W(1)-O(10) 2.3243(2) S(3)-C(4) 1.706(5) W(2)-O(2) 1.910(4) S(3)-C(6) 1.736(5) W(2)-O(4) 1.922(3) S(4)-C(4) 1.718(5) W(2)-O(5) 1.709(4) S(4)-C(7) 1.739(5) W(2)-O(6) 1.931(3) C(1)-C(4) 1.391(6) W(2)-O(7) 1.924(3) C(2)-C(3) 1.332(8) W(2)-O(10) 2.3221(3) C(5)-C(6) 1.502(7) W(3)-O(3) 1.910(3) C(6)-C(7) 1.336(7) W(3)-O(4) 1.931(3) C(7)-C(S) 1.499(7) W(3)-O(7) 1.911(3)

  with the formula (o-Me2TTF)2[W6O19] A Q =-17.92 ?23.43 9 A (A/B) from Ref.

  2b

  . The head-to-tail orientation of the o-Me2TTF radicals for (o-Me2TTF)2[W6O19] is similar to that seen in (o-Me-2TTF)[I3], (o-Me2TTF)[BF4] and

Table 1

 1

	X-ray crystallographic and refinement data for (o-Me2TTF)2
	[W6O19]
	Compound (o-Me2TTT)2[W6019]
	CCDC code 890234
	Formula C16H16S8W6O19
	Formula weight 1871.51
	Temp. 110(2)
	Space group P1
	a, A7.6211(7)
	°
	b, A9.5231(9)
	°
	c, A12.2148(11)
	°
	a, .
	105.5870(10)
	b, .
	106.8340(10)
	c, .
	95.6950(10)
	°3
	Volume, A802.10(13)
	Z1
	Density (calculated), mg/m3 3.845
	l,mm -1 22.026
	Scan x scan
	h range for data collection, .
	1.83-28.66
	Reflections measured 4092
	Independent observed reflns. 3707
	Independent reflns. [I [ 2r] 3613
	Data/restraints/parameters 3707/0/216
	Rint 0.0296

Acknowledgments This work was supported by the Polish-French Polonium 2009-2010 Program No. 20083YF, the CNRS-PAN program, and by the French Agence Nationale de la Recherche (Contract No. BLAN08-3 317277) for financial support. MF also thanks the ANR for additional financial support (Contract No. 08-BLAN-0140 3/4-Filled).

EWR thanks the French Government for financial support in the form of a Chateaubriand Fellowship (Dossier #634148J). We also thank Thierry Roisnel (CDIFX Rennes) for the X-ray data collection.