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November in Washington, D. The only characterisation of words that will be taken into account is that words are not, as de Saussure would have them, seen as pairs of form and meaning, but words are seen as more abstract entities, where the same word can have various meanings. Karl Heinz Berger Pseudonyme:

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Because the system of FCA is completely symmetrical with respect to objects and attributes, we can do the same for attributes: If we use this convention on the lattice for our toy model, we get the diagram represented in figure 2. Often you will see mixed versions of these two systems, where both concept labels and intent and extent are represented. This of course will be redundant information, but helps to make the lattice intelligible.

Also the notion of a concept that is defined by FCA has as such nothing to do with concepts. Formal concepts are no more than what they are defined to be: The system acquires a meaning only by giving an interpretation to these entities; making these entities stand for something.

Concept Lattice with Intent and Extent pretation of Formal Concept Analysis is to interpret the formal objects as representing real objects in the external world, and the formal attributes as representing the properties that these objects have. But that is by no means the only possible interpretation.

Another possible interpretation however pointless is the following: In other words, interchange the interpretation of objects and attributes. That would lead to formal concepts where the extent of the formal concept is the intention of the real world concept; in which the sub-concept relation indicates a real-world super-concept, etc. The resulting system would be almost identical to the more natural one isomorphic when the order is reversed , since in FCA objects and attributes are defined completely symmetrical.

The only difference would be that all the names of the formal properties would be confusing at least. A more interesting alternative interpretation of FCA, and the one that this thesis will focus on, is one relating to words and word meanings.

She also describes a different kind of context, which she calls connotative contexts. In a connotative context, the formal objects are word meanings, and the formal attributes are the attributes related to these word meanings: A connotative context K K: The set of particular meanings is denoted by M W , the set of features of the particular meanings by A K, and a relation that assigns features to particular meanings by I K.

The concept lattic K K of a connotative context K K. On top of the denotative structure and the connotative structure both of which are defined in terms of FCA , there is also a lexical structure, in which the different aspects of the word itself are modelled. This lexical structure is not defined in term of FCA. The structure in figure 2.

The denotation of a noun is the set of objects in the world denoted by the word, whereas the connotation is the way in which this object is denoted. It describes these aspects in a very strict and precise way, which has both an advantage and a disadvantage: But on the negative side, this also makes it very clear where the theory assumes too much structure on the world and on words. By the very fact that there is a denotative context, the theory assumes the world to be nicely ordered in objects and attributes.

Also, by the fact that the denotative word concepts form a subset of the denotative concepts, it assumes our lexicon to correctly follow this inherent structure of the world. Since there is a mapping dnt from the connotative word concepts to denotative word concepts, the intentional meanings are assumed to uniquely determine their denotation.

And all the other mappings create similar predictions. If there is one thing that lexical semantic theories have shown, it is that almost no relation between words and anything else is completely tenable.

Connotative concepts often represent what linguists for example Saussure mean when they say the meaning of a word is a concept. The definitional attributes in connotative context are all those aspects of word senses that are not linked to the denotation.

This involves on the one hand those things that would belong to the Sinn of the word: But on the other hand it also involves the more pragmatic aspects of words, such as common language for common dog, vs. Connotative contexts are explicitly designed to be able to deal with lexicographic data. Despite this applicability to dictionaries, connotative contexts do not directly lead to a multilingual lexical database as described in the previous chapter. The main reason for this is that the definitional attributes of connotative contexts are not worked out in sufficient detail for an analysis of the actual definitions in dictionaries to lead to the kind of structure that was argued for in the previous chapter.

This thesis will present a different approach to FCA in combination with words and lexical definitions, that hopefully does provide such a structure. We will not start from the elaborate structure in figure 2.

The idea is that they are closer to the semantic part of dictionary definitions. To distinguish them from con-. Their nature is best shown using some actual dictionary definitions, such as the definitions of English words for kinds of horses as found in the Longman Dictionary of Contemporary English LDOCE. The definitions are displayed in table 2. They are not words but rather disambiguated words word-forms; their exact status will be discussed in the next chapter , since only one of their meanings is given: The right-hand side of the definition can be easily interpreted as describing features of these word meanings.

Take for instance the definition of filly: So the differentiae specificae young and female are features, distinguishing the meaning of filly from the meaning of other hyponyms of horse. We will call these features definitional attributes, and those definitional attributes will be the formal attributes of lexicographic contexts.

But the differentiae do not form the entire definition of filly: This genus is again a disambiguated word; it relates to a specific meaning of the word horse, since it is not supposed to refer to a kind of gymnastic equipment. This word meaning again has a definition in the dictionary, consisting of genus proximum et differentiae specificae. So we can further unfold the definition of colt into more definitional attributes with a new genus term.

The idea behind SIMuLLDA is, that if you unfold the definitions in this way, you completely reduce the dictionary definition of a word to a set of definitional attributes 6.

There are, of course, many problems related to this process of unfolding: These problems will be discussed later on in this thesis. But in our current example it works nicely. For the sake of the example, we will make a few simplifications to the definitions in table 2. Firstly, the genus term horse will not be treated as a genus term, but as most specific term: Such a top-word is necessary, since otherwise the process of unfolding would never stop; in a complete system, this would be an empty word like entity or thing, but here we will take horse to only have a differentiam: Also, the definitions of mare and stallion will be slightly modified for simplicity: Secondly, the asymmetry that stallions have to be fully-grown, whereas mares do not is resolved, by opting that both should be adult horses.

We will get back to these definitions without simplifications in chapter 4. In this way, we can reduce the definitions in table 2. The result of this is given in table 2. Analysis of Definitions for Horses If we view this as a lexicographic context, we will get formal concepts related to these definitions, resulting in a lattice of the ordered set of lexicographic concepts. This lattice can be graphically displayed as in figure 2.

Following convention, the labels under the nodes represent the formal objects word meanings at the lowest node where they appear in the extent of the related concept, while the labels above the nodes represent the formal attributes definitional attributes at the highest node where they appear in the intent. In building a lattice out of the definitions in table 2. There is even new information: So at least in a monolingual setting, with simple examples, FCA yields a nice structure for dictionary definitions.

Concept Lattice for Horses Multilinguality Since this is a thesis on a multilingual lexical database, we want to use lexicographic contexts in a multilingual setting. In principle, this can be done in a very simple and straightforward way: Now virtually every language has a word for a male adult horse, as well as for most of the other meanings in table 2.

A small sample of the different words in some language is listed in table 2. And in the same way all the other words in table 2. This will lead to a multilingual lexicographic context, in which all the words in table 2.

And that in turn yields a multilingual concept lattice. This way, it is even simple to predict what this interlingual concept lattice will look like: The reason is that formal objects that have exactly the same formal attributes will always appear together in every extent; compare the coinciding of the identical objects 5 and 6 and 7, 8, and 9 in figure 2.

With this method, the words stallion and hengst would be indistinguishable; but there is an important difference between them: This information could easily be resolved by adding language as an additional definitional attribute. This would lead to the much more complex concept lattice in figure In principle, the network in figure 2. And there is no direct access to the word itself, so the entire lattice has to be searched in order to get all the different meanings for a single word.

Therefore, it would be very convenient to have a single entry for a word that directly gives access to all its various meanings 8. The reason for that is rather simple: So the reason why it is not a good multilingual lexical database is because the basic element of dictionaries, the words are absent from the system.

Also, it is not an interlingual lattice, since the lattice explicitly contains elements that are language-dependent: Therefore, this thesis will opt for a slightly different approach, and remove all languagedependent elements from the central system and into language-dependent structures, where they are linked to words.

In that way, the lattice can truly operate as an interlingua, linking the various languages. Thus removing the word-forms, the formal objects in lexicographic contexts will no longer be language-dependent disambiguated words, but rather language-independent meanings, which are related to, but not identified with, the word-forms of the various languages. The word-forms themselves are situated outside of the lattice, and related to the meanings in the sense that every word-form expresses one or more meanings.

The word-forms are grouped into languages, where languages are taken as little more than lists of word-forms. This gives us a set-up as exemplified in figure 2. In this figure, the boxes in the figure represent the languages, containing word-forms.

These word-forms refer to language-independent meanings in the interlingua, and this relation is indicated with a grey line. The language-independent meanings are the formal objects in the FCA context, and hence by convention represented below the lowest formal concept in the extent of which they appear.

Above the nodes are the definitional attributes; the context producing the lattice is only implicitly present in this figure. The definitional attributes in the figure are not yet linked to the different languages. Definitional attributes are, in a way, less obviously language- 8 This is, of course, not a result of the multilinguality, but already present in the monolingual lexicographic contexts.

Partial Multilingual Set-up dependent than words: That fact itself is not language related what exactly definitional attributes are will be discussed in section 3. So in the SIMuLLDA set-up, also the definitional attributes are related to expressions of the various languages, leaving the definitional attributes themselves completely language-independent. Of course, the language-independent items themselves meanings and definitional attributes , do not have a written form.

For clarity, they will be given the names of their English lexicalisations, indexed with a number where necessary. This naming is arbitrary, and has some exceptions: Logically, it holds that ext colt COLT. As discussed earlier, not all meanings have to be lexicalised in every language; there may be lexical gaps such as the lexical gap in French for colt. Definitional attributes, however, will be required to be lexicalised in every language. The reason for this will be explained in the next section, since it involves the process of lexical gap filling.

Since the relation between words and interlingual meanings is an important one, two functions will be introduced: Where necessary, wfs will be indexed with the language in which the word-forms should be given.

This interlingual meaning can be lexicalised in French, since the meaning HORSE is related to the word cheval in the French language module. This yields a direct translational relation between the English word horse and the French word cheval. For the English word colt however, this does not work: This implies that there is a lexical gap in French for the English word colt.

So a lexical gap can be defined in the following way: Lexical Gap There is a lexical gap in a languages Y for the word x in language X, if one of the interlingual meaning s expressed by x does not have a lexicalisation in language Y. In terms of the functions wfs and mng: However, because of the structure of the SIMuLLDA interlingua, even meanings that are not lexicalised in a particular language, can be translated into that language by the following mechanism: By definition, there will be a definitional surplus for the subconcept, which can be calculated in the following way: This definitional surplus, together with the lexicalisation of the superconcept, will give a complete translation of the meaning of the concept in the desired target language.

So the French word poulain is an incomplete translation of the English word colt. The process of lexical gap filling is exactly the same process as would take place when one tries to recover a monolingual definition from the system. So the lexical definition for colt, which is male foal, is construed in exactly the same way as its translation in French. From this, we can deduce what the process of lexical gap filling renders: Although lexical gap filling renders the same result as generating a monolingual lexical definition, this is not necessarily the monolingual lexical definition that was put into the system in the first place.

In fact, in our example it is not: The reason for this is that the generation of a meaning description with the method described above is not a decisive process. It uses the notion of a lexicalised superconcept, but there can be various lexicalised superconcepts.

If we sharpen the process of lexical gap filling by saying that we should take the smallest superconcept, it would still not be decisive. Given the fact that the structure of FCA is not a hierarchy but a lattice, there does not have to be a unique smallest superconcept; not even a unique smallest lexicalised superconcept.

This non-uniqueness of result is not problematic: These properties will be discussed in this section. Part of this logical analysis will be a calculation of the maximum number of formal concepts for a given context. The rest of this thesis will use, but not depend on, the properties described and proved in this section FCA and Lattices One of the crucial properties of formal concepts is the fact that they form a complete lattice. All of these follow from the fact that by 2.

Since we already know that A, A is a concept for any A see 2. Then A X by double application of 2. To prove this, firstly, define a set of filled rectangles F: Any rectangle C, D that extends a concept A, B has to be identical to that concept. The property that is most important for this thesis is that concept lattices are atomic: This property is important, since it is the underlying principle used for computing the formal concepts in JaLaBA see section 2.

Atomic Lattices Let obpro be the set of all sets of attributes that are projections by of individual objects: This set obpro represents a set of concepts the concepts obpro, obbpro. And this set of concepts encloses the set of most specific concepts atoms, which are most specific in the following sense The second claim is that the entire set of intents of all formal concepts except possibly for is simply the set of all intersections of the elements of atoms.

So let us define a set atoms of all inductive intersections of members of atoms: So we can conclude that concept 13 A complete lattice, in which every element is a supremum of atoms, is called atomic. The set obpro is not most specific in any sense: Distributivity A distributive lattice has the following property: So we need to prove the following: A 0 A B 0 B This is shown very easily.

Since the new concept has to include the old one, it can only be the concept A 0, A 0 with the function from the larger context. This, as we have seen, will always be a concept. Furthermore, since yields the attributes that all objects in A 0 have, and I 0 I, it logically follows that all the attributes in B 0 which is A 0 with the function over the smaller context , will still be in A 0.

And since by definition 2. However, it would be interesting to know the following given a number of objects, and a number of attributes, what would be the minimum and maximum number of formal concepts, with a free choice of relation I? The minimum is trivial: And given the fact that X, X will be a concept for any X, there will always be at least one concept. The maximum number of formal concepts is less easy to see.

There are two cases that would have to be treated separately: But given the symmetry of the system, these cases run completely parallel, so let us assume that M G. Given the fact that any formal concept has a unique intent, and every intent is an arbitrary combination of attributes, it immediately follows that there is a total of 2 M possible intents. To show that this worst case of 2 M concepts also occurs, we need a context in which for every B M , B, B is a concept.

We get such a context if we take in injection e: With this, we have: This maximum number depends on the fact that M G. Given the symmetry of the system, we can conclude that: But more often than not, formal attributes are not independent: In these cases, we can see all colours as values for the same attribute. The disadvantage of this method is that there is one big, unordered set of values. However, in normal circumstances, every attribute will have its own set of values.

An alternative is therefore to partition M into a set of sets, where every set contains the values of a specific attribute: Notice that such a partition does not need to take semantics into account, but merely needs to obey mutual exclusiveness as in 2. There is, however, exactly one extent in which all attributes, even from the same element of the partition, can coincide, namely. So the maximal number of formal concepts will be: That the actual number is lower namely 13 , is because this model accidentally has no non-white triangles, no white squares and no black or white triangles.

Notice also that where the basis of FCA is completely symmetrical in the sense that objects and attributes can switch places without much consequence, restrictions such as these are natural only for attributes, hence making it asymmetrical. Additional Restrictions Besides mutual exclusiveness, there are more properties that attributes can have, that limit the number of possible concepts.

Notice that all these restrictions could in principle also be applied to objects, though that is less natural. Subordinateness Some attributes are subordinate: Such an attribute will contribute exactly one additional concept m 2 is not also subordinate to m 1, for then they would be coextensive Partial Ordering on Attributes The FCA framework assumes attributes to be completely independent.

But sometimes, the attributes we use to characterise objects are just as interdependent as the concepts themselves. To give an example, if we have a model in which there are, amongst others, squares and rectangles, they would be the extent of different formal concepts, distinguishable by the fact that the squares are square and the rectangles are rectangular.

But being square is just a special case of being rectangular. It is being rectangular with the additional constraints that all the sides have to be of an equal length. So it is possible to divide the attribute of being square into two sub-attributes, namely being rectangular and having equally sized sides. But if we were to follow this line, we would have to further divide the attribute of being rectangular into having parallel sides that make corners of 90 o, in order not to get into conflict with the attribute of being rhomboid diamond-shaped; a square is also a special kind of rhomboid, where the sides make corners of 90 o.

Although this line of action would solve the problem, it would have us end up with attributes that are way too mathematical in nature to be useful in much everyday practice. We often just want to say that some objects are square, while others are merely rectangular, knowing that the first implies the second.

In that case, we have not simply a set of attributes, but a partially ordered set of attributes. This has consequences for the notion of being a sub-concept, since a concept can now also be a sub-concept of another if it has more specific, rather than simply more attributes.

Adjusting the FCA framework for this new situation is rather trivial. In this definition, it is not strictly necessary, though useful, to first define the relation of more restricting set of attributes, and redefine the relation with this new relation: The new relation in 2. We merely know that if for two sets we know for both that the first is more restrictive than the second, they are equivalent, but not necessarily identical: But the second is also less restrictive not in the strong sense, but equally restrictive than the first.

The reason for this is that the additional attributes do not give additional information, since they are implied by the stronger attributes.

But as a result, two non-identical sets can mutually be smaller in the relation A B A B. It is clear that this problem is created by the fact that there are useless attributes in the set, and that if we would forbid those, the problem would go away. The best way to formally realise this, is not to remove all useless attributes, but instead to add them all.

The downward closure of a set X notated as X is exactly the set where we have added all the less informative elements, and we can redefine equivalence in term of downward closure: A special case of more restrictive attributes that we will encounter in the next chapter is one with disjunctive attributes; for instance, flowing to the sea or another river is less restrictive than flowing to the sea. This could be modelled logically: As part of this thesis, I developed an online application that does this: JaLaBA an online tool that takes contexts as input, and renders their concept lattices.

An illustration of the JaLaBA application is given in figure 2. JaLaBA consists of two parts: These two parts will be discussed in turn here. The Online Java Lattice Building Application Construing Formal Concepts The first requirement for building a concept lattice is having a definition of the underlying context. Since a context is little more than a cross-table, it can best be entered by means of a cross-table.

But the number of formal objects and formal attributes is not predefined, therefore there is no way of telling the size of this cross-table. The method for dealing with this problem is to use HTML tables. By using an HTML based system, there is no need to worry about the interface itself, since the browser will take care of that:.

And it has extendable windows that will adjust to the size of the table it has to display. The HTML table has to be generated on the basis of the desired number of objects and attributes, and this is done using a cgi-script. Since Perl is a very useful language for building cgi-scripts, the formal concept creation part of JaLaBA is written in Perl. JaLaBA starts out by prompting for three arrays: It then puts the names of the objects and attributes aside, and treats the relation as relations between numbers.

From this, it builds a comma-separated list for every object, of all the attributes that object has. These comma-separated lists form the basis for the generation of the lattice; this set of comma-separated lists is simply the set atoms, which constitutes the foundation of a lattice by means of equation 2. Since atoms is a set, we can throw away all duplicate elements, by means of the following algorithm: The reason for this is that the procedure only yield intersections of pairs of set.

But these intersections only form the first superconcepts of atoms; in a multi-layered lattice, there will be further superconcepts of the intersection. These have to be found by further taking the intersections between the new concepts generated in the previous step with the existing with the existing.

Small Layered Context elements This because the newly added sets of attributes in the last step can in principle lead to more FC s, but only if two conditions are met: Therefore it is impossible to prove that this procedure actually yields the complete set of formal concepts. After the formal concepts have been found, the partial order on these concepts has to be established Given the fact that all concepts are simply comma-separated lists of numbers, this is easy: All elements of atoms are larger than, and all intersections are larger than both their composing parts.

The abstract representation of the lattice in figure 2. This abstract representation of the lattice is formatted in such a way that it can serve as the input for the second part of the lattice building: Drawing a Hasse diagram has an inherent problem: There is no specification about their horizontal configuration.

Drawing a diagram that simply obeys the specifications is easy. But drawing a good diagram is less so: On the other hand, a poorly chosen diagram of even a small, well known lattice can render it unrecognizable Abstract Representation of Lattice in Figure 2. More importantly, JaLaBA adds the Formal Concept Analysis labels for formal objects and formal attributes above and below the appropriate nodes.

Lattice Draw organises the elements in the lattice by building a 3D model of it. This is done in three steps: The point is then moved by these forces over a distance proportional to n, where n is the size of the lattice. This process is repeated until an equilibrium is reached.

These three steps yield a 3-dimensional arrangement of all the nodes in the lattice. This 3-D object can of course be projected onto a 2-D space, giving a Hasse diagram of the lattice. But there is no guarantee that this projection. However, the object can be rotated and projected under any angle. This gives a large range of accessible representations for the lattice, amongst which a nice lattice is bound to appear. Furthermore, the rotating object itself does not merely look good, but has an additional advantage: Before After Figure 2.

Adaptation of a Lattice There are of course more direct ways of assuring a nice representation of a lattice. For instance, Ganter has argued that for a nice representation, two things should be minimized: This last point maximizes the number of parallel lines, which is one of the main criteria for niceness. The JaLaBA applet does not include such rules, since it builds its lattice in 3D, while such rules always apply to 2D images.

Furthermore, the rotatability of the Lattice Draw set-up usually yields a nice image, and otherwise the nodes can be moved manually.

In both cases, formal concepts are pairs of objects and their attributes, that belong together because the objects share all the attributes and vice versa. But intuitively, we can also say that formal concepts are maximal areas of filled squares in a cross-table. The logical system of Formal Concept Analysis can be applied to dictionaries in the following way: With this set-up translations for words can be found: If so, this will be the translation of the word.

More interestingly, a translation can also be construed if no lexicalisation of the interlingual meaning exists in the target language i. Then, find the definitional surplus of the smallest common concept with respect to this superconcept. The lexicalisation of the superconcept, together with the lexicalisation of the definitional surplus will be the desired translation; it will be an explanatory equivalent and not a translational equivalent.

The monolingual definitions of words can be found in the same way, by taking source and target language to be the same. In the formal-properties section, I have shown that in a worst case scenario, the number of formal concepts grows exponentially with the number of attributes 2 n. But in lexicographic contexts, attributes normally have features that restrict the number of formal concepts: Therefore, the next chapter will be dedicated to a thorough analysis of the appropriate interpretation of these elements.

Therefore, in order to have a precise meaning for the system, a precise description of these basic elements is required. Of course, fine-tuning the interpretation will not affect the system as such: And these bilingual definitions will not change by careful consideration of the nature of the basic elements that played a role in their conception.

But there are two reasons for a detailed analysis of the basic elements. Firstly, word-meaning is a very difficult field, and there are many problematic cases. To reach a proper analysis of these problematic cases, it is necessary to have a precise perception of what every analysis implies. And secondly, there are many aspects and elements of dictionaries that are not accounted for by the basic set-up explained in the previous chapter. Given the complexity of semantics, I believe it is only possible to deal with additional elements properly and at the appropriate level.

This chapter will be dedicated to the analysis of the nature of the basic elements of the theory: Words in the SIMuLLDA set-up are language-dependent elements, grouped into languages, and linked to one or more interlingual meanings in the concept lattice.

To answer this question, I will give an overview of what types and aspects of words we can distinguish. This is a matter that has been discussed at length in. The discussion in this section will follow the standard literature on the subject, but deviate from it where need arises. As part of this general specification of words, I will turn to the question which of these aspects of words should be part of the SIMuLLDA set-up, and where and how they should be modelled.

It is, however, much too vague a notion to be used in a formal context. The word word does not have a clear, single meaning: Sometimes, we even refer to entire sentences as words In one word: I will, therefore, avoid in the following the use of the word word as a technical term and use it only informally. Instead, I will use intuitively less clear, though formally much better defined notions.

This section will introduce some common terminology, most of which is drawn from the standard work on semantics by Lyons ; , with the addition of a few more technical terms. On top of the terminology, some notational conventions will be introduced, that will be used throughout this thesis, to keep the different kinds of words apart. When just the abstract word-as-such is meant, it will be underlined.

The definition of a word that usually springs to mind first is that of a sequence of letters, also called a string or an orthographic word. Strings will be typeset in courier. The identity of strings is straightforward: Strings come in types and tokens: On the second point, the Court of Appeals noted that, as a matter of fact, the Bankruptcy Court denied the break-up fee in the midst of the auction on the basis of its determination that the stalking horse would not abandon the purchase even without the break-up fee.

Moreover, the Court of Appeals noted, the auction actually resulted in a higher bid for the power plant than had existed before the auction, such that a break-up fee was not required to raise the purchase price at auction. Notably, the Court of Appeals was not swayed by the fact that only the other competitive bidder, and not any creditor of the seller, objected to the break-up fee, nor that the seller was solvent even though in bankruptcy and therefore able to pay the break-up fee without harming creditors.

The Court of Appeals stated that the O'Brien standard is applicable notwithstanding these circumstances, and also discarded other theories espoused by the stalking horse regarding the alleged unfairness of the result. The Bottom Line Potential buyers of distressed assets out of bankruptcy, especially in the active Delaware Bankruptcy Court, should be aware of the Reliant decision and its potential impact on future requests for break-up fees.

Stalking horse bidders should consider making the approval of their break-up fees at the outset a firm condition to their proceeding with the sale, because the lack of such an express contractual condition precedent was a crucial fact in the Reliant case.

Subject to issues surrounding standing and the circumstances of any particular case, other competitive bidders should consider whether there are ways to challenge a break-up fee of a stalking horse in order to make a Section auction more competitive and reduce the end price for the distressed assets on the block. While Section sales can present opportunities for potential buyers of distressed assets, seizing those opportunities and successfully consummating transactions require an understanding of the legal rules regarding break-up fees and similar conventions of the bankruptcy sale process.

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