An analysis of an inflected word form is produced by mapping the input form to a sequence of lexical forms through the transducers and by composing some out­put from the annotations on the leaf nodes of the lexical paths that were traversed.

Comprehensive morphological descrip­tions of this type have been developed for several languages including Finnish, Swedish, Russian, English, Swahili, and Arabic. Although they have several good features, these Kimmo-sys terns also have some limitations. The ones we want to ad­dress in this paper are the following:

(1) Lexical representations tend to be arbitrary. Because it is difficult to write and test two-level systems that map between pairs of radically dissimilar forms, lexical representations in existing two-level analyzers tend to stay close to the surface forms.

This is not a problem for morpho­logically simple languages like English because, for most words, inflected forms are very similar to the canonical dictionary entry. Except for a small number of irregular verbs and nouns, it is not difficult to create a two-level description for English in which lexical forms coincide with the canonical citation forms found in a dictionary.

However, current analyzers for mor­phologically more complex languages (Finnish and Russian, for example) are not as satisfying in this respect. In these systems, lexical forms typically contain diacritic markers and special symbols; they are not real words in the language. For example, in Finnish the lexical counterpart of of in T took' might be rendered as otTallln, where T, at, and II are an arbitrary encoding of morpho­logical alternations that determine the allomorphs of the stem and the past tense morpheme. The canonical citation form ottaa 'to take' is composed from annotations on the leaf nodes of the letter trees that are linked to match the input. It is not in any direct way related to the lexical form produced by the transducers.

(2) Morphological categories are not directly encoded as part of the lexical form. Instead of morphemes like Plural or Past, we typically see suffix strings like +s, and +ed, which do not by themselves indi­cate what morpheme they express. Different realizations of the same morpho­logical category are often represented as different even on the lexical side.

These characteristics lead to some un­desirable consequences:

We follow two simple principles:

(1) Inflected forms of the same word are mapped to the same canonical dictio­nary form. This applies to both regular and irregular forms. For example, in our English analyzer the surface forms happier and better are directly matched with the lexical forms happy and good, respectively, rather than some nonwords.

(2) Morphological categories are repre­sented as part of the lexical form. Instead of encoding morphological categories such as Plural, Comparative, IstPerson as annota­tions on strings that realize them, we in­clude them directly in the lexical represen­tation. Consequently, our two-level repre­sentation of happier and better are:

lexical level happy +Comp +Adj surface level happi er 0 lexical level good +Comp +Adj surface level bett er 0

Figure 1

The stems are presented as the lemmas found in a dictionary, followed by mor­phological tags. 0 serves here as the epsilon symbol. Because there is no need to have other annotations on the lexicon trees, problems I and II in Section 1 have been eliminated. Lexical forms are always sequences of morphemes in their canonical representation.

The only obstacle to this approach is that the rules that constrain the surface realization of lexical forms become more difficult to write when there is little or no similarity between the two levels of repre­sentation. Designing such rules and under­standing their interactions is a hard task even with the computational assistance provided by a complete compiler for the two-level formalism (Karttunen et al. [6]).

As the distance between lexical and sur­face form increases, the mapping is easier to describe by allowing one or more intermediate levels of representation. The solution we adopted combines the two-level rule formalism with the cascade model of finite-state morphology dis­cussed by Kaplan & Kay [7].

Our formal understanding of finite-state morphology is based on the demonstrations that both rewriting rules and two-level rules denote regular relations on strings (Kaplan [9]). The correspondence between regular relations and finite-state transducers and the closure properties of regular relations provide the computational and mathematical tools that our approach depends on. One of the earliest results of finite-state morphology is the observation that regular relations are closed under composition (Johnson [8], Kaplan&Kay [7], Kaplan [9]). Consequently, a single transducer can be constructed whose behavior is exactly the same as a set of transducers arranged in an ordered feeding cascade:

lexical string intermediate string ♦ surface string surface string

Figure 2

This observation was originally made about transducers corresponding to phonological rewrite rules, but it applies to regular relations or transducers no matter how they are specified. Although regular relations in general are not closed under intersection, the subclass of relations denoted by standard two-level rules is closed under this operation (Kaplan [9]). Thus fstj and fst2 in Figure 2 may represent either a single two-level rule or the intersection of any number of rules.

When the relationship between lexical and surface forms is complex, the descrip­tive task of setting up rules that relate the two levels can be simplified by decompos­ing the complex relation to a series of less opaque matches. For efficient recognition and generation, the resulting cascade can be reduced to a single transducer. Al­though it would be possible in principle to produce the same single transducer directly from two-level rules, we have found many cases in our descriptions of English and French where the composition approach is not only easier but also linguistically more justified. We describe one such case in detail.

French plurals can be formed in a vari­ety of ways. Some of the most common patterns are illustrated in Figure 3.

We omit here the actual two-level rules; what Figure 3 illustrates is simply the joint effect of several rules that con­strain the realization of the plural mor­pheme and the shape of the stem in regu­lar nouns. Note that the constraints here are local; the stem and the plural morpheme are in a fixed position with respect to each other.

'message ' message +masc +pl

Figure 3

In compound nouns and adjectives, several patterns are possible: (1) only the first part of the compound is marked for the plural, (2) both are, (3) none are or (4) only the last is. The possible patterns and some examples are given in Figure 4.

The interesting cases are those in which the first part needs to be pluralized. In a simple two-level system, the information about plural formation summarized in Figure 3 would have to be rewritten and adapted so that the rules could apply over a. distance in the position just before the hyphen.

No plural marking at all un je-ne-sais-quoi 'a certain something' des je-ne-sais-quoi Plural marking on the first compound un chef-d'oeuvre 'masterpiece' des chefs-d'oeuvre Plural marking on the second compound une mini-jupe 'mini-skirt' des mini-jupes Plural marking on both compounds une porte-fenêtre Trench window' des portes-fenêtres

Figure 4

The simple rules for regular plural for­mation illustrated in Figure 3 do not work for first parts of compounds because the affected elements are not in the same con­figuration relative to each other. Although it is possible to modify the rules, the new versions would be rather complicated and do not capture the simple fact that the plu­rals portes and fenêtres in portes-fenêtres in themselves are regular, the only thing that is special about the word is that plurality is expressed in both parts of the com­pound.

We avoid these complications by creat­ing a cascade of two-level rules in which the first stage is only concerned with the plurals of compounds. It starts from a lexi­cal form in which the words are marked for the pattern that they take and creates an intermediate level in which the information about number and gender is distributed over the agreeing parts. This is illustrated in Figure 5 for the masculine plural of social-démocrate, a word in which both parts get pluralized.

social+masc+pl-démocrate 0 +masc+pl Figure 5

The effect of the first stage of rules is to copy the morphological tags from the end of the compound to the middle whenever the +dpl (double plural) diacritic is pre­sent.

The second layer of rules applies uni­formly to simple nouns as well as com­pounds. In the case at hand, the two plu­rals in sociaux-démocrates are realized in the regular way, as shown in Figure 6.

social +masc +pl -démocrate +masc +pl

Figure 6

By first intersecting the rules in each set and then composing the results in the way shown in Figure 2, we end up with a transducer that eliminates the intermedi­ate level altogether and maps the lexical representation directly to the correct sur­face form, and vice versa. Figure 7 illus­trates the final result.

social 0 -démocrate +DPL +masc +pl sociau x -démocrate 0 Os

Figure 7

The representation in Figure 7 fulfills the desiderata laid out in Section 2 except that it contains a special diacritic +dpl that marks the behavior of social-démocrate, with respect to plural formation. In the next section, we show how that diacritic can be eliminated.

By choosing the canonical dictionary form as the lexical form in our English and French analyzers and by including mor­phological categories directly as part of that representation, we have eliminated the need for additional annotations in the lexical structure that are common in exist­ing Kimmo systems. We can treat the letter tree as a simple finite-state network in which all morphological information is carried on the branches of the tree and not on the leaves.

Taking this idea one step further, we may think of the lexicon as a trivial first stage in a cascade of transducers that maps between the lexical and the surface levels. The second stage is the two-level rule sys­tem. In the case of our analyzers for English and French, the rule system starts out with three levels but reduces to two by intersection and composition. The final stage is the composition of the rule system with the lexicon.

This progression of pushing the original Kaplan & Kay [7] program to its logical conclusion is depicted in Figure 8.

fsl.

surface string Cejrical string LEXICON o

Figure 8

Figure 8 sketches the construction of our morphological analyzers for English and French. Arrows labeled with & repre­sent intersection, arrows marked with o stand for composition. (We have simpli­fied this picture slightly by omitting the composition of small bookkeeping rela­tions that are necessary to model properly the interpretation of epsilon transitions in two-level rules.)

Stage 1 consists of two parallel two-level rule systems arranged in a cascade, as illustrated in Section 4. In Stage 2, the rules on each level have been intersected to a single transducer. Stage 3 shows the composition of the two-level rule systems to a single transducer and Stage 4 repre­sents the final result: a transducer that maps sequences of canonical dictionary forms and morphological categories to the corresponding surface forms, and vice versa. Although the conceptual picture is quite straightforward, the actual com­putations to produce the structures can be resource intensive, in some cases quite impractical.

At the last stage, when the idiosyncratic behavior of particular lexical items has been taken into account in the composition of the lexicon with the rule transducers, all morphological diacritics such as the +DPL tag for French nouns with double plurals can be eliminated because the rules that depend on them have been applied. In full compliance with our desiderata in Section 2, the resulting transducer maps, among other things, social-démocrate+masc+pl directly to sociaux-démocrates, and vice versa.

Finite-state morphology rests on the ob­servation that ordinary morphological al­ternations involve regular relations. This is the basis of the early work by Kaplan and Kay [7] on converting ordered rewrite rules to a cascade of transducers and the parallel transducers of Koskenniemi's two-level model [1]. In recent times the two-level model has been more popular. It has turned out (Karttunen [10]) that parallel two-level constraints are even expressive enough to account for phenomena that require rule ordering in the classical phonological rule formalism. But there is no computational or theoretical reason to insist on two-level descriptions. Because the mathematical properties of rewrite and two-level rules are now well-understood (Kaplan [9], Ritchie [11]), we can compose any n-level description to just two levels. In our work on English and French mor­phology we came across many instances in which the introduction of an extra level is both practical and linguistically motivated. The case of French compound plurals is a typical example.

Our success in composing the rule sys­tem with the lexicon (Stage 4 in Figure 8) is due to a number of fortunate characteristics that morphological alternations and lexicons of natural languages seem to have even though they are not necessary or even probable from a formal point of view. We at least were surprised by some of our results. The most important of these delightful discoveries are:

(1) Small case studies can be mis­leading. The composition of a rule trans­ducer against a lexicon containing a hand­ful of words is so much larger than the input lexicon that one is tempted to con­clude that the method can never succeed on a large scale. However, this blowup seems not to occur when the lexicon is already large.

(2) Intersections and compositions of rule transducers tend to be large, but not nearly as large as they might be. The result of intersecting a few dozen two-level rules may have thousands or tens of thousands of states, but not trillions as the worst case scenario predicts. Many rules tend to apply either in quite similar or in quite different environments. The finite state machinery can represent such pat­terns without multiplying state sets.

(3) Composition with the lexicons reduces the complexity of rule inter­actions. It might turn out that the composition of a large lexicon with an even larger rule transducer is bigger than either one of the input structures. In reality, the size of the result seems to be somewhere in the middle. The rules constrain the realization of all possible lexical forms. In the composition, their scope is restricted to just the forms that actually exist in the language. It turns out that this restriction makes the result smaller rather than larger even though the lexicon itself is a very irregular collection of forms.

The fact that it is possible to construct a lexical transducer for the whole language raises interesting theoretical issues. In lin­guistics it is commonly assumed that lexical entries and the rules for realizing them exist independently from one another. That assumption is, of course, also the starting point of the work that we are re­porting about in this paper. The initial separation between the lexicon and the rules is useful in constructing a system for word recognition and generation. The rules are, in a sense, a decomposition of a very complex mapping between lexical and surface forms to a set of simpler rela­tions that we can comprehend and ma­nipulate. But in the construction of the fi­nal result individual rules and the distinct lexicon disappear. The rules play no role at all in the actual generation and recognition process. They are needed only for the pur­pose of enlarging the lexicon, although other acquisition methods can be envi­sioned. The rules are true generalizations about the two-level lexicon constructed with them but they are not a part of it.

In linguistics the psychological reality of rules is often taken to be established by the observation that a simple listing of all forms would be not only implausible but even impossible, given that the brain must have some storage limitations. The general organization of the system like the one we have described suggests that the role of rules might be quite different. Instead of being essential for the production and comprehension of speech, the rules that linguists are trying to discover may be—if they exist in the mind at all—only sec­ondary reflections on the generalizations that can be encoded in the finite-state lexi­cal structure itself.