The metarules widiin the theory of Generalized Phrase Structure Grammar (GPSG) are a very interesting device to express generalizations on a basic set of immediate dominance (ID) rules. A short introduction to the theory of metarules is given in the following section. Metarules arc used to generate an object grammar from a set of basic ID rules. One of the first discussions about the application time of metarules within the GPSG theory, which has strongly influenced the succeeding research in this field, lias been published by [Thompson 82|. In his article Thompson argued for applying the metarules all at once in a preprocessing step (compile-time application). How­ever, our parser applies die metarules during the parsing process (run-time application or direct application). A discussion why we prefer die direct application is given in section 3. No matter when the metarules arc applied u> the basic set of ID rules, we have to care for the termination of the recursive application of metarules. [Thompson 82] made a proposal to guarantee termination by the definition of a Finite Closure (FC). This approach restricts the application of metarules such that one metarule can only be applied once in the derivation of an ID rule and prevents it from recursive application. This restriction has been taken over by the authors of [Gazdar et at. 85] and they gave a further restriction. They restricted the application of metarules to lexical ID rules, even though they knew that Ulis restriction may prove to 1» incompatible with the descriptive power needed for natural language grammars (p. 59). But we think that there is no need to restrict the application of metarules only to lexical ID rules, even if there are proposals to eliminate die use of metarules and to use lexical rules like in I.F'G and HPSG (see [Jacobson 87J). But to do so with GPSG would involve crucial changes to the theory, and therefore we preserved the metarule component in our machine translation (MT) system and tried to find an adequate criterion for the termination of metarules. In our approach the grammar writer is free to decide whether a metarule is to be applied to lexical ID rules or to anoüicr type of rule. The Finite Closure (FC) is loo rcstricive, because in some cases (sec the examples of |Uszkoreit 87] and [Gazdar et al. 85) in the sections 3 and 4, respectively) recursive application is needed. In section 4 wc present an alternative constraint on the basis of results in the field of term-rewriting. This constraint is less restrictive than the FC. It allows for die definition of recursive metarules which may be applied freely and guarantees the termination. In section 5 the metarules of the German GPSG grammar used in our MT system are outlined and, with die help of the constraint in section 4, wc show that their application is terminating. In section 6 wc give an outline of how to modify the parsing process in a way that metarules can be applied directly rather dian at compile-time. In the last section wc draw some conclusions for future work in this field.

Metarules arc one of die most criticized devices of the GPSG formalism. GPSG is a grammar formalism that states most of its generalizations on the level of local trees. Metarules were introduced to capture generalizations on the set of ID rules. An ID rule slates the dominance relation between the mother category and a multiset of (laughter categories in a local tree without fixing die linear precedence relation of the daughters. ID rules have the following format:

Co - > Cu C2,CnMetarules define a relation on ID rules. They have the following format:

'input ID rule scheme' 'output ID rule scheme' and can be read as: If the set of ID rules contains an ID rule which is matched by 'input ID rule scheme', then it also contains an ID rule that matches 'output ID rule scheme', where the feature specifications of die input ID iule are carried over to die output ID rule if not specified oüicrwise by the metarule. For example the metarule VP[-PAS] -> W, NP[accl => VP[+PAS] W, (PP|byl) states the connection between active and passive, where W is a variable ranging over a (possibly empty) multiset of categories. The major point of criticism against metarules is that diey increase the generative ]iowcr of GPSG in an undesirable way when they are recursively applicable, because this may lead to an infinite set of ID rules. The resulting grammar need not Ik context free. In order to remedy the situation, suggestions of varying radicality were made. The proposal of [Thompson 82] and [Gazdar et al. 85], which tries to maintain metarules, was simply to apply a metarule at most once in the generation of an ID rule. This stipulation is somewhat strange, because it allows lor recursive metarules and just prévenus them from being applied recursively.

[Kilbury 86] suggested to eliminate metarules by using category cooccurrence restrictions. The most radical proposal was to dispense with metarules. But our aim was to stay within the framework of GPSG, and it would be a loss to dispense widi metarules, because GPSG formulates for example valency of verbs and other constituents on the level of ID rules and metarules are the means to capture generalisations on that level.

For this reason we formalize the properties of metarules that terminate recursive application and slate them as a condition that a set of metarules must fulfil. Metarules can then be applied freely.

There are two possibilities for the time to apply the metarules. The first is to compile the basic set of ID rules (compile-time application) in a preprocessing step. Thompson calls it all-at-once approach. The other possibility is to apply die metarules during the parsing process (run-time application or direct applica­tion), which Thompson calls as-needed approach. Thompson argued lor the compile-time application because the direct application of metarules has the following disadvantages (sec [Thompson 82]: p.5):

(1) If a metarule can be applied to an ID rule during the parsing process, the metarule has to be applied again when the same ID rule is involved in the same or a subsequent parse.

(2) To store the structures generated by ID rules which are the result of the application of a metarule is just another instance of the compile-time application.

(3) Derivations of ID rules of length greater than one, i.e. ID rules which are the results of applying more than one metarule to one basic ID rule, will rapidly expand the search space.

In order to look a little bit closer to Thompson's arguments and to stay on his line, we presuppose that a kind of top-down parsing mediod is used and there are n basic ID rules and m ID rules, generated by the application of the metarules.

When looking to argument (1), we see diat it is an argument for the run time approach. If the metarules are applied at compile-time a huge set of ID rules is compiled from the basic set. For example if we would apply the metarules of our MT system (see section 5) to our basic set of 8Ü ID rules at compile-time, wc would get about 240 ID rules in the object grammar. Let us assume that some category C has to be expanded and there arc i ID rules in our grammar with mother category C. In the compile-time approach the parser would have to check (n+m)/n*i ID rules on average, whereas in the run-time approach i ID rules and (n+m)/n metarules ((n+m)/n+i rules) have to be checked for application to diese ID rules. In the normal case that are less than in the compile-time approach. Argument (2) is indeed an argument for the run-time approach. Let us again consider the above example and each of the i ID rules has d daughters on average. If the category C is expanded according to all ID rules, in the worst case (n+m)/n*d*i (partial) structures have to be stored on average in the compile-time approach. These structures arc very similar, because in general the melarulcs modify the ID rules slightly. The run­time approach can make use of Ulis fact and stores only approximately d*i (partial) structures and additionally (n+m)/n*i structures after the application of the metarules. That makes ((n+m)/n+d)*i structures to be stored in the run-time approach. The common parts of die ID rules generated by metarules need not to be stored, that arc (n+m)/n*(d-l)*i-d*i parüal structures less. For example if n = 80, m = 160, d = 3 and i = 10 then diis would mean that on average 30 partial structures less have to be stored for the corresponding constituent.

Concerning argument (3), [Barton et al. 87]: p.226 showed dial die computation of the Finite Closure (FC) of a GPSG G with x ID rules and y metarules can increase the number of the ID rules worse than exponentially, namely from x to more than x*, i.e. there is no difference between the compile-time and run-time approach.

In order to sum up this discussion, we can say that there is no difference in complexity between the com-plilc-time and the run-time approach with respect to the arguments in [Thompson 82]. The direct approach is even preferable to the complile time approach when looking at the arguments (1) and (2), which are indeed arguments for direct application of metarules. There is another argument for direct application of metarules. The FC in [Thompson 82] states that every metarule can apply at most once in the derivation of any given object grammar rule from one basic rule. An example for the recursive application of a metarule has been proposed in [Uszkoreit 87]: p.145 in his German grammar. It makes the adverbial phrase (AdvP) a sister node of die verb and its arguments:

V[-AUX] h> V, W => V[-AUX] -> V, W, AdvP This metarule is to solve the problem that adverbial phrases can be interspersed freely among the arguments of the verb and the number of AdvPs in a verb phrase is not limited (but it is finite) and the metarule has to be applied recursively. This fact would rule out the definition of this metarule with respect to the definition of the FC which has been adopted also in [Gazdar et al. 85] to avoid the production of an infinite number of ID rules. Uszkoreit tried to circumvent the problem in redefining the above metarule such that it fits the requirements of the FC. It employs the Klecne star operator:

V[-AUX] -> V, W => V[-AUX] -> V, W, AdvP* This change of the metarule is not necessary if the metarules are applied directly during the parsing process and the above metarule without die Kleene star can be applied freely, because the termination is determined by die finiteness of the input string.

No matter whether the metarules are applied directly or at compile-time, we have to care for die termination. Wc think that the restriction on the application of metarules imposed by the Finite Closure (FC) is too strong. That may have crucial consequences for some metarules. Look for example at the Complement Omission Metarule from the English grammar in [Gazdar ct al. 85], which is [+N, BAR 1] -> N°, W => [+N, BAR 1] -> N°. This metarule deletes optional complements of nouns. For example the noun gift may have two optional prepositional phrases as comple­ments, i.e. N -> N°, (PP[ofJ), (PP[to]). The prepositions o/and to are fixed but either of the PPs or both may be left out: the gift, the gift of John, the gift to Paul, the gift of John to Paul. The above metarule only allows for the gift and the gift of John to Paul, because it deletes all complements of the noun. The correct metarule would be:

[+N, BAR 1] N°, X, W => [+N, BAR 1] -> N°, W [Gazdar et al. 85] have not been able to define this metarule, because it has to be applied recursively and the FC does not allow recursive application. For this reason we decided to define another constraint which is not so restrictive.

As it lias turned out, most of die metarules which have been defined within the fragment of English in [Gazdar ct al. 85] or of German (see the section below for a more detailed discussion) arc or can be defined in a way that guarantees the termination of the recursive application of metarules. In order to prove the termination, some research results within die field of term-rewriting can lie applied (sec [Dershowitz 82 and 851 for general results and [Weisweber 89], [Weiswc-ber/ Hauenschild 90] and TWeiswebcr 92] for an application to mappings widiin machine translation). ID rules can be viewed as terms and metarules can be viewed as term-rewrite rules, because they derive one ID rule from anotiier. A set of term-rewrite ndes terminates if an ordering '>' on the terms of the left-hand and right-hand sides (lhs and rhs, respectively) of die rewrite rules can be defined. This may lie a quantitative ordering, e.g. a category occuring on the lhs of a metarule is deleted on its rhs, or a qualitative ordering, e.g. an operator precedence. We think that a mixture of bodi types of ordcrings is needed to prove the termination of sets of metarules. If a qualitative and a quantitative ordering are merged, the resulting ordering guarantees termination (sec [Dershowitz 82 and 85]). The operator precedence that is used in our MT system is in fact a precedence ordering on feature values occurring at die categories of the lhs and rhs of the metarules.

Terminalion condition for metarule application For every metarule lhs rhs, lhs > rhs. lhs > rhs <=>

(i) a daughter category occurring on the lhs is deleted on the rhs and/or

(ii) an operator precedence >up on feature values occurring at die categories of the lhs and rhs can lie defined, which is not contradictory for die whole set of metarules and

(iii) every variable for (multisets of) categories occurring on the rhs occurs on the lhs.

Metarules have to fulfil the conditions (i) and (iii) or (ii) and (iii). The condition (i) is a quantitative ordering and the termination of metarules which fulfil (i) is obvious, because evcrytime such a metarule is applied one category is deleted and the number of daughter categories in an ID rule is finite. The condition (ii) is a qualitative ordering. The termi­nation of metarules which fulfil (ii) is not as obvious as in (i). It means that a feature value of a category has to be changed and there must not be another metarule, which reverses the change of this feature value. It allows for not having to delete categories, adding categories or adding values to a list, which may be a feature value, on the rhs of a metarule, if a feature value is changed on another category. This is the reason why we decided to impose control on the definition of metarules and not to move away from such devices as recent research in computational lin­guistics docs. If a metarule fulfils condition (ii) it can­not be applied for a second time in a derivation of an ID rule, because once a feature value has been changed it will never be reversed and the metarule will not be applicable again. This part of die termination condition simulates the termination condition of the FC. The condition (iii) prohibits the introduction or doubling of variables for (multisets of) categories on the rhs.

Thus the termination of a certain set of metarules can be guaranteed, iff for all metarules cither the metarule deletes a category occuring on its lhs and/or a non-con­tradicting precedence on operators (feature values of the categories occurring in the metarule) can be defined and all variables occurring on die rhs occur on the lhs. The application of die termination condition is demonstrated with die metarules for German in the next section.

Our GPSG syntax of German is based on the grammar developed in [Uszkoreit 87]. Wc assume a flat structure for the verb and its complements including the subject. Subcatcgorization of verbs is stated in ID rules of the following form:

V -> V°[SUBCAT a,,], Q.....C„.

V is a sentential category and SÜBCAT 0„ means the fixed value for die subcatcgorization feature in the ID rule with n arguments, but for every subcatcgorization there is a scperate rule. The subject of main verbs is included in the rhs of the rule. Unlike Uszkoreit's approach we do not add die subject to the complements of a verb phrase via metarule application but reduce a sentential category to a verb phrase and delete the subject. The following Subject Deletion Metarule fulfils the termination conditions (i) and (iii), because it deletes the category DPfnoin]. Subject Deletion Metarule:

Additionally, the operator precedence 'BAR 3,' >OP'BAR 2' has to be defined, because the feature BAR at the mother category is changed on the rhs. This additional definition is needed in order to get a noii-contradicting set of operator precedence definitions out of the whole set of metarules in the grammar. The Slash termination Metarule is responsible for ending (or from the bottom-up view, for the introduc­tion ol) a long distance relationship that is handled in GPSG via the category-valued feature SLASH. Unlike [Gazdar et al. 85] we do not have a trace. Traces cause problems in flat structures widiout fixed word order, because there are multiple analyses that arc only different with respect to the position of die trace.

Slash Termination Metarules:

VJ|-AUX, SLASH V] -> V°, W.

Here the termination conditions (i) and (iii) arc also fulfilled, because a category of the rhs of the ID rule is deleted. The operator precedence definitions are 'SLASH -V >01, 'SLASH X' and 'SLASH -.' >0p 'SLASH V', respectively.

"Hie Extraposition Metarule handles complement sen­tences and infinitive constructions that we treat as dislocated when they appear in die final position of a sentence. Another category-valued feature, SLASH1, is

The category DP is a determiner phrase according to die X-Bar-Sclicma in the Government and Binding Theory.

'F means that the feature F has the value '-i' (see (Busemann/ Hauenschikl 88] and [Busemann/ Hauen­schild 89]). This is equivalent to the notation ~[F] of [Gazdar el al. 85] and means that the value for F is always undefined, i.e. the corresponding category docs not take a value for F. The value '->' is specially treated by the unification and the feature instantiation principles.

In our parser, which is a part of an experimental machine translation system (see [Weisweber 87] for a

Treating passive, one has to say a word about semantics. We do not adopt the semantics of [Gazdar ct al. 85] because of its shortcomings (sec [Umbach 87]), but developed a semantic level of our own that captures die functor argument structures (FAS, see [Busemann/ Hau­enschild 89] and [Hauenschild/ Umbach 88]) of sentences and is derived from the syntactic stmcmre via term-rewrite rules. Here an explicit assignment of semantic roles to complements of verbs takes place that is dependent on the subcatcgorization of the verb and its voice.

In this case wc have to define a precedence for all feature values which are changed.

detailed description of die parser without direct application of metarules), the metarules are defined according to the following scheme:

C0 -> Cc, W, Cd => CÖ —> q, W, (Ci) C0, Cc and Cd arc categories and W is a variable for a (possibly empty) multiset of categories. The categories CJ and Q correspond to C0 and Cc, respectively, in terms of [Gazdar ct al. 85]. The category q can be viewed as a condition category for the application of the metarule. Q is the category which is to be deleted or modified. This is indicated by the brackets arround Cj. If Cd is to be deleted, Cd is left out on the rhs of the melarulc. If Cd is to be modified, Cd is replaced by q. The feature values of the categories are cospecified on the lhs and rhs of a metarule, if not specified otherwise. This causes the values to be carried over to the rhs. If the metarule should only be applied to lexical ID rules as proposed in [Gazdar et al. 85], the category Cc has to be the lexical head with respect to Co-The proof for the termination of sets of such metarules is simple. At first we look at the case in which Cj is deleted, then the termination condition (i) holds. The second case is that the category Cd is replaced by the category Cd and die number of categories is not reduced. Now the termination condition (ii) has to he applied and at least one feature value of the categories (Co, Cc]m Um W has to be changed, which must not be reversed by another metarule.

The termination condition (iii) holds in every case, because the variable W occurs on bodi sides of the metarule.

In order to apply die metarules direcUy during the parsing process, all the categories of an ID rule, which are matched by die multiset (Cc}m Um W, have to be collected by the parser. This is done for example by the Completer of the modified Earlcy algorithm (see [Earley 70], [Shiebcr 84], [Kilbury 84] and [Dörre/ Momma 85]). Suppose the Completer tries to complete with the inactive edge (q, i, j, Y-Om). which is spanning from node i to node j of the chart, where y is a multiset of daughter categories which have already been analysed and the remainder, i.e. the multiset of daughter categories still to be analysed, is empty and Cj is the mother category of the ID rule, which is licensing this edge. M is the set of metarules.

then the Completer introduces a new inactive edge (Co, h, j, a-()m) and computes its closure. The category Cc in a Um ß is replaced by q.

Cj is consistent to the categories in ß/(Cd)m with

respect to linear precedence then the Completer introduces a new edge (C0, h, j, a Um (Cd]m-ß/(q) J and the category Cc in a Um ß is replaced by q. If the remainder ß/(Cd)m = ()m then die closure of this edge has to be computed.

The advantage of direct parsing with metarules is an increase of efficiency, because all the inactive edges which are licensed by ID rules indroduccd by metarules need not to be stored scperately and the number of inactive edges generated by the Earley parser is reduced considerably.

Metarules are an interesting device to express some important generalizations on phrase structure rules of a natural language grammar. If they arc used in preprocessing to compile a huge set of rules out of a small set of basic ones, die parsing process may become very inefficient, because it has to care for the set of basic rules and additionally for the rules which have been derived from them and are very similar to die basic ones. When metarules arc applied directly during the parsing process, only the set of basic rules in connection with the metarules have to he considered by the parser. This reduces the set of intermediate solutions (inactive edges) to be stored considerably. In order to apply metarules directly, it has to be guaranteed that the given set of metarules will terminate if all metarules fulfil the termination criterion in section 4. Wc gave die termination proof for the metarules of our German grammar in section 5. Wc think that with die help of this criterion the termination of every relevant set of metarules can be proven, because a metarule is defined to change something in an ID rule, either to delete a category, to modify some feature values, to add a category or to do a combination of all. With the termination criterion it is possible to construct a device which automatically proves the termination of a given set of metarules. This algorithm computes the set of operator precedences from the feature values which are changed on the categories of a metarule.

To enable the parser to process metarules like Uszko-reit's lor AdvPs in section 3, which add categories to basic ID rules and for which the termination can be proven, will be subject to future work.