Formal Constraints on Metarules* 
Stuart M. Shieber, Susan U. Stucky, Hans Uszkoreit, and Jane J. Robinson 
SRI International 
333 Ravenswood Avenue 
Menlo Park, California 
Abstract 
Metagrammaticai formalisms that combine context-free 
phrase structure rules and metarules (MPS grammars) allow con- 
cise statement of generalizations about the syntax of natural lan- 
guages. Unconstrained MPS grammars, tmfortunately, are not 
cornputationally "safe." We evaluate several proposals for con- 
straining them, basing our amae~ment on computational trac- 
tability and explanatory adequacy. We show that none of them 
satisfies both criteria, and suggest new directions for research on 
alternative metagrammatical formalisms. 
1. Introduction 
The computational-linguistics community has recently 
shown interest in a variety of metagrammatical formalisms for 
encoding grammars of natural language. A common technique 
found in these formalisms involves the notion of a metarule, 
which, in its most common conception, is a device used to 
generate grammar rules from other given grammar rules. 1 A 
metarule is essentially a statement declaring that, if a grammar 
contains rules that match one specified pattern, it also contains 
rules that match some other specified pattern. For example, the 
following metarule 
(1) VP -..V VP ~ VP-*Y ADVP VP \[+/;-I 
\[+o.~i 
states that, if there is a rule that expands a finite VP into a 
finite auxiliary and a nonfinite VP, there will also be a rule 
that expands the VP as before except for an additional adverb 
between the auxiliary and the nnnfinite VP. 2 The patterns may 
contain variables, in which case they characterize "families ~ of 
related rules rather than individual pairs. 
*This reeearch was supported by the National Science Foundation grant No. 
IST-8103550. The views and conclusions expressed in this document are 
those of the authors and should not be interpreted as represent,.tive of the 
views of the National Science Foundation or the United States government. 
We are indebted to Fernando Pereira, Stanley Peters, and Stanley 
Roscnscheln for many helpful discun~ons leading to the writing of this paper. 
IMetarules were first utilized for natural-language research and are most 
extensively developed within the theory of Generalized Phrase Structure 
Grammar (GPSG) \[Ga2dar end Pullum, 1082; Gawron et ~., 1982; 
Thompson. 1082 I. 
2A metarule similar to our example was proposed by Gazdar, Pullum, and 
sag \[10s2, p. oorl. 
The metarule notion is a seductive one, intuitively allowing 
generalizations about the grammar of a language to be stated 
concisely. However, unconstrained metarule formalisms may 
possess more expressive power than is apparently needed, and, 
moreover, they are not always compatationally "safe." For ex- 
ample, they may generate infinite sew of rules and describe ar- 
bitrary languages, lu this paper we examine both the formal 
and linguistic implications of various constraints on metagram- 
matical formalisms consisting of a combination of context-free 
phrase structure rules and metarules, which we will call metarule 
phrase.structure (MPS\] grammars. 
The term "MPS grammar" is used in two ways in this 
paper. An MPS grammar can be viewed as a grammar in its 
own right that characterizes a language directly. Alternatively, 
it can be viewed as a metagrammar, that is, as a generator of 
a phrase structure obiect grammar, the characterized language 
being defined as the language of the object grammar. 
Uszkoreit and Peters \[1982\] have developed a formal 
definition of MPS grammars and have shown that an uncon- 
strained MPS grammar can encode any recursively enumerable 
language. As long am the framework for grammatical descrip- 
tion is not seen am part of a theory of natural language, this 
fact may not alt'ect the usefulness of MPS grammars am tools for 
purely descriptive linguistics research; however, it has direct and 
obvious impact on those doing research in a computational or 
theoretical linguistic paradigm. Clearly, some way of constrain- 
ing the power of MPS grammars is necessary to enable their use 
for encoding grammars in a ¢omputationally feasible way. In 
the sections that follow, we consider several formal proposals for 
constraining their power and discuss some of their computational 
and linguistic ramifications. 
In our discussion of the computational ramifications of the 
proposed constraints, we will use the notion of weak-generative 
capacity as a barometer of the expressive power of a formalism. 
Other notions of expre~ivity are possible, although some of 
the traditional ones may not be applicable to MPS grammars. 
Strong*generative capacity, for instance, though well-defined, 
seems to be an inadequate notion for comparison of MPS gram- 
mars, since it would have to be extended to include informa- 
tion about rule derivations am well am tree derivations. Similarly, 
we do not mean to imply by our arguments that the class of 
natural languages corresponds to some class that ranks low in 
the Chomsky hierarchy merely because the higher classes are less 
constrained in weak-generative power. The appropriate charac- 
terization of possible natural languages may not coincide at all 
22 
with the divisions in the Chomsky hierarchy. Nevertheless weak- 
generative capacity--the weakest useful metric of capacity--will 
be the primary concern of this paper as a well-defined and 
relevant standard for measuring constraints. 
2. Constraints by Change of Perspective 
Peters and Ritchie \[1973\] have pointed out that context- 
sensitive grammars have no more than context-free power when 
their rules are viewed as node-admissibility conditions. This 
suggests that MPS grammars might be analogously constrained 
by regarding the metarules as something other than phruse- 
structure grammar generators. A brief examination of three 
alternative approaches indicates, however, that none of them 
clearly yields any useful constraints on weak-generative capacity. 
Two of the alternatives discussed below consider metarules to be 
part of the grammar itself, rather than as part of the metagramo 
mar. The third views them as a set of redundant generalizations 
about the grammar. 
Stucky \[forthcoming\] investigates the possibility of defining 
metarules as complex node-admissibility conditions, which she 
calls meta, node-admissibility conditions. Two computationally 
desirable results could ensue, were this reinterpretation possible. 
Because the metarules do not generate rules under the meta, 
node-admissibility interpretation, it follows that there will be 
neither a combinatorial explosion of rules nor any derivation 
resulting in an infinite set of rules (both of which are potential 
problems that could arise under the original generative inter- 
pretation). 
For this reinterpretation to have a computationally tract- 
able implementation, however, two preconditions must be met. 
First, an independent mechanism must be provided that assig~ 
to any string a finite set of trees, including those admitted by 
the metarules together with the bmm rules. Second, a procedure 
must be defined that checks node admissibilities according to the 
base rules and metarules of the grammar--and that terminates. 
\[t is this latter condition that we snspect will not be possible 
without constraining the weak-generative capacity of MPS gram- 
mars. Thus, this perspective does not seem to change the basic 
expressivity problems of the formalism by itself. 
A second alternative, proposed by Kay \[1982\], is one in 
which metarules are viewed as chart-manipulating operators on 
a chart parser. Here too, the metarules are not part of a 
metagrammar that generates a context-free grammar; rather, 
they constitute a second kind of rule in the grammar. Just 
like the meta-node-admissibility interpretation, Kay's explics- 
t, ion seems to retain the basic problem of expressive power, 
though Kay hints at a gain in efficiency if the metarules are 
compiled into a finite-state transducer. 
Finally, an alternative that does not integrate metarules 
into the object grammar but, on the other hand, does not as- 
sign them a role in generating an object grammar either, is to 
view them as redundancy statements describing the relationships 
that hold among rules in the full grammar. This interpretation 
eliminates the problem of generating infinite rule sets that gave 
rise to the Uszkoreit and Peters results. However, it is difficult 
to see how the solution supports a computationally useful notion 
of metarules, since it requires that all rules of the grammar be 
stated explicitly. Confining the role of metarules to that of stat- 
ing redundancies prevent~ their productive application, so that 
the metarules serve no clear computational purpose for grammar 
implementation. 3 
We thus conclude that, in contrust to context-sensltive 
grammar, in which an alternative interpretation of the phruse 
structure rules makes a difference in weak-generative capacity, 
MPS grammars do not seem to benefit from the reinterpretations 
we have investigated. 
3. For:hal Constraints 
~. a, e it appears unlikely that a reinterpretation of MPS 
grammars can be found that solves their complexity problem, 
formal constraints on the MPS formalism itself have to be ex- 
plored if we want to salvage the basic concept of metarules. In 
the following examination of currently proposed constraints, the 
two criteria for evaluation are their effects on computational trac- 
tability and on the ezplanatory adcquaeltof the formalism. 
As an example of constraints that satisfy the criterion of 
computational tractability but not that of explanatory adequacy, 
we examine the issue of essential variables. These are variables in 
the metarule pattern that can match an arbitrary string of items 
in a phrase structure rule. Uszkoreit and Peters have shown that, 
contrary to an initial conjecture by Jcehi (see \[Gazdar, 1982, 
fn. 28\]), allowing even one such variable per metarule extends 
the power of the formalism to recursive enumerability. Gazdar 
has recommended \[1982, p.160\] that the power of metarules be 
controlled by eliminating essential variables, exchanging them 
for abbreviatory variables that can stand only for strings in a 
finite and cztrinsieally determined range. This constraint yields 
a computationslly tractable system with only context-free power. 
Exchanging essential for abbreviatory variables is not, 
however, as attractive a prospect as it appears at first blush. 
Uszkoreit and Peters \[1982\[ show that by restricting MFS gram- 
mars to using abbreviatory variables only, some significant 
generalizations are lost. Consider the following metarule that 
is proposed and motivated in \[Gazdar 1982\] for endowing VSO 
languages with the category VP. The metarule generates fiat 
VSO sentence rules from VP rules. 
(2) VP-.V U~ S-.V NPU 
Since U is an abbreviatory variable, its range needs to be stated 
explicitly. Let us imagine 'h:,t the VSO language in question has 
the follo~ ;~ small set of VF rules: 
(3) w ,'~ 
VP -- V NP 
vP-. V-~ 
VP -. V VP 
VP -. V NP V-P 
Therefore, the range of U has to be {e, NP, ~, \]77~, NP V'P}. 
3As statements about the object ~'~mmar, however, metxrules might play 
s role in language acquisition or in dia~hronie processes. 
23 
If these VP rules are the only rules that satisfy the left- 
hand side of (2), then (2) generates exactly the same rules am it 
would if we declared U to be an essential variable--i.e., let its 
range be (Vr O VN) °. But now imagine that the language adopts 
a new subcategorizatiun frame for verbs, 4 e.g., a verb that takes 
an NP and an S am complements. VP rule (4) is added: 
(4) VP -- I/" NP -S 
Metarule (2) predicts that VPs headed by this verb do not have 
a corresponding fiat V$O sentence rule. We will have to change 
the metarule by extending the range of U in order to retain the 
generalization originally intended by the metarule. Obviously, 
our metarule did not encode the right generalization (a simple 
intension-extensiun problem). 
This shortcoming nun also surface in cases where the input 
to a metarule is the output of another metaruh. It might be 
that metarule (2) not only applies to basic verb rules but also 
includes the output of, say, a passive rule. The range of the 
variable \[.r would have to be extended to cover these tames too, 
and, moreover, might have to be altered if its feeding metarules 
change. 
Thus, if the restriction to abbreviatury variables is to have 
no effect on the weak-gensrative capacity of a grammar, the 
range assigned to each variable must include the range that 
would have actually instantiated the variable on an expansion of 
the MPS grammar in which the variable was treated as essential. 
The assignment of a range to the variable can only be done po,t 
/actum. This would be a satisfactory result, were it not for the 
fact that finding the necessary range of a variable in this way 
is an undecidable problem in general. Thus, to exchange essen- 
tial for abbreviatory variables is to risk affecting the generative 
capacity of the grammar~with quite unintultive and unpredict- 
able results. In short, the choice is among three options: to affect 
the language of the grammar in ways that are linguistically un- 
moti~at4ed and arbitrary, to solve an undecidable problem, or 
to discard the notion of exchanging essential for abbreviatory 
variables--in effect, a Hobsun's choice. 
An example of a constraint that satisfies the second 
criterion, that of explanatory adequacy, hut not the first, com- 
putational tractability, is the leziesl-head constraint of GPSG 
\[Gazdar and Pullum, 1982\[. This constraint allows metarules 
to operate only on rules whose stipulated head is a lexical 
(preterminal) category. Since the Uszkoreit and Peters results are 
achieved even under this restriction to the formalism, the cow 
straint does not provide a solution to the problem of expressive 
power. Of course, this is no criticism of the proposal, since it was 
never intended as a formal restriction on the class of languages, 
but rather ~ a restriction un linguistically motivated grammars. 
Unfortunal,ely, the motivation behind even this use of the lexical- 
head constraint may be lacking. One of the few analyses that 
relies on the lexical-head constraint is a recent GPSG analysis of 
coordination and extraction in English (Gazdar, 1981\]. In this 
ease--indeed, in general-one could achieve the desired effect 
simply by specifying that the coefficient of the bar feature be 
lezical. It remains to be seen whether the constraint must be 
imposed for enough metarules so as to justify its incorporation 
as a general principle. 
Even with such motivation one might raise a question 
about the advisability of the lexical-head constraint on a meta- 
theoretical level. The linguistic intuition behind the constraint 
is that the role of metarules is to "express generalizations about 
possibilities of subeategorizatiun" exclusively \[Gaadar, Klein, 
Pullum, and Sag, 1982, p.391, e.g., to express the p~mive-active 
relation. This result is said to follow from principles of ~ syntax 
\[Jackendoff, 1077\], in which just those categories that are sub- 
categorized for are siblings of a lexieal head. However, in a lan- 
guage with freer word order than English, categories other than 
those subcategorized for will be siblings of lexieal heads; they 
would, thus, be affected by metarules even under the lexical-head 
constraint. This result will certainly follow from the liberation 
rule approach to free word order \[Pullum, 1982\]. The original 
linguistic generalization intended by the hxical-head constraint, 
therefore, will not hold cross-linguistically. 
Finally, there is the current proposal of the GPSG com- 
munity for constraining the formal powers of metarules by al- 
lowing each metaruh to apply only once in a derivation of a 
rule. Originally dubbed the once.through hgpothe~is, this con- 
straint is now incorporated" into GPSG under the name finite 
closure \[Gazdar and Pullum, 1982\]. Although linguistic evidence 
for the constraint has never been provided, the formal motiva- 
tion is quite strong because, under this constraint, the metarule 
formalism would have only context-free power. 
Several linguistic constructions present problems with 
respect to the adequacy of the finite-closure hypothesis. For in- 
stance, the liberation rule technique for handling free-word-order 
languages {Pullum, 1982\] would require ffi noun-phrase liberation 
rule to be applied twice in a derivation of a rule with sibling 
noun phrases that permute their subconstituents freely among 
one another. As a hypothetical example of this phenomenon, let 
us suppose that English allowed relative clauses to be extraposed 
in general from noun phrases, instead of allowing just one ex- 
traposifion. For instance, in this quasi-English, the sentence 
(5) Two children are chasing the dog who are small that is 
here. 
would he a grammatical paraphrase of 
(0) Two children who are small axe chasing the dog that is 
here. 
Let us suppose further that the analysis of this phenomenon 
involved liberation of the NP-S substructure of the noun phrases 
for incorporation into the main sentence. Then the noun-phrase 
liberation rule would apply once to liberate the subject noun 
phrase, once again to liberate the object noun phrase. That these 
are not idle concerns is demonstrated by the following sentence 
in the free-word-order Australian aboriginal language Warlpiri. s 
4Note that it does not matter whether the grammar writer discovers an 
additional subcateKorization, or the language develops one diachronically; 
the same problem obtains. 5This example is t,.ken from \[van Riemsdijk, 1981\]. 
24 
(7) Kutdu-jarra-rlu ks-pals maliki wita-jarra-rlu 
chiId-DUAL-ERG AUX:DUAL dog-ABS smalI-DUAL-ERG 
yalumpu wajilipi-nyi 
that-ABS chase=NONPAST 
Two 8mall children are cha,ing that dog. 
The Warlpiri example is analogous to the quasi-English 
example in that both sentences have two discontinuous NPs in 
the same distribution. Furthermore, the liberation rule approach 
has been proposed as a method of modeling the free word order 
of Waripiri. Thus, it appears that finite closure is not consistent 
with the liberation rule approach to free word order. 
Adverb distribution presents another problem for the 
hypothesis. In German, for example, and to a lesser extent in 
Engiish, an unbounded number of adverbs can be quite freely 
interspersed with the complements of a verb. The following 
German sentence is an extreme example of this phenomenon 
\[Uszkoreit, 1982\]. The sequence of its major constituents is given 
under (9). 
(8) Gestern hatte in dec Mittagspause 
yesterday had during lunch break 
der Brigadier in dec Werkzeugkammer 
the foreman (NOM) in the tool shop 
dam Labeling au~ Boehaftigkeit lancaam 
the apprentice (DAT) maliciously slowly 
zehn schmierige Gasseisenscbeiben unbemerkt 
ten greasy cast iron disks (ACC) unnoticed 
in die Hosentasche gesteckt 
in the pocket put 
)'*aerdav, durin~ lunch break in the tool shop, the 
foreman, malicioedy and unnoticed, put ten grea,y caJt 
iron disks tlowist into the apprentice's pocket. 
(9) ADVP VrrN ADVP NPsuuJ ADVP NProaJ ADVP 
ADVP NPDoa.t ADVP PP VIN e 
A metarule might therefore be proposed that inserts a 
single adverb in a verb-phrase rule. Repeated application 
of this rule (in contradiction to the finite-closure hypothesis) 
would achieve the desired effect. To maintain the finite-closure 
hypothesis, we could merely extend the notion of context-free 
rule to allow regular expressions on the right-hand side of a 
rule. The verb phrase rule would then be accurately, albeit 
clumsily, expressed as, say, VP -.* V NP ADVP* or VP -* 
V NP ADVP* PP ADVP* for ditransitives. 
Similar constructions in free-word-order languages do not 
permit such naive solutions. As an example, let us consider 
the Japanese causative. In this construction, the verb sutRx 
"-sase" signals the causativization of the verb, allowing an extra 
NP argument. The process is putatively unbounded (ignoring 
performance limitations). Furthermore, Japanese allows the NPs 
to order freely relative to one another (subject to considerations 
of ambiguity and focus), so that a fiat structure with some kind 
of extrinsic ordering is presumably preferable. 
One means of achieving a fiat structure with extrinsic 
ordering is by using the ID/LP formalism, a subformalism of 
GPSG that allows immediate dominance (ID) information to be 
specified separately from linear precedence (LP) notions. (Cf. 
context-free phrase structure grammar, which forces a strict one- 
to-one correlation between the two types of information.) ID 
information is specified by context-free style rules with unordered 
right-hand sides, notated, e.g., .4 ~ B, C, D. LP informa,Aon is 
specified as a partial order over the nonterminals in the ..orr-,m max, 
notated, e.g., B < C (read B precedes C). These two rules 
can be viewed as schematizing a set of three context-free rules, 
namely, A -- B C D, A -- B D C, and A -- D B C. 
Without a causativization metarule that can operate more 
than once, we might attempt to use the regular expression nota- 
tion that solved the adverb problem. For example, we might 
postulate the ID rule VP -, NP*, V, sane* with the LP rela- 
tion NP < V < sase, but no matching of NPs with sases 
is achieved. We might attempt to write a liberation rule that 
pulls NP.saee pairs from a nested structure into a flat one, 
but this would violate the finite-closure hypothesis (as well as 
Pullum's requirement precluding liberation through a recursive 
category). We could attempt to use even more of the power of 
regular-expression rules with ID/LP, i.e., VP -, {NP, 8a,e} °, V 
under the same LP relation. The formalism presupposed by this 
analysis, however, has greater than context-free power, ° so that 
this solution may not be desirable. Nevertheless, it should not 
be ruled out before the parsing properties of such a formalism 
are understood. T Gunji's analysis of Japanese, which attempts 
to solve such problems with the multiple application of a tlash 
introduction metarule \[Gunji, 1980 l, again raises the problem of 
violating the 6nite-closure hypothesis (as well as being incom- 
patible with the current version of GPSG which disallows mul- 
tiple slashes). Finally, we could always move ca~ativization into 
the lexicon as a lexical rule. Such a move, though it does cir- 
cumvent the difficulty in the syntax, merely serves to move it 
elsewhere without resolving the basic problem. 
Yet another alternative involves treating the right-hand 
~ides of phrase structure rules as sets, rather than multisets as is 
implicit in the ID/LP format. Since the nonterminal vocabulary 
is finite, right-hand sides of ID rules must be subsets of a finite 
set and therefore finite sets themselves. This hypothesis is quite 
similar in effect to the finite-closure hypothesis, albeit even more 
limited, and thus inherits the same problems aa were discussed 
above. 
4. The Ultimate Solution 
An obvious way to constrain MPS grammar, is to eliminate 
metarules entirely and replace them with other mechanisms. In 
fact, within the GPSG paradigm, several of the functions of 
metarules have been replaced by other metagrammatical devices. 
Other functions have not, as of the writing of this paper, though 
8For instance, the grammar $ ~ {a,b,e} e with a < b < • generates 
anb~en" 
7Shieber \[forthcoming\] provides an ~l&orithm for parsing ID/LP grammars 
directly that includes a method for utilizing the Kleene star device. It 
could be extended to even more of the regular expression notation, though 
the effect of such extenslon-on the time complexity of the algorithm is an 
open question. 
25 
it i$ instructive ~.o co=ider ~.he c~es covered ~y this cia~s. In 
the discussion to follow we have isolated thxee of the primary 
functions of metarules. This is not intended az an exhaustive 
taxonomy, and certain metarules may manifest more than one 
of these functions. 
First, we consider generalizations over linear order. If 
metarules are metagrammatical statements about rules encod- 
ing linear order, they may relate rules that differ only in the 
linear order of categories. With the introduction of ID/LP for- 
mat, however, the hypothesis i, that this latter metagrammatical 
device will suffice to account for the linear order among the cat- 
egories within rules. For instance, the problematic adverb and 
causative metarnles could be replaced by extended contex.t-free 
rules with \[D/LP, as was suggested in Section 3 above. Shieber 
\[forthcoming\[ has shown that a pure ID/LP formalism (without 
metarules, Kleene star, or the like) is no le~ computationally 
tractable than context-free grammars themselves. Although we 
do not yet know what the consequences of incorporating the 
extended context-free rules would be for computational com- 
plexity, ID/LP format can be used to replace certain word-order- 
variation metarules. 
A second function of metarnles wa~ to relate sets of rules 
that differed only in the values of certain specifed features. It 
has been suggested \[Gat~iar and Pullum 1982\] that such features 
are distributed according to certain general principles. For in- 
stance, the slash-propagation metarule haz been replaced by the 
distribution of slash features in accord with such a principle. 
A third function of metarules under the original interpreta- 
tion has not been relegated to other metagr~nmatical devices. 
\Ve have no single device to suggest, though we axe exploring 
alternative ways r,o account for the phenomena. Formally, this 
third class can be characterized as comprising those metacules 
that relate sets of rules in which the number of categories on the 
right- and left-hand sides of rules differ. It is this sort of metarule 
that is essential for the extension of GPSGs beyond context-free 
power in the Uszkoreit and Peters proofs {1982\]. Simply requiring 
that such metarules be disallowed would not resolve the linguistic 
issues, however, since this constraint would inherit the problems 
connected with the regular expression and set notations discussed 
in Section 3 above. This third cl~s further breaks down into two 
cases: those that have different parent categories on the right- 
and left-hand sides of the metarule and those that have the same 
category on both sides. The ~rst c~e includes those liberation 
rules that figure in analyses of free-word-order phenomena, plus 
such other rules as the subject-auxiliary-inversion metarule in 
English. Uszkoreit \[forthcoming\] is exploring a method for isolat- 
ing liberation rules in a separate metagrammaticul formalism. It 
also appears that the subject-auxiliary inversion may be analyzed 
by already existing principles governing the distribution of fea- 
tures. The second case (those in which the categories on the 
right- and left-hand sides are the same) includes such analyses 
as the passive in English. This instance, at least, might be re- 
placed by a lexicai-redundancy rule. Thus, no uniform solution 
has yet been found for this third function of metarules. 
We conclude that it may be possible to replace MPS-style 
metagrammatical formalisms entirely without losing generaliza- 
tion~. '~Ve ~re consequently pursuing re~eaxcu tu ~u,o o~,,. 
5. Conclusion 
The formal power of metaxule formalisms is clearly an 
important consideration for computational linguists. Uszkoreit 
and Pet.era \[1982\] have shown that the potential exists for 
defining metarule formalisms that are computationally "unsafe." 
However, these results do not sound a death knell for metarules. 
On the contrary, the safety of metarule formalisms is still an 
open question. We have merely shown that the constraints on 
metarules necessary to make them formally tractable will have to 
be based on empirical linguiaic evidence as well as solid formal 
research. The solutions to constraining metarules analyzed here 
seem to be either formally or linguistically inadequate. Further 
research is needed in the actual uses of metarules and in con- 
structions that axe problematic for metarules so ~ to develop 
either linguistically motivated and computationally interesting 
constraints on the formalisms, or alternative formalisms that axe 
linguistically adequate but not heir to the problems of metaxules. 
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